The (literally) steepest slope: spatial, temporal, and elevation variance gradients in urban spatial modelling

The (literally) steepest slope: spatial, temporal, and elevation variance gradients in urban... Abstract This paper presents an analysis of elevation gradient and temporal future-station effects in urban real estate markets. Using a novel dataset from the Hong Kong publicly constructed housing sector, we find enormous housing price effects caused by levels of terrain incline between apartments and subway stations. Ceteris paribus, two similar apartments with closest metro stations of the same walking distance may sell at a difference of up to 20% because of differences in the apartment-station slope alone. Anticipatory effects are similarly robust: apartment buyers regard a future, closer metro station as being 60% present when making purchases 2 years prior to its opening. 1. Introduction This is a story of three gradients. The urban spatial modelling literature focuses almost exclusively on distance—a discussion that developed from the original Alonso-Muth-Mills (AMM) model (Alonso, 1964; Muth, 1969; Mills, 1972) to models with tiered transit systems (LeRoy and Sonstelie, 1981) and, more recently, to effects of mass public transit (Glaeser et al., 2008). Yet the underlying premise of investigations on the effect of distance remains unchanged: cities are largely regarded as featureless, flat plains with fixed, immutable transit nodes. Reality is neither flat nor featureless. Cities are often built on or against mountainsides and over rolling hills, the economic influence of which have thus far lacked in-depth investigation. Cities also are dynamic, continuously receiving structural upgrades in the form of new roads and transit lines. Utilizing extensive geospatial modelling methods, including the use of Geographical Information Systems (GISs) and high-resolution Digital Elevation Models (DEMs), as well as a novel dataset of apartment transactions from the Hong Kong government-subsidized real estate market, this paper explores these factors—elevation and temporal gradients—from the housing market perspective. In the context of recent literature on elevation effects (Saiz, 2010; Ye and Becker, 2016), this paper eschews a multi-city framework in favor of a single market, transaction-level approach that focuses on one particular economic aspect of elevation: its influence on utility derived from public transit access. Taking advantage of the strongly homogeneous quality of government-subsidized housing and the long, 17-year timespan of our data, we create a ‘near-perfect’ spatial and temporal hedonic housing model, moving beyond identification to estimate the magnitude with which hilliness influences housing prices. We demonstrate that the value of public transit hubs varies not only by distance but also by altitude differentials: holding other factors constant, apartment selling price may differ up to 20% because of difference in the slope between the apartment and the closest metro station. Buyers also seem to robustly but incompletely value the existence of future, close-by stations: we estimate that stations are perceived as being 58–60% ‘present’ to buyers 2 years prior to opening, and appreciate in perceived value at an average rate of 1.1% per month within a 6-year window before opening. This rate resembles interest returns on capital invested locally during the corresponding time period. Contributing to the tradition of penalized distance gradient models (Yinger, 1974; Anas and Kim, 1996), our work introduces and evaluates a new distance–steepness interaction penalty associated with public transit use: the undesirability of walking up or down slopes to transit hubs. By allowing for differences in locations resulting from both z-axis variations and future amenities of a particular location, our modelling approach allows cities with significant unevenness (San Francisco, Hong Kong) to behave fundamentally and qualitatively differently from cities with little to no variance (Beijing, Chicago). It also provides a more comprehensive perspective for rapidly developing cities that experience major public infrastructure development. Given the robustness and magnitude of these effects, spatial models of cities with significant altitude variance are expected to suffer from bias if elevation gradients are not specifically accounted for. For such cities, large, hidden costs exist in developing public transit: residents of a more ‘uneven’ city will receive less utility from a given level of transit infrastructure. Temporal models of rapidly developing cities suffer if the effect of major, future infrastructure developments are not considered in the time-frame of the study. Hence, rigorous investigations of housing and other urban markets in modern economies require advanced modelling methods with spatial and temporal elements that are not currently widely utilized. Section 2 discusses previous research on the effects of transit and elevation on urban housing. Section 3 provides descriptive and graphical evidence for elevation effects. Section 4 presents a data summary and outlines conceptual and empirical methodologies involved in constructing the model. Section 5 analyzes model robustness, presents results of the regression model and visualizes anticipatory and elevation gradient effects. Section 6 discusses broader implications of these effects; Section 7 concludes. 2. Prior literature We believe this is the first paper to formally investigate elevation variance effects in a monocentric, transaction-level housing framework. Past literature on elevation in urban economics is sparse: research on elevation effects in housing markets concentrates on price effects of low elevation in coastal regions and, in particular, flood risk (Scawthorn et al., 1982; Shilling et al., 1989; Husby et al., 2014). Elevation effects have also been utilized recently in broader, geographical perspectives to explain the locational choice of cities (Bosker and Buring, 2015) and in discussions about features that contribute to land value (Kok et al., 2014). None of these papers links elevation gradients to transit or uses transaction-level data at the city level. Saiz (2010) analyses elevation gradient effects in housing markets using data from 73 major MSAs, concluding that undevelopable land, approximated by peripheral areas with slopes exceeding 15%, is a strong predictor of low housing supply elasticity. Although highlighting the importance of elevation as a factor of urban housing systems, his approach has significant shortcomings: the 15% slope cutoff assumes a somewhat arbitrary boundary at which development becomes constrained by elevation, yet our results indicate that elevation gradient effects apply at far more moderate slopes and small increments of unevenness. Furthermore, using cities as observations precludes controls for city-level fixed effects: the model cannot infer the specific methods through which elevation exerts influence at units smaller than cities. Ye and Becker (2016) address both concerns using census-tract level data from 17 US cities. In addition to establishing a connection between elevation effects and income gradients, they present evidence of high terrain unevenness decreasing use of public transit and commuting by walking. However, their approach is limited by imperfect controls for exogenous spatial factors and aggregation inherent in using tract-level data. By utilizing observations at the transaction level, we expand on these effects and demonstrate specifically how the degree of incline between metro stations and apartments are priced at different apartment-station walking distances. With regard to urban gradients in Chinese and other East Asian cities, a number of salient characteristics are discussed in literature. Chinese cities generally face considerably higher income elasticities of housing demand than do US cities (Gong et al., 2013). Compared with US counterparts, Chinese urban residents seem to be willing to devote significantly larger portions of income to housing consumption: this pattern may reflect greater land scarcity in China. Chinese cities also feature immature housing markets and, consequentially, inefficiencies in housing allocation and spatial distributions. Using panel data from five Chinese cities, Zheng et al. (2006) find evidence of sub-optimal spatial outcomes in housing markets; inefficiencies are attributed to poorly defined property rights, historical housing regulations, and home inheritance. Although commercialized real estate has a much longer history in Hong Kong than other Chinese cities, the city also has a long history of large public housing programs. Apartments of the same program enjoy substantially different amenities depending on their location, a factor for which traditional distance-gradient models fails to account. Another significant feature of Chinese cities is the dominance of public transit. As of 2008, 52% of Beijing workers employed in the city centre commute by public transit (Zheng and Kahn, 2008). In contrast, only 18% of workers in areas serviced by Washington DC’s WMATA regularly use public transportation (Nelson et al., 2007). The average commuting time is correspondingly higher: one-way commutes average 37.8 min in Beijing, considerably longer than comparable US cities such as Boston, Minneapolis, and Chicago (Zheng et al., 2009). In our data, the average commute to central Hong Kong is 22.9 min by car and 52.6 min by public transit. However, driving times also likely reflect the effects of traffic and the tendency for the average subsidized homes to be further from downtown than commercially developed residencies. Specific to Hong Kong, Chan and Tse (2003) estimate rent and distance gradients by measuring monetary commuting cost and time. Their findings suggest that cost of commuting to central Hong Kong is significantly correlated with property values. However, commute time does not display similar significance. In contrast, our spatial analysis suggests robust significance in transit time effects, even after adjusting for commute time by car and walking distance to the closest metro station. Chan and Tse further estimate that the value of commuting time in Hong Kong is approximately 48% of the wage rate, within the established range of 25–50% in the urban economics literature (Small, 1983). Yiu and Tam (2004) develop the discussion with a contrast of rent gradient analysis methods applied to the Hong Kong housing market. A four-way comparison is made between among methodologies, divided by the use of hedonic versus repeat-sale models and the inclusion versus exclusion of monocentric assumptions. The authors confirm monocentricity but challenge the reliance on monocentric approaches, finding mis-estimations of prices attributed to district sub-centres and variations in spatial amenities: we address both concerns in our modelling process. More recently, Chow (2011) finds a statistically significant link between distance to metro stations and the selling price of commercial apartments in Hong Kong. Other factors constant, a doubling of walking distance to the closest station lowers the expected selling price by approximately 3.7%. Although the scope of data involved in the paper is limited, the fairly large coefficient underpins the importance of the Hong Kong metro system, suggesting the possibility of price effects beyond that of conventional distance gradients. 3. Visualizing elevation effects Before discussing modelling and statistical methodologies in detail, we first provide intuition regarding the existence of elevation effects in Hong Kong. It is difficult to directly present descriptive evidence of elevation effects: spatial factors are generally highly correlated, and effects such as centricity are potentially more robust predictors of price. However, the presence of a preference for flatness conceptually should have a large impact on price residuals, given that other spatial factors were rigorously controlled for. To this end, we perform an OLS regression on the full 121-variable dataset as outlined in Section 4.1, but with all variables related to the slope between the apartment and the nearest metro station removed. Standardized residuals of predicted selling prices of apartments, averaged at the apartment-group-level, are plotted spatially with metro station locations on the local altitude contour, with one example provided in Figure 1.1 Figure 1 View largeDownload slide Filled altitude contour plot with spatially mapped apartment-group average price residuals and metro station locations, Hong Kong, 1998–2014. Figure 1 View largeDownload slide Filled altitude contour plot with spatially mapped apartment-group average price residuals and metro station locations, Hong Kong, 1998–2014. We note that visually, large negative residuals appear to be distributed at higher altitude levels. As our analysis in Section 5 suggests, the discrepancy is most likely caused by selling prices of apartments with large elevation differentials from the closest station being systematically over-predicted. This trend can be observed more clearly by binning price residuals by a fixed range of slope levels and walking distances. Figure 2 summarizes price residuals for walking distance intervals of 0–2000, 500–1000 and 1000–1500 m, respectively. We find that for a given range of walking distances, sale prices of apartments with steeper slopes to the nearest station are generally overestimated, with the amount of over-estimation increasing on average with slope. Figure 2 View largeDownload slide Average regression price residual summarized by walking distance interval and range of slope in degrees, compared. Figure 2 View largeDownload slide Average regression price residual summarized by walking distance interval and range of slope in degrees, compared. We note that the trend is weaker when summarized from 0 to 2000 m, since apartments with steep walks also tend to have shorter walks; we explicitly control for this interaction effect in our model in Section 4.1. Also, consistent with our findings of substantial elevation effects in Section 5, a gap of 0.12–0.15 in average price residuals translates to a fairly large difference of approximately 0.45–0.6 standard deviations. 4. Data and methodology 4.1. Conceptual and empirical framework Neither of the two gradient effects we investigate is conceptually complicated. Knowing that a positive relationship exists between proximity to metro stations and apartment prices, we introduce two terms representing the walking distance and decimal degree of incline between the location of the apartment and that of the station. Both effects ought to be distinctly non-linear since transportation modes are expected to change: people will not use the metro if accessing the nearest station is sufficiently difficult. Similarly, slopes only slightly steeper could translate to significantly less walkable conditions. The anticipatory effect of a future, closer station on current prices is expected to vary not only with the time gap between the transaction date and the opening of the future station, but also with the gain in metro accessibility provided by a closer station. Conceptually, future stations should be able to influence current prices as Hong Kong regularly announces metro projects decades before construction starts, at which point future station locations become public knowledge. This does not necessarily imply information symmetry: certain individuals might learn about such events before others do. The precise opening date also would be uncertain from the perspective of the buyer. These intuitions suggest non-constant anticipatory effects caused by the gradual diffusion of information throughout the city’s population. Independent of information diffusion effects, the value of a future subway station should be discounted by its ‘futureness’, i.e. the amount of time a buyer has to wait until he or she can access the station. After adjusting for uncertainty and discounting perceived future gains, buyers should regard future stations as substitutes, albeit imperfect, for currently available ones in the surrounding area. Specifically, a buyer might pay a significant premium for an apartment currently far from any subway station, with the knowledge that a close-by station would open in the near future. This effect is bounded from spatial and temporal directions: a station under construction can never be exactly as desirable as the same station opened, and a station that is sufficiently ‘distant’ in the future should not have much effect on current sales. Therefore, aside from announcement and opening effects, we expect no sudden jumps in the perceived premium: price effects ought to move gradually from zero to a fixed value at the date of opening. The size of the perceived premium of a future station should also vary by how much utility it brings. If the current-best station is far away and the future station close by, the price effect of the future station should be greater than that of a scenario where walking distance savings of moving to the new station is small. To obtain stable estimates of elevation and distance gradient effects, we first restrict our dataset to 30,528 observations where the closest metro station did not change for the duration of the data.2 These observations are used to generate housing price effect estimates for stations with a given walking distance/slope combination (part I). Using these estimates and the remaining 7017 observations that experience metro expansion, a second, separate regression analysis (part II) is performed to determine the relationship between the equivalent anticipatory station value, denoted as a percentage of the station’s value post-opening, and the time gap. However, predictions on the price effects of metro station-apartment slope and distance are limited in range. In our part I data, 93.8% of all observations (28,634) report a closest station walking distance of less than 5000 m. Moreover, 98.5% of observations (30,059) report a closest station average slope of less than 9 decimal degrees. Given that little information is available on stations that do not fit these criteria, it is unfeasible to reliably extend inference to price effects of stations with distances and steepness that exceed these values. On the other hand, most ‘current’ stations in the anticipatory effect subset have much greater walking distances. 62.2% of all observations (4367) in the part II subset have a current station distance that exceeds 5 km. This is not unexpected, since subway line development is most robust in districts that have a dire need for extra public transit options. Without discarding a majority of observations, estimations of equivalent prices of current stations in this dataset will be problematic. Results from the regression in part I perform well when estimating ‘final’ station price effects in the data for part II, but suffer from severe range mismatch when estimating current station price effects. Our solution to this issue is 2-fold: first, we select a two-stage regression approach with cross-validation (CV) for part I with the goal of optimizing inference quality of elevation-price and distance-price effects. Variable-selection is conducted with a first-stage LASSO regression (Tibshirani, 1996) with 50-fold CV. The LASSO process assumes a penalty parameter selected via CV and, by fitting for a biased least squares estimator subjected to the constraints of the penalty term, reduces the size of coefficients accordingly. Ideally, coefficients of variables that do not significantly contribute to the model’s predictive ability are reduced to exactly zero, and are effectively removed from the regression. However, over-penalization is a general concern with such methods: significant coefficients might be reduced too much by a large penalty parameter. In response, we introduce a secondary OLS stage to the regression process. Variables that survive the initial selection process are regressed with a large number of OLS models on randomized and reduced subsets of the dataset, and results are averaged across models. More specifically, for an apartment transaction j within the subset of apartment sale prices yi, a vector of corresponding elevation and distance gradient terms xije and a vector of covariates xijc, we fit:   yîj=βe*xije*+βc*xijc*+εij, (1) where ^yij is the predicted selling price, and xije* and xijc* are subsets of xije and xijc whose coefficients are not reduced to exactly zero after fitting:   ̂yk=βexek+βcxck+ɛk (2) over observation k in the full dataset, subjected to a criterion of minimizing the sum of squares Pk(yk − y^k)2 subjected to the constraint (Pm|βem|+Pn|βcn|) ≤ s for a total of m elevation and distance terms and a total of n covariates, given tuning parameter s as determined by CV. Here, s controls the amount of penalization: when s is very large relative to model size, the constraint has no effect and the results are identical to an identically specified OLS fit. When s is small, a large proportion of terms in βe and βc are reduced to exactly zero. We subset the data by drawing fewer than the original number of observations for three purposes. First, we seek to prevent ‘data-point chasing’ behavior introduced by multiple higher-order terms on slope and distance effects: drawing fewer observations for each iteration without replacement decreases the potential structural similarity between each draw and the full dataset, providing conservative estimates of higher-order effects. Second, we are concerned by potential outliers in a standard sample-with-replacement bootstrapping approach. Using less data sampled over a greater number of regression iterations further reduces the risk of an iterative approach over-estimating confidence. Third, the subsetting of data inflates uncertainty by a factor inversely proportional to the fraction of data used. Since our results are intended to be presented graphically with bootstrapped standard errors, obtaining extremely conservative estimations of prediction boundaries is crucial. The data in part II are further selected for observations with a future station closer than 4000 m in distance and a current station further than 4000 m. This reduces the part II dataset to 4683 observation, all of which are subjected to final-station price effects that can be reliably estimated with results from part I.3 Conceptually, we do not expect current stations that are more than 4 km’ distance away to have any significant price effect and model them as being nonexistent. Four alternative models of a curve between the time gap to opening and relative value of the future station are fitted using an iterative scheme similar to that of part I, which we discuss in Section 4.2. 4.2. Data An in-depth overview of subsidized housing in Hong Kong can be found in Ye (2015). Semi-commercial housing programs in Hong Kong were originally devised to bridge the gap between commercial housing and public rental programs. Apartments are built by developers under guidelines of the Hong Kong Housing Authority (HKHA) and are purchased by eligible residents at a government-subsidized, ‘discounted’ rate. Eligibility is determined by a combination of factors including family size, income, and duration of residency in public rental units. The subsidy program provides an open-market platform for current owners, either to sell to other eligible buyers or to non-eligible buyers willing to ‘refund’ the original subsidy to the HKHA. Several features of subsidized apartments are noteworthy in the context of this paper. First, subsidized apartment residents are typically lower-middle to middle class in terms of income: as of 2003, approximately 54% of Hong Kong residents live in heavily subsidized, low-cost public rental units, and 31% live in commercial apartments or houses (Liu, 2003). Second, most subsidized apartments are sold to families of four or more individuals. Hence, residents are very likely to be working couples with at least one school-age child (Er and Li, 2008). Third, subsidized apartments generally are not located in the downtown area but rather are in suburbs and district sub-centres. Only 2.7% of all transactions in the dataset are located in downtown Hong Kong.4 These features present significant advantages for our data analysis. The strong homogeneity of subsidized apartments and their residents is an extraordinary benefit to the modelling of spatial amenities. Traditional predictors of price differences in homes, such as furnishing, number of bathrooms and number of bedrooms, can be assumed to be virtually identical across observations. Heterogeneity in non-spatial parameters is likely trivial after controlling for size, floor level and age, leading to models with high explanatory power: an OLS fit on the full model in part I generates an R2 value of 0.94. The expected income level and family size requirements of residents also suggest a reliance on public transit: individuals who qualify for subsidized apartments are most likely not wealthy enough to afford regular car use in Hong Kong. Transaction data are obtained from HKHA public records (HKHA, 2015). The data consist of 37,750 transactions between August 1997 and July 2014, accurate to the month of transaction with information on size, location and other characteristics of the apartment. Data from Google Maps, the OpenStreeMap project and Microsoft Representational State Transfer Services are pooled to create a detail geospatial profile of the HKSAR region that incorporates local amenities, road networks and transit availability. A 30-m resolution DEM is used to obtain altitude information at each latitude and longitude combination.5 We note that bus system data are not included. Data on bus lines in Hong Kong are difficult to obtain: there are multiple service providers, and the fact that not all stop names have consistent English transliterations presents a major challenge for Geocoding. Also, given Hong Kong’s traffic conditions we believe it unlikely that individual bus lines are viewed by the buyer as significant amenities. Bus accessibility in aggregate could be valuable: this should be highly correlated with local commercial activity, which we account for through a variety of geospatial control terms. Subway station entrances are approximated by intersecting a buffer of 50 m, centred on the point estimate of the station’s location, with a network layer of walkable roads.6 Apartment-to-station walking distances are calculated via the Closest Facility Analysis tool in the ArcGIS Network Analyst package, with routes also limited to walkable roads. If an apartment group’s centre is within the buffer zone of a station, a minimum distance of 50 m is assumed. Elevation gradients are approximated by the average, absolute decimal degree of incline between the location of the apartment and metro station. No distinction is made between positive and negative elevation differentials, under the assumption that trips to the station must go both ways. An estimate by evaluating the slope along a walking route would perhaps be preferable, but is not used because available road network profiles for Hong Kong are not designed to accommodate z-axis information or to be used in a three-dimension GIS frame. To measure anticipatory effects, five different network layers of metro stations are created in ArcGIS to reflect metro system development between 1997 and 2014. Each additional layer includes new stations from one major metro expansion. Two sets of distances are created for the apartment in each transaction: walking distance to the closest currently available station at the transaction date, and walking distance to the closest future station, if one exists between the transaction date and 2014. We also collect data on each transaction’s time gap between the transaction date and the future station opening date. Note that metro stations are assumed to be perfectly substitutable. We justify this decision for three reasons. First, the majority of apartments lies further away from central Hong Kong and residents typically only have access to one metro line connecting to the downtown area. This suggests that substitution effects are generally not between stations on different metro lines but between adjacent stations on the same line. Hence, conditional on distance from the CBD, the main commuting costs are incurred by travel time and effort to a given metro station, rather than differential on-rail commuting time depending on a particular metro stop. Second, the amenity advantage of specific stations is partially controlled for in the regression model via district boundary distance, housing density and a variety of centricity-based metrics. Although we cannot directly measure commercial activity along the apartment-station walking route, these variables will be highly correlated with the supply of local amenities and largely capture such effects. Third, assuming station homogeneity provides key advantages in analyzing anticipatory effects. With this assumption, price effects of stations on apartment sales can be reduced to a fixed premium defined by walking distance and slope, greatly reducing model complexity. In Section 5, we show that these two variables reasonably encapsulate the full price effect of stations. 5. Model checks and results Our analysis consists of two parts. Part I estimates price effects of elevation variance and distance gradients; part II utilizes these estimates to investigate future-station effects. For part I, a two-stage penalized regression model is fitted on the no-anticipatory-effect dataset with apartment-specific and geospatial controls as well as year, month and district fixed effects. Altitude from sea level and interactions among altitude, coastline distance and floor level are also controlled for as a proxy for scenery effects. For the full model, terms for slope and walking distance up to the third power and interaction terms up to quadratic effects are included for a total of 121 variables. A list of variables used in the full regression and summary information can be found in Table A3. The initial stage utilizes a LASSO regression as described in Section 4.1 with standardization and 50-fold CV training for variable selection. Using the resulting penalty term, 98 out of the 121 full-model variables, including 6 out of the 10 metro distance, slope and distance-slope interaction terms, survive the penalization scheme and are included in the second-stage OLS. A log-lambda traceplot of the CV procedure is provided in Figure A3. We note that conceptually there is no guarantee that this process consistently selects the same terms: if there are two weakly explanatory yet highly correlated variables in the full model, LASSO may be ambiguous between reducing the coefficient on one or the other. What the process does guarantee is stability: surviving terms are optimized for predictive performance. We provide a check for our LASSO procedure by comparing it with a conceptually equivalent alternative: a Bayesian LASSO model (Park and Casella, 2008) implemented via Markov Chain Monte Carlo (MCMC). Details and results are discussed in the Appendix A. In stage 2 we employ an iterative, reduced-data bootstrap regression approach using randomized data. 1000 OLS iterations are performed on the reduced model with a randomized, 20% subset of the no-anticipatory-effect data (6105 observations). Output is summarized in Figure A4 with null-hypothesis tests estimated by one-tail credible intervals derived from the draws. Bootstrap densities of select coefficients are displayed in Figure A7. Distributions of model R-square and adjusted R-square values are presented in Figure A6. A simple residual check is performed against age, log apartment size, selling price and walking distance, and results are provided in Figure A15. Predicted curves between slope and expected selling price per square meter are plotted for fixed walking distances of 500, 1000, 1500 and 2000 m, using 100 random samples from the 1000 sets of regression coefficients. As shown in Figure 3, the negative relationship between steepness and predicted selling price is extremely robust for walking distances up to 1500 m.7 The area covered by fitted curves can be considered as describing an interval where the ‘true’ fitted slope-price curve would fall with at least 99% certainty.8 Hence, we conclude that there is strong evidence that elevation gradient differentials influence selling prices of apartments at a range of walking distances. Figure 3 View largeDownload slide Fitted slope-estimated price curves with 99% credible intervals for fixed walking distance of 500–2000 m to metro station, degrees/HKD. Figure 3 View largeDownload slide Fitted slope-estimated price curves with 99% credible intervals for fixed walking distance of 500–2000 m to metro station, degrees/HKD. Holding other factors constant and at respective dataset averages, an apartments that is 500 m away from the closest metro station at a 10-degree average incline is expected to sell for 19.8% less than a similar apartment but on the same altitude level as the station. The corresponding expected discount is 19.9% for a 10-degree difference at 1000 m and 13.8% for an 8-degree difference at 1500 m. These effects are huge when taking walking distance into perspective: an apartment with a 500-m closest station at a 10-degree slope is expected to sell for 8% less than an otherwise similar apartment with a zero-slope closest station 2000 m away.9 These effects are likely to be exaggerated for two reasons: first, the average apartment-station slope is typically much smaller than the maximum slope. An average slope of 7–8 degrees likely suggests a route maximum of more than 15 degrees and hence significant difficulties in terms of pedestrian access. Second, the model in part I controls for a range of spatial benefits associated with higher altitude: apartments with higher altitude enjoy better views, better air quality and less noise and there is some evidence for ‘pure’ altitude and elevation-floor level interaction effects.10 Hence, the lack of metro station walkability is at least partially offset by other amenities for higher-altitude apartments. For part II, we use coefficients on slope and walking distance effects to generate an ‘equivalent’ future station value variable for each observation in the anticipatory-effect data. If the slope–distance combination of the future station lies within the boundaries of the dataset in part I, this estimated value ought to describe the real, perceived value of an opened station with reasonable accuracy. Unopened stations are expected to be comparatively of less value, with perceived worth decreasing proportionally to the time gap, Δt, between the transaction date and station opening. To explore this relationship, we fit four alternative models using modified versions of the full model from part I. Note that we do not perform variable selection for this part since predictive performance is not of primary concern: we care about the ‘true’ shape of the time-station value relationship as opposed to its most predictive shape given our data. Considering the sensitivity of treatment effects to time and the reduced temporal range of part II data (89 months), we replace year and month indicators in the full model with indicators of the specific time period of the transaction, 88 in total.11 All variables involving metro station gradients are replaced with the equivalent future station value and its interactions with Δt in months. Δt as a stand-alone effect is not included in the model, since there ought not to be a premium simply for being close to a certain point in time per se. Models are fitted against price with different numbers of higher-order interaction terms between the time gap and estimated future station value. The largest model includes interactions up to station value (Δt7); the other three models correspondingly limit Δt to the fifth, third and second power terms. To assess quality-of-fit, we present ‘pseudo’ credible intervals obtained by an n-fold bootstrap procedure analogous to that of part I.12 1000 OLS regressions are performed with all four models, with 20% of the data randomly omitted for each iteration. We summarize regression output in Tables A6, A7, A8 and A9. Table A10 contrasts coefficients on the future station value variable and interaction terms, and Figure A10 plots R2 and adjusted R2 distributions of the models. Note that there is little meaning in drawing inferences on the spatial control covariates in Tables A6–A9 individually. This is not only because of the large number of variables but also because of terms designed to approximate spatial factors that are more difficult to measure (distance to district boundary as proxy for local commercial buildup). With geospatial quantities inevitably being correlated (different estimates of centricity, for example), interpreting a particular spatial control’s coefficient in isolation will not result in useful results. This also holds true for the comparison between models in parts I and II because part II does not involve penalization. Our evidence for consistency between the regressions of the two parts is 3-fold. First, the overall explanatory power is highly robust for models of both parts. The part I bootstrap iterations have an average R2 value of 0.93, and the part II average R2 values are between 0.87 and 0.88 for all models. Second, we note that non-spatial terms in both parts—size, floor level indicators, discount rate—are consistent in terms of direction and significance. This suggests that both sets of models consistently evaluate basic preferences for housing consumption and amenities, and that preferences are likely similar between buyers in the data in the two parts. Third, all models in part II predict a ‘future’ station value of approximately 67–80% of the opened station value at a Δt of 0 months. Although we discuss at length the possible causes of failing to converge to full value at Δt = 0, the key finding is that after controlling for ‘futureness’, the parameters given by part I describe reasonably well how much individuals in the data of part II are willing to pay for a given, present subway station. This is our strongest case for consistency between models: conceptually, other discrepancies between the two datasets are not of significant concern if one part provides robust information about perceived metro station values of the other. Fitted results from 100 random coefficient set samples are presented in Figure 4. Similar to the estimated boundaries in Figure 3, regions defined by multiple curves approximate the range in which the ‘true’ fit lies with at least 99% statistical certainty. All four models suggest a significant, negative relationship between Δt and the equivalent future station value. The station effect does not seem to converge to zero for a sufficiently large Δt in any of the four models: the equivalent value is estimated to be approximately 25–45% of the opened station at 80 months prior to opening. Figure 4 View largeDownload slide Fitted slope (w/99% CI) between months to opening and perceived metro station value as % of opened station value, Hong Kong, 1997–2004. Figure 4 View largeDownload slide Fitted slope (w/99% CI) between months to opening and perceived metro station value as % of opened station value, Hong Kong, 1997–2004. The presence of higher-order effects is evident: best-fit curves for the first–seventh and first–fifth order power term models distinctly differ from those with only lower-order Δt terms. However, the inclusion of the sixth and seventh order terms appears to reduce certainty at large Δt values. Pseudo-error bounds increase marginally for larger models at all Δt, but the smallest model do not appear to be substantially more stable than its larger alternatives.13 Beyond comprehensively providing strong evidence of a negative, non-linear trend, none of the four model alternatives stands out as a clear choice.14 In response, we perform unweighted averages over these fitted curves for the following analyses. Failure to realize full value at zero months is likely caused by transactions being recorded at the time of finalization. Observed sale prices usually are determined prior to finalization, suggesting that the 0-month value does not describe that of an opened station but that of a station shortly before opening. This suggests an observed station opening ‘premium’ of approximately 20–30% of the station’s full, perceived price value. This is not unreasonable, given the fact that any period without metro access could be highly undesirable if commuting depend crucially on subway availability. Failure to converge to zero value at a large Δt of 70–80 months could be caused by several reasons. First, station value may come from expected future gains: the buyer may not be able to use the metro system during his or her tenure of ownership but may expect the apartment to nonetheless appreciate in price. Second, an announced location of a future station will generate commercial development, much of which will precede the opening of the actual station. Although access to the station is only possible after a given date, secondary effects of ongoing development can be enjoyed prior to station completion. Third, buyers may not have identical preferences. For example, some buyers may prefer transit by bus or to work close to home. An open-market scenario could mostly contain such buyers when a nearby station is not available in the near future. If strongly metro-preferring buyers do not have a major presence until the completion date draws close, prices of earlier transaction may reflect a premium paid by metro-indifferent (or even metro-disliking) buyers. Using the averaged fitted coefficients from each model, we estimate the relative appreciation of a station’s value for different time intervals. Moving from 6 years prior to opening to 4 years prior increases the station’s perceived value by approximately 82–92%. Moving from 4 years to 2 further increases its value by approximately 25–46%. At 2 years prior to opening, all four models suggest the station to be valued at 58–60% of the full, opened value. The estimated value increases to 61–67% at 6 months prior to opening, and to 66–80% at 0 months. Performing a simple average over the four models, we estimate the gain in perceived value of station at an average compound rate of 1.1% per month or 14% per year over a 6-year interval. If the future station is considered as an investment, given a 6-year-period the buyer can expect earnings of approximately 122.5% of the principal. In comparison, the 6-year earnings given local, real, annually compounded interest rates for years 1999–2004 is approximately 81.4%.15 It is unsurprising that these figures are roughly comparable: the possibility of choosing an apartment without anticipatory effects prevents futures stations from being priced too robustly, and arbitrage would prevent them from being priced too cheaply. The comparison also suggests that a future, close-by subway station, given the time period, is arguably an investment vehicle competitive with other financial assets. If this remains true today, there may be a case for purchasing apartments (with the intent to sell in the foreseeable future) in locations that are anticipating future subway access, with the current sale price premium serving as the principal of investment. However, the high ‘return’ of the future station may also reflect expect gains of commercial development near the station and expected future utility of using it, both of which will bias the rate of return to the infrastructure investment upwards. 6. Broader implications Neither elevation gradient nor anticipatory effects are trivial from the urban economic perspective. Anticipatory effects are particularly relevant for cities in developing economies experiencing high rates of infrastructure development. As an example, Beijing has added 17 new metro lines in the past 15 years, with 6 more planned for completion in the next 5 years. The Hong Kong metro system is scheduled to add three new lines and expand on six existing lines by the end of 2026. The extremely rapid public transit system buildup of these cities suggest that local housing markets are not only privy to information about anticipatory effects, but are likely to be actively pricing future stations into current transactions, subject to constraints on real rates of return. Time-trend modelling of such housing markets suffer if the overall price effect of new, major subway expansions is assumed to be applied at opening instead of accruing gradually for extended periods prior to completion. Similarly, stationary spatial models lose inference accuracy by severely under-fitting prices of homes in areas close to future stations. Hence, models that explicitly account for future conditions of urban systems are expected to perform significantly better in terms of analyzing current prices of spatial amenities. Beyond metro systems, the concept of future amenities exerting influence on current prices indicates the necessity of dynamic, temporal modelling processes, as outlined in this paper, in urban economics research. Since almost all investigations of housing and other price gradients are conducted in hindsight, information about what occurs after the period of investigative interest can be harnessed to address anticipatory effects. These effects may be of diverse origins: announcements of urban renewal, future schools, commercial centres or road network expansions all could affect current housing markets. Typically, hilliness is not explicitly controlled for in urban economic modelling. However, our results from Hong Kong demonstrate that elevation is a potentially huge determinant of housing prices. Spatial models that involve high elevation–variance cities, especially those with unevenness somewhat comparable to Hong Kong, are susceptible to large residual issues over spatial parameters: home price observations will be systemically over-estimated or under-estimated depending on the local degree of hilliness. The more uneven a city, the more severe such mis-estimations are likely to be. Furthermore, elevation gradient effects demonstrated in this paper suggest that cities with high levels of elevation variance, ceteris paribus, will benefit less from the same level of public transit infrastructure. Assuming that individuals have a fixed level of tolerance for difficulty of walking to a station for daily commuting, fewer individuals will use public transit in the area with a given proximity to a station in highly uneven areas. Walk-oriented public transit hubs, if located in areas with significant terrain unevenness, should correspondingly be valued less by local home-owners. These effects suggest the potential for optimization: hilly, high-unevenness cities face different constraints when developing public transit networks. Because providing the same level of consumed public transit infrastructure is more costly in uneven cities, such cities may have public transit amenity equilibria below those of flat cities. By better understanding the influence of elevation gradients on preferences and behavior of urban markets, solutions could be designed to minimize the adverse impact of unevenness. Elevation gradient effects for metro stations also indicate similar behavior for other common destinations reached by walking: for example, schools, parks, and bus stops. With neighboring amenities decreasing in perceived value as elevation differentials increase, elevation variance of cities are expected to predict other, more important urban gradients such as population and housing density. As an addition to existing literature (Saiz, 2010; Ye and Becker, 2016), we present a simple analysis of the relationship between density gradients and unevenness of urban areas in the Appendix C. 7. Conclusion Using extensive geospatial methods, we provide evidence of significant anticipatory future-station effects in the price influence of metro stations on urban housing. We estimate that for the Hong Kong subsidized market, the average growth trend of the perceived price value is approximately 1.1% of the full station value per month, starting from 6 years prior to completion. Paying for a future, close-by metro station yields returns on investment that are roughly consistent with and marginally superior to period interest rates. We also present evidence for highly significant, non-linear elevation gradient effects on the walking distance between metro stations and apartments. We find that elevation differentials between stations and apartments are a huge determining factor in selling prices of apartments. Although in reality these effects are partially offset by benefits associated with higher elevation, they nonetheless introduce significant implications in terms of geospatial modelling methodologies, investigations of urban housing and public transit systems, as well as relationships between elevation and urban gradients in general. This paper reveals two significant shortcomings of the conventional, bi-dimensional spatial modelling approach: the omission of temporal and elevation-variance-related elements. Models of rapidly developing cities suffer if not considered in a dynamic framework inclusive of anticipatory effects of future amenities; models of uneven cities suffer without the augmentation of elevation profiles in spatial parameters. We conclude that without explicitly addressing these effects in a multi-dimensional approach, investigations of spatially sensitive economic effects are highly susceptible to significant and systematic residual issues. In light of recent literature on geographical effects in urban systems which suggest that a large range of urban gradients are influenced by elevation patterns, the transaction-level analysis presented in this paper allows for a better understanding of how elevation effects at the district and neighborhood-level, as well as at the multi-city level, originate. By estimating the aversion of elevation disparities caused by difficulty in accessing public transit, our work provides a conceptual basis for the role that terrain unevenness plays in urban density and population gradients. Footnotes 1 Two other price residual-contour plots are available in Figure A1 and Figure A2. 2 Table A1 summarizes descriptive figures of transactions in this data subset. 3 Table A2 summarizes descriptive figures of transactions in this dataset. 4 Central Hong Kong is defined as the following five districts: Wan Chai, Central, Yau Tsim Mong, Sham Shui Po and Kowloon City. 5 A three-dimensional contour plot demonstrating the DEM resolution and the extreme level of terrain unevenness of the HKSAR region is provided in Figure A12; Figure A13 presents a map of Hong Kong with locations of apartment groups, select amenities and walkable roads. 6 The 50-m buffer radius is re-applied to the final station-to-apartment distance estimates. 7 A comparison of fitted slope-sale price curves of the four walking distances is available in Figure A14. Densities of predicted selling price for select combinations of slope and distance corresponding to Figure 3 are presented in Figure A9. 8 Given that 20% of the data are used, under normality assumptions we can estimate that the standard deviation, as reflected in Figure 3 is inflated by a factor of 2.2. 9 Conceptually, it should be possible that elevation effects vary by time of year since perceived difficulty of walking would be greater during certain seasons. However, we find that this is likely not the case. A detailed discussion is provided in the Appendix B. 10 The pure altitude term reports p = 0.034. p = 0.08 for the interaction term between the indicator for floor levels ≥27 and elevation. 11 Transactions that occur after 2004 do not experience metro system expansion and are therefore not subjected to a change in the distance of the closest station. 12 This is because we have no knowledge of what the ‘true’ curve shape is under these circumstances. For instance, we do not know if housing market shocks may change perceived values of future spatial amenities, perhaps leading to non-linear fits with multiple local maxima. 13 An alternative visualization to Figure 4 is presented in Figure A11 with densities of predicted equivalent station values for each model contrasted between several time gaps. 14 The first–fifth order power term model has the highest overall number of highly significant interaction terms (all p < 0.001). However, this is only evidence that this model most robustly rejects the null hypothesis of there being no anticipatory effect at all. 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Table A1 Summary statistics on select characteristics of apartments, Part I no-anticipatory effect data Name  Mean  SD  Min  Max  Age (years)  12.53  5.34  1  33  Size (m2)  49.78  7.81  12  86  Driving distance to central Hong Kong (km)  19.53  10.36  6.9  59  Driving time to central Hong Kong (min)  21.65  8.49  9  63  Price per m2 ($HKD)  28,798.9  14,595.37  2398  105428  Walking distance to closest station (m)  1422.6  1353.22  52.49  13,274.3  Name  Mean  SD  Min  Max  Age (years)  12.53  5.34  1  33  Size (m2)  49.78  7.81  12  86  Driving distance to central Hong Kong (km)  19.53  10.36  6.9  59  Driving time to central Hong Kong (min)  21.65  8.49  9  63  Price per m2 ($HKD)  28,798.9  14,595.37  2398  105428  Walking distance to closest station (m)  1422.6  1353.22  52.49  13,274.3  Table A2 Summary statistics on select characteristics of apartments, Part II anticipatory-effect data Name  Mean  SD  Min  Max  Age (years)  8.41  2.70  2  20  Size (m2)  49.92  7.19  20  90  Driving distance to central Hong Kong (km)  29.46  7.92  10  39  Driving time to central Hong Kong (min)  30.04  5.79  13  37  Price per m2 ($HKD)  19,410.14  5653.74  3337  54,896  Walking distance to closest (final) station (m)  1429.15  745.56  52.49  3613.87  Name  Mean  SD  Min  Max  Age (years)  8.41  2.70  2  20  Size (m2)  49.92  7.19  20  90  Driving distance to central Hong Kong (km)  29.46  7.92  10  39  Driving time to central Hong Kong (min)  30.04  5.79  13  37  Price per m2 ($HKD)  19,410.14  5653.74  3337  54,896  Walking distance to closest (final) station (m)  1429.15  745.56  52.49  3613.87  Because the classic Bayesian LASSO as implemented by Park and Casella does not reduce any of the coefficients to exactly zero, we choose an alternative Reversible Jump MCMC-based model modified by Gramacy (2014) from Hans (2009) specifically for the purpose of variable selection. Using 10,000 MCMC iterations and a burn-in of 2000 draws, the algorithm returns an average post-penalization model size of 103 variables. Convergence of the MCMC process is demonstrated by trace plots of key variables in Figure A4, and a trace plot of the size of the best model selected in each iteration is provided in A5. Although the two approaches do not perfectly agree on the number of variables that should be removed, evidence from the Bayesian approach strongly agrees with the regular LASSO in terms of post-penalization model size: both procedures suggest eliminating approximately 20 variables from the full model. This result suggests that the explanatory power gains to be had from a significantly larger model would not compensate for the loss of predictive stability and that the stability gains of a significantly smaller model would not sufficiently compensate for the loss of explanatory power. Here, we also illustrate the distinction between optimizing for general model performance versus predictive ability by presenting a comparison between the regular LASSO and forward–backward stepwise variable selection. Stepwise AIC yields a larger model size of 117 variables and stepwise BIC a model size of 101 variables, comparable in size to the LASSO models. Models selected by AIC and BIC are also regressed with reduced-data bootstrapping. As shown in Figure A8, stability of key variables in models determined by stepwise processes are not significantly better than that of the full model. In contrast, LASSO omits variables from the full model in a way that pulls surviving coefficients into substantially denser distributions. Therefore, given new data with different combinations of slope and walking distance, the LASSO will return predicted price effects that are much more consistent. Table A3 Names and information of variables in Part I full model Name  Variable meaning  Priceadjtwothird  Selling price of apartment per m2, transformed to 1.5th power  Walkmtractual  Walking distance to closest metro station, m  Walkmtractualsqr  Walking distance to closest metro station^2, m  walkmtractual3  Walking distance to closest metro station^3, m  Walkmtrangle  Slope between apartment and closest station, degrees  walkmtranglesqr  Slope between apartment and closest station^2, degrees  walkmtrangle3  Slope between apartment and closest station^3, degrees  mtrwalkINTangle1  Closest station slope * station walking distance  mtrwalk2INTangle2  Closest station slope^2 * station walking distance^2  mtrwalk1INTangle2  Closest station slope^2 * station walking distance  mtrwalk2INTangle1  Closest station slope * station walking distance^2  Feb  Indicator for transaction being in February  Mar  Indicator for transaction being in March  Apr  Indicator for transaction being in April  May  Indicator for transaction being in May  Jun  Indicator for transaction being in June  Jul  Indicator for transaction being in July  Aug  Indicator for transaction being in August  Sep  Indicator for transaction being in September  Oct  Indicator for transaction being in October  Nov  Indicator for transaction being in November  Dec  Indicator for transaction being in December  yd1998  Year indicator for being in 1998  yd1999  Year indicator for being in 1999  yd2000  Year indicator for being in 2000  yd2001  Year indicator for being in 2001  yd2002  Year indicator for being in 2002  yd2003  Year indicator for being in 2003  yd2004  Year indicator for being in 2004  yd2005  Year indicator for being in 2005  yd2006  Year indicator for being in 2006  yd2007  Year indicator for being in 2007  yd2008  Year indicator for being in 2008  yd2009  Year indicator for being in 2009  yd2010  Year indicator for being in 2010  yd2011  Year indicator for being in 2011  yd2012  Year indicator for being in 2012  yd2013  Year indicator for being in 2013  yd2014  Year indicator for being in 2014  Age  Age of apartment, years  Sqrage  Age of apartment^2, years  Age3  Age of apartment^3, years  Age4  Age of apartment^4, years  Age5  Age of apartment^5, years  Age6  Age of apartment^6, years  age7  Age of apartment^7, years  age8  Age of apartment^8, years  Lgsize  Size of apartment, m2, logged  Floorm  14–26 floors in height  Floorh  >26 floors in height  discountrate  Percentage amount which apartment is subsidized  intdiscount1998  discountrate * yd1998  intdiscount1999  discountrate * yd1999  intdiscount2000  discountrate * yd2000  intdiscount2001  discountrate * yd2001  intdiscount2002  discountrate * yd2002  intdiscount2003  discountrate * yd2003  intdiscount2004  discountrate * yd2004  intdiscount2005  discountrate * yd2005  intdiscount2006  discountrate * yd2006  intdiscount2007  discountrate * yd2007  intdiscount2008  discountrate * yd2008  intdiscount2009  discountrate * yd2009  intdiscount2010  discountrate * yd2010  intdiscount2011  discountrate * yd2011  intdiscount2012  discountrate * yd2012  intdiscount2013  discountrate * yd2013  intdiscount2014  discountrate * yd2014  Name  Variable meaning  Unluckynum  Indicator for apartment with number 4 or 13 in address  luckynum  Indicator for apartment with number 8 in address  lgdtoNoSec  Distance to nearest major road, de ned as highways or four lanes and above, km, logged  Lgdtocentralcar  Logged driving distance to central Hong Kong by automobile, km  INTperioddtocentral  Time period of transaction in months * Logged driving distance to central  INTperioddtocentral2  Time period of transaction in months * Logged driving distance to central^2  Lgttocentralcar  Logged driving time to central Hong Kong by automobile, minutes  INTperiodttocentral  Time period of transaction in months * Logged driving time to central  INTperiodttocentral2  Time period of transaction in months * Logged driving time to central^2  INTperiodttocentral3  Time period of transaction in months * Logged driving time to central^3  Lgttocentralpublic  Logged public transit time to central Hong Kong, minutes  INTperiodcentralpublic  Time period of transaction in months * Logged time by public transit  INTperiodcentralpublic2  Time period of transaction in months * Logged time by public transit^2  INTperiodcentralpublic3  Time period of transaction in months * Logged time by public transit^3  lgdtoborder  Distance to nearest major road, de ned as highways or four lanes and above, km, logged  lgincome  Logged income of district where apartment is located in, $HKD  INTincomeborder  Distance from apartment to district border * logged income level of district of apartment  lgelev  Logged elevation level of apartment, m  lgcoast  Logged distance to coast, m  intcoastelev  Coastline distance * elevation  intcoast floorh  Coastline distance * floorh  intcoast floorm  Coastline distance * floorm  intelev floorh  elevation * floorh  intelev floorm  elevation * floorm  countdss  Number of (Direct Subsidy Scheme) DSS middle schools in a 3 km radius of the apartment  INTperiodDSS  Time period of transaction * number of DSS middle schools  INTperiodDSS2  Time period of transaction * number of DSS middle schools^2  INTperiodDSS3  Time period of transaction * number of DSS middle schools^3  lgwalkelementary  Logged walking distance to nearest elementary school, m  INTperiodelementary  Time period of transaction in months * logged walking distance to nearest elementary school  INTperiodelementary2  Time period of transaction in months * logged walking distance to nearest elementary school^2  slopeelementary  Average slope between apartment and nearest elementary school, degrees  INTslopewalkelem  Average slope between apartment and nearest elementary school * logged walking distance to nearest elementary school  airport1km  Indicator for being closer than 1 km to the airport  Courtsize  Size of apartment group of apartment, number of units  courtsizesqr  Size of apartment group^2  totaldensity3km10000  Total number of apartments in 3-km radius from apartment  densityprivate3km  Number of commercial apartments in 3-km    radius from apartment  densitypublic3km  Number of public rental apartments in 3-km    radius from apartment  subdensityper3km  Proportion of apartments in 3-km radius    that are subsidized-for-purchase, %  districtcode  Indicators for district-fixed effects (15 in total)  Name  Variable meaning  Priceadjtwothird  Selling price of apartment per m2, transformed to 1.5th power  Walkmtractual  Walking distance to closest metro station, m  Walkmtractualsqr  Walking distance to closest metro station^2, m  walkmtractual3  Walking distance to closest metro station^3, m  Walkmtrangle  Slope between apartment and closest station, degrees  walkmtranglesqr  Slope between apartment and closest station^2, degrees  walkmtrangle3  Slope between apartment and closest station^3, degrees  mtrwalkINTangle1  Closest station slope * station walking distance  mtrwalk2INTangle2  Closest station slope^2 * station walking distance^2  mtrwalk1INTangle2  Closest station slope^2 * station walking distance  mtrwalk2INTangle1  Closest station slope * station walking distance^2  Feb  Indicator for transaction being in February  Mar  Indicator for transaction being in March  Apr  Indicator for transaction being in April  May  Indicator for transaction being in May  Jun  Indicator for transaction being in June  Jul  Indicator for transaction being in July  Aug  Indicator for transaction being in August  Sep  Indicator for transaction being in September  Oct  Indicator for transaction being in October  Nov  Indicator for transaction being in November  Dec  Indicator for transaction being in December  yd1998  Year indicator for being in 1998  yd1999  Year indicator for being in 1999  yd2000  Year indicator for being in 2000  yd2001  Year indicator for being in 2001  yd2002  Year indicator for being in 2002  yd2003  Year indicator for being in 2003  yd2004  Year indicator for being in 2004  yd2005  Year indicator for being in 2005  yd2006  Year indicator for being in 2006  yd2007  Year indicator for being in 2007  yd2008  Year indicator for being in 2008  yd2009  Year indicator for being in 2009  yd2010  Year indicator for being in 2010  yd2011  Year indicator for being in 2011  yd2012  Year indicator for being in 2012  yd2013  Year indicator for being in 2013  yd2014  Year indicator for being in 2014  Age  Age of apartment, years  Sqrage  Age of apartment^2, years  Age3  Age of apartment^3, years  Age4  Age of apartment^4, years  Age5  Age of apartment^5, years  Age6  Age of apartment^6, years  age7  Age of apartment^7, years  age8  Age of apartment^8, years  Lgsize  Size of apartment, m2, logged  Floorm  14–26 floors in height  Floorh  >26 floors in height  discountrate  Percentage amount which apartment is subsidized  intdiscount1998  discountrate * yd1998  intdiscount1999  discountrate * yd1999  intdiscount2000  discountrate * yd2000  intdiscount2001  discountrate * yd2001  intdiscount2002  discountrate * yd2002  intdiscount2003  discountrate * yd2003  intdiscount2004  discountrate * yd2004  intdiscount2005  discountrate * yd2005  intdiscount2006  discountrate * yd2006  intdiscount2007  discountrate * yd2007  intdiscount2008  discountrate * yd2008  intdiscount2009  discountrate * yd2009  intdiscount2010  discountrate * yd2010  intdiscount2011  discountrate * yd2011  intdiscount2012  discountrate * yd2012  intdiscount2013  discountrate * yd2013  intdiscount2014  discountrate * yd2014  Name  Variable meaning  Unluckynum  Indicator for apartment with number 4 or 13 in address  luckynum  Indicator for apartment with number 8 in address  lgdtoNoSec  Distance to nearest major road, de ned as highways or four lanes and above, km, logged  Lgdtocentralcar  Logged driving distance to central Hong Kong by automobile, km  INTperioddtocentral  Time period of transaction in months * Logged driving distance to central  INTperioddtocentral2  Time period of transaction in months * Logged driving distance to central^2  Lgttocentralcar  Logged driving time to central Hong Kong by automobile, minutes  INTperiodttocentral  Time period of transaction in months * Logged driving time to central  INTperiodttocentral2  Time period of transaction in months * Logged driving time to central^2  INTperiodttocentral3  Time period of transaction in months * Logged driving time to central^3  Lgttocentralpublic  Logged public transit time to central Hong Kong, minutes  INTperiodcentralpublic  Time period of transaction in months * Logged time by public transit  INTperiodcentralpublic2  Time period of transaction in months * Logged time by public transit^2  INTperiodcentralpublic3  Time period of transaction in months * Logged time by public transit^3  lgdtoborder  Distance to nearest major road, de ned as highways or four lanes and above, km, logged  lgincome  Logged income of district where apartment is located in, $HKD  INTincomeborder  Distance from apartment to district border * logged income level of district of apartment  lgelev  Logged elevation level of apartment, m  lgcoast  Logged distance to coast, m  intcoastelev  Coastline distance * elevation  intcoast floorh  Coastline distance * floorh  intcoast floorm  Coastline distance * floorm  intelev floorh  elevation * floorh  intelev floorm  elevation * floorm  countdss  Number of (Direct Subsidy Scheme) DSS middle schools in a 3 km radius of the apartment  INTperiodDSS  Time period of transaction * number of DSS middle schools  INTperiodDSS2  Time period of transaction * number of DSS middle schools^2  INTperiodDSS3  Time period of transaction * number of DSS middle schools^3  lgwalkelementary  Logged walking distance to nearest elementary school, m  INTperiodelementary  Time period of transaction in months * logged walking distance to nearest elementary school  INTperiodelementary2  Time period of transaction in months * logged walking distance to nearest elementary school^2  slopeelementary  Average slope between apartment and nearest elementary school, degrees  INTslopewalkelem  Average slope between apartment and nearest elementary school * logged walking distance to nearest elementary school  airport1km  Indicator for being closer than 1 km to the airport  Courtsize  Size of apartment group of apartment, number of units  courtsizesqr  Size of apartment group^2  totaldensity3km10000  Total number of apartments in 3-km radius from apartment  densityprivate3km  Number of commercial apartments in 3-km    radius from apartment  densitypublic3km  Number of public rental apartments in 3-km    radius from apartment  subdensityper3km  Proportion of apartments in 3-km radius    that are subsidized-for-purchase, %  districtcode  Indicators for district-fixed effects (15 in total)  Table A4 Coefficients and significance of Part I regression, excluding year, month, district indicators and year-discount rate interactions Variable  Coefficient  Bootstrap p-value  Variable  Coefficient  Bootstrap P-val  (Intercept)  −1.33E+03  0.008  INTperiodcentralpublic3  4.73E−05  <0.001  walkmtractual  −5.89E−02  <0.001  lgdtoborder  1.74E+02  0.026  walkmtractual3  3.42E−10  <0.001  lgincome  2.98E+02  <0.001  walkmtrangle  −9.83E+00  <0.001  INTincomeborder  −2.01E+01  0.014  mtrwalkINTangle1  −1.14E−02  <0.001  lgelev  5.77E+00  0.034  mtrwalk2INTangle2  1.90E−07  0.028  lgcoast  7.91E+00  0.004  mtrwalk2INTangle1  6.12E−06  <0.001  intelev floorh  4.15E+00  0.08  Age  −1.41E+01  <0.001  intelev floorm  2.86E+00  0.145  age8  7.31E−11  0.002  countdss  −9.35E−01  0.027  Lgsize  1.45E+02  <0.001  INTperiodDSS  4.93E−03  0.199  Floorm  2.90E+01  0.002  INTperiodDSS3  −6.48E−08  0.315  Floorh  4.00E+01  <0.001  lgwalkelementary  3.65E+01  <0.001  discountrate  −5.40E+00  <0.001  INTperiodelementary  −4.08E−01  <0.001  unluckynum  −1.89E+01  0.002  INTperiodelementary2  1.96E−03  <0.001  Luckynum  1.35E+01  <0.001  slopeelementary  3.89E+01  <0.001  lgdtoNoSec  4.15E+00  <0.001  INTslopewalkelem  −7.09E+00  <0.001  lgdtocentralcar  −6.69E+01  0.009  airport1km  9.64E+01  0.221  INTperioddtocentral  1.73E+00  <0.001  courtsize  1.19E−02  <0.001  INTperioddtocentral2  −8.46E−03  <0.001  courtsizesqr  −2.75E−06  <0.001  lgttocentralcar  −8.62E+00  0.399  totaldensity3km10000  −6.95E+00  <0.001  INTperiodttocentral  −1.19E+00  <0.001  subdensityper3km  −1.28E+02  <0.001  lgttocentralpublic  −1.53E+02  <0.001  densityprivate3km  6.89E−04  <0.001  INTperiodcentralpublic2  −5.57E−03  0.001        Variable  Coefficient  Bootstrap p-value  Variable  Coefficient  Bootstrap P-val  (Intercept)  −1.33E+03  0.008  INTperiodcentralpublic3  4.73E−05  <0.001  walkmtractual  −5.89E−02  <0.001  lgdtoborder  1.74E+02  0.026  walkmtractual3  3.42E−10  <0.001  lgincome  2.98E+02  <0.001  walkmtrangle  −9.83E+00  <0.001  INTincomeborder  −2.01E+01  0.014  mtrwalkINTangle1  −1.14E−02  <0.001  lgelev  5.77E+00  0.034  mtrwalk2INTangle2  1.90E−07  0.028  lgcoast  7.91E+00  0.004  mtrwalk2INTangle1  6.12E−06  <0.001  intelev floorh  4.15E+00  0.08  Age  −1.41E+01  <0.001  intelev floorm  2.86E+00  0.145  age8  7.31E−11  0.002  countdss  −9.35E−01  0.027  Lgsize  1.45E+02  <0.001  INTperiodDSS  4.93E−03  0.199  Floorm  2.90E+01  0.002  INTperiodDSS3  −6.48E−08  0.315  Floorh  4.00E+01  <0.001  lgwalkelementary  3.65E+01  <0.001  discountrate  −5.40E+00  <0.001  INTperiodelementary  −4.08E−01  <0.001  unluckynum  −1.89E+01  0.002  INTperiodelementary2  1.96E−03  <0.001  Luckynum  1.35E+01  <0.001  slopeelementary  3.89E+01  <0.001  lgdtoNoSec  4.15E+00  <0.001  INTslopewalkelem  −7.09E+00  <0.001  lgdtocentralcar  −6.69E+01  0.009  airport1km  9.64E+01  0.221  INTperioddtocentral  1.73E+00  <0.001  courtsize  1.19E−02  <0.001  INTperioddtocentral2  −8.46E−03  <0.001  courtsizesqr  −2.75E−06  <0.001  lgttocentralcar  −8.62E+00  0.399  totaldensity3km10000  −6.95E+00  <0.001  INTperiodttocentral  −1.19E+00  <0.001  subdensityper3km  −1.28E+02  <0.001  lgttocentralpublic  −1.53E+02  <0.001  densityprivate3km  6.89E−04  <0.001  INTperiodcentralpublic2  −5.57E−03  0.001        Table A5 Names and information of additional variables in Part II models Name  Variable meaning  mtrpremt1  The equivalent premium of future station in $HKD, obtained by predicting the price premium on the apartment of the future station given its walking distance, slope, and Coefficients from the Part I regression Model  mtrpremINTtime(n)  mtrpremt1 interacted with the time gap (months) to the nth power.  gcode(n)  Indicator for the transaction being in the nth time period (month) of the dataset time span  Name  Variable meaning  mtrpremt1  The equivalent premium of future station in $HKD, obtained by predicting the price premium on the apartment of the future station given its walking distance, slope, and Coefficients from the Part I regression Model  mtrpremINTtime(n)  mtrpremt1 interacted with the time gap (months) to the nth power.  gcode(n)  Indicator for the transaction being in the nth time period (month) of the dataset time span  Table A6 Coefficients and significance of Part II regression for model with interaction terms up to t7, excluding district and time period indicator effects Variable  Coefficient  Bootstrap p-val  Variable  Coefficient  Bootstrap p-val  (Intercept)  4.06E+03  0.018  lgttocentralpublic  −3.08E+02  <0.001  mtrpremt1  7.79E−01  <0.001  INTperiodcentralpublic  −6.34E−01  0.424  mtrpremINTtime1  −5.20E−02  0.03  INTperiodcentralpublic2  9.35E−02  0.125  mtrpremINTtime2  5.04E−03  0.069  INTperiodcentralpublic3  −8.16E−04  0.103  mtrpremINTtime3  −2.58E−04  0.085  lgdtoborder  −1.10E+02  0.301  mtrpremINTtime4  7.82E−06  0.078  lgincome  −2.10E+02  0.115  mtrpremINTtime5  −1.38E−07  0.054  INTincomeborder  9.44E+00  0.326  mtrpremINTtime6  1.26E−09  0.032  lgelev  3.46E+01  0.14  mtrpremINTtime7  −4.56E−12  0.019  lgcoast  −2.84E+01  0.031  Age  1.59E+02  0.291  intcoastelev  −4.20E+00  0.155  Sqrage  −8.28E+01  0.25  intcoast floorh  −6.53E+00  <0.001  age3  2.20E+01  0.219  intcoast floorm  −6.60E+00  <0.001  age4  −3.46E+00  0.186  intelev floorh  −5.95E+00  <0.001  age5  3.26E−01  0.164  intelev floorm  −5.04E+00  <0.001  age6  −1.80E−02  0.148  countdss  −2.02E+01  <0.001  age7  5.37E−04  0.143  INTperiodDSS  6.65E−01  <0.001  age8  −6.64E−06  0.13  INTperiodDSS2  −1.14E−02  <0.001  Lgsize  1.28E+02  <0.001  INTperiodDSS3  6.88E−05  0.007  Floorm  8.74E+01  <0.001  lgwalkelementary  4.01E+00  0.171  Floorh  1.00E+02  <0.001  INTperiodelementary  5.41E−02  0.359  discountrate  −1.47E+00  <0.001  INTperiodelementary2  −7.47E−04  0.336  unluckynum  7.27E+01  <0.001  slopeelementary  9.23E+00  0.022  Luckynum  −2.73E+00  0.255  INTslopewalkelem  −1.24E+00  0.055  lgdtoNoSec  2.25E+00  0.023  airport1km  —  —  lgdtocentralcar  8.00E+01  0.215  courtsize  5.10E−02  <0.001  INTperioddtocentral  2.95E+01  <0.001  courtsizesqr  −9.63E−06  <0.001  INTperioddtocentral2  −2.74E−01  <0.001  totaldensity3km  7.40E−03  <0.001  lgttocentralcar  5.03E+01  0.359  subdensityper3km  −2.11E+02  <0.001  INTperiodttocentral  −1.34E+01  0.027  densitypublic3km  −1.27E−02  <0.001  INTperiodttocentral2  −7.80E−02  0.231  densityprivate3km  −8.65E−03  <0.001  INTperiodttocentral3  2.42E−03  <0.001        Variable  Coefficient  Bootstrap p-val  Variable  Coefficient  Bootstrap p-val  (Intercept)  4.06E+03  0.018  lgttocentralpublic  −3.08E+02  <0.001  mtrpremt1  7.79E−01  <0.001  INTperiodcentralpublic  −6.34E−01  0.424  mtrpremINTtime1  −5.20E−02  0.03  INTperiodcentralpublic2  9.35E−02  0.125  mtrpremINTtime2  5.04E−03  0.069  INTperiodcentralpublic3  −8.16E−04  0.103  mtrpremINTtime3  −2.58E−04  0.085  lgdtoborder  −1.10E+02  0.301  mtrpremINTtime4  7.82E−06  0.078  lgincome  −2.10E+02  0.115  mtrpremINTtime5  −1.38E−07  0.054  INTincomeborder  9.44E+00  0.326  mtrpremINTtime6  1.26E−09  0.032  lgelev  3.46E+01  0.14  mtrpremINTtime7  −4.56E−12  0.019  lgcoast  −2.84E+01  0.031  Age  1.59E+02  0.291  intcoastelev  −4.20E+00  0.155  Sqrage  −8.28E+01  0.25  intcoast floorh  −6.53E+00  <0.001  age3  2.20E+01  0.219  intcoast floorm  −6.60E+00  <0.001  age4  −3.46E+00  0.186  intelev floorh  −5.95E+00  <0.001  age5  3.26E−01  0.164  intelev floorm  −5.04E+00  <0.001  age6  −1.80E−02  0.148  countdss  −2.02E+01  <0.001  age7  5.37E−04  0.143  INTperiodDSS  6.65E−01  <0.001  age8  −6.64E−06  0.13  INTperiodDSS2  −1.14E−02  <0.001  Lgsize  1.28E+02  <0.001  INTperiodDSS3  6.88E−05  0.007  Floorm  8.74E+01  <0.001  lgwalkelementary  4.01E+00  0.171  Floorh  1.00E+02  <0.001  INTperiodelementary  5.41E−02  0.359  discountrate  −1.47E+00  <0.001  INTperiodelementary2  −7.47E−04  0.336  unluckynum  7.27E+01  <0.001  slopeelementary  9.23E+00  0.022  Luckynum  −2.73E+00  0.255  INTslopewalkelem  −1.24E+00  0.055  lgdtoNoSec  2.25E+00  0.023  airport1km  —  —  lgdtocentralcar  8.00E+01  0.215  courtsize  5.10E−02  <0.001  INTperioddtocentral  2.95E+01  <0.001  courtsizesqr  −9.63E−06  <0.001  INTperioddtocentral2  −2.74E−01  <0.001  totaldensity3km  7.40E−03  <0.001  lgttocentralcar  5.03E+01  0.359  subdensityper3km  −2.11E+02  <0.001  INTperiodttocentral  −1.34E+01  0.027  densitypublic3km  −1.27E−02  <0.001  INTperiodttocentral2  −7.80E−02  0.231  densityprivate3km  −8.65E−03  <0.001  INTperiodttocentral3  2.42E−03  <0.001        Table A7 Coefficients and significance of Part II regression for model with interaction terms up to t5, excluding district and time period indicator effects Variable  Coefficient  Bootstrap p-val  Variable  Coefficient  Bootstrap p-val  (Intercept)  3.93E+03  0.024  INTperiodcentralpublic  −1.57E+00  0.338  mtrpremt1v3  8.02E−01  <0.001  INTperiodcentralpublic2  1.20E−01  0.077  mtrpremINTtime1  −5.07E−02  <0.001  INTperiodcentralpublic3  −1.01E−03  0.053  mtrpremINTtime2  3.45E−03  <0.001  lgdtoborder  −7.41E+01  0.357  mtrpremINTtime3  −9.29E−05  <0.001  lgincome  −1.96E+02  0.136  mtrpremINTtime4  1.00E−06  <0.001  INTincomeborder  5.70E+00  0.387  mtrpremINTtime5  −3.66E−09  0.001    3.44E+01  0.143  Age  1.71E+02  0.291    −2.82E+01  0.037  Sqrage  −8.50E+01  0.249  intcoastelev  −4.17E+00  0.161  age3  2.20E+01  0.227  intcoast floorh  −6.56E+00  <0.001  age4  −3.40E+00  0.201  intcoast floorm  −6.54E+00  <0.001  age5  3.19E−01  0.173  intelev floorh  −5.94E+00  <0.001  age6  −1.76E−02    intelev floorm  −4.98E+00  <0.001  age7  5.24E−04    countdss  −2.07E+01  <0.001  age8  −6.51E−06  0.131  INTperiodDSS  6.88E−01  <0.001  Lgsize  1.27E+02  <0.001  INTperiodDSS2  −1.20E−02  <0.001  Floorm  8.69E+01  <0.001  INTperiodDSS3  7.36E−05  0.003  Floorh  1.01E+02  <0.001  lgwalkelementary  4.42E+00  0.16  discountrate  −1.48E+00  <0.001  INTperiodelementary  3.97E−02  0.411  unluckynum  7.32E+01  <0.001  INTperiodelementary2  −6.04E−04  0.388  Luckynum  −2.48E+00  0.274  slopeelementary  9.28E+00  0.022  lgdtoNoSec  2.35E+00  0.016  INTslopewalkelem  −1.24E+00  0.054  lgdtocentralcar  8.81E+01  0.175  airport1km  —  —  INTperioddtocentral  2.92E+01  <0.001  courtsize  5.15E−02  <0.001  INTperioddtocentral2  −2.69E−01  <0.001  courtsizesqr  −9.76E−06  <0.001  lgttocentralcar  3.27E+01  0.385  totaldensity3km  7.48E−03  <0.001  INTperiodttocentral  −1.17E+01  0.047  subdensityper3km  −2.10E+02  <0.001  INTperiodttocentral2  −1.19E−01  0.125  densitypublic3km  −1.29E−02  <0.001  INTperiodttocentral3  2.69E−03  <0.001  densityprivate3km  −8.73E−03  <0.001  lgttocentralpublic  −3.03E+02  <0.001        Variable  Coefficient  Bootstrap p-val  Variable  Coefficient  Bootstrap p-val  (Intercept)  3.93E+03  0.024  INTperiodcentralpublic  −1.57E+00  0.338  mtrpremt1v3  8.02E−01  <0.001  INTperiodcentralpublic2  1.20E−01  0.077  mtrpremINTtime1  −5.07E−02  <0.001  INTperiodcentralpublic3  −1.01E−03  0.053  mtrpremINTtime2  3.45E−03  <0.001  lgdtoborder  −7.41E+01  0.357  mtrpremINTtime3  −9.29E−05  <0.001  lgincome  −1.96E+02  0.136  mtrpremINTtime4  1.00E−06  <0.001  INTincomeborder  5.70E+00  0.387  mtrpremINTtime5  −3.66E−09  0.001    3.44E+01  0.143  Age  1.71E+02  0.291    −2.82E+01  0.037  Sqrage  −8.50E+01  0.249  intcoastelev  −4.17E+00  0.161  age3  2.20E+01  0.227  intcoast floorh  −6.56E+00  <0.001  age4  −3.40E+00  0.201  intcoast floorm  −6.54E+00  <0.001  age5  3.19E−01  0.173  intelev floorh  −5.94E+00  <0.001  age6  −1.76E−02    intelev floorm  −4.98E+00  <0.001  age7  5.24E−04    countdss  −2.07E+01  <0.001  age8  −6.51E−06  0.131  INTperiodDSS  6.88E−01  <0.001  Lgsize  1.27E+02  <0.001  INTperiodDSS2  −1.20E−02  <0.001  Floorm  8.69E+01  <0.001  INTperiodDSS3  7.36E−05  0.003  Floorh  1.01E+02  <0.001  lgwalkelementary  4.42E+00  0.16  discountrate  −1.48E+00  <0.001  INTperiodelementary  3.97E−02  0.411  unluckynum  7.32E+01  <0.001  INTperiodelementary2  −6.04E−04  0.388  Luckynum  −2.48E+00  0.274  slopeelementary  9.28E+00  0.022  lgdtoNoSec  2.35E+00  0.016  INTslopewalkelem  −1.24E+00  0.054  lgdtocentralcar  8.81E+01  0.175  airport1km  —  —  INTperioddtocentral  2.92E+01  <0.001  courtsize  5.15E−02  <0.001  INTperioddtocentral2  −2.69E−01  <0.001  courtsizesqr  −9.76E−06  <0.001  lgttocentralcar  3.27E+01  0.385  totaldensity3km  7.48E−03  <0.001  INTperiodttocentral  −1.17E+01  0.047  subdensityper3km  −2.10E+02  <0.001  INTperiodttocentral2  −1.19E−01  0.125  densitypublic3km  −1.29E−02  <0.001  INTperiodttocentral3  2.69E−03  <0.001  densityprivate3km  −8.73E−03  <0.001  lgttocentralpublic  −3.03E+02  <0.001        B. Seasonal variation in elevation effects We test for seasonal variance in elevation effects by introducing additional interaction effects of all walking distance, slope and distance-slope interactions with indicators for summer (May, June, July), autumn (August, September, October) and winter (November, December, January). The regression specifications are otherwise identical to the part I two-stage model outlined in Section 4.1, but uses a single OLS regression on the full dataset with selected variables from the LASSO for simplicity. Table A8 Coefficients and significance of Part II regression for model with interaction terms up to t3, excluding district and time period indicator effects Variable  Coefficient  Bootstrap p-val  Variable  Coefficient  Bootstrap p-val  (Intercept)  3.71E+03  0.037  INTperiodcentralpublic2  −4.57E−02  0.272  mtrpremt1v3  6.72E−01  <0.001  INTperiodcentralpublic3  2.19E−04  0.333  mtrpremINTtime1  1.43E−03  0.383  Lgdtoborder  −1.42E+02  0.263  mtrpremINTtime2  −2.25E−04  0.072  Lgincome  −1.68E+02  0.174  mtrpremINTtime3  1.66E−06  0.129  INTincomeborder  1.24E+01  0.285  Age  −1.89E+01  0.418  Lgelev  2.65E+01  0.188  Sqrage  1.68E+00  0.459  Lgcoast  −2.88E+01  0.033  age3  1.06E+00  0.495  Intcoastelev  −3.06E+00  0.217  age4  −4.49E−01  0.457  intcoast floorh  −7.05E+00  <0.001  age5  6.63E−02  0.419  intcoast floorm  −7.01E+00  <0.001  age6  −4.79E−03  0.386  intelev floorh  −6.09E+00  <0.001  age7  1.71E−04  0.363  intelev floorm  −5.31E+00  <0.001  age8  −2.41E−06  0.336  Countdss  −1.78E+01  <0.001  Lgsize  1.27E+02  <0.001  INTperiodDSS  4.21E−01  <0.001  Floorm  9.11E+01  <0.001  INTperiodDSS2  −4.17E−03  0.089  Floorh  1.04E+02  <0.001  INTperiodDSS3  6.20E−06  0.4  discountrate  −1.46E+00  <0.001  Lgwalkelementary  7.29E+00  0.054  unluckynum  7.44E+01  <0.001  INTperiodelementary  −4.63E−02  0.38  Luckynum  −2.47E+00  0.274  INTperiodelementary2  3.29E−04  0.435  lgdtoNoSec  2.58E+00  0.012  Slopeelementary  1.09E+01  0.011  lgdtocentralcar  1.02E+02  0.142  INTslopewalkelem  −1.51E+00  0.031  INTperioddtocentral  2.92E+01  <0.001  airport1km  —  —  INTperioddtocentral2  −2.70E−01  <0.001  Courtsize  5.31E−02  <0.001  lgttocentralcar  1.30E+02  0.152  Courtsizesqr  −1.01E−05  <0.001  INTperiodttocentral  −2.05E+01  0.001  totaldensity3km  7.60E−03  <0.001  INTperiodttocentral2  9.48E−02  0.157  subdensityper3km  −2.17E+02  <0.001  INTperiodttocentral3  1.20E−03  0.003  densitypublic3km  −1.32E−02  <0.001  lgttocentralpublic  −3.78E+02  <0.001  densityprivate3km  −8.84E−03  <0.001  INTperiodcentralpublic  4.98E+00  0.072          Variable  Coefficient  Bootstrap p-val  Variable  Coefficient  Bootstrap p-val  (Intercept)  3.71E+03  0.037  INTperiodcentralpublic2  −4.57E−02  0.272  mtrpremt1v3  6.72E−01  <0.001  INTperiodcentralpublic3  2.19E−04  0.333  mtrpremINTtime1  1.43E−03  0.383  Lgdtoborder  −1.42E+02  0.263  mtrpremINTtime2  −2.25E−04  0.072  Lgincome  −1.68E+02  0.174  mtrpremINTtime3  1.66E−06  0.129  INTincomeborder  1.24E+01  0.285  Age  −1.89E+01  0.418  Lgelev  2.65E+01  0.188  Sqrage  1.68E+00  0.459  Lgcoast  −2.88E+01  0.033  age3  1.06E+00  0.495  Intcoastelev  −3.06E+00  0.217  age4  −4.49E−01  0.457  intcoast floorh  −7.05E+00  <0.001  age5  6.63E−02  0.419  intcoast floorm  −7.01E+00  <0.001  age6  −4.79E−03  0.386  intelev floorh  −6.09E+00  <0.001  age7  1.71E−04  0.363  intelev floorm  −5.31E+00  <0.001  age8  −2.41E−06  0.336  Countdss  −1.78E+01  <0.001  Lgsize  1.27E+02  <0.001  INTperiodDSS  4.21E−01  <0.001  Floorm  9.11E+01  <0.001  INTperiodDSS2  −4.17E−03  0.089  Floorh  1.04E+02  <0.001  INTperiodDSS3  6.20E−06  0.4  discountrate  −1.46E+00  <0.001  Lgwalkelementary  7.29E+00  0.054  unluckynum  7.44E+01  <0.001  INTperiodelementary  −4.63E−02  0.38  Luckynum  −2.47E+00  0.274  INTperiodelementary2  3.29E−04  0.435  lgdtoNoSec  2.58E+00  0.012  Slopeelementary  1.09E+01  0.011  lgdtocentralcar  1.02E+02  0.142  INTslopewalkelem  −1.51E+00  0.031  INTperioddtocentral  2.92E+01  <0.001  airport1km  —  —  INTperioddtocentral2  −2.70E−01  <0.001  Courtsize  5.31E−02  <0.001  lgttocentralcar  1.30E+02  0.152  Courtsizesqr  −1.01E−05  <0.001  INTperiodttocentral  −2.05E+01  0.001  totaldensity3km  7.60E−03  <0.001  INTperiodttocentral2  9.48E−02  0.157  subdensityper3km  −2.17E+02  <0.001  INTperiodttocentral3  1.20E−03  0.003  densitypublic3km  −1.32E−02  <0.001  lgttocentralpublic  −3.78E+02  <0.001  densityprivate3km  −8.84E−03  <0.001  INTperiodcentralpublic  4.98E+00  0.072          Of the 30 season–slope–distance interaction terms, only 10 survive the variable selection and only three are significant with p < 0.05. Although this may not seem insignificant, in contrast a majority of the non-metro variables in the second stage model are significant at beyond the 99.5% level. Five of the six metro-related variables in the part I regression model are significant at the 99.9% level. Also, coefficients of the seasonal interactions are small when compared with the base metro station effects: the only term significant at p < 0.01, winter walkingdistance, has a coefficient size of approximately 5% that of the base walking distance term. The second stage OLS regression output is summarized in Table A12. Finally, we note that introducing the seasonal effects do not significantly change the explanatory power of the model: R2 and AR2 values increase by less than 0.001 after adding seasonal interaction. We therefore conclude that it is unlikely that there are seasonal effects in the perceived value of station proximity in terms of walking distance and slope. Table A9 Coefficients and significance of Part II regression for model with interaction terms up to t2, excluding district and time period indicator effects Variable  Coefficient  Bootstrap p-val  Variable  Coefficient  Bootstrap p-val  (Intercept)  3.90E+03  0.026  INTperiodcentralpublic2  −6.23E−02  0.179  mtrpremt1v3  7.16E−01  <0.001  INTperiodcentralpublic3  3.04E−04  0.295  mtrpremINTtime1  −4.55E−03  0.046  Lgdtoborder  −1.99E+02  0.171  mtrpremINTtime2  −3.58E−05  0.178  Lgincome  −1.92E+02  0.137  Age  −2.28E+01  0.385  INTincomeborder  1.83E+01  0.189  Sqrage  5.12E+00  0.415  Lgelev  2.42E+01  0.203  age3  −9.78E−02  0.442  Lgcoast  −2.98E+01  0.033  age4  −2.46E−01  0.477  intcoastelev  −2.80E+00  0.233  age5  4.65E−02  0.485  intcoast floorh  −7.06E+00  <0.001  age6  −3.69E−03  0.449  intcoast floorm  −6.98E+00  <0.001  age7  1.39E−04  0.401  intelev floorh  −6.00E+00  <0.001  age8  −2.03E−06  0.37  intelev floorm  −5.25E+00  <0.001  Lgsize  1.27E+02  <0.001  countdss  −1.70E+01  <0.001  Floorm  9.07E+01  <0.001  INTperiodDSS  3.49E−01  <0.001  Floorh  1.04E+02  <0.001  INTperiodDSS2  −2.60E−03  0.179  discountrate  −1.44E+00  <0.001  INTperiodDSS3  −4.17E−06  0.417  unluckynum  7.49E+01  <0.001  lgwalkelementary  8.83E+00  0.014  luckynum  −2.19E+00  0.283  INTperiodelementary  −1.13E−01  0.212  lgdtoNoSec  2.68E+00  0.008  INTperiodelementary2  1.08E−03  0.254  lgdtocentralcar  1.03E+02  0.15  slopeelementary  1.18E+01  0.003  INTperioddtocentral  2.89E+01  <0.001  INTslopewalkelem  −1.64E+00  0.014  INTperioddtocentral2  −2.66E−01  <0.001  airport1km  —  —  lgttocentralcar  1.60E+02  0.112  courtsize  5.34E−02  <0.001  INTperiodttocentral  −2.15E+01  0.002  courtsizesqr  −1.02E−05  <0.001  INTperiodttocentral2  1.10E−01  0.134  totaldensity3km  7.62E−03  <0.001  INTperiodttocentral3  1.12E−03  0.005  subdensityper3km  −2.19E+02  <0.001  lgttocentralpublic  −3.93E+02  <0.001  densitypublic3km  −1.32E−02  <0.001  INTperiodcentralpublic  5.95E+00  0.019  densityprivate3km  −8.88E−03  <0.001  Variable  Coefficient  Bootstrap p-val  Variable  Coefficient  Bootstrap p-val  (Intercept)  3.90E+03  0.026  INTperiodcentralpublic2  −6.23E−02  0.179  mtrpremt1v3  7.16E−01  <0.001  INTperiodcentralpublic3  3.04E−04  0.295  mtrpremINTtime1  −4.55E−03  0.046  Lgdtoborder  −1.99E+02  0.171  mtrpremINTtime2  −3.58E−05  0.178  Lgincome  −1.92E+02  0.137  Age  −2.28E+01  0.385  INTincomeborder  1.83E+01  0.189  Sqrage  5.12E+00  0.415  Lgelev  2.42E+01  0.203  age3  −9.78E−02  0.442  Lgcoast  −2.98E+01  0.033  age4  −2.46E−01  0.477  intcoastelev  −2.80E+00  0.233  age5  4.65E−02  0.485  intcoast floorh  −7.06E+00  <0.001  age6  −3.69E−03  0.449  intcoast floorm  −6.98E+00  <0.001  age7  1.39E−04  0.401  intelev floorh  −6.00E+00  <0.001  age8  −2.03E−06  0.37  intelev floorm  −5.25E+00  <0.001  Lgsize  1.27E+02  <0.001  countdss  −1.70E+01  <0.001  Floorm  9.07E+01  <0.001  INTperiodDSS  3.49E−01  <0.001  Floorh  1.04E+02  <0.001  INTperiodDSS2  −2.60E−03  0.179  discountrate  −1.44E+00  <0.001  INTperiodDSS3  −4.17E−06  0.417  unluckynum  7.49E+01  <0.001  lgwalkelementary  8.83E+00  0.014  luckynum  −2.19E+00  0.283  INTperiodelementary  −1.13E−01  0.212  lgdtoNoSec  2.68E+00  0.008  INTperiodelementary2  1.08E−03  0.254  lgdtocentralcar  1.03E+02  0.15  slopeelementary  1.18E+01  0.003  INTperioddtocentral  2.89E+01  <0.001  INTslopewalkelem  −1.64E+00  0.014  INTperioddtocentral2  −2.66E−01  <0.001  airport1km  —  —  lgttocentralcar  1.60E+02  0.112  courtsize  5.34E−02  <0.001  INTperiodttocentral  −2.15E+01  0.002  courtsizesqr  −1.02E−05  <0.001  INTperiodttocentral2  1.10E−01  0.134  totaldensity3km  7.62E−03  <0.001  INTperiodttocentral3  1.12E−03  0.005  subdensityper3km  −2.19E+02  <0.001  lgttocentralpublic  −3.93E+02  <0.001  densitypublic3km  −1.32E−02  <0.001  INTperiodcentralpublic  5.95E+00  0.019  densityprivate3km  −8.88E−03  <0.001  This is not surprising. Hong Kong has a fairly mild temperature cycle among major cities with an annual average temperature spread of 13–14 °C.16 Summers are usually not extremely hot (i.e., above 35 °C) and there is almost no possibility of ice in winter. However, it may be of interest to replicate certain aspects of our study for a Northern city and testing for whether the effect of icy slopes is priced into summer transactions. C. Unevenness and Density Gradients Our approach in this subsection has two major distinctions from the approach of Saiz (2010). First, we model unevenness not as a dichotomy of developable versus un-developable land but as a continuous ‘elevation variance’ variable. Second, we select an international perspective using major cities worldwide instead of major US metro areas. Using figures formatted from United Nations population data, we create scatterplots of log-transformed ‘elevation variance’ estimates against log-transformed population density of 93 major world cities.17 City-level elevation variance estimates are obtained using the Microsoft Bing REST Services. Altitude estimates are sampled over a 30-by-30 point grid of fixed size over each city, centered on respective downtown areas, with zero or negative elevation samples discarded as potential errors or points on a body of water. Two alternative area sizes are used for each city: a 0.2-by-0.2 decimal degree square approximating a ‘city-and-near-suburb’ elevation variance value and a smaller, 0.1-by-0.1 degree square representing a ‘city-only’ variance value. Table A10 Equivalent station value and interaction coefficient of Part II regression, compared   Model with in-teraction terms up to t7  Model with in-teraction terms up to t5  Model with in-teraction terms up to t3  Model with in-teraction terms up to t2  Variable  Coefficient  Bootstrap p-val  Coefficient  Bootstrap p-val  Coefficient  Bootstrap p-val  Coefficient  Bootstrap p-val  mtrpremt1v3  7.79E−01  <0.001  8.02E−01  <0.001  6.72E−01  <0.001  7.16E−01  <0.001  mtrpremINTtime1  −5.20E−02  0.03  −5.07E−02  <0.001  1.43E−03  0.383  −4.55E−03  0.046  mtrpremINTtime2  5.04E−03  0.069  3.45E−03  <0.001  −2.25E−04  0.072  −3.58E−05  0.178  mtrpremINTtime3  −2.58E−04  0.085  −9.29E−05  <0.001  1.66E−06  0.129  —  —  mtrpremINTtime4  7.82E−06  0.078  1.00E−06  <0.001  —  —  —  —  mtrpremINTtime5  −1.38E−07  0.054  −3.66E−09  0.001  —  —  —  —  mtrpremINTtime6  1.26E−09  0.032  —  —  —  —  —  —  mtrpremINTtime7  −4.56E−12  0.019  —  —  —  —  —  —    Model with in-teraction terms up to t7  Model with in-teraction terms up to t5  Model with in-teraction terms up to t3  Model with in-teraction terms up to t2  Variable  Coefficient  Bootstrap p-val  Coefficient  Bootstrap p-val  Coefficient  Bootstrap p-val  Coefficient  Bootstrap p-val  mtrpremt1v3  7.79E−01  <0.001  8.02E−01  <0.001  6.72E−01  <0.001  7.16E−01  <0.001  mtrpremINTtime1  −5.20E−02  0.03  −5.07E−02  <0.001  1.43E−03  0.383  −4.55E−03  0.046  mtrpremINTtime2  5.04E−03  0.069  3.45E−03  <0.001  −2.25E−04  0.072  −3.58E−05  0.178  mtrpremINTtime3  −2.58E−04  0.085  −9.29E−05  <0.001  1.66E−06  0.129  —  —  mtrpremINTtime4  7.82E−06  0.078  1.00E−06  <0.001  —  —  —  —  mtrpremINTtime5  −1.38E−07  0.054  −3.66E−09  0.001  —  —  —  —  mtrpremINTtime6  1.26E−09  0.032  —  —  —  —  —  —  mtrpremINTtime7  −4.56E−12  0.019  —  —  —  —  —  —  Results for both variance estimates are contrasted in Figure A1. Both estimates display a positive association with population density: the correlation coefficient between the city-and-suburb elevation variance and population density is approximately 0.117, and the coefficient for the city-only variance 0.143. Although there are distinct low-variance, high-density outliers (Surat, Dhaka) and relatively high-variance, low-density outliers (Los Angeles, Phoenix), the two linear trends clearly indicate that from an international perspective, cities that have more uneven terrain tend to also have generally higher population densities. Furthermore, both Dhaka and Surat are highly susceptible to flooding (Ramirez and Rajasekar, 2015) and therefore one would expect the population to be concentrated in less risky locations, which our estimation procedure does not explicitly incorporate. Figure A1 View largeDownload slide Unevenness (log)–population density (log) scatterplots of major urban areas with fitted linear trend. Figure A1 View largeDownload slide Unevenness (log)–population density (log) scatterplots of major urban areas with fitted linear trend. We can also contrast the relationship between elevation and city size by population count. The same elevation variance variables are plotted against population count figures in Figure A2. The relationship between population and elevation variance is generally negative at the city level: the correlation coefficient is − 0.138 for log-transformed population and city-and-suburb elevation variance, and −0.105 for population and city-only elevation variance. Figure A2 View largeDownload slide Unevenness (log)–population count (log) scatterplots of major urban areas with fitted linear trend. Figure A2 View largeDownload slide Unevenness (log)–population count (log) scatterplots of major urban areas with fitted linear trend. Note that the comparative size of the correlations is opposite for population count versus density. This is consistent with literature and can be explained by the nature of the influence of elevation variance: high-unevenness downtown areas increase costs of central-city infrastructure and housing development, leading to inflated bid-rent curves and higher density. However, high-unevenness suburbs dis-incentivize development and are conceivably a greater constraint on the absolute size of a city, forcing bid-rent curves to drop more rapidly as distance to the city center increases. Although somewhat low in statistical significance, these relationships are nonetheless illuminating.18 Viewed in the context of existing literature on elevation gradient effects, the fitted trends suggestively connect the transaction level elevation-price effects outlined in Section 4 to differences in urban markets at the multi-city-level caused by elevation patterns, bridged by empirical work on the same effects at the sub-city or district-level. It ought to be noted that because the analysis treats cities as identical in size for altitude sampling, the majority of estimates among the observations likely suffer from size mismatch issues. If elevation could be accurately sampled from the specific areas from which population figures were collected, we would expect the relationships in Figure A1 and Figure A2 to be significantly stronger. Table A11 List of major urban areas and respective locations Location  Urban area  Location  Urban area  Japan  Tokyo- Yokohama  Indonesia  Jakarta (Jab- otabek)  India  Delhi  Philippines  Manila  China  Shanghai  Pakistan  Karachi  USA  New York  Mexico  Mexico City  Brazil  Sao Paulo  China  Beijing  India  Mumbai  Russia  Moscow  USA  Los Angeles  Egypt  Cairo  Thailand  Bangkok  India  Kolkota  Bangladesh  Dhaka  Argentina  Buenos Aires  Iran  Tehran  Turkey  Istanbul  China  Shenzhen  Nigeria  Lagos  Brazil  Rio de Janeiro  France  Paris  Japan  Nagoya  UK  London  Congo (Dem. Rep.)  Kinshasa  China  Quanzhou  Peru  Lima  China  Tianjin  India  Chennai  India  Bangalore  USA  Chicago  Viet Nam  Ho Chi Minh        City  China  Chengdu  China  Dongguan  India  Hyderabad  Pakistan  Lahore  Colombia  Bogota  China  Wuhan  China: Taiwan  Taipei  China: HKSAR  Hong Kong  India  Ahmedabad  China  Chongqing  China  Hangzhou  Malaysia  Kuala Lumpur  Iraq  Baghdad  Canada  Toronto  Chile  Santiago  Spain  Madrid  China  Nanjing  USA  Miami  China  Shenyang  Indonesia  Bandung  Angola  Luanda  United States  Houston  USA  Philadelphia  China  Xi'an  Singapore  Singapore  China  Qingdao  India  Pune  Italy  Milan  Saudi Arabia  Riyadh  Russia  St. Petersburg  Sudan  Khartoum  Indonesia  Surabaya  India  Surat  USA  Atlanta  USA  Washington  Ivory Coast  Abidjan  Myanmar  Yangon  Spain  Barcelona  Kenya  Nairobi  China  Harbin  Egypt  Alexandria  China  Suzhou  USA  Boston  Brazil  Belo Horizonte  Turkey  Ankara  China  Zhengzhou  Ghana  Accra  USA  Phoenix  Germany  Berlin  Indonesia  Medan  Australia  Sydney  South Korea  Busan  Kuwait  Kuwait  Tanzania  Dar es Salaam  Mexico  Monterey  China  Dalian  Italy  Rome  Australia  Melbourne  Mexico  Guadalajara      Location  Urban area  Location  Urban area  Japan  Tokyo- Yokohama  Indonesia  Jakarta (Jab- otabek)  India  Delhi  Philippines  Manila  China  Shanghai  Pakistan  Karachi  USA  New York  Mexico  Mexico City  Brazil  Sao Paulo  China  Beijing  India  Mumbai  Russia  Moscow  USA  Los Angeles  Egypt  Cairo  Thailand  Bangkok  India  Kolkota  Bangladesh  Dhaka  Argentina  Buenos Aires  Iran  Tehran  Turkey  Istanbul  China  Shenzhen  Nigeria  Lagos  Brazil  Rio de Janeiro  France  Paris  Japan  Nagoya  UK  London  Congo (Dem. Rep.)  Kinshasa  China  Quanzhou  Peru  Lima  China  Tianjin  India  Chennai  India  Bangalore  USA  Chicago  Viet Nam  Ho Chi Minh        City  China  Chengdu  China  Dongguan  India  Hyderabad  Pakistan  Lahore  Colombia  Bogota  China  Wuhan  China: Taiwan  Taipei  China: HKSAR  Hong Kong  India  Ahmedabad  China  Chongqing  China  Hangzhou  Malaysia  Kuala Lumpur  Iraq  Baghdad  Canada  Toronto  Chile  Santiago  Spain  Madrid  China  Nanjing  USA  Miami  China  Shenyang  Indonesia  Bandung  Angola  Luanda  United States  Houston  USA  Philadelphia  China  Xi'an  Singapore  Singapore  China  Qingdao  India  Pune  Italy  Milan  Saudi Arabia  Riyadh  Russia  St. Petersburg  Sudan  Khartoum  Indonesia  Surabaya  India  Surat  USA  Atlanta  USA  Washington  Ivory Coast  Abidjan  Myanmar  Yangon  Spain  Barcelona  Kenya  Nairobi  China  Harbin  Egypt  Alexandria  China  Suzhou  USA  Boston  Brazil  Belo Horizonte  Turkey  Ankara  China  Zhengzhou  Ghana  Accra  USA  Phoenix  Germany  Berlin  Indonesia  Medan  Australia  Sydney  South Korea  Busan  Kuwait  Kuwait  Tanzania  Dar es Salaam  Mexico  Monterey  China  Dalian  Italy  Rome  Australia  Melbourne  Mexico  Guadalajara      Table A12 Summary of regression for OLS model with season–distance–slope interaction terms, excluding district and time period indicator effects   Estimate  Std. error  t-value  Pr(>t)  (Intercept)  −0.0000  0.0015  −0.00  1.0000  sum walkmtractual3a  0.0015  0.0024  0.61  0.5422  sum walkmtrangle  −0.0021  0.0072  −0.29  0.7682  sum walkmtranglesqr  0.0102  0.0054  1.91  0.0559  sum mtrwalkINTangle1  −0.0088  0.0038  −2.29  0.0220  aut walkmtractual  0.0087  0.0054  1.61  0.1067  aut walkmtractual3  −0.0028  0.0048  −0.58  0.5627  aut mtrwalkINTangle1  0.0003  0.0027  0.11  0.9150  wint walkmtractual  −0.0135  0.0047  −2.89  0.0039  wint walkmtractual3  0.0100  0.0039  2.57  0.0102  wint mtrwalk2INTangle1  0.0009  0.0027  0.33  0.7449  walkmtractual  −0.2646  0.0080  −32.87  0.0000  walkmtractual3  0.0675  0.0070  9.72  0.0000  walkmtrangle  −0.0791  0.0049  −16.15  0.0000  mtrwalkINTangle1  −0.1084  0.0115  −9.42  0.0000  mtrwalk2INTangle2  0.0242  0.0064  3.78  0.0002  mtrwalk2INTangle1  0.1267  0.0072  17.49  0.0000  age  −0.2494  0.0034  −74.41  0.0000  age8  0.0160  0.0019  8.27  0.0000  lgsize  0.0802  0.0017  47.63  0.0000  floorm  0.0468  0.0076  6.16  0.0000  floorh  0.0590  0.0074  7.96  0.0000  discountrate  −0.1799  0.0067  −26.93  0.0000  intdiscount1998  0.0772  0.0082  9.42  0.0000  intdiscount1999  0.1235  0.0112  11.05  0.0000  intdiscount2000  0.1166  0.0105  11.06  0.0000  intdiscount2001  0.0888  0.0107  8.33  0.0000  intdiscount2002  0.0818  0.0104  7.85  0.0000  intdiscount2003  0.0871  0.0113  7.68  0.0000  intdiscount2004  0.0571  0.0126  4.55  0.0000  intdiscount2005  0.0199  0.0091  2.19  0.0289  intdiscount2006  0.0348  0.0129  2.70  0.0069  intdiscount2007  0.1204  0.0129  9.34  0.0000  intdiscount2009  0.0617  0.0137  4.50  0.0000  intdiscount2010  0.0140  0.0134  1.04  0.2969  intdiscount2011  0.0055  0.0124  0.45  0.6562  intdiscount2012  −0.0116  0.0121  −0.96  0.3363  intdiscount2013  −0.0168  0.0115  −1.46  0.1440  intdiscount2014  0.0282  0.0118  2.39  0.0170  unluckynum  −0.0145  0.0023  −6.35  0.0000  luckynum  0.0179  0.0019  9.42  0.0000  lgdtoNoSec  0.0193  0.0023  8.36  0.0000  lgdtocentralcar  −0.1110  0.0239  −4.64  0.0000  INTperioddtocentral  1.0034  0.0842  11.92  0.0000  INTperioddtocentral2  −1.0231  0.0570  −17.94  0.0000  lgttocentralcar  −0.0047  0.0211  −0.22  0.8222  INTperiodttocentral  −0.7402  0.0718  −10.31  0.0000  lgttocentralpublic  −0.1334  0.0066  −20.33  0.0000  INTperiodcentralpublic2  −0.9738  0.1217  −8.00  0.0000  INTperiodcentralpublic3  1.5768  0.1114  14.16  0.0000  lgdtoborder  0.5792  0.1365  4.24  0.0000  lgincome  0.1907  0.0183  10.41  0.0000  INTincomeborder  −0.6482  0.1362  −4.76  0.0000  lgelev  0.0158  0.0038  4.11  0.0000  lgcoast  0.0294  0.0048  6.16  0.0000  intelev floorh  0.0217  0.0075  2.90  0.0037  intelev floorm  0.0159  0.0077  2.07  0.0385  countdss  −0.0318  0.0075  −4.21  0.0000  INTperiodDSS  0.0206  0.0138  1.49  0.1356  INTperiodDSS3  −0.0084  0.0098  −0.86  0.3918  lgwalkelementary  0.1033  0.0059  17.40  0.0000  INTperiodelementary  −0.5272  0.0451  −11.68  0.0000  INTperiodelementary2  0.5438  0.0438  12.42  0.0000  slopeelementary  0.3974  0.0180  22.09  0.0000  INTslopewalkelem  −0.3988  0.0177  −22.48  0.0000  airport1km  0.0463  0.0117  3.94  0.0001  courtsize  0.0437  0.0046  9.48  0.0000  courtsizesqr  −0.0633  0.0045  −14.21  0.0000  totaldensity3km10000  −0.1587  0.0073  −21.63  0.0000  subdensityper3km  −0.0523  0.0036  −14.35  0.0000  densityprivate3km  0.0759  0.0062  12.18  0.0000    Estimate  Std. error  t-value  Pr(>t)  (Intercept)  −0.0000  0.0015  −0.00  1.0000  sum walkmtractual3a  0.0015  0.0024  0.61  0.5422  sum walkmtrangle  −0.0021  0.0072  −0.29  0.7682  sum walkmtranglesqr  0.0102  0.0054  1.91  0.0559  sum mtrwalkINTangle1  −0.0088  0.0038  −2.29  0.0220  aut walkmtractual  0.0087  0.0054  1.61  0.1067  aut walkmtractual3  −0.0028  0.0048  −0.58  0.5627  aut mtrwalkINTangle1  0.0003  0.0027  0.11  0.9150  wint walkmtractual  −0.0135  0.0047  −2.89  0.0039  wint walkmtractual3  0.0100  0.0039  2.57  0.0102  wint mtrwalk2INTangle1  0.0009  0.0027  0.33  0.7449  walkmtractual  −0.2646  0.0080  −32.87  0.0000  walkmtractual3  0.0675  0.0070  9.72  0.0000  walkmtrangle  −0.0791  0.0049  −16.15  0.0000  mtrwalkINTangle1  −0.1084  0.0115  −9.42  0.0000  mtrwalk2INTangle2  0.0242  0.0064  3.78  0.0002  mtrwalk2INTangle1  0.1267  0.0072  17.49  0.0000  age  −0.2494  0.0034  −74.41  0.0000  age8  0.0160  0.0019  8.27  0.0000  lgsize  0.0802  0.0017  47.63  0.0000  floorm  0.0468  0.0076  6.16  0.0000  floorh  0.0590  0.0074  7.96  0.0000  discountrate  −0.1799  0.0067  −26.93  0.0000  intdiscount1998  0.0772  0.0082  9.42  0.0000  intdiscount1999  0.1235  0.0112  11.05  0.0000  intdiscount2000  0.1166  0.0105  11.06  0.0000  intdiscount2001  0.0888  0.0107  8.33  0.0000  intdiscount2002  0.0818  0.0104  7.85  0.0000  intdiscount2003  0.0871  0.0113  7.68  0.0000  intdiscount2004  0.0571  0.0126  4.55  0.0000  intdiscount2005  0.0199  0.0091  2.19  0.0289  intdiscount2006  0.0348  0.0129  2.70  0.0069  intdiscount2007  0.1204  0.0129  9.34  0.0000  intdiscount2009  0.0617  0.0137  4.50  0.0000  intdiscount2010  0.0140  0.0134  1.04  0.2969  intdiscount2011  0.0055  0.0124  0.45  0.6562  intdiscount2012  −0.0116  0.0121  −0.96  0.3363  intdiscount2013  −0.0168  0.0115  −1.46  0.1440  intdiscount2014  0.0282  0.0118  2.39  0.0170  unluckynum  −0.0145  0.0023  −6.35  0.0000  luckynum  0.0179  0.0019  9.42  0.0000  lgdtoNoSec  0.0193  0.0023  8.36  0.0000  lgdtocentralcar  −0.1110  0.0239  −4.64  0.0000  INTperioddtocentral  1.0034  0.0842  11.92  0.0000  INTperioddtocentral2  −1.0231  0.0570  −17.94  0.0000  lgttocentralcar  −0.0047  0.0211  −0.22  0.8222  INTperiodttocentral  −0.7402  0.0718  −10.31  0.0000  lgttocentralpublic  −0.1334  0.0066  −20.33  0.0000  INTperiodcentralpublic2  −0.9738  0.1217  −8.00  0.0000  INTperiodcentralpublic3  1.5768  0.1114  14.16  0.0000  lgdtoborder  0.5792  0.1365  4.24  0.0000  lgincome  0.1907  0.0183  10.41  0.0000  INTincomeborder  −0.6482  0.1362  −4.76  0.0000  lgelev  0.0158  0.0038  4.11  0.0000  lgcoast  0.0294  0.0048  6.16  0.0000  intelev floorh  0.0217  0.0075  2.90  0.0037  intelev floorm  0.0159  0.0077  2.07  0.0385  countdss  −0.0318  0.0075  −4.21  0.0000  INTperiodDSS  0.0206  0.0138  1.49  0.1356  INTperiodDSS3  −0.0084  0.0098  −0.86  0.3918  lgwalkelementary  0.1033  0.0059  17.40  0.0000  INTperiodelementary  −0.5272  0.0451  −11.68  0.0000  INTperiodelementary2  0.5438  0.0438  12.42  0.0000  slopeelementary  0.3974  0.0180  22.09  0.0000  INTslopewalkelem  −0.3988  0.0177  −22.48  0.0000  airport1km  0.0463  0.0117  3.94  0.0001  courtsize  0.0437  0.0046  9.48  0.0000  courtsizesqr  −0.0633  0.0045  −14.21  0.0000  totaldensity3km10000  −0.1587  0.0073  −21.63  0.0000  subdensityper3km  −0.0523  0.0036  −14.35  0.0000  densityprivate3km  0.0759  0.0062  12.18  0.0000  a Interactions with the summer indicator (May, June, July) is denoted with the pre x ‘sum’, interactions with the autumn indicator (August, September, October) is denoted with the pre x ‘aut’, and interactions with the winter indicator (November, December, January) is denoted with the pre x ‘wint’. Figure A3 View largeDownload slide Log-lambda trace plot of LASSO Cross-validation process. Figure A3 View largeDownload slide Log-lambda trace plot of LASSO Cross-validation process. Figure A4 View largeDownload slide Trace plots of select regression variables in Bayesian Lasso model MCMC process. Figure A4 View largeDownload slide Trace plots of select regression variables in Bayesian Lasso model MCMC process. Figure A5 View largeDownload slide Trace plots of key regression variables in Bayesian Lasso model MCMC process. Figure A5 View largeDownload slide Trace plots of key regression variables in Bayesian Lasso model MCMC process. Figure A6 View largeDownload slide R-square and adjusted R-square value distributions with bootstrapping, part I model. Figure A6 View largeDownload slide R-square and adjusted R-square value distributions with bootstrapping, part I model. Figure A7 View largeDownload slide Bootstrap densities of select regression coefficients, part I model. Figure A7 View largeDownload slide Bootstrap densities of select regression coefficients, part I model. Figure A8 View largeDownload slide Density of select slope/distance variable coefficients, compared. Figure A8 View largeDownload slide Density of select slope/distance variable coefficients, compared. Figure A9 View largeDownload slide Densities of selling price predictions given fixed slope and by walking distance. Figure A9 View largeDownload slide Densities of selling price predictions given fixed slope and by walking distance. Figure A10 View largeDownload slide R-square and adjusted R-square value distributions with bootstrapping, Part II models. Figure A10 View largeDownload slide R-square and adjusted R-square value distributions with bootstrapping, Part II models. Figure A11 View largeDownload slide Densities of equivalent station value predictions given fixed time gap and by model specificities. Figure A11 View largeDownload slide Densities of equivalent station value predictions given fixed time gap and by model specificities. Figure A12 View largeDownload slide Elevation contour plot of the HKSAR. Figure A12 View largeDownload slide Elevation contour plot of the HKSAR. Figure A13 View largeDownload slide Map of HKSAR with locations of courts and select amenities. Figure A13 View largeDownload slide Map of HKSAR with locations of courts and select amenities. Figure A14 View largeDownload slide Elevation gradients at different walking distances, compared. Figure A14 View largeDownload slide Elevation gradients at different walking distances, compared. Figure A15 View largeDownload slide Residual scatterplots with selected parameters and subsetted data for Part I model. Figure A15 View largeDownload slide Residual scatterplots with selected parameters and subsetted data for Part I model. Figure A16 View largeDownload slide Filled altitude contour plot with spatially mapped apartment-group average price residuals and metro station locations, Hong Kong, 1998–2014 (2 of 3). Figure A16 View largeDownload slide Filled altitude contour plot with spatially mapped apartment-group average price residuals and metro station locations, Hong Kong, 1998–2014 (2 of 3). Figure A17 View largeDownload slide Filled altitude contour plot with spatially mapped apartment-group average price residuals and metro station locations, 1998–2014 (3 of 3). Figure A17 View largeDownload slide Filled altitude contour plot with spatially mapped apartment-group average price residuals and metro station locations, 1998–2014 (3 of 3). © The Author (2017). Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Economic Geography Oxford University Press

The (literally) steepest slope: spatial, temporal, and elevation variance gradients in urban spatial modelling

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© The Author (2017). Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com
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Abstract

Abstract This paper presents an analysis of elevation gradient and temporal future-station effects in urban real estate markets. Using a novel dataset from the Hong Kong publicly constructed housing sector, we find enormous housing price effects caused by levels of terrain incline between apartments and subway stations. Ceteris paribus, two similar apartments with closest metro stations of the same walking distance may sell at a difference of up to 20% because of differences in the apartment-station slope alone. Anticipatory effects are similarly robust: apartment buyers regard a future, closer metro station as being 60% present when making purchases 2 years prior to its opening. 1. Introduction This is a story of three gradients. The urban spatial modelling literature focuses almost exclusively on distance—a discussion that developed from the original Alonso-Muth-Mills (AMM) model (Alonso, 1964; Muth, 1969; Mills, 1972) to models with tiered transit systems (LeRoy and Sonstelie, 1981) and, more recently, to effects of mass public transit (Glaeser et al., 2008). Yet the underlying premise of investigations on the effect of distance remains unchanged: cities are largely regarded as featureless, flat plains with fixed, immutable transit nodes. Reality is neither flat nor featureless. Cities are often built on or against mountainsides and over rolling hills, the economic influence of which have thus far lacked in-depth investigation. Cities also are dynamic, continuously receiving structural upgrades in the form of new roads and transit lines. Utilizing extensive geospatial modelling methods, including the use of Geographical Information Systems (GISs) and high-resolution Digital Elevation Models (DEMs), as well as a novel dataset of apartment transactions from the Hong Kong government-subsidized real estate market, this paper explores these factors—elevation and temporal gradients—from the housing market perspective. In the context of recent literature on elevation effects (Saiz, 2010; Ye and Becker, 2016), this paper eschews a multi-city framework in favor of a single market, transaction-level approach that focuses on one particular economic aspect of elevation: its influence on utility derived from public transit access. Taking advantage of the strongly homogeneous quality of government-subsidized housing and the long, 17-year timespan of our data, we create a ‘near-perfect’ spatial and temporal hedonic housing model, moving beyond identification to estimate the magnitude with which hilliness influences housing prices. We demonstrate that the value of public transit hubs varies not only by distance but also by altitude differentials: holding other factors constant, apartment selling price may differ up to 20% because of difference in the slope between the apartment and the closest metro station. Buyers also seem to robustly but incompletely value the existence of future, close-by stations: we estimate that stations are perceived as being 58–60% ‘present’ to buyers 2 years prior to opening, and appreciate in perceived value at an average rate of 1.1% per month within a 6-year window before opening. This rate resembles interest returns on capital invested locally during the corresponding time period. Contributing to the tradition of penalized distance gradient models (Yinger, 1974; Anas and Kim, 1996), our work introduces and evaluates a new distance–steepness interaction penalty associated with public transit use: the undesirability of walking up or down slopes to transit hubs. By allowing for differences in locations resulting from both z-axis variations and future amenities of a particular location, our modelling approach allows cities with significant unevenness (San Francisco, Hong Kong) to behave fundamentally and qualitatively differently from cities with little to no variance (Beijing, Chicago). It also provides a more comprehensive perspective for rapidly developing cities that experience major public infrastructure development. Given the robustness and magnitude of these effects, spatial models of cities with significant altitude variance are expected to suffer from bias if elevation gradients are not specifically accounted for. For such cities, large, hidden costs exist in developing public transit: residents of a more ‘uneven’ city will receive less utility from a given level of transit infrastructure. Temporal models of rapidly developing cities suffer if the effect of major, future infrastructure developments are not considered in the time-frame of the study. Hence, rigorous investigations of housing and other urban markets in modern economies require advanced modelling methods with spatial and temporal elements that are not currently widely utilized. Section 2 discusses previous research on the effects of transit and elevation on urban housing. Section 3 provides descriptive and graphical evidence for elevation effects. Section 4 presents a data summary and outlines conceptual and empirical methodologies involved in constructing the model. Section 5 analyzes model robustness, presents results of the regression model and visualizes anticipatory and elevation gradient effects. Section 6 discusses broader implications of these effects; Section 7 concludes. 2. Prior literature We believe this is the first paper to formally investigate elevation variance effects in a monocentric, transaction-level housing framework. Past literature on elevation in urban economics is sparse: research on elevation effects in housing markets concentrates on price effects of low elevation in coastal regions and, in particular, flood risk (Scawthorn et al., 1982; Shilling et al., 1989; Husby et al., 2014). Elevation effects have also been utilized recently in broader, geographical perspectives to explain the locational choice of cities (Bosker and Buring, 2015) and in discussions about features that contribute to land value (Kok et al., 2014). None of these papers links elevation gradients to transit or uses transaction-level data at the city level. Saiz (2010) analyses elevation gradient effects in housing markets using data from 73 major MSAs, concluding that undevelopable land, approximated by peripheral areas with slopes exceeding 15%, is a strong predictor of low housing supply elasticity. Although highlighting the importance of elevation as a factor of urban housing systems, his approach has significant shortcomings: the 15% slope cutoff assumes a somewhat arbitrary boundary at which development becomes constrained by elevation, yet our results indicate that elevation gradient effects apply at far more moderate slopes and small increments of unevenness. Furthermore, using cities as observations precludes controls for city-level fixed effects: the model cannot infer the specific methods through which elevation exerts influence at units smaller than cities. Ye and Becker (2016) address both concerns using census-tract level data from 17 US cities. In addition to establishing a connection between elevation effects and income gradients, they present evidence of high terrain unevenness decreasing use of public transit and commuting by walking. However, their approach is limited by imperfect controls for exogenous spatial factors and aggregation inherent in using tract-level data. By utilizing observations at the transaction level, we expand on these effects and demonstrate specifically how the degree of incline between metro stations and apartments are priced at different apartment-station walking distances. With regard to urban gradients in Chinese and other East Asian cities, a number of salient characteristics are discussed in literature. Chinese cities generally face considerably higher income elasticities of housing demand than do US cities (Gong et al., 2013). Compared with US counterparts, Chinese urban residents seem to be willing to devote significantly larger portions of income to housing consumption: this pattern may reflect greater land scarcity in China. Chinese cities also feature immature housing markets and, consequentially, inefficiencies in housing allocation and spatial distributions. Using panel data from five Chinese cities, Zheng et al. (2006) find evidence of sub-optimal spatial outcomes in housing markets; inefficiencies are attributed to poorly defined property rights, historical housing regulations, and home inheritance. Although commercialized real estate has a much longer history in Hong Kong than other Chinese cities, the city also has a long history of large public housing programs. Apartments of the same program enjoy substantially different amenities depending on their location, a factor for which traditional distance-gradient models fails to account. Another significant feature of Chinese cities is the dominance of public transit. As of 2008, 52% of Beijing workers employed in the city centre commute by public transit (Zheng and Kahn, 2008). In contrast, only 18% of workers in areas serviced by Washington DC’s WMATA regularly use public transportation (Nelson et al., 2007). The average commuting time is correspondingly higher: one-way commutes average 37.8 min in Beijing, considerably longer than comparable US cities such as Boston, Minneapolis, and Chicago (Zheng et al., 2009). In our data, the average commute to central Hong Kong is 22.9 min by car and 52.6 min by public transit. However, driving times also likely reflect the effects of traffic and the tendency for the average subsidized homes to be further from downtown than commercially developed residencies. Specific to Hong Kong, Chan and Tse (2003) estimate rent and distance gradients by measuring monetary commuting cost and time. Their findings suggest that cost of commuting to central Hong Kong is significantly correlated with property values. However, commute time does not display similar significance. In contrast, our spatial analysis suggests robust significance in transit time effects, even after adjusting for commute time by car and walking distance to the closest metro station. Chan and Tse further estimate that the value of commuting time in Hong Kong is approximately 48% of the wage rate, within the established range of 25–50% in the urban economics literature (Small, 1983). Yiu and Tam (2004) develop the discussion with a contrast of rent gradient analysis methods applied to the Hong Kong housing market. A four-way comparison is made between among methodologies, divided by the use of hedonic versus repeat-sale models and the inclusion versus exclusion of monocentric assumptions. The authors confirm monocentricity but challenge the reliance on monocentric approaches, finding mis-estimations of prices attributed to district sub-centres and variations in spatial amenities: we address both concerns in our modelling process. More recently, Chow (2011) finds a statistically significant link between distance to metro stations and the selling price of commercial apartments in Hong Kong. Other factors constant, a doubling of walking distance to the closest station lowers the expected selling price by approximately 3.7%. Although the scope of data involved in the paper is limited, the fairly large coefficient underpins the importance of the Hong Kong metro system, suggesting the possibility of price effects beyond that of conventional distance gradients. 3. Visualizing elevation effects Before discussing modelling and statistical methodologies in detail, we first provide intuition regarding the existence of elevation effects in Hong Kong. It is difficult to directly present descriptive evidence of elevation effects: spatial factors are generally highly correlated, and effects such as centricity are potentially more robust predictors of price. However, the presence of a preference for flatness conceptually should have a large impact on price residuals, given that other spatial factors were rigorously controlled for. To this end, we perform an OLS regression on the full 121-variable dataset as outlined in Section 4.1, but with all variables related to the slope between the apartment and the nearest metro station removed. Standardized residuals of predicted selling prices of apartments, averaged at the apartment-group-level, are plotted spatially with metro station locations on the local altitude contour, with one example provided in Figure 1.1 Figure 1 View largeDownload slide Filled altitude contour plot with spatially mapped apartment-group average price residuals and metro station locations, Hong Kong, 1998–2014. Figure 1 View largeDownload slide Filled altitude contour plot with spatially mapped apartment-group average price residuals and metro station locations, Hong Kong, 1998–2014. We note that visually, large negative residuals appear to be distributed at higher altitude levels. As our analysis in Section 5 suggests, the discrepancy is most likely caused by selling prices of apartments with large elevation differentials from the closest station being systematically over-predicted. This trend can be observed more clearly by binning price residuals by a fixed range of slope levels and walking distances. Figure 2 summarizes price residuals for walking distance intervals of 0–2000, 500–1000 and 1000–1500 m, respectively. We find that for a given range of walking distances, sale prices of apartments with steeper slopes to the nearest station are generally overestimated, with the amount of over-estimation increasing on average with slope. Figure 2 View largeDownload slide Average regression price residual summarized by walking distance interval and range of slope in degrees, compared. Figure 2 View largeDownload slide Average regression price residual summarized by walking distance interval and range of slope in degrees, compared. We note that the trend is weaker when summarized from 0 to 2000 m, since apartments with steep walks also tend to have shorter walks; we explicitly control for this interaction effect in our model in Section 4.1. Also, consistent with our findings of substantial elevation effects in Section 5, a gap of 0.12–0.15 in average price residuals translates to a fairly large difference of approximately 0.45–0.6 standard deviations. 4. Data and methodology 4.1. Conceptual and empirical framework Neither of the two gradient effects we investigate is conceptually complicated. Knowing that a positive relationship exists between proximity to metro stations and apartment prices, we introduce two terms representing the walking distance and decimal degree of incline between the location of the apartment and that of the station. Both effects ought to be distinctly non-linear since transportation modes are expected to change: people will not use the metro if accessing the nearest station is sufficiently difficult. Similarly, slopes only slightly steeper could translate to significantly less walkable conditions. The anticipatory effect of a future, closer station on current prices is expected to vary not only with the time gap between the transaction date and the opening of the future station, but also with the gain in metro accessibility provided by a closer station. Conceptually, future stations should be able to influence current prices as Hong Kong regularly announces metro projects decades before construction starts, at which point future station locations become public knowledge. This does not necessarily imply information symmetry: certain individuals might learn about such events before others do. The precise opening date also would be uncertain from the perspective of the buyer. These intuitions suggest non-constant anticipatory effects caused by the gradual diffusion of information throughout the city’s population. Independent of information diffusion effects, the value of a future subway station should be discounted by its ‘futureness’, i.e. the amount of time a buyer has to wait until he or she can access the station. After adjusting for uncertainty and discounting perceived future gains, buyers should regard future stations as substitutes, albeit imperfect, for currently available ones in the surrounding area. Specifically, a buyer might pay a significant premium for an apartment currently far from any subway station, with the knowledge that a close-by station would open in the near future. This effect is bounded from spatial and temporal directions: a station under construction can never be exactly as desirable as the same station opened, and a station that is sufficiently ‘distant’ in the future should not have much effect on current sales. Therefore, aside from announcement and opening effects, we expect no sudden jumps in the perceived premium: price effects ought to move gradually from zero to a fixed value at the date of opening. The size of the perceived premium of a future station should also vary by how much utility it brings. If the current-best station is far away and the future station close by, the price effect of the future station should be greater than that of a scenario where walking distance savings of moving to the new station is small. To obtain stable estimates of elevation and distance gradient effects, we first restrict our dataset to 30,528 observations where the closest metro station did not change for the duration of the data.2 These observations are used to generate housing price effect estimates for stations with a given walking distance/slope combination (part I). Using these estimates and the remaining 7017 observations that experience metro expansion, a second, separate regression analysis (part II) is performed to determine the relationship between the equivalent anticipatory station value, denoted as a percentage of the station’s value post-opening, and the time gap. However, predictions on the price effects of metro station-apartment slope and distance are limited in range. In our part I data, 93.8% of all observations (28,634) report a closest station walking distance of less than 5000 m. Moreover, 98.5% of observations (30,059) report a closest station average slope of less than 9 decimal degrees. Given that little information is available on stations that do not fit these criteria, it is unfeasible to reliably extend inference to price effects of stations with distances and steepness that exceed these values. On the other hand, most ‘current’ stations in the anticipatory effect subset have much greater walking distances. 62.2% of all observations (4367) in the part II subset have a current station distance that exceeds 5 km. This is not unexpected, since subway line development is most robust in districts that have a dire need for extra public transit options. Without discarding a majority of observations, estimations of equivalent prices of current stations in this dataset will be problematic. Results from the regression in part I perform well when estimating ‘final’ station price effects in the data for part II, but suffer from severe range mismatch when estimating current station price effects. Our solution to this issue is 2-fold: first, we select a two-stage regression approach with cross-validation (CV) for part I with the goal of optimizing inference quality of elevation-price and distance-price effects. Variable-selection is conducted with a first-stage LASSO regression (Tibshirani, 1996) with 50-fold CV. The LASSO process assumes a penalty parameter selected via CV and, by fitting for a biased least squares estimator subjected to the constraints of the penalty term, reduces the size of coefficients accordingly. Ideally, coefficients of variables that do not significantly contribute to the model’s predictive ability are reduced to exactly zero, and are effectively removed from the regression. However, over-penalization is a general concern with such methods: significant coefficients might be reduced too much by a large penalty parameter. In response, we introduce a secondary OLS stage to the regression process. Variables that survive the initial selection process are regressed with a large number of OLS models on randomized and reduced subsets of the dataset, and results are averaged across models. More specifically, for an apartment transaction j within the subset of apartment sale prices yi, a vector of corresponding elevation and distance gradient terms xije and a vector of covariates xijc, we fit:   yîj=βe*xije*+βc*xijc*+εij, (1) where ^yij is the predicted selling price, and xije* and xijc* are subsets of xije and xijc whose coefficients are not reduced to exactly zero after fitting:   ̂yk=βexek+βcxck+ɛk (2) over observation k in the full dataset, subjected to a criterion of minimizing the sum of squares Pk(yk − y^k)2 subjected to the constraint (Pm|βem|+Pn|βcn|) ≤ s for a total of m elevation and distance terms and a total of n covariates, given tuning parameter s as determined by CV. Here, s controls the amount of penalization: when s is very large relative to model size, the constraint has no effect and the results are identical to an identically specified OLS fit. When s is small, a large proportion of terms in βe and βc are reduced to exactly zero. We subset the data by drawing fewer than the original number of observations for three purposes. First, we seek to prevent ‘data-point chasing’ behavior introduced by multiple higher-order terms on slope and distance effects: drawing fewer observations for each iteration without replacement decreases the potential structural similarity between each draw and the full dataset, providing conservative estimates of higher-order effects. Second, we are concerned by potential outliers in a standard sample-with-replacement bootstrapping approach. Using less data sampled over a greater number of regression iterations further reduces the risk of an iterative approach over-estimating confidence. Third, the subsetting of data inflates uncertainty by a factor inversely proportional to the fraction of data used. Since our results are intended to be presented graphically with bootstrapped standard errors, obtaining extremely conservative estimations of prediction boundaries is crucial. The data in part II are further selected for observations with a future station closer than 4000 m in distance and a current station further than 4000 m. This reduces the part II dataset to 4683 observation, all of which are subjected to final-station price effects that can be reliably estimated with results from part I.3 Conceptually, we do not expect current stations that are more than 4 km’ distance away to have any significant price effect and model them as being nonexistent. Four alternative models of a curve between the time gap to opening and relative value of the future station are fitted using an iterative scheme similar to that of part I, which we discuss in Section 4.2. 4.2. Data An in-depth overview of subsidized housing in Hong Kong can be found in Ye (2015). Semi-commercial housing programs in Hong Kong were originally devised to bridge the gap between commercial housing and public rental programs. Apartments are built by developers under guidelines of the Hong Kong Housing Authority (HKHA) and are purchased by eligible residents at a government-subsidized, ‘discounted’ rate. Eligibility is determined by a combination of factors including family size, income, and duration of residency in public rental units. The subsidy program provides an open-market platform for current owners, either to sell to other eligible buyers or to non-eligible buyers willing to ‘refund’ the original subsidy to the HKHA. Several features of subsidized apartments are noteworthy in the context of this paper. First, subsidized apartment residents are typically lower-middle to middle class in terms of income: as of 2003, approximately 54% of Hong Kong residents live in heavily subsidized, low-cost public rental units, and 31% live in commercial apartments or houses (Liu, 2003). Second, most subsidized apartments are sold to families of four or more individuals. Hence, residents are very likely to be working couples with at least one school-age child (Er and Li, 2008). Third, subsidized apartments generally are not located in the downtown area but rather are in suburbs and district sub-centres. Only 2.7% of all transactions in the dataset are located in downtown Hong Kong.4 These features present significant advantages for our data analysis. The strong homogeneity of subsidized apartments and their residents is an extraordinary benefit to the modelling of spatial amenities. Traditional predictors of price differences in homes, such as furnishing, number of bathrooms and number of bedrooms, can be assumed to be virtually identical across observations. Heterogeneity in non-spatial parameters is likely trivial after controlling for size, floor level and age, leading to models with high explanatory power: an OLS fit on the full model in part I generates an R2 value of 0.94. The expected income level and family size requirements of residents also suggest a reliance on public transit: individuals who qualify for subsidized apartments are most likely not wealthy enough to afford regular car use in Hong Kong. Transaction data are obtained from HKHA public records (HKHA, 2015). The data consist of 37,750 transactions between August 1997 and July 2014, accurate to the month of transaction with information on size, location and other characteristics of the apartment. Data from Google Maps, the OpenStreeMap project and Microsoft Representational State Transfer Services are pooled to create a detail geospatial profile of the HKSAR region that incorporates local amenities, road networks and transit availability. A 30-m resolution DEM is used to obtain altitude information at each latitude and longitude combination.5 We note that bus system data are not included. Data on bus lines in Hong Kong are difficult to obtain: there are multiple service providers, and the fact that not all stop names have consistent English transliterations presents a major challenge for Geocoding. Also, given Hong Kong’s traffic conditions we believe it unlikely that individual bus lines are viewed by the buyer as significant amenities. Bus accessibility in aggregate could be valuable: this should be highly correlated with local commercial activity, which we account for through a variety of geospatial control terms. Subway station entrances are approximated by intersecting a buffer of 50 m, centred on the point estimate of the station’s location, with a network layer of walkable roads.6 Apartment-to-station walking distances are calculated via the Closest Facility Analysis tool in the ArcGIS Network Analyst package, with routes also limited to walkable roads. If an apartment group’s centre is within the buffer zone of a station, a minimum distance of 50 m is assumed. Elevation gradients are approximated by the average, absolute decimal degree of incline between the location of the apartment and metro station. No distinction is made between positive and negative elevation differentials, under the assumption that trips to the station must go both ways. An estimate by evaluating the slope along a walking route would perhaps be preferable, but is not used because available road network profiles for Hong Kong are not designed to accommodate z-axis information or to be used in a three-dimension GIS frame. To measure anticipatory effects, five different network layers of metro stations are created in ArcGIS to reflect metro system development between 1997 and 2014. Each additional layer includes new stations from one major metro expansion. Two sets of distances are created for the apartment in each transaction: walking distance to the closest currently available station at the transaction date, and walking distance to the closest future station, if one exists between the transaction date and 2014. We also collect data on each transaction’s time gap between the transaction date and the future station opening date. Note that metro stations are assumed to be perfectly substitutable. We justify this decision for three reasons. First, the majority of apartments lies further away from central Hong Kong and residents typically only have access to one metro line connecting to the downtown area. This suggests that substitution effects are generally not between stations on different metro lines but between adjacent stations on the same line. Hence, conditional on distance from the CBD, the main commuting costs are incurred by travel time and effort to a given metro station, rather than differential on-rail commuting time depending on a particular metro stop. Second, the amenity advantage of specific stations is partially controlled for in the regression model via district boundary distance, housing density and a variety of centricity-based metrics. Although we cannot directly measure commercial activity along the apartment-station walking route, these variables will be highly correlated with the supply of local amenities and largely capture such effects. Third, assuming station homogeneity provides key advantages in analyzing anticipatory effects. With this assumption, price effects of stations on apartment sales can be reduced to a fixed premium defined by walking distance and slope, greatly reducing model complexity. In Section 5, we show that these two variables reasonably encapsulate the full price effect of stations. 5. Model checks and results Our analysis consists of two parts. Part I estimates price effects of elevation variance and distance gradients; part II utilizes these estimates to investigate future-station effects. For part I, a two-stage penalized regression model is fitted on the no-anticipatory-effect dataset with apartment-specific and geospatial controls as well as year, month and district fixed effects. Altitude from sea level and interactions among altitude, coastline distance and floor level are also controlled for as a proxy for scenery effects. For the full model, terms for slope and walking distance up to the third power and interaction terms up to quadratic effects are included for a total of 121 variables. A list of variables used in the full regression and summary information can be found in Table A3. The initial stage utilizes a LASSO regression as described in Section 4.1 with standardization and 50-fold CV training for variable selection. Using the resulting penalty term, 98 out of the 121 full-model variables, including 6 out of the 10 metro distance, slope and distance-slope interaction terms, survive the penalization scheme and are included in the second-stage OLS. A log-lambda traceplot of the CV procedure is provided in Figure A3. We note that conceptually there is no guarantee that this process consistently selects the same terms: if there are two weakly explanatory yet highly correlated variables in the full model, LASSO may be ambiguous between reducing the coefficient on one or the other. What the process does guarantee is stability: surviving terms are optimized for predictive performance. We provide a check for our LASSO procedure by comparing it with a conceptually equivalent alternative: a Bayesian LASSO model (Park and Casella, 2008) implemented via Markov Chain Monte Carlo (MCMC). Details and results are discussed in the Appendix A. In stage 2 we employ an iterative, reduced-data bootstrap regression approach using randomized data. 1000 OLS iterations are performed on the reduced model with a randomized, 20% subset of the no-anticipatory-effect data (6105 observations). Output is summarized in Figure A4 with null-hypothesis tests estimated by one-tail credible intervals derived from the draws. Bootstrap densities of select coefficients are displayed in Figure A7. Distributions of model R-square and adjusted R-square values are presented in Figure A6. A simple residual check is performed against age, log apartment size, selling price and walking distance, and results are provided in Figure A15. Predicted curves between slope and expected selling price per square meter are plotted for fixed walking distances of 500, 1000, 1500 and 2000 m, using 100 random samples from the 1000 sets of regression coefficients. As shown in Figure 3, the negative relationship between steepness and predicted selling price is extremely robust for walking distances up to 1500 m.7 The area covered by fitted curves can be considered as describing an interval where the ‘true’ fitted slope-price curve would fall with at least 99% certainty.8 Hence, we conclude that there is strong evidence that elevation gradient differentials influence selling prices of apartments at a range of walking distances. Figure 3 View largeDownload slide Fitted slope-estimated price curves with 99% credible intervals for fixed walking distance of 500–2000 m to metro station, degrees/HKD. Figure 3 View largeDownload slide Fitted slope-estimated price curves with 99% credible intervals for fixed walking distance of 500–2000 m to metro station, degrees/HKD. Holding other factors constant and at respective dataset averages, an apartments that is 500 m away from the closest metro station at a 10-degree average incline is expected to sell for 19.8% less than a similar apartment but on the same altitude level as the station. The corresponding expected discount is 19.9% for a 10-degree difference at 1000 m and 13.8% for an 8-degree difference at 1500 m. These effects are huge when taking walking distance into perspective: an apartment with a 500-m closest station at a 10-degree slope is expected to sell for 8% less than an otherwise similar apartment with a zero-slope closest station 2000 m away.9 These effects are likely to be exaggerated for two reasons: first, the average apartment-station slope is typically much smaller than the maximum slope. An average slope of 7–8 degrees likely suggests a route maximum of more than 15 degrees and hence significant difficulties in terms of pedestrian access. Second, the model in part I controls for a range of spatial benefits associated with higher altitude: apartments with higher altitude enjoy better views, better air quality and less noise and there is some evidence for ‘pure’ altitude and elevation-floor level interaction effects.10 Hence, the lack of metro station walkability is at least partially offset by other amenities for higher-altitude apartments. For part II, we use coefficients on slope and walking distance effects to generate an ‘equivalent’ future station value variable for each observation in the anticipatory-effect data. If the slope–distance combination of the future station lies within the boundaries of the dataset in part I, this estimated value ought to describe the real, perceived value of an opened station with reasonable accuracy. Unopened stations are expected to be comparatively of less value, with perceived worth decreasing proportionally to the time gap, Δt, between the transaction date and station opening. To explore this relationship, we fit four alternative models using modified versions of the full model from part I. Note that we do not perform variable selection for this part since predictive performance is not of primary concern: we care about the ‘true’ shape of the time-station value relationship as opposed to its most predictive shape given our data. Considering the sensitivity of treatment effects to time and the reduced temporal range of part II data (89 months), we replace year and month indicators in the full model with indicators of the specific time period of the transaction, 88 in total.11 All variables involving metro station gradients are replaced with the equivalent future station value and its interactions with Δt in months. Δt as a stand-alone effect is not included in the model, since there ought not to be a premium simply for being close to a certain point in time per se. Models are fitted against price with different numbers of higher-order interaction terms between the time gap and estimated future station value. The largest model includes interactions up to station value (Δt7); the other three models correspondingly limit Δt to the fifth, third and second power terms. To assess quality-of-fit, we present ‘pseudo’ credible intervals obtained by an n-fold bootstrap procedure analogous to that of part I.12 1000 OLS regressions are performed with all four models, with 20% of the data randomly omitted for each iteration. We summarize regression output in Tables A6, A7, A8 and A9. Table A10 contrasts coefficients on the future station value variable and interaction terms, and Figure A10 plots R2 and adjusted R2 distributions of the models. Note that there is little meaning in drawing inferences on the spatial control covariates in Tables A6–A9 individually. This is not only because of the large number of variables but also because of terms designed to approximate spatial factors that are more difficult to measure (distance to district boundary as proxy for local commercial buildup). With geospatial quantities inevitably being correlated (different estimates of centricity, for example), interpreting a particular spatial control’s coefficient in isolation will not result in useful results. This also holds true for the comparison between models in parts I and II because part II does not involve penalization. Our evidence for consistency between the regressions of the two parts is 3-fold. First, the overall explanatory power is highly robust for models of both parts. The part I bootstrap iterations have an average R2 value of 0.93, and the part II average R2 values are between 0.87 and 0.88 for all models. Second, we note that non-spatial terms in both parts—size, floor level indicators, discount rate—are consistent in terms of direction and significance. This suggests that both sets of models consistently evaluate basic preferences for housing consumption and amenities, and that preferences are likely similar between buyers in the data in the two parts. Third, all models in part II predict a ‘future’ station value of approximately 67–80% of the opened station value at a Δt of 0 months. Although we discuss at length the possible causes of failing to converge to full value at Δt = 0, the key finding is that after controlling for ‘futureness’, the parameters given by part I describe reasonably well how much individuals in the data of part II are willing to pay for a given, present subway station. This is our strongest case for consistency between models: conceptually, other discrepancies between the two datasets are not of significant concern if one part provides robust information about perceived metro station values of the other. Fitted results from 100 random coefficient set samples are presented in Figure 4. Similar to the estimated boundaries in Figure 3, regions defined by multiple curves approximate the range in which the ‘true’ fit lies with at least 99% statistical certainty. All four models suggest a significant, negative relationship between Δt and the equivalent future station value. The station effect does not seem to converge to zero for a sufficiently large Δt in any of the four models: the equivalent value is estimated to be approximately 25–45% of the opened station at 80 months prior to opening. Figure 4 View largeDownload slide Fitted slope (w/99% CI) between months to opening and perceived metro station value as % of opened station value, Hong Kong, 1997–2004. Figure 4 View largeDownload slide Fitted slope (w/99% CI) between months to opening and perceived metro station value as % of opened station value, Hong Kong, 1997–2004. The presence of higher-order effects is evident: best-fit curves for the first–seventh and first–fifth order power term models distinctly differ from those with only lower-order Δt terms. However, the inclusion of the sixth and seventh order terms appears to reduce certainty at large Δt values. Pseudo-error bounds increase marginally for larger models at all Δt, but the smallest model do not appear to be substantially more stable than its larger alternatives.13 Beyond comprehensively providing strong evidence of a negative, non-linear trend, none of the four model alternatives stands out as a clear choice.14 In response, we perform unweighted averages over these fitted curves for the following analyses. Failure to realize full value at zero months is likely caused by transactions being recorded at the time of finalization. Observed sale prices usually are determined prior to finalization, suggesting that the 0-month value does not describe that of an opened station but that of a station shortly before opening. This suggests an observed station opening ‘premium’ of approximately 20–30% of the station’s full, perceived price value. This is not unreasonable, given the fact that any period without metro access could be highly undesirable if commuting depend crucially on subway availability. Failure to converge to zero value at a large Δt of 70–80 months could be caused by several reasons. First, station value may come from expected future gains: the buyer may not be able to use the metro system during his or her tenure of ownership but may expect the apartment to nonetheless appreciate in price. Second, an announced location of a future station will generate commercial development, much of which will precede the opening of the actual station. Although access to the station is only possible after a given date, secondary effects of ongoing development can be enjoyed prior to station completion. Third, buyers may not have identical preferences. For example, some buyers may prefer transit by bus or to work close to home. An open-market scenario could mostly contain such buyers when a nearby station is not available in the near future. If strongly metro-preferring buyers do not have a major presence until the completion date draws close, prices of earlier transaction may reflect a premium paid by metro-indifferent (or even metro-disliking) buyers. Using the averaged fitted coefficients from each model, we estimate the relative appreciation of a station’s value for different time intervals. Moving from 6 years prior to opening to 4 years prior increases the station’s perceived value by approximately 82–92%. Moving from 4 years to 2 further increases its value by approximately 25–46%. At 2 years prior to opening, all four models suggest the station to be valued at 58–60% of the full, opened value. The estimated value increases to 61–67% at 6 months prior to opening, and to 66–80% at 0 months. Performing a simple average over the four models, we estimate the gain in perceived value of station at an average compound rate of 1.1% per month or 14% per year over a 6-year interval. If the future station is considered as an investment, given a 6-year-period the buyer can expect earnings of approximately 122.5% of the principal. In comparison, the 6-year earnings given local, real, annually compounded interest rates for years 1999–2004 is approximately 81.4%.15 It is unsurprising that these figures are roughly comparable: the possibility of choosing an apartment without anticipatory effects prevents futures stations from being priced too robustly, and arbitrage would prevent them from being priced too cheaply. The comparison also suggests that a future, close-by subway station, given the time period, is arguably an investment vehicle competitive with other financial assets. If this remains true today, there may be a case for purchasing apartments (with the intent to sell in the foreseeable future) in locations that are anticipating future subway access, with the current sale price premium serving as the principal of investment. However, the high ‘return’ of the future station may also reflect expect gains of commercial development near the station and expected future utility of using it, both of which will bias the rate of return to the infrastructure investment upwards. 6. Broader implications Neither elevation gradient nor anticipatory effects are trivial from the urban economic perspective. Anticipatory effects are particularly relevant for cities in developing economies experiencing high rates of infrastructure development. As an example, Beijing has added 17 new metro lines in the past 15 years, with 6 more planned for completion in the next 5 years. The Hong Kong metro system is scheduled to add three new lines and expand on six existing lines by the end of 2026. The extremely rapid public transit system buildup of these cities suggest that local housing markets are not only privy to information about anticipatory effects, but are likely to be actively pricing future stations into current transactions, subject to constraints on real rates of return. Time-trend modelling of such housing markets suffer if the overall price effect of new, major subway expansions is assumed to be applied at opening instead of accruing gradually for extended periods prior to completion. Similarly, stationary spatial models lose inference accuracy by severely under-fitting prices of homes in areas close to future stations. Hence, models that explicitly account for future conditions of urban systems are expected to perform significantly better in terms of analyzing current prices of spatial amenities. Beyond metro systems, the concept of future amenities exerting influence on current prices indicates the necessity of dynamic, temporal modelling processes, as outlined in this paper, in urban economics research. Since almost all investigations of housing and other price gradients are conducted in hindsight, information about what occurs after the period of investigative interest can be harnessed to address anticipatory effects. These effects may be of diverse origins: announcements of urban renewal, future schools, commercial centres or road network expansions all could affect current housing markets. Typically, hilliness is not explicitly controlled for in urban economic modelling. However, our results from Hong Kong demonstrate that elevation is a potentially huge determinant of housing prices. Spatial models that involve high elevation–variance cities, especially those with unevenness somewhat comparable to Hong Kong, are susceptible to large residual issues over spatial parameters: home price observations will be systemically over-estimated or under-estimated depending on the local degree of hilliness. The more uneven a city, the more severe such mis-estimations are likely to be. Furthermore, elevation gradient effects demonstrated in this paper suggest that cities with high levels of elevation variance, ceteris paribus, will benefit less from the same level of public transit infrastructure. Assuming that individuals have a fixed level of tolerance for difficulty of walking to a station for daily commuting, fewer individuals will use public transit in the area with a given proximity to a station in highly uneven areas. Walk-oriented public transit hubs, if located in areas with significant terrain unevenness, should correspondingly be valued less by local home-owners. These effects suggest the potential for optimization: hilly, high-unevenness cities face different constraints when developing public transit networks. Because providing the same level of consumed public transit infrastructure is more costly in uneven cities, such cities may have public transit amenity equilibria below those of flat cities. By better understanding the influence of elevation gradients on preferences and behavior of urban markets, solutions could be designed to minimize the adverse impact of unevenness. Elevation gradient effects for metro stations also indicate similar behavior for other common destinations reached by walking: for example, schools, parks, and bus stops. With neighboring amenities decreasing in perceived value as elevation differentials increase, elevation variance of cities are expected to predict other, more important urban gradients such as population and housing density. As an addition to existing literature (Saiz, 2010; Ye and Becker, 2016), we present a simple analysis of the relationship between density gradients and unevenness of urban areas in the Appendix C. 7. Conclusion Using extensive geospatial methods, we provide evidence of significant anticipatory future-station effects in the price influence of metro stations on urban housing. We estimate that for the Hong Kong subsidized market, the average growth trend of the perceived price value is approximately 1.1% of the full station value per month, starting from 6 years prior to completion. Paying for a future, close-by metro station yields returns on investment that are roughly consistent with and marginally superior to period interest rates. We also present evidence for highly significant, non-linear elevation gradient effects on the walking distance between metro stations and apartments. We find that elevation differentials between stations and apartments are a huge determining factor in selling prices of apartments. Although in reality these effects are partially offset by benefits associated with higher elevation, they nonetheless introduce significant implications in terms of geospatial modelling methodologies, investigations of urban housing and public transit systems, as well as relationships between elevation and urban gradients in general. This paper reveals two significant shortcomings of the conventional, bi-dimensional spatial modelling approach: the omission of temporal and elevation-variance-related elements. Models of rapidly developing cities suffer if not considered in a dynamic framework inclusive of anticipatory effects of future amenities; models of uneven cities suffer without the augmentation of elevation profiles in spatial parameters. We conclude that without explicitly addressing these effects in a multi-dimensional approach, investigations of spatially sensitive economic effects are highly susceptible to significant and systematic residual issues. In light of recent literature on geographical effects in urban systems which suggest that a large range of urban gradients are influenced by elevation patterns, the transaction-level analysis presented in this paper allows for a better understanding of how elevation effects at the district and neighborhood-level, as well as at the multi-city level, originate. By estimating the aversion of elevation disparities caused by difficulty in accessing public transit, our work provides a conceptual basis for the role that terrain unevenness plays in urban density and population gradients. Footnotes 1 Two other price residual-contour plots are available in Figure A1 and Figure A2. 2 Table A1 summarizes descriptive figures of transactions in this data subset. 3 Table A2 summarizes descriptive figures of transactions in this dataset. 4 Central Hong Kong is defined as the following five districts: Wan Chai, Central, Yau Tsim Mong, Sham Shui Po and Kowloon City. 5 A three-dimensional contour plot demonstrating the DEM resolution and the extreme level of terrain unevenness of the HKSAR region is provided in Figure A12; Figure A13 presents a map of Hong Kong with locations of apartment groups, select amenities and walkable roads. 6 The 50-m buffer radius is re-applied to the final station-to-apartment distance estimates. 7 A comparison of fitted slope-sale price curves of the four walking distances is available in Figure A14. Densities of predicted selling price for select combinations of slope and distance corresponding to Figure 3 are presented in Figure A9. 8 Given that 20% of the data are used, under normality assumptions we can estimate that the standard deviation, as reflected in Figure 3 is inflated by a factor of 2.2. 9 Conceptually, it should be possible that elevation effects vary by time of year since perceived difficulty of walking would be greater during certain seasons. However, we find that this is likely not the case. A detailed discussion is provided in the Appendix B. 10 The pure altitude term reports p = 0.034. p = 0.08 for the interaction term between the indicator for floor levels ≥27 and elevation. 11 Transactions that occur after 2004 do not experience metro system expansion and are therefore not subjected to a change in the distance of the closest station. 12 This is because we have no knowledge of what the ‘true’ curve shape is under these circumstances. For instance, we do not know if housing market shocks may change perceived values of future spatial amenities, perhaps leading to non-linear fits with multiple local maxima. 13 An alternative visualization to Figure 4 is presented in Figure A11 with densities of predicted equivalent station values for each model contrasted between several time gaps. 14 The first–fifth order power term model has the highest overall number of highly significant interaction terms (all p < 0.001). However, this is only evidence that this model most robustly rejects the null hypothesis of there being no anticipatory effect at all. 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Table A1 Summary statistics on select characteristics of apartments, Part I no-anticipatory effect data Name  Mean  SD  Min  Max  Age (years)  12.53  5.34  1  33  Size (m2)  49.78  7.81  12  86  Driving distance to central Hong Kong (km)  19.53  10.36  6.9  59  Driving time to central Hong Kong (min)  21.65  8.49  9  63  Price per m2 ($HKD)  28,798.9  14,595.37  2398  105428  Walking distance to closest station (m)  1422.6  1353.22  52.49  13,274.3  Name  Mean  SD  Min  Max  Age (years)  12.53  5.34  1  33  Size (m2)  49.78  7.81  12  86  Driving distance to central Hong Kong (km)  19.53  10.36  6.9  59  Driving time to central Hong Kong (min)  21.65  8.49  9  63  Price per m2 ($HKD)  28,798.9  14,595.37  2398  105428  Walking distance to closest station (m)  1422.6  1353.22  52.49  13,274.3  Table A2 Summary statistics on select characteristics of apartments, Part II anticipatory-effect data Name  Mean  SD  Min  Max  Age (years)  8.41  2.70  2  20  Size (m2)  49.92  7.19  20  90  Driving distance to central Hong Kong (km)  29.46  7.92  10  39  Driving time to central Hong Kong (min)  30.04  5.79  13  37  Price per m2 ($HKD)  19,410.14  5653.74  3337  54,896  Walking distance to closest (final) station (m)  1429.15  745.56  52.49  3613.87  Name  Mean  SD  Min  Max  Age (years)  8.41  2.70  2  20  Size (m2)  49.92  7.19  20  90  Driving distance to central Hong Kong (km)  29.46  7.92  10  39  Driving time to central Hong Kong (min)  30.04  5.79  13  37  Price per m2 ($HKD)  19,410.14  5653.74  3337  54,896  Walking distance to closest (final) station (m)  1429.15  745.56  52.49  3613.87  Because the classic Bayesian LASSO as implemented by Park and Casella does not reduce any of the coefficients to exactly zero, we choose an alternative Reversible Jump MCMC-based model modified by Gramacy (2014) from Hans (2009) specifically for the purpose of variable selection. Using 10,000 MCMC iterations and a burn-in of 2000 draws, the algorithm returns an average post-penalization model size of 103 variables. Convergence of the MCMC process is demonstrated by trace plots of key variables in Figure A4, and a trace plot of the size of the best model selected in each iteration is provided in A5. Although the two approaches do not perfectly agree on the number of variables that should be removed, evidence from the Bayesian approach strongly agrees with the regular LASSO in terms of post-penalization model size: both procedures suggest eliminating approximately 20 variables from the full model. This result suggests that the explanatory power gains to be had from a significantly larger model would not compensate for the loss of predictive stability and that the stability gains of a significantly smaller model would not sufficiently compensate for the loss of explanatory power. Here, we also illustrate the distinction between optimizing for general model performance versus predictive ability by presenting a comparison between the regular LASSO and forward–backward stepwise variable selection. Stepwise AIC yields a larger model size of 117 variables and stepwise BIC a model size of 101 variables, comparable in size to the LASSO models. Models selected by AIC and BIC are also regressed with reduced-data bootstrapping. As shown in Figure A8, stability of key variables in models determined by stepwise processes are not significantly better than that of the full model. In contrast, LASSO omits variables from the full model in a way that pulls surviving coefficients into substantially denser distributions. Therefore, given new data with different combinations of slope and walking distance, the LASSO will return predicted price effects that are much more consistent. Table A3 Names and information of variables in Part I full model Name  Variable meaning  Priceadjtwothird  Selling price of apartment per m2, transformed to 1.5th power  Walkmtractual  Walking distance to closest metro station, m  Walkmtractualsqr  Walking distance to closest metro station^2, m  walkmtractual3  Walking distance to closest metro station^3, m  Walkmtrangle  Slope between apartment and closest station, degrees  walkmtranglesqr  Slope between apartment and closest station^2, degrees  walkmtrangle3  Slope between apartment and closest station^3, degrees  mtrwalkINTangle1  Closest station slope * station walking distance  mtrwalk2INTangle2  Closest station slope^2 * station walking distance^2  mtrwalk1INTangle2  Closest station slope^2 * station walking distance  mtrwalk2INTangle1  Closest station slope * station walking distance^2  Feb  Indicator for transaction being in February  Mar  Indicator for transaction being in March  Apr  Indicator for transaction being in April  May  Indicator for transaction being in May  Jun  Indicator for transaction being in June  Jul  Indicator for transaction being in July  Aug  Indicator for transaction being in August  Sep  Indicator for transaction being in September  Oct  Indicator for transaction being in October  Nov  Indicator for transaction being in November  Dec  Indicator for transaction being in December  yd1998  Year indicator for being in 1998  yd1999  Year indicator for being in 1999  yd2000  Year indicator for being in 2000  yd2001  Year indicator for being in 2001  yd2002  Year indicator for being in 2002  yd2003  Year indicator for being in 2003  yd2004  Year indicator for being in 2004  yd2005  Year indicator for being in 2005  yd2006  Year indicator for being in 2006  yd2007  Year indicator for being in 2007  yd2008  Year indicator for being in 2008  yd2009  Year indicator for being in 2009  yd2010  Year indicator for being in 2010  yd2011  Year indicator for being in 2011  yd2012  Year indicator for being in 2012  yd2013  Year indicator for being in 2013  yd2014  Year indicator for being in 2014  Age  Age of apartment, years  Sqrage  Age of apartment^2, years  Age3  Age of apartment^3, years  Age4  Age of apartment^4, years  Age5  Age of apartment^5, years  Age6  Age of apartment^6, years  age7  Age of apartment^7, years  age8  Age of apartment^8, years  Lgsize  Size of apartment, m2, logged  Floorm  14–26 floors in height  Floorh  >26 floors in height  discountrate  Percentage amount which apartment is subsidized  intdiscount1998  discountrate * yd1998  intdiscount1999  discountrate * yd1999  intdiscount2000  discountrate * yd2000  intdiscount2001  discountrate * yd2001  intdiscount2002  discountrate * yd2002  intdiscount2003  discountrate * yd2003  intdiscount2004  discountrate * yd2004  intdiscount2005  discountrate * yd2005  intdiscount2006  discountrate * yd2006  intdiscount2007  discountrate * yd2007  intdiscount2008  discountrate * yd2008  intdiscount2009  discountrate * yd2009  intdiscount2010  discountrate * yd2010  intdiscount2011  discountrate * yd2011  intdiscount2012  discountrate * yd2012  intdiscount2013  discountrate * yd2013  intdiscount2014  discountrate * yd2014  Name  Variable meaning  Unluckynum  Indicator for apartment with number 4 or 13 in address  luckynum  Indicator for apartment with number 8 in address  lgdtoNoSec  Distance to nearest major road, de ned as highways or four lanes and above, km, logged  Lgdtocentralcar  Logged driving distance to central Hong Kong by automobile, km  INTperioddtocentral  Time period of transaction in months * Logged driving distance to central  INTperioddtocentral2  Time period of transaction in months * Logged driving distance to central^2  Lgttocentralcar  Logged driving time to central Hong Kong by automobile, minutes  INTperiodttocentral  Time period of transaction in months * Logged driving time to central  INTperiodttocentral2  Time period of transaction in months * Logged driving time to central^2  INTperiodttocentral3  Time period of transaction in months * Logged driving time to central^3  Lgttocentralpublic  Logged public transit time to central Hong Kong, minutes  INTperiodcentralpublic  Time period of transaction in months * Logged time by public transit  INTperiodcentralpublic2  Time period of transaction in months * Logged time by public transit^2  INTperiodcentralpublic3  Time period of transaction in months * Logged time by public transit^3  lgdtoborder  Distance to nearest major road, de ned as highways or four lanes and above, km, logged  lgincome  Logged income of district where apartment is located in, $HKD  INTincomeborder  Distance from apartment to district border * logged income level of district of apartment  lgelev  Logged elevation level of apartment, m  lgcoast  Logged distance to coast, m  intcoastelev  Coastline distance * elevation  intcoast floorh  Coastline distance * floorh  intcoast floorm  Coastline distance * floorm  intelev floorh  elevation * floorh  intelev floorm  elevation * floorm  countdss  Number of (Direct Subsidy Scheme) DSS middle schools in a 3 km radius of the apartment  INTperiodDSS  Time period of transaction * number of DSS middle schools  INTperiodDSS2  Time period of transaction * number of DSS middle schools^2  INTperiodDSS3  Time period of transaction * number of DSS middle schools^3  lgwalkelementary  Logged walking distance to nearest elementary school, m  INTperiodelementary  Time period of transaction in months * logged walking distance to nearest elementary school  INTperiodelementary2  Time period of transaction in months * logged walking distance to nearest elementary school^2  slopeelementary  Average slope between apartment and nearest elementary school, degrees  INTslopewalkelem  Average slope between apartment and nearest elementary school * logged walking distance to nearest elementary school  airport1km  Indicator for being closer than 1 km to the airport  Courtsize  Size of apartment group of apartment, number of units  courtsizesqr  Size of apartment group^2  totaldensity3km10000  Total number of apartments in 3-km radius from apartment  densityprivate3km  Number of commercial apartments in 3-km    radius from apartment  densitypublic3km  Number of public rental apartments in 3-km    radius from apartment  subdensityper3km  Proportion of apartments in 3-km radius    that are subsidized-for-purchase, %  districtcode  Indicators for district-fixed effects (15 in total)  Name  Variable meaning  Priceadjtwothird  Selling price of apartment per m2, transformed to 1.5th power  Walkmtractual  Walking distance to closest metro station, m  Walkmtractualsqr  Walking distance to closest metro station^2, m  walkmtractual3  Walking distance to closest metro station^3, m  Walkmtrangle  Slope between apartment and closest station, degrees  walkmtranglesqr  Slope between apartment and closest station^2, degrees  walkmtrangle3  Slope between apartment and closest station^3, degrees  mtrwalkINTangle1  Closest station slope * station walking distance  mtrwalk2INTangle2  Closest station slope^2 * station walking distance^2  mtrwalk1INTangle2  Closest station slope^2 * station walking distance  mtrwalk2INTangle1  Closest station slope * station walking distance^2  Feb  Indicator for transaction being in February  Mar  Indicator for transaction being in March  Apr  Indicator for transaction being in April  May  Indicator for transaction being in May  Jun  Indicator for transaction being in June  Jul  Indicator for transaction being in July  Aug  Indicator for transaction being in August  Sep  Indicator for transaction being in September  Oct  Indicator for transaction being in October  Nov  Indicator for transaction being in November  Dec  Indicator for transaction being in December  yd1998  Year indicator for being in 1998  yd1999  Year indicator for being in 1999  yd2000  Year indicator for being in 2000  yd2001  Year indicator for being in 2001  yd2002  Year indicator for being in 2002  yd2003  Year indicator for being in 2003  yd2004  Year indicator for being in 2004  yd2005  Year indicator for being in 2005  yd2006  Year indicator for being in 2006  yd2007  Year indicator for being in 2007  yd2008  Year indicator for being in 2008  yd2009  Year indicator for being in 2009  yd2010  Year indicator for being in 2010  yd2011  Year indicator for being in 2011  yd2012  Year indicator for being in 2012  yd2013  Year indicator for being in 2013  yd2014  Year indicator for being in 2014  Age  Age of apartment, years  Sqrage  Age of apartment^2, years  Age3  Age of apartment^3, years  Age4  Age of apartment^4, years  Age5  Age of apartment^5, years  Age6  Age of apartment^6, years  age7  Age of apartment^7, years  age8  Age of apartment^8, years  Lgsize  Size of apartment, m2, logged  Floorm  14–26 floors in height  Floorh  >26 floors in height  discountrate  Percentage amount which apartment is subsidized  intdiscount1998  discountrate * yd1998  intdiscount1999  discountrate * yd1999  intdiscount2000  discountrate * yd2000  intdiscount2001  discountrate * yd2001  intdiscount2002  discountrate * yd2002  intdiscount2003  discountrate * yd2003  intdiscount2004  discountrate * yd2004  intdiscount2005  discountrate * yd2005  intdiscount2006  discountrate * yd2006  intdiscount2007  discountrate * yd2007  intdiscount2008  discountrate * yd2008  intdiscount2009  discountrate * yd2009  intdiscount2010  discountrate * yd2010  intdiscount2011  discountrate * yd2011  intdiscount2012  discountrate * yd2012  intdiscount2013  discountrate * yd2013  intdiscount2014  discountrate * yd2014  Name  Variable meaning  Unluckynum  Indicator for apartment with number 4 or 13 in address  luckynum  Indicator for apartment with number 8 in address  lgdtoNoSec  Distance to nearest major road, de ned as highways or four lanes and above, km, logged  Lgdtocentralcar  Logged driving distance to central Hong Kong by automobile, km  INTperioddtocentral  Time period of transaction in months * Logged driving distance to central  INTperioddtocentral2  Time period of transaction in months * Logged driving distance to central^2  Lgttocentralcar  Logged driving time to central Hong Kong by automobile, minutes  INTperiodttocentral  Time period of transaction in months * Logged driving time to central  INTperiodttocentral2  Time period of transaction in months * Logged driving time to central^2  INTperiodttocentral3  Time period of transaction in months * Logged driving time to central^3  Lgttocentralpublic  Logged public transit time to central Hong Kong, minutes  INTperiodcentralpublic  Time period of transaction in months * Logged time by public transit  INTperiodcentralpublic2  Time period of transaction in months * Logged time by public transit^2  INTperiodcentralpublic3  Time period of transaction in months * Logged time by public transit^3  lgdtoborder  Distance to nearest major road, de ned as highways or four lanes and above, km, logged  lgincome  Logged income of district where apartment is located in, $HKD  INTincomeborder  Distance from apartment to district border * logged income level of district of apartment  lgelev  Logged elevation level of apartment, m  lgcoast  Logged distance to coast, m  intcoastelev  Coastline distance * elevation  intcoast floorh  Coastline distance * floorh  intcoast floorm  Coastline distance * floorm  intelev floorh  elevation * floorh  intelev floorm  elevation * floorm  countdss  Number of (Direct Subsidy Scheme) DSS middle schools in a 3 km radius of the apartment  INTperiodDSS  Time period of transaction * number of DSS middle schools  INTperiodDSS2  Time period of transaction * number of DSS middle schools^2  INTperiodDSS3  Time period of transaction * number of DSS middle schools^3  lgwalkelementary  Logged walking distance to nearest elementary school, m  INTperiodelementary  Time period of transaction in months * logged walking distance to nearest elementary school  INTperiodelementary2  Time period of transaction in months * logged walking distance to nearest elementary school^2  slopeelementary  Average slope between apartment and nearest elementary school, degrees  INTslopewalkelem  Average slope between apartment and nearest elementary school * logged walking distance to nearest elementary school  airport1km  Indicator for being closer than 1 km to the airport  Courtsize  Size of apartment group of apartment, number of units  courtsizesqr  Size of apartment group^2  totaldensity3km10000  Total number of apartments in 3-km radius from apartment  densityprivate3km  Number of commercial apartments in 3-km    radius from apartment  densitypublic3km  Number of public rental apartments in 3-km    radius from apartment  subdensityper3km  Proportion of apartments in 3-km radius    that are subsidized-for-purchase, %  districtcode  Indicators for district-fixed effects (15 in total)  Table A4 Coefficients and significance of Part I regression, excluding year, month, district indicators and year-discount rate interactions Variable  Coefficient  Bootstrap p-value  Variable  Coefficient  Bootstrap P-val  (Intercept)  −1.33E+03  0.008  INTperiodcentralpublic3  4.73E−05  <0.001  walkmtractual  −5.89E−02  <0.001  lgdtoborder  1.74E+02  0.026  walkmtractual3  3.42E−10  <0.001  lgincome  2.98E+02  <0.001  walkmtrangle  −9.83E+00  <0.001  INTincomeborder  −2.01E+01  0.014  mtrwalkINTangle1  −1.14E−02  <0.001  lgelev  5.77E+00  0.034  mtrwalk2INTangle2  1.90E−07  0.028  lgcoast  7.91E+00  0.004  mtrwalk2INTangle1  6.12E−06  <0.001  intelev floorh  4.15E+00  0.08  Age  −1.41E+01  <0.001  intelev floorm  2.86E+00  0.145  age8  7.31E−11  0.002  countdss  −9.35E−01  0.027  Lgsize  1.45E+02  <0.001  INTperiodDSS  4.93E−03  0.199  Floorm  2.90E+01  0.002  INTperiodDSS3  −6.48E−08  0.315  Floorh  4.00E+01  <0.001  lgwalkelementary  3.65E+01  <0.001  discountrate  −5.40E+00  <0.001  INTperiodelementary  −4.08E−01  <0.001  unluckynum  −1.89E+01  0.002  INTperiodelementary2  1.96E−03  <0.001  Luckynum  1.35E+01  <0.001  slopeelementary  3.89E+01  <0.001  lgdtoNoSec  4.15E+00  <0.001  INTslopewalkelem  −7.09E+00  <0.001  lgdtocentralcar  −6.69E+01  0.009  airport1km  9.64E+01  0.221  INTperioddtocentral  1.73E+00  <0.001  courtsize  1.19E−02  <0.001  INTperioddtocentral2  −8.46E−03  <0.001  courtsizesqr  −2.75E−06  <0.001  lgttocentralcar  −8.62E+00  0.399  totaldensity3km10000  −6.95E+00  <0.001  INTperiodttocentral  −1.19E+00  <0.001  subdensityper3km  −1.28E+02  <0.001  lgttocentralpublic  −1.53E+02  <0.001  densityprivate3km  6.89E−04  <0.001  INTperiodcentralpublic2  −5.57E−03  0.001        Variable  Coefficient  Bootstrap p-value  Variable  Coefficient  Bootstrap P-val  (Intercept)  −1.33E+03  0.008  INTperiodcentralpublic3  4.73E−05  <0.001  walkmtractual  −5.89E−02  <0.001  lgdtoborder  1.74E+02  0.026  walkmtractual3  3.42E−10  <0.001  lgincome  2.98E+02  <0.001  walkmtrangle  −9.83E+00  <0.001  INTincomeborder  −2.01E+01  0.014  mtrwalkINTangle1  −1.14E−02  <0.001  lgelev  5.77E+00  0.034  mtrwalk2INTangle2  1.90E−07  0.028  lgcoast  7.91E+00  0.004  mtrwalk2INTangle1  6.12E−06  <0.001  intelev floorh  4.15E+00  0.08  Age  −1.41E+01  <0.001  intelev floorm  2.86E+00  0.145  age8  7.31E−11  0.002  countdss  −9.35E−01  0.027  Lgsize  1.45E+02  <0.001  INTperiodDSS  4.93E−03  0.199  Floorm  2.90E+01  0.002  INTperiodDSS3  −6.48E−08  0.315  Floorh  4.00E+01  <0.001  lgwalkelementary  3.65E+01  <0.001  discountrate  −5.40E+00  <0.001  INTperiodelementary  −4.08E−01  <0.001  unluckynum  −1.89E+01  0.002  INTperiodelementary2  1.96E−03  <0.001  Luckynum  1.35E+01  <0.001  slopeelementary  3.89E+01  <0.001  lgdtoNoSec  4.15E+00  <0.001  INTslopewalkelem  −7.09E+00  <0.001  lgdtocentralcar  −6.69E+01  0.009  airport1km  9.64E+01  0.221  INTperioddtocentral  1.73E+00  <0.001  courtsize  1.19E−02  <0.001  INTperioddtocentral2  −8.46E−03  <0.001  courtsizesqr  −2.75E−06  <0.001  lgttocentralcar  −8.62E+00  0.399  totaldensity3km10000  −6.95E+00  <0.001  INTperiodttocentral  −1.19E+00  <0.001  subdensityper3km  −1.28E+02  <0.001  lgttocentralpublic  −1.53E+02  <0.001  densityprivate3km  6.89E−04  <0.001  INTperiodcentralpublic2  −5.57E−03  0.001        Table A5 Names and information of additional variables in Part II models Name  Variable meaning  mtrpremt1  The equivalent premium of future station in $HKD, obtained by predicting the price premium on the apartment of the future station given its walking distance, slope, and Coefficients from the Part I regression Model  mtrpremINTtime(n)  mtrpremt1 interacted with the time gap (months) to the nth power.  gcode(n)  Indicator for the transaction being in the nth time period (month) of the dataset time span  Name  Variable meaning  mtrpremt1  The equivalent premium of future station in $HKD, obtained by predicting the price premium on the apartment of the future station given its walking distance, slope, and Coefficients from the Part I regression Model  mtrpremINTtime(n)  mtrpremt1 interacted with the time gap (months) to the nth power.  gcode(n)  Indicator for the transaction being in the nth time period (month) of the dataset time span  Table A6 Coefficients and significance of Part II regression for model with interaction terms up to t7, excluding district and time period indicator effects Variable  Coefficient  Bootstrap p-val  Variable  Coefficient  Bootstrap p-val  (Intercept)  4.06E+03  0.018  lgttocentralpublic  −3.08E+02  <0.001  mtrpremt1  7.79E−01  <0.001  INTperiodcentralpublic  −6.34E−01  0.424  mtrpremINTtime1  −5.20E−02  0.03  INTperiodcentralpublic2  9.35E−02  0.125  mtrpremINTtime2  5.04E−03  0.069  INTperiodcentralpublic3  −8.16E−04  0.103  mtrpremINTtime3  −2.58E−04  0.085  lgdtoborder  −1.10E+02  0.301  mtrpremINTtime4  7.82E−06  0.078  lgincome  −2.10E+02  0.115  mtrpremINTtime5  −1.38E−07  0.054  INTincomeborder  9.44E+00  0.326  mtrpremINTtime6  1.26E−09  0.032  lgelev  3.46E+01  0.14  mtrpremINTtime7  −4.56E−12  0.019  lgcoast  −2.84E+01  0.031  Age  1.59E+02  0.291  intcoastelev  −4.20E+00  0.155  Sqrage  −8.28E+01  0.25  intcoast floorh  −6.53E+00  <0.001  age3  2.20E+01  0.219  intcoast floorm  −6.60E+00  <0.001  age4  −3.46E+00  0.186  intelev floorh  −5.95E+00  <0.001  age5  3.26E−01  0.164  intelev floorm  −5.04E+00  <0.001  age6  −1.80E−02  0.148  countdss  −2.02E+01  <0.001  age7  5.37E−04  0.143  INTperiodDSS  6.65E−01  <0.001  age8  −6.64E−06  0.13  INTperiodDSS2  −1.14E−02  <0.001  Lgsize  1.28E+02  <0.001  INTperiodDSS3  6.88E−05  0.007  Floorm  8.74E+01  <0.001  lgwalkelementary  4.01E+00  0.171  Floorh  1.00E+02  <0.001  INTperiodelementary  5.41E−02  0.359  discountrate  −1.47E+00  <0.001  INTperiodelementary2  −7.47E−04  0.336  unluckynum  7.27E+01  <0.001  slopeelementary  9.23E+00  0.022  Luckynum  −2.73E+00  0.255  INTslopewalkelem  −1.24E+00  0.055  lgdtoNoSec  2.25E+00  0.023  airport1km  —  —  lgdtocentralcar  8.00E+01  0.215  courtsize  5.10E−02  <0.001  INTperioddtocentral  2.95E+01  <0.001  courtsizesqr  −9.63E−06  <0.001  INTperioddtocentral2  −2.74E−01  <0.001  totaldensity3km  7.40E−03  <0.001  lgttocentralcar  5.03E+01  0.359  subdensityper3km  −2.11E+02  <0.001  INTperiodttocentral  −1.34E+01  0.027  densitypublic3km  −1.27E−02  <0.001  INTperiodttocentral2  −7.80E−02  0.231  densityprivate3km  −8.65E−03  <0.001  INTperiodttocentral3  2.42E−03  <0.001        Variable  Coefficient  Bootstrap p-val  Variable  Coefficient  Bootstrap p-val  (Intercept)  4.06E+03  0.018  lgttocentralpublic  −3.08E+02  <0.001  mtrpremt1  7.79E−01  <0.001  INTperiodcentralpublic  −6.34E−01  0.424  mtrpremINTtime1  −5.20E−02  0.03  INTperiodcentralpublic2  9.35E−02  0.125  mtrpremINTtime2  5.04E−03  0.069  INTperiodcentralpublic3  −8.16E−04  0.103  mtrpremINTtime3  −2.58E−04  0.085  lgdtoborder  −1.10E+02  0.301  mtrpremINTtime4  7.82E−06  0.078  lgincome  −2.10E+02  0.115  mtrpremINTtime5  −1.38E−07  0.054  INTincomeborder  9.44E+00  0.326  mtrpremINTtime6  1.26E−09  0.032  lgelev  3.46E+01  0.14  mtrpremINTtime7  −4.56E−12  0.019  lgcoast  −2.84E+01  0.031  Age  1.59E+02  0.291  intcoastelev  −4.20E+00  0.155  Sqrage  −8.28E+01  0.25  intcoast floorh  −6.53E+00  <0.001  age3  2.20E+01  0.219  intcoast floorm  −6.60E+00  <0.001  age4  −3.46E+00  0.186  intelev floorh  −5.95E+00  <0.001  age5  3.26E−01  0.164  intelev floorm  −5.04E+00  <0.001  age6  −1.80E−02  0.148  countdss  −2.02E+01  <0.001  age7  5.37E−04  0.143  INTperiodDSS  6.65E−01  <0.001  age8  −6.64E−06  0.13  INTperiodDSS2  −1.14E−02  <0.001  Lgsize  1.28E+02  <0.001  INTperiodDSS3  6.88E−05  0.007  Floorm  8.74E+01  <0.001  lgwalkelementary  4.01E+00  0.171  Floorh  1.00E+02  <0.001  INTperiodelementary  5.41E−02  0.359  discountrate  −1.47E+00  <0.001  INTperiodelementary2  −7.47E−04  0.336  unluckynum  7.27E+01  <0.001  slopeelementary  9.23E+00  0.022  Luckynum  −2.73E+00  0.255  INTslopewalkelem  −1.24E+00  0.055  lgdtoNoSec  2.25E+00  0.023  airport1km  —  —  lgdtocentralcar  8.00E+01  0.215  courtsize  5.10E−02  <0.001  INTperioddtocentral  2.95E+01  <0.001  courtsizesqr  −9.63E−06  <0.001  INTperioddtocentral2  −2.74E−01  <0.001  totaldensity3km  7.40E−03  <0.001  lgttocentralcar  5.03E+01  0.359  subdensityper3km  −2.11E+02  <0.001  INTperiodttocentral  −1.34E+01  0.027  densitypublic3km  −1.27E−02  <0.001  INTperiodttocentral2  −7.80E−02  0.231  densityprivate3km  −8.65E−03  <0.001  INTperiodttocentral3  2.42E−03  <0.001        Table A7 Coefficients and significance of Part II regression for model with interaction terms up to t5, excluding district and time period indicator effects Variable  Coefficient  Bootstrap p-val  Variable  Coefficient  Bootstrap p-val  (Intercept)  3.93E+03  0.024  INTperiodcentralpublic  −1.57E+00  0.338  mtrpremt1v3  8.02E−01  <0.001  INTperiodcentralpublic2  1.20E−01  0.077  mtrpremINTtime1  −5.07E−02  <0.001  INTperiodcentralpublic3  −1.01E−03  0.053  mtrpremINTtime2  3.45E−03  <0.001  lgdtoborder  −7.41E+01  0.357  mtrpremINTtime3  −9.29E−05  <0.001  lgincome  −1.96E+02  0.136  mtrpremINTtime4  1.00E−06  <0.001  INTincomeborder  5.70E+00  0.387  mtrpremINTtime5  −3.66E−09  0.001    3.44E+01  0.143  Age  1.71E+02  0.291    −2.82E+01  0.037  Sqrage  −8.50E+01  0.249  intcoastelev  −4.17E+00  0.161  age3  2.20E+01  0.227  intcoast floorh  −6.56E+00  <0.001  age4  −3.40E+00  0.201  intcoast floorm  −6.54E+00  <0.001  age5  3.19E−01  0.173  intelev floorh  −5.94E+00  <0.001  age6  −1.76E−02    intelev floorm  −4.98E+00  <0.001  age7  5.24E−04    countdss  −2.07E+01  <0.001  age8  −6.51E−06  0.131  INTperiodDSS  6.88E−01  <0.001  Lgsize  1.27E+02  <0.001  INTperiodDSS2  −1.20E−02  <0.001  Floorm  8.69E+01  <0.001  INTperiodDSS3  7.36E−05  0.003  Floorh  1.01E+02  <0.001  lgwalkelementary  4.42E+00  0.16  discountrate  −1.48E+00  <0.001  INTperiodelementary  3.97E−02  0.411  unluckynum  7.32E+01  <0.001  INTperiodelementary2  −6.04E−04  0.388  Luckynum  −2.48E+00  0.274  slopeelementary  9.28E+00  0.022  lgdtoNoSec  2.35E+00  0.016  INTslopewalkelem  −1.24E+00  0.054  lgdtocentralcar  8.81E+01  0.175  airport1km  —  —  INTperioddtocentral  2.92E+01  <0.001  courtsize  5.15E−02  <0.001  INTperioddtocentral2  −2.69E−01  <0.001  courtsizesqr  −9.76E−06  <0.001  lgttocentralcar  3.27E+01  0.385  totaldensity3km  7.48E−03  <0.001  INTperiodttocentral  −1.17E+01  0.047  subdensityper3km  −2.10E+02  <0.001  INTperiodttocentral2  −1.19E−01  0.125  densitypublic3km  −1.29E−02  <0.001  INTperiodttocentral3  2.69E−03  <0.001  densityprivate3km  −8.73E−03  <0.001  lgttocentralpublic  −3.03E+02  <0.001        Variable  Coefficient  Bootstrap p-val  Variable  Coefficient  Bootstrap p-val  (Intercept)  3.93E+03  0.024  INTperiodcentralpublic  −1.57E+00  0.338  mtrpremt1v3  8.02E−01  <0.001  INTperiodcentralpublic2  1.20E−01  0.077  mtrpremINTtime1  −5.07E−02  <0.001  INTperiodcentralpublic3  −1.01E−03  0.053  mtrpremINTtime2  3.45E−03  <0.001  lgdtoborder  −7.41E+01  0.357  mtrpremINTtime3  −9.29E−05  <0.001  lgincome  −1.96E+02  0.136  mtrpremINTtime4  1.00E−06  <0.001  INTincomeborder  5.70E+00  0.387  mtrpremINTtime5  −3.66E−09  0.001    3.44E+01  0.143  Age  1.71E+02  0.291    −2.82E+01  0.037  Sqrage  −8.50E+01  0.249  intcoastelev  −4.17E+00  0.161  age3  2.20E+01  0.227  intcoast floorh  −6.56E+00  <0.001  age4  −3.40E+00  0.201  intcoast floorm  −6.54E+00  <0.001  age5  3.19E−01  0.173  intelev floorh  −5.94E+00  <0.001  age6  −1.76E−02    intelev floorm  −4.98E+00  <0.001  age7  5.24E−04    countdss  −2.07E+01  <0.001  age8  −6.51E−06  0.131  INTperiodDSS  6.88E−01  <0.001  Lgsize  1.27E+02  <0.001  INTperiodDSS2  −1.20E−02  <0.001  Floorm  8.69E+01  <0.001  INTperiodDSS3  7.36E−05  0.003  Floorh  1.01E+02  <0.001  lgwalkelementary  4.42E+00  0.16  discountrate  −1.48E+00  <0.001  INTperiodelementary  3.97E−02  0.411  unluckynum  7.32E+01  <0.001  INTperiodelementary2  −6.04E−04  0.388  Luckynum  −2.48E+00  0.274  slopeelementary  9.28E+00  0.022  lgdtoNoSec  2.35E+00  0.016  INTslopewalkelem  −1.24E+00  0.054  lgdtocentralcar  8.81E+01  0.175  airport1km  —  —  INTperioddtocentral  2.92E+01  <0.001  courtsize  5.15E−02  <0.001  INTperioddtocentral2  −2.69E−01  <0.001  courtsizesqr  −9.76E−06  <0.001  lgttocentralcar  3.27E+01  0.385  totaldensity3km  7.48E−03  <0.001  INTperiodttocentral  −1.17E+01  0.047  subdensityper3km  −2.10E+02  <0.001  INTperiodttocentral2  −1.19E−01  0.125  densitypublic3km  −1.29E−02  <0.001  INTperiodttocentral3  2.69E−03  <0.001  densityprivate3km  −8.73E−03  <0.001  lgttocentralpublic  −3.03E+02  <0.001        B. Seasonal variation in elevation effects We test for seasonal variance in elevation effects by introducing additional interaction effects of all walking distance, slope and distance-slope interactions with indicators for summer (May, June, July), autumn (August, September, October) and winter (November, December, January). The regression specifications are otherwise identical to the part I two-stage model outlined in Section 4.1, but uses a single OLS regression on the full dataset with selected variables from the LASSO for simplicity. Table A8 Coefficients and significance of Part II regression for model with interaction terms up to t3, excluding district and time period indicator effects Variable  Coefficient  Bootstrap p-val  Variable  Coefficient  Bootstrap p-val  (Intercept)  3.71E+03  0.037  INTperiodcentralpublic2  −4.57E−02  0.272  mtrpremt1v3  6.72E−01  <0.001  INTperiodcentralpublic3  2.19E−04  0.333  mtrpremINTtime1  1.43E−03  0.383  Lgdtoborder  −1.42E+02  0.263  mtrpremINTtime2  −2.25E−04  0.072  Lgincome  −1.68E+02  0.174  mtrpremINTtime3  1.66E−06  0.129  INTincomeborder  1.24E+01  0.285  Age  −1.89E+01  0.418  Lgelev  2.65E+01  0.188  Sqrage  1.68E+00  0.459  Lgcoast  −2.88E+01  0.033  age3  1.06E+00  0.495  Intcoastelev  −3.06E+00  0.217  age4  −4.49E−01  0.457  intcoast floorh  −7.05E+00  <0.001  age5  6.63E−02  0.419  intcoast floorm  −7.01E+00  <0.001  age6  −4.79E−03  0.386  intelev floorh  −6.09E+00  <0.001  age7  1.71E−04  0.363  intelev floorm  −5.31E+00  <0.001  age8  −2.41E−06  0.336  Countdss  −1.78E+01  <0.001  Lgsize  1.27E+02  <0.001  INTperiodDSS  4.21E−01  <0.001  Floorm  9.11E+01  <0.001  INTperiodDSS2  −4.17E−03  0.089  Floorh  1.04E+02  <0.001  INTperiodDSS3  6.20E−06  0.4  discountrate  −1.46E+00  <0.001  Lgwalkelementary  7.29E+00  0.054  unluckynum  7.44E+01  <0.001  INTperiodelementary  −4.63E−02  0.38  Luckynum  −2.47E+00  0.274  INTperiodelementary2  3.29E−04  0.435  lgdtoNoSec  2.58E+00  0.012  Slopeelementary  1.09E+01  0.011  lgdtocentralcar  1.02E+02  0.142  INTslopewalkelem  −1.51E+00  0.031  INTperioddtocentral  2.92E+01  <0.001  airport1km  —  —  INTperioddtocentral2  −2.70E−01  <0.001  Courtsize  5.31E−02  <0.001  lgttocentralcar  1.30E+02  0.152  Courtsizesqr  −1.01E−05  <0.001  INTperiodttocentral  −2.05E+01  0.001  totaldensity3km  7.60E−03  <0.001  INTperiodttocentral2  9.48E−02  0.157  subdensityper3km  −2.17E+02  <0.001  INTperiodttocentral3  1.20E−03  0.003  densitypublic3km  −1.32E−02  <0.001  lgttocentralpublic  −3.78E+02  <0.001  densityprivate3km  −8.84E−03  <0.001  INTperiodcentralpublic  4.98E+00  0.072          Variable  Coefficient  Bootstrap p-val  Variable  Coefficient  Bootstrap p-val  (Intercept)  3.71E+03  0.037  INTperiodcentralpublic2  −4.57E−02  0.272  mtrpremt1v3  6.72E−01  <0.001  INTperiodcentralpublic3  2.19E−04  0.333  mtrpremINTtime1  1.43E−03  0.383  Lgdtoborder  −1.42E+02  0.263  mtrpremINTtime2  −2.25E−04  0.072  Lgincome  −1.68E+02  0.174  mtrpremINTtime3  1.66E−06  0.129  INTincomeborder  1.24E+01  0.285  Age  −1.89E+01  0.418  Lgelev  2.65E+01  0.188  Sqrage  1.68E+00  0.459  Lgcoast  −2.88E+01  0.033  age3  1.06E+00  0.495  Intcoastelev  −3.06E+00  0.217  age4  −4.49E−01  0.457  intcoast floorh  −7.05E+00  <0.001  age5  6.63E−02  0.419  intcoast floorm  −7.01E+00  <0.001  age6  −4.79E−03  0.386  intelev floorh  −6.09E+00  <0.001  age7  1.71E−04  0.363  intelev floorm  −5.31E+00  <0.001  age8  −2.41E−06  0.336  Countdss  −1.78E+01  <0.001  Lgsize  1.27E+02  <0.001  INTperiodDSS  4.21E−01  <0.001  Floorm  9.11E+01  <0.001  INTperiodDSS2  −4.17E−03  0.089  Floorh  1.04E+02  <0.001  INTperiodDSS3  6.20E−06  0.4  discountrate  −1.46E+00  <0.001  Lgwalkelementary  7.29E+00  0.054  unluckynum  7.44E+01  <0.001  INTperiodelementary  −4.63E−02  0.38  Luckynum  −2.47E+00  0.274  INTperiodelementary2  3.29E−04  0.435  lgdtoNoSec  2.58E+00  0.012  Slopeelementary  1.09E+01  0.011  lgdtocentralcar  1.02E+02  0.142  INTslopewalkelem  −1.51E+00  0.031  INTperioddtocentral  2.92E+01  <0.001  airport1km  —  —  INTperioddtocentral2  −2.70E−01  <0.001  Courtsize  5.31E−02  <0.001  lgttocentralcar  1.30E+02  0.152  Courtsizesqr  −1.01E−05  <0.001  INTperiodttocentral  −2.05E+01  0.001  totaldensity3km  7.60E−03  <0.001  INTperiodttocentral2  9.48E−02  0.157  subdensityper3km  −2.17E+02  <0.001  INTperiodttocentral3  1.20E−03  0.003  densitypublic3km  −1.32E−02  <0.001  lgttocentralpublic  −3.78E+02  <0.001  densityprivate3km  −8.84E−03  <0.001  INTperiodcentralpublic  4.98E+00  0.072          Of the 30 season–slope–distance interaction terms, only 10 survive the variable selection and only three are significant with p < 0.05. Although this may not seem insignificant, in contrast a majority of the non-metro variables in the second stage model are significant at beyond the 99.5% level. Five of the six metro-related variables in the part I regression model are significant at the 99.9% level. Also, coefficients of the seasonal interactions are small when compared with the base metro station effects: the only term significant at p < 0.01, winter walkingdistance, has a coefficient size of approximately 5% that of the base walking distance term. The second stage OLS regression output is summarized in Table A12. Finally, we note that introducing the seasonal effects do not significantly change the explanatory power of the model: R2 and AR2 values increase by less than 0.001 after adding seasonal interaction. We therefore conclude that it is unlikely that there are seasonal effects in the perceived value of station proximity in terms of walking distance and slope. Table A9 Coefficients and significance of Part II regression for model with interaction terms up to t2, excluding district and time period indicator effects Variable  Coefficient  Bootstrap p-val  Variable  Coefficient  Bootstrap p-val  (Intercept)  3.90E+03  0.026  INTperiodcentralpublic2  −6.23E−02  0.179  mtrpremt1v3  7.16E−01  <0.001  INTperiodcentralpublic3  3.04E−04  0.295  mtrpremINTtime1  −4.55E−03  0.046  Lgdtoborder  −1.99E+02  0.171  mtrpremINTtime2  −3.58E−05  0.178  Lgincome  −1.92E+02  0.137  Age  −2.28E+01  0.385  INTincomeborder  1.83E+01  0.189  Sqrage  5.12E+00  0.415  Lgelev  2.42E+01  0.203  age3  −9.78E−02  0.442  Lgcoast  −2.98E+01  0.033  age4  −2.46E−01  0.477  intcoastelev  −2.80E+00  0.233  age5  4.65E−02  0.485  intcoast floorh  −7.06E+00  <0.001  age6  −3.69E−03  0.449  intcoast floorm  −6.98E+00  <0.001  age7  1.39E−04  0.401  intelev floorh  −6.00E+00  <0.001  age8  −2.03E−06  0.37  intelev floorm  −5.25E+00  <0.001  Lgsize  1.27E+02  <0.001  countdss  −1.70E+01  <0.001  Floorm  9.07E+01  <0.001  INTperiodDSS  3.49E−01  <0.001  Floorh  1.04E+02  <0.001  INTperiodDSS2  −2.60E−03  0.179  discountrate  −1.44E+00  <0.001  INTperiodDSS3  −4.17E−06  0.417  unluckynum  7.49E+01  <0.001  lgwalkelementary  8.83E+00  0.014  luckynum  −2.19E+00  0.283  INTperiodelementary  −1.13E−01  0.212  lgdtoNoSec  2.68E+00  0.008  INTperiodelementary2  1.08E−03  0.254  lgdtocentralcar  1.03E+02  0.15  slopeelementary  1.18E+01  0.003  INTperioddtocentral  2.89E+01  <0.001  INTslopewalkelem  −1.64E+00  0.014  INTperioddtocentral2  −2.66E−01  <0.001  airport1km  —  —  lgttocentralcar  1.60E+02  0.112  courtsize  5.34E−02  <0.001  INTperiodttocentral  −2.15E+01  0.002  courtsizesqr  −1.02E−05  <0.001  INTperiodttocentral2  1.10E−01  0.134  totaldensity3km  7.62E−03  <0.001  INTperiodttocentral3  1.12E−03  0.005  subdensityper3km  −2.19E+02  <0.001  lgttocentralpublic  −3.93E+02  <0.001  densitypublic3km  −1.32E−02  <0.001  INTperiodcentralpublic  5.95E+00  0.019  densityprivate3km  −8.88E−03  <0.001  Variable  Coefficient  Bootstrap p-val  Variable  Coefficient  Bootstrap p-val  (Intercept)  3.90E+03  0.026  INTperiodcentralpublic2  −6.23E−02  0.179  mtrpremt1v3  7.16E−01  <0.001  INTperiodcentralpublic3  3.04E−04  0.295  mtrpremINTtime1  −4.55E−03  0.046  Lgdtoborder  −1.99E+02  0.171  mtrpremINTtime2  −3.58E−05  0.178  Lgincome  −1.92E+02  0.137  Age  −2.28E+01  0.385  INTincomeborder  1.83E+01  0.189  Sqrage  5.12E+00  0.415  Lgelev  2.42E+01  0.203  age3  −9.78E−02  0.442  Lgcoast  −2.98E+01  0.033  age4  −2.46E−01  0.477  intcoastelev  −2.80E+00  0.233  age5  4.65E−02  0.485  intcoast floorh  −7.06E+00  <0.001  age6  −3.69E−03  0.449  intcoast floorm  −6.98E+00  <0.001  age7  1.39E−04  0.401  intelev floorh  −6.00E+00  <0.001  age8  −2.03E−06  0.37  intelev floorm  −5.25E+00  <0.001  Lgsize  1.27E+02  <0.001  countdss  −1.70E+01  <0.001  Floorm  9.07E+01  <0.001  INTperiodDSS  3.49E−01  <0.001  Floorh  1.04E+02  <0.001  INTperiodDSS2  −2.60E−03  0.179  discountrate  −1.44E+00  <0.001  INTperiodDSS3  −4.17E−06  0.417  unluckynum  7.49E+01  <0.001  lgwalkelementary  8.83E+00  0.014  luckynum  −2.19E+00  0.283  INTperiodelementary  −1.13E−01  0.212  lgdtoNoSec  2.68E+00  0.008  INTperiodelementary2  1.08E−03  0.254  lgdtocentralcar  1.03E+02  0.15  slopeelementary  1.18E+01  0.003  INTperioddtocentral  2.89E+01  <0.001  INTslopewalkelem  −1.64E+00  0.014  INTperioddtocentral2  −2.66E−01  <0.001  airport1km  —  —  lgttocentralcar  1.60E+02  0.112  courtsize  5.34E−02  <0.001  INTperiodttocentral  −2.15E+01  0.002  courtsizesqr  −1.02E−05  <0.001  INTperiodttocentral2  1.10E−01  0.134  totaldensity3km  7.62E−03  <0.001  INTperiodttocentral3  1.12E−03  0.005  subdensityper3km  −2.19E+02  <0.001  lgttocentralpublic  −3.93E+02  <0.001  densitypublic3km  −1.32E−02  <0.001  INTperiodcentralpublic  5.95E+00  0.019  densityprivate3km  −8.88E−03  <0.001  This is not surprising. Hong Kong has a fairly mild temperature cycle among major cities with an annual average temperature spread of 13–14 °C.16 Summers are usually not extremely hot (i.e., above 35 °C) and there is almost no possibility of ice in winter. However, it may be of interest to replicate certain aspects of our study for a Northern city and testing for whether the effect of icy slopes is priced into summer transactions. C. Unevenness and Density Gradients Our approach in this subsection has two major distinctions from the approach of Saiz (2010). First, we model unevenness not as a dichotomy of developable versus un-developable land but as a continuous ‘elevation variance’ variable. Second, we select an international perspective using major cities worldwide instead of major US metro areas. Using figures formatted from United Nations population data, we create scatterplots of log-transformed ‘elevation variance’ estimates against log-transformed population density of 93 major world cities.17 City-level elevation variance estimates are obtained using the Microsoft Bing REST Services. Altitude estimates are sampled over a 30-by-30 point grid of fixed size over each city, centered on respective downtown areas, with zero or negative elevation samples discarded as potential errors or points on a body of water. Two alternative area sizes are used for each city: a 0.2-by-0.2 decimal degree square approximating a ‘city-and-near-suburb’ elevation variance value and a smaller, 0.1-by-0.1 degree square representing a ‘city-only’ variance value. Table A10 Equivalent station value and interaction coefficient of Part II regression, compared   Model with in-teraction terms up to t7  Model with in-teraction terms up to t5  Model with in-teraction terms up to t3  Model with in-teraction terms up to t2  Variable  Coefficient  Bootstrap p-val  Coefficient  Bootstrap p-val  Coefficient  Bootstrap p-val  Coefficient  Bootstrap p-val  mtrpremt1v3  7.79E−01  <0.001  8.02E−01  <0.001  6.72E−01  <0.001  7.16E−01  <0.001  mtrpremINTtime1  −5.20E−02  0.03  −5.07E−02  <0.001  1.43E−03  0.383  −4.55E−03  0.046  mtrpremINTtime2  5.04E−03  0.069  3.45E−03  <0.001  −2.25E−04  0.072  −3.58E−05  0.178  mtrpremINTtime3  −2.58E−04  0.085  −9.29E−05  <0.001  1.66E−06  0.129  —  —  mtrpremINTtime4  7.82E−06  0.078  1.00E−06  <0.001  —  —  —  —  mtrpremINTtime5  −1.38E−07  0.054  −3.66E−09  0.001  —  —  —  —  mtrpremINTtime6  1.26E−09  0.032  —  —  —  —  —  —  mtrpremINTtime7  −4.56E−12  0.019  —  —  —  —  —  —    Model with in-teraction terms up to t7  Model with in-teraction terms up to t5  Model with in-teraction terms up to t3  Model with in-teraction terms up to t2  Variable  Coefficient  Bootstrap p-val  Coefficient  Bootstrap p-val  Coefficient  Bootstrap p-val  Coefficient  Bootstrap p-val  mtrpremt1v3  7.79E−01  <0.001  8.02E−01  <0.001  6.72E−01  <0.001  7.16E−01  <0.001  mtrpremINTtime1  −5.20E−02  0.03  −5.07E−02  <0.001  1.43E−03  0.383  −4.55E−03  0.046  mtrpremINTtime2  5.04E−03  0.069  3.45E−03  <0.001  −2.25E−04  0.072  −3.58E−05  0.178  mtrpremINTtime3  −2.58E−04  0.085  −9.29E−05  <0.001  1.66E−06  0.129  —  —  mtrpremINTtime4  7.82E−06  0.078  1.00E−06  <0.001  —  —  —  —  mtrpremINTtime5  −1.38E−07  0.054  −3.66E−09  0.001  —  —  —  —  mtrpremINTtime6  1.26E−09  0.032  —  —  —  —  —  —  mtrpremINTtime7  −4.56E−12  0.019  —  —  —  —  —  —  Results for both variance estimates are contrasted in Figure A1. Both estimates display a positive association with population density: the correlation coefficient between the city-and-suburb elevation variance and population density is approximately 0.117, and the coefficient for the city-only variance 0.143. Although there are distinct low-variance, high-density outliers (Surat, Dhaka) and relatively high-variance, low-density outliers (Los Angeles, Phoenix), the two linear trends clearly indicate that from an international perspective, cities that have more uneven terrain tend to also have generally higher population densities. Furthermore, both Dhaka and Surat are highly susceptible to flooding (Ramirez and Rajasekar, 2015) and therefore one would expect the population to be concentrated in less risky locations, which our estimation procedure does not explicitly incorporate. Figure A1 View largeDownload slide Unevenness (log)–population density (log) scatterplots of major urban areas with fitted linear trend. Figure A1 View largeDownload slide Unevenness (log)–population density (log) scatterplots of major urban areas with fitted linear trend. We can also contrast the relationship between elevation and city size by population count. The same elevation variance variables are plotted against population count figures in Figure A2. The relationship between population and elevation variance is generally negative at the city level: the correlation coefficient is − 0.138 for log-transformed population and city-and-suburb elevation variance, and −0.105 for population and city-only elevation variance. Figure A2 View largeDownload slide Unevenness (log)–population count (log) scatterplots of major urban areas with fitted linear trend. Figure A2 View largeDownload slide Unevenness (log)–population count (log) scatterplots of major urban areas with fitted linear trend. Note that the comparative size of the correlations is opposite for population count versus density. This is consistent with literature and can be explained by the nature of the influence of elevation variance: high-unevenness downtown areas increase costs of central-city infrastructure and housing development, leading to inflated bid-rent curves and higher density. However, high-unevenness suburbs dis-incentivize development and are conceivably a greater constraint on the absolute size of a city, forcing bid-rent curves to drop more rapidly as distance to the city center increases. Although somewhat low in statistical significance, these relationships are nonetheless illuminating.18 Viewed in the context of existing literature on elevation gradient effects, the fitted trends suggestively connect the transaction level elevation-price effects outlined in Section 4 to differences in urban markets at the multi-city-level caused by elevation patterns, bridged by empirical work on the same effects at the sub-city or district-level. It ought to be noted that because the analysis treats cities as identical in size for altitude sampling, the majority of estimates among the observations likely suffer from size mismatch issues. If elevation could be accurately sampled from the specific areas from which population figures were collected, we would expect the relationships in Figure A1 and Figure A2 to be significantly stronger. Table A11 List of major urban areas and respective locations Location  Urban area  Location  Urban area  Japan  Tokyo- Yokohama  Indonesia  Jakarta (Jab- otabek)  India  Delhi  Philippines  Manila  China  Shanghai  Pakistan  Karachi  USA  New York  Mexico  Mexico City  Brazil  Sao Paulo  China  Beijing  India  Mumbai  Russia  Moscow  USA  Los Angeles  Egypt  Cairo  Thailand  Bangkok  India  Kolkota  Bangladesh  Dhaka  Argentina  Buenos Aires  Iran  Tehran  Turkey  Istanbul  China  Shenzhen  Nigeria  Lagos  Brazil  Rio de Janeiro  France  Paris  Japan  Nagoya  UK  London  Congo (Dem. Rep.)  Kinshasa  China  Quanzhou  Peru  Lima  China  Tianjin  India  Chennai  India  Bangalore  USA  Chicago  Viet Nam  Ho Chi Minh        City  China  Chengdu  China  Dongguan  India  Hyderabad  Pakistan  Lahore  Colombia  Bogota  China  Wuhan  China: Taiwan  Taipei  China: HKSAR  Hong Kong  India  Ahmedabad  China  Chongqing  China  Hangzhou  Malaysia  Kuala Lumpur  Iraq  Baghdad  Canada  Toronto  Chile  Santiago  Spain  Madrid  China  Nanjing  USA  Miami  China  Shenyang  Indonesia  Bandung  Angola  Luanda  United States  Houston  USA  Philadelphia  China  Xi'an  Singapore  Singapore  China  Qingdao  India  Pune  Italy  Milan  Saudi Arabia  Riyadh  Russia  St. Petersburg  Sudan  Khartoum  Indonesia  Surabaya  India  Surat  USA  Atlanta  USA  Washington  Ivory Coast  Abidjan  Myanmar  Yangon  Spain  Barcelona  Kenya  Nairobi  China  Harbin  Egypt  Alexandria  China  Suzhou  USA  Boston  Brazil  Belo Horizonte  Turkey  Ankara  China  Zhengzhou  Ghana  Accra  USA  Phoenix  Germany  Berlin  Indonesia  Medan  Australia  Sydney  South Korea  Busan  Kuwait  Kuwait  Tanzania  Dar es Salaam  Mexico  Monterey  China  Dalian  Italy  Rome  Australia  Melbourne  Mexico  Guadalajara      Location  Urban area  Location  Urban area  Japan  Tokyo- Yokohama  Indonesia  Jakarta (Jab- otabek)  India  Delhi  Philippines  Manila  China  Shanghai  Pakistan  Karachi  USA  New York  Mexico  Mexico City  Brazil  Sao Paulo  China  Beijing  India  Mumbai  Russia  Moscow  USA  Los Angeles  Egypt  Cairo  Thailand  Bangkok  India  Kolkota  Bangladesh  Dhaka  Argentina  Buenos Aires  Iran  Tehran  Turkey  Istanbul  China  Shenzhen  Nigeria  Lagos  Brazil  Rio de Janeiro  France  Paris  Japan  Nagoya  UK  London  Congo (Dem. Rep.)  Kinshasa  China  Quanzhou  Peru  Lima  China  Tianjin  India  Chennai  India  Bangalore  USA  Chicago  Viet Nam  Ho Chi Minh        City  China  Chengdu  China  Dongguan  India  Hyderabad  Pakistan  Lahore  Colombia  Bogota  China  Wuhan  China: Taiwan  Taipei  China: HKSAR  Hong Kong  India  Ahmedabad  China  Chongqing  China  Hangzhou  Malaysia  Kuala Lumpur  Iraq  Baghdad  Canada  Toronto  Chile  Santiago  Spain  Madrid  China  Nanjing  USA  Miami  China  Shenyang  Indonesia  Bandung  Angola  Luanda  United States  Houston  USA  Philadelphia  China  Xi'an  Singapore  Singapore  China  Qingdao  India  Pune  Italy  Milan  Saudi Arabia  Riyadh  Russia  St. Petersburg  Sudan  Khartoum  Indonesia  Surabaya  India  Surat  USA  Atlanta  USA  Washington  Ivory Coast  Abidjan  Myanmar  Yangon  Spain  Barcelona  Kenya  Nairobi  China  Harbin  Egypt  Alexandria  China  Suzhou  USA  Boston  Brazil  Belo Horizonte  Turkey  Ankara  China  Zhengzhou  Ghana  Accra  USA  Phoenix  Germany  Berlin  Indonesia  Medan  Australia  Sydney  South Korea  Busan  Kuwait  Kuwait  Tanzania  Dar es Salaam  Mexico  Monterey  China  Dalian  Italy  Rome  Australia  Melbourne  Mexico  Guadalajara      Table A12 Summary of regression for OLS model with season–distance–slope interaction terms, excluding district and time period indicator effects   Estimate  Std. error  t-value  Pr(>t)  (Intercept)  −0.0000  0.0015  −0.00  1.0000  sum walkmtractual3a  0.0015  0.0024  0.61  0.5422  sum walkmtrangle  −0.0021  0.0072  −0.29  0.7682  sum walkmtranglesqr  0.0102  0.0054  1.91  0.0559  sum mtrwalkINTangle1  −0.0088  0.0038  −2.29  0.0220  aut walkmtractual  0.0087  0.0054  1.61  0.1067  aut walkmtractual3  −0.0028  0.0048  −0.58  0.5627  aut mtrwalkINTangle1  0.0003  0.0027  0.11  0.9150  wint walkmtractual  −0.0135  0.0047  −2.89  0.0039  wint walkmtractual3  0.0100  0.0039  2.57  0.0102  wint mtrwalk2INTangle1  0.0009  0.0027  0.33  0.7449  walkmtractual  −0.2646  0.0080  −32.87  0.0000  walkmtractual3  0.0675  0.0070  9.72  0.0000  walkmtrangle  −0.0791  0.0049  −16.15  0.0000  mtrwalkINTangle1  −0.1084  0.0115  −9.42  0.0000  mtrwalk2INTangle2  0.0242  0.0064  3.78  0.0002  mtrwalk2INTangle1  0.1267  0.0072  17.49  0.0000  age  −0.2494  0.0034  −74.41  0.0000  age8  0.0160  0.0019  8.27  0.0000  lgsize  0.0802  0.0017  47.63  0.0000  floorm  0.0468  0.0076  6.16  0.0000  floorh  0.0590  0.0074  7.96  0.0000  discountrate  −0.1799  0.0067  −26.93  0.0000  intdiscount1998  0.0772  0.0082  9.42  0.0000  intdiscount1999  0.1235  0.0112  11.05  0.0000  intdiscount2000  0.1166  0.0105  11.06  0.0000  intdiscount2001  0.0888  0.0107  8.33  0.0000  intdiscount2002  0.0818  0.0104  7.85  0.0000  intdiscount2003  0.0871  0.0113  7.68  0.0000  intdiscount2004  0.0571  0.0126  4.55  0.0000  intdiscount2005  0.0199  0.0091  2.19  0.0289  intdiscount2006  0.0348  0.0129  2.70  0.0069  intdiscount2007  0.1204  0.0129  9.34  0.0000  intdiscount2009  0.0617  0.0137  4.50  0.0000  intdiscount2010  0.0140  0.0134  1.04  0.2969  intdiscount2011  0.0055  0.0124  0.45  0.6562  intdiscount2012  −0.0116  0.0121  −0.96  0.3363  intdiscount2013  −0.0168  0.0115  −1.46  0.1440  intdiscount2014  0.0282  0.0118  2.39  0.0170  unluckynum  −0.0145  0.0023  −6.35  0.0000  luckynum  0.0179  0.0019  9.42  0.0000  lgdtoNoSec  0.0193  0.0023  8.36  0.0000  lgdtocentralcar  −0.1110  0.0239  −4.64  0.0000  INTperioddtocentral  1.0034  0.0842  11.92  0.0000  INTperioddtocentral2  −1.0231  0.0570  −17.94  0.0000  lgttocentralcar  −0.0047  0.0211  −0.22  0.8222  INTperiodttocentral  −0.7402  0.0718  −10.31  0.0000  lgttocentralpublic  −0.1334  0.0066  −20.33  0.0000  INTperiodcentralpublic2  −0.9738  0.1217  −8.00  0.0000  INTperiodcentralpublic3  1.5768  0.1114  14.16  0.0000  lgdtoborder  0.5792  0.1365  4.24  0.0000  lgincome  0.1907  0.0183  10.41  0.0000  INTincomeborder  −0.6482  0.1362  −4.76  0.0000  lgelev  0.0158  0.0038  4.11  0.0000  lgcoast  0.0294  0.0048  6.16  0.0000  intelev floorh  0.0217  0.0075  2.90  0.0037  intelev floorm  0.0159  0.0077  2.07  0.0385  countdss  −0.0318  0.0075  −4.21  0.0000  INTperiodDSS  0.0206  0.0138  1.49  0.1356  INTperiodDSS3  −0.0084  0.0098  −0.86  0.3918  lgwalkelementary  0.1033  0.0059  17.40  0.0000  INTperiodelementary  −0.5272  0.0451  −11.68  0.0000  INTperiodelementary2  0.5438  0.0438  12.42  0.0000  slopeelementary  0.3974  0.0180  22.09  0.0000  INTslopewalkelem  −0.3988  0.0177  −22.48  0.0000  airport1km  0.0463  0.0117  3.94  0.0001  courtsize  0.0437  0.0046  9.48  0.0000  courtsizesqr  −0.0633  0.0045  −14.21  0.0000  totaldensity3km10000  −0.1587  0.0073  −21.63  0.0000  subdensityper3km  −0.0523  0.0036  −14.35  0.0000  densityprivate3km  0.0759  0.0062  12.18  0.0000    Estimate  Std. error  t-value  Pr(>t)  (Intercept)  −0.0000  0.0015  −0.00  1.0000  sum walkmtractual3a  0.0015  0.0024  0.61  0.5422  sum walkmtrangle  −0.0021  0.0072  −0.29  0.7682  sum walkmtranglesqr  0.0102  0.0054  1.91  0.0559  sum mtrwalkINTangle1  −0.0088  0.0038  −2.29  0.0220  aut walkmtractual  0.0087  0.0054  1.61  0.1067  aut walkmtractual3  −0.0028  0.0048  −0.58  0.5627  aut mtrwalkINTangle1  0.0003  0.0027  0.11  0.9150  wint walkmtractual  −0.0135  0.0047  −2.89  0.0039  wint walkmtractual3  0.0100  0.0039  2.57  0.0102  wint mtrwalk2INTangle1  0.0009  0.0027  0.33  0.7449  walkmtractual  −0.2646  0.0080  −32.87  0.0000  walkmtractual3  0.0675  0.0070  9.72  0.0000  walkmtrangle  −0.0791  0.0049  −16.15  0.0000  mtrwalkINTangle1  −0.1084  0.0115  −9.42  0.0000  mtrwalk2INTangle2  0.0242  0.0064  3.78  0.0002  mtrwalk2INTangle1  0.1267  0.0072  17.49  0.0000  age  −0.2494  0.0034  −74.41  0.0000  age8  0.0160  0.0019  8.27  0.0000  lgsize  0.0802  0.0017  47.63  0.0000  floorm  0.0468  0.0076  6.16  0.0000  floorh  0.0590  0.0074  7.96  0.0000  discountrate  −0.1799  0.0067  −26.93  0.0000  intdiscount1998  0.0772  0.0082  9.42  0.0000  intdiscount1999  0.1235  0.0112  11.05  0.0000  intdiscount2000  0.1166  0.0105  11.06  0.0000  intdiscount2001  0.0888  0.0107  8.33  0.0000  intdiscount2002  0.0818  0.0104  7.85  0.0000  intdiscount2003  0.0871  0.0113  7.68  0.0000  intdiscount2004  0.0571  0.0126  4.55  0.0000  intdiscount2005  0.0199  0.0091  2.19  0.0289  intdiscount2006  0.0348  0.0129  2.70  0.0069  intdiscount2007  0.1204  0.0129  9.34  0.0000  intdiscount2009  0.0617  0.0137  4.50  0.0000  intdiscount2010  0.0140  0.0134  1.04  0.2969  intdiscount2011  0.0055  0.0124  0.45  0.6562  intdiscount2012  −0.0116  0.0121  −0.96  0.3363  intdiscount2013  −0.0168  0.0115  −1.46  0.1440  intdiscount2014  0.0282  0.0118  2.39  0.0170  unluckynum  −0.0145  0.0023  −6.35  0.0000  luckynum  0.0179  0.0019  9.42  0.0000  lgdtoNoSec  0.0193  0.0023  8.36  0.0000  lgdtocentralcar  −0.1110  0.0239  −4.64  0.0000  INTperioddtocentral  1.0034  0.0842  11.92  0.0000  INTperioddtocentral2  −1.0231  0.0570  −17.94  0.0000  lgttocentralcar  −0.0047  0.0211  −0.22  0.8222  INTperiodttocentral  −0.7402  0.0718  −10.31  0.0000  lgttocentralpublic  −0.1334  0.0066  −20.33  0.0000  INTperiodcentralpublic2  −0.9738  0.1217  −8.00  0.0000  INTperiodcentralpublic3  1.5768  0.1114  14.16  0.0000  lgdtoborder  0.5792  0.1365  4.24  0.0000  lgincome  0.1907  0.0183  10.41  0.0000  INTincomeborder  −0.6482  0.1362  −4.76  0.0000  lgelev  0.0158  0.0038  4.11  0.0000  lgcoast  0.0294  0.0048  6.16  0.0000  intelev floorh  0.0217  0.0075  2.90  0.0037  intelev floorm  0.0159  0.0077  2.07  0.0385  countdss  −0.0318  0.0075  −4.21  0.0000  INTperiodDSS  0.0206  0.0138  1.49  0.1356  INTperiodDSS3  −0.0084  0.0098  −0.86  0.3918  lgwalkelementary  0.1033  0.0059  17.40  0.0000  INTperiodelementary  −0.5272  0.0451  −11.68  0.0000  INTperiodelementary2  0.5438  0.0438  12.42  0.0000  slopeelementary  0.3974  0.0180  22.09  0.0000  INTslopewalkelem  −0.3988  0.0177  −22.48  0.0000  airport1km  0.0463  0.0117  3.94  0.0001  courtsize  0.0437  0.0046  9.48  0.0000  courtsizesqr  −0.0633  0.0045  −14.21  0.0000  totaldensity3km10000  −0.1587  0.0073  −21.63  0.0000  subdensityper3km  −0.0523  0.0036  −14.35  0.0000  densityprivate3km  0.0759  0.0062  12.18  0.0000  a Interactions with the summer indicator (May, June, July) is denoted with the pre x ‘sum’, interactions with the autumn indicator (August, September, October) is denoted with the pre x ‘aut’, and interactions with the winter indicator (November, December, January) is denoted with the pre x ‘wint’. Figure A3 View largeDownload slide Log-lambda trace plot of LASSO Cross-validation process. Figure A3 View largeDownload slide Log-lambda trace plot of LASSO Cross-validation process. Figure A4 View largeDownload slide Trace plots of select regression variables in Bayesian Lasso model MCMC process. Figure A4 View largeDownload slide Trace plots of select regression variables in Bayesian Lasso model MCMC process. Figure A5 View largeDownload slide Trace plots of key regression variables in Bayesian Lasso model MCMC process. Figure A5 View largeDownload slide Trace plots of key regression variables in Bayesian Lasso model MCMC process. Figure A6 View largeDownload slide R-square and adjusted R-square value distributions with bootstrapping, part I model. Figure A6 View largeDownload slide R-square and adjusted R-square value distributions with bootstrapping, part I model. Figure A7 View largeDownload slide Bootstrap densities of select regression coefficients, part I model. Figure A7 View largeDownload slide Bootstrap densities of select regression coefficients, part I model. Figure A8 View largeDownload slide Density of select slope/distance variable coefficients, compared. Figure A8 View largeDownload slide Density of select slope/distance variable coefficients, compared. Figure A9 View largeDownload slide Densities of selling price predictions given fixed slope and by walking distance. Figure A9 View largeDownload slide Densities of selling price predictions given fixed slope and by walking distance. Figure A10 View largeDownload slide R-square and adjusted R-square value distributions with bootstrapping, Part II models. Figure A10 View largeDownload slide R-square and adjusted R-square value distributions with bootstrapping, Part II models. Figure A11 View largeDownload slide Densities of equivalent station value predictions given fixed time gap and by model specificities. Figure A11 View largeDownload slide Densities of equivalent station value predictions given fixed time gap and by model specificities. Figure A12 View largeDownload slide Elevation contour plot of the HKSAR. Figure A12 View largeDownload slide Elevation contour plot of the HKSAR. Figure A13 View largeDownload slide Map of HKSAR with locations of courts and select amenities. Figure A13 View largeDownload slide Map of HKSAR with locations of courts and select amenities. Figure A14 View largeDownload slide Elevation gradients at different walking distances, compared. Figure A14 View largeDownload slide Elevation gradients at different walking distances, compared. Figure A15 View largeDownload slide Residual scatterplots with selected parameters and subsetted data for Part I model. Figure A15 View largeDownload slide Residual scatterplots with selected parameters and subsetted data for Part I model. Figure A16 View largeDownload slide Filled altitude contour plot with spatially mapped apartment-group average price residuals and metro station locations, Hong Kong, 1998–2014 (2 of 3). Figure A16 View largeDownload slide Filled altitude contour plot with spatially mapped apartment-group average price residuals and metro station locations, Hong Kong, 1998–2014 (2 of 3). Figure A17 View largeDownload slide Filled altitude contour plot with spatially mapped apartment-group average price residuals and metro station locations, 1998–2014 (3 of 3). Figure A17 View largeDownload slide Filled altitude contour plot with spatially mapped apartment-group average price residuals and metro station locations, 1998–2014 (3 of 3). © The Author (2017). Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

Journal

Journal of Economic GeographyOxford University Press

Published: Mar 1, 2018

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