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The Review of Economic Studies
, Volume Advance Article – Jan 22, 2018

37 pages

/lp/ou_press/volume-volatility-and-public-news-announcements-0BqiQozy4N

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press on behalf of The Review of Economic Studies Limited.
- ISSN
- 0034-6527
- eISSN
- 1467-937X
- D.O.I.
- 10.1093/restud/rdy003
- Publisher site
- See Article on Publisher Site

Abstract We provide new empirical evidence for the way in which financial markets process information. Our results rely critically on high-frequency intraday price and volume data for the S&P 500 equity portfolio and U.S. Treasury bonds, along with new econometric techniques, for making inference on the relationship between trading intensity and spot volatility around public news announcements. Consistent with the predictions derived from a theoretical model in which investors agree to disagree, our estimates for the intraday volume-volatility elasticity around important news announcements are systematically below unity. Our elasticity estimates also decrease significantly with measures of disagreements in beliefs, economic uncertainty, and textual-based sentiment, further highlighting the key role played by differences-of-opinion. 1. Introduction Trading volume and return volatility in financial markets typically, but not always, move in tandem. By studying the strength of this relationship around important public news announcements, we shed new light on the way in which financial markets function and process new information. Our empirical investigations rely critically on the use of high-frequency intraday price and volume data for the S&P 500 equity portfolio and U.S. Treasury bonds, together with new econometric inference procedures explicitly designed to deal with the unique complications that arise in the high-frequency data setting. Consistent with the implications derived from a stylized theoretical model in which investors agree to disagree, our estimates for the intraday volume-volatility elasticity around important news announcements are systematically below unity, the benchmark case that is obtained in the absence of any disagreement among investors. In line with the theoretical predictions from the same model, our estimates for the elasticity also decrease significantly with proxies for disagreements in beliefs, economic uncertainty, and textual-based sentiment, further corroborating the key role played by differences-of-opinion. An extensive empirical literature has documented the existence of an on-average strong contemporaneous relation between trading volume and volatility; see Karpoff (1987) for a survey of some of the earliest empirical evidence. The mixture-of-distributions hypothesis (MDH) (see, e.g., Clark, 1973; Tauchen and Pitts, 1983; Andersen, 1996) provides a possible statistical explanation for the positive volume–volatility relationship based on the idea of a common news arrival process driving both the magnitude of returns and trading volume.1 The MDH, however, remains silent about the underlying economic mechanisms that link the actual trades and price adjustments to the news. Meanwhile, a variety of equilibrium-based economic models have been developed to help better understand how prices and volume respond to new information. This includes the rational-expectations type models of Kyle (1985) and Kim and Verrecchia (1991, 1994) among many others, in which investors agree on the interpretation of the news, but their information sets differ. In this class of models, the trading volume is mainly determined by liquidity trading and portfolio rebalancing needs. Although this is able to explain the on-average positive correlation between volume and volatility, the underlying trading motives would seem too small to account for the large trading volume observed empirically, especially when the price changes are close to zero (see, e.g., Hong and Stein, 2007, for additional arguments along these lines). Instead, models that feature differences-of-opinion, including those by Harrison and Kreps (1978), Harris and Raviv (1993), Kandel and Pearson (1995), Scheinkman and Xiong (2003) and Banerjee and Kremer (2010) among others, in which investors agree to disagree, may help explain these oft observed empirical phenomena. In the differences-of-opinion class of models, the investors’ interpretation of the news and their updated valuations of the assets do not necessarily coincide, thereby providing an additional trading motive that is not directly tied to changes in the equilibrium price. Much of empirical evidence presented in the literature in regards to the economic models discussed above, and the volume–volatility relationship in particular, have been based on daily or coarser frequency data (see, e.g., Tauchen and Pitts, 1983; Andersen, 1996).2 Meanwhile, another large and growing strand of literature has emphasized the advantages of the use of high-frequency intraday data for analysing the way in which financial markets respond to new information and more accurately identifying price jumps. Harvey and Huang (1991) and Ederington and Lee (1993), in particular, both find that the strong intraday volatility patterns observed in most financial markets may in part be attributed to regularly scheduled macroeconomic news announcements. Correspondingly large price jumps are often readily associated with specific news announcements; see, e.g., Andersen et al. (2007), Lee and Mykland (2008) and Lee (2012).3 This naturally suggests that by “ zooming in” and analysing how not only prices but also trading volume and volatility evolve around important news announcements, a deeper understanding of the economic mechanisms at work and the functioning of markets may be forthcoming. Set against this background, we provide new empirical evidence on the volume–volatility relationship around various macroeconomic news announcements. Our leading empirical investigations are based on high-frequency one-minute data for the aggregate S&P 500 equity market portfolio, but we also provide complimentary results for U.S. interest rates using one-minute Treasury bond futures data. We begin by documenting the occurrence of large increases in trading volume intensity around Federal Open Market Committee (FOMC) meetings without accompanying large price jumps. As noted above, this presents a challenge for models in which investors rationally update their beliefs based on the same interpretation of the news, and instead points to the importance of models allowing for disagreements or, differences-of-opinion, among investors. To help further explore this thesis and guide our more in-depth empirical investigations, we derive an explicit expression for the elasticity of expected trading volume with respect to price volatility within the Kandel and Pearson (1995) differences-of-opinion model. We purposely focus our analysis on the elasticity as it may be conveniently estimated with high-frequency data and, importantly, has a clear economic interpretation in terms of model primitives. In particular, we show theoretically that the volume–volatility elasticity is monotonically decreasing in a well-defined measure of relative disagreement. Moreover, the elasticity is generally below one and reaches its upper bound of unity only in the benchmark case without disagreement. The theoretical model underlying these predictions is inevitably stylized, focusing exclusively on the impact of public news announcements. As such, the theory mainly speaks to the “abnormal” movements in volume and volatility observed around these news events. To identify the abnormal movements, and thus help mitigate the effects of other confounding forces, we rely on the “jumps” in the volume intensity and volatility around the news announcements. Our estimation of the jumps is based on the differences between the post- and pre-event levels of the instantaneous volume intensity and volatility, which we recover non-parametrically using high-frequency data. Even though the differencing step used in identifying the jumps effectively removes low-frequency dynamics in the volatility and volume series (including daily and lower frequency trending behaviour) that might otherwise confound the estimates, the jump estimates are still affected by the well-documented strong intraday periodic patterns that exist in both volume and volatility (for some of the earliest empirical evidence, see Wood et al., 1985; Jain and Joh, 1988). In an effort to remove this additional confounding effect, we apply a second difference with respect to a control group of non-announcement days. The resulting “doubly-differenced” jump estimates in turn serve as our empirical analogues of the abnormal volume and volatility movements that we use in our regression-based analysis of the theoretical predictions. Our empirical strategy for estimating the jumps may be viewed as a Difference-in-Difference (DID) type approach, as commonly used in empirical microeconomic studies (see, e.g., Ashenfelter and Card, 1985). The subsequent regression involving the jumps is similar in spirit to the “jump regressions” studied by Li et al. (2017), which in turn resembles the non-parametric estimation in (fuzzy) Regression Discontinuity Designs (RDD) (see, e.g., Lee and Lemieux, 2010).4 We hence refer to our new econometric method as a DID jump regression. However, our setup is distinctly different from conventional econometric settings, and the usual justification for the use of DID or RDD does not apply in the high-frequency data setting. Correspondingly, our new econometric procedures and the justification thereof entail two important distinctions. First, to accommodate the strong dynamic dependencies in the volatility and volume intensity, we provide a rigorous theoretical justification based on a continuous-time infill asymptotic framework allowing for essentially unrestricted non-stationarity. Secondly, we provide an easy-to-implement local bootstrap method for conducting valid inference. By randomly resampling only locally in time (separately before and after each announcement), the method provides a simple solution to the issue of data heterogeneity, which otherwise presents a formidable challenge in the high-frequency data setting (see, e.g., Gonçalves and Meddahi, 2009). Our actual empirical findings are closely in line with the theoretical predictions derived from the Kandel and Pearson (1995) model and the differences-of-opinion class of models more generally. In particular, we first document that the estimated volume–volatility elasticity around FOMC announcements is significantly below unity. This finding carries over to other important intraday public news announcements closely monitored by market participants. Interestingly, the volume–volatility elasticity estimates are lower for announcements that are released earlier in the monthly news cycle (see, e.g., Andersen et al., 2003), such as the ISM Manufacturing Index and the Consumer Confidence Index, reflecting the importance of the timing across the different announcements and the effect of learning. Going one step further, we show that the intraday volume–volatility elasticity around news announcements decreases significantly in response to increases in measures of dispersions-in-beliefs (based on the survey of professional forecasters as in, e.g., Van Nieuwerburgh and Veldkamp, 2006; Pasquariello and Vega, 2007) and economic uncertainty (based on the economic policy uncertainty index of Baker et al., 2015). This holds true for the S&P 500 aggregate equity portfolio as well as the U.S. Treasury bond market, and again corroborates our theoretical predictions and the key role played by differences-of-opinion. black Our more detailed analysis of FOMC announcements, in which we employ an additional textual-based measure for the negative sentiment in the accompanying FOMC statements (based on the methodology of Loughran and McDonald, 2011), further underscores the time-varying nature of the high-frequency volume–volatility relationship and the way in which the market processes new information: when the textual sentiment in the FOMC statement is more negative, the relative disagreement among investors also tends to be higher, pushing down the volume–volatility elasticity. In contrast to prior empirical studies related to the volume–volatility relationship (see, e.g., Tauchen and Pitts, 1983; Andersen, 1996), our analysis is much more closely guided by an economic model. In particular, based on the implications derived from the Kandel–Pearson model, we provide a new perspective on the volume–volatility relationship by directly linking the strength of the relationship to notions of investors’ disagreement. Since the Kandel–Pearson model explicitly concerns “abnormal” variations in volume and volatility around news announcements, we also derive the new DID jump regression framework to rigorously analyse the joint behaviour of volume and volatility jumps. The use of high-frequency data are crucial in this regard, as jumps are invariably short-lived in nature and would be difficult/impossible to accurately identify using data sampled at coarser, say daily, frequencies. Putting the empirical results in the article into a broader perspective, there is a large literature in market microstructure finance on the price impact of trades, and correspondingly the development of “ optimal” trade execution strategies; empirical work along these lines include Hasbrouck (1991), Madhavan et al. (1997), and Chordia et al. (2002). In contrast to the volume–volatility relationship analysed here, the price impact literature is explicitly concerned with directional price changes predicted by “signed” trading volume, or order flow. Green (2004), in particular, documents a significant increase in the impact of changes in order flow in the U.S. Treasury bond market on intraday bond prices immediately following macroeconomic news announcements, and goes on to suggest that this heightened price impact of trades may be attributed to increased informational asymmetry at the time of the announcements.5 Further corroborating this idea, Pasquariello and Vega (2007) find that the regression-based estimate for the impact of unanticipated daily order flow in the U.S. Treasury bond market is higher when the dispersion in beliefs among market participants, as measured by the standard deviation of the forecasts from professional forecasters, is high and when the public news announcement is more “noisy”. Our main empirical findings based on high-frequency intraday data for the S&P 500 aggregate market portfolio are generally in line with these existing results about the price impact of order flow in the U.S. Treasury bond market. At the same time, however, our focus on trading volume as opposed to order flow presents an important distinction from the aforementioned studies.6 In line with the insights of Hong and Stein (2007), by focusing on trading volume our empirical findings map more closely into the theoretical predictions from differences-of-opinion class of models in which the disagreement among traders provides an important trading motive beyond conventional rational-expectation type models (as, e.g., the classical model by Kyle, 1985). In addition, our empirical strategy relies critically on the use of high-frequency data coupled with new econometric techniques for non-parametrically uncovering both the instantaneous trading intensity and volatility under minimal statistical assumptions. The rest of the article is organized as follows. Section 2 presents the basic economic arguments and theoretical model that guide our empirical investigations. Section 3 describes the high-frequency intraday data and news announcements used in our empirical analysis. To help set the stage for our more detailed subsequent empirical investigations, Section 4 discusses some preliminary findings specifically related to the behavior of the aggregate stock market around the FOMC announcements. Section 5 describes the new inference procedures necessitated by our more in-depth high-frequency empirical analysis. Section 6 presents our main empirical findings based on the full set of news announcements, followed by our more detailed analysis of FOMC announcements. In addition to our main results based on data for the S&P 500 aggregate equity portfolio, we also present complementary results for the U.S. Treasury bond market. Section 7 concludes. Technical details concerning the new econometric inference procedure are provided in Appendix A. Appendix B contains further data descriptions. Additional empirical results and robustness checks are relegated to a (not-for-publication) Supplementary Appendix. 2. Theoretical motivation We rely on the theoretical volume–volatility relations derived from the differences-of-opinion model of Kandel and Pearson (1995) to help guide our empirical investigations. We purposely focus on a simplified version of the model designed to highlight the specific features that we are after, and the volume–volatility elasticity around news arrivals in particular. We begin by discussing the basic setup and assumptions. 2.1. New information and differences-of-opinion Following Kandel and Pearson (1995), henceforth KP, we assume that a continuum of traders trade a risky asset and a risk-free asset in a competitive market. The random payoff of the risky asset, denoted $$\tilde{u}$$, is unknown to the traders. The risk-free rate is normalized to be zero. The traders’ utility functions have constant absolute risk aversion with risk tolerance $$r$$. There are only two types of traders, $$i\in \{1,2\}$$, with the proportion of type 1 traders denoted $$\alpha $$. The traders have different prior beliefs about the payoff before the announcement, and they also disagree about the interpretation of the public signal at the time of the announcement. Type $$i$$ trader’s prior is given by a normal distribution with precision $$s_{i}$$. After the announcement, the traders observe the same public signal $$\tilde{u} +\tilde{\varepsilon}$$, where the noise term $$\tilde{\varepsilon}$$ is normally distributed. The traders then update their beliefs about $$\tilde{u}$$ and optimally re-balance their positions. The key feature of the KP model is that the two types of traders agree to disagree on how to interpret the public signal when updating their beliefs about the asset value: type $$i$$ traders believe that $$\tilde{\varepsilon}$$ is drawn from the $$N(\mu_{i},h^{-1})$$ distribution. Differences-of-opinion regarding the public signal among the traders thus corresponds to $$\mu _{1}\neq \mu _{2}$$. Following KP it is possible to show that in equilibrium7 \begin{equation} \text{Volume }=\,\left\vert \beta _{0}+\beta _{1}\cdot \text{Price Change} \right\vert , \label{kp1} \end{equation} (2.1) where \begin{equation} \beta _{0}=r\alpha \left( 1-\alpha \right) h\left( \mu _{1}-\mu _{2}\right) ,\quad \beta _{1}=r\alpha \left( 1-\alpha \right) \left( s_{1}-s_{2}\right) . \label{kpbetas} \end{equation} (2.2) The coefficient $$\beta _{0}$$ is directly associated with the degree of differences-of-opinion concerning the interpretation of the public signal (i.e.$$\mu _{1}-\mu _{2}$$), while $$\beta_{1}$$ depends on the dispersion in the precisions of the prior beliefs about the payoff (i.e.$$s_{1}-s_{2}$$). Both of the coefficients are increasing in the degree of risk tolerance $$r$$, and the degree of heterogeneity among the traders as measured by $$\alpha (1-\alpha )$$. Looking at the equilibrium relationship in equation (2.1), the first $$\beta _{0}$$ term represents the “disagreement component”. This term becomes increasingly more important for higher levels of disagreement (i.e.$$|\mu _{1}-\mu _{2}|$$ is large) and/or when the traders are more confident about their interpretation of the public signal (i.e., $$h$$ is large). Hence, other things being equal, the higher the level of disagreement, the weaker the relationship between trading activity and price changes. In the extreme case, when the equilibrium price does not change as a result of the announcement and the second term on the right-hand side of (2.1) equals zero, there can still be large trading volume arising from disagreement among the traders because of the $$\beta _{0}$$ term. In this sense, disagreement generates an additional endogenous trading motive that is effectively “ orthogonal” to any revision in the equilibrium price. 2.2. Expected volume and volatility The implication of the KP model for the relationship between price adjustment and trading volume in response to new information is succinctly summarized by equations (2.1) and (2.2). These equations, however, depict an exact functional relationship between (observed) random quantities. A weaker, but empirically more realistic, implication can be obtained by thinking of this equilibrium relationship as only holding “on average”. Moment conditions corresponding to the stochastic version (2.1) formally capture this idea. Specifically, let $$m(\sigma )$$ denote the expected volume as a function of the volatility $$\sigma $$ (i.e. the standard deviation of price change). We assume that the price changes are normally distributed with mean zero and standard deviation $$\sigma $$. In particular, we note that the zero-mean assumption is empirically sound for high-frequency returns (Table 1). It follows by direct integration of (2.1) that \begin{equation} m\left( \sigma \right) =\sqrt{\frac{2}{\pi }}|\beta _{1}|\sigma \exp \left( - \frac{\beta _{0}^{2}}{2\beta _{1}^{2}\sigma ^{2}}\right) +|\beta _{0}|\left( 2\Phi \left( \frac{|\beta _{0}|}{|\beta _{1}|\sigma }\right) -1\right) , \label{mkvmkp} \end{equation} (2.3) where $$\Phi $$ denotes the cumulative distribution function of the standard normal distribution. The expected volume $$m(\sigma )$$ depends on $$\sigma $$ and the $$(\beta _{0},\beta _{1})$$ coefficients in a somewhat complicated fashion. However, it is straightforward to show that $$m(\sigma )$$ is increasing in $$\sigma $$. Table 1 Summary statistics of high-frequency price returns and volume data Mean Min 1% 25% 50% 75% 99% Max Return (percent) 0.000 $$-$$2.014 $$-$$0.149 $$-$$0.021 0.000 0.021 0.149 2.542 Volume (1,000 shares) 289.5 0.0 2.4 55.4 147.6 350.7 2005 31153 Mean Min 1% 25% 50% 75% 99% Max Return (percent) 0.000 $$-$$2.014 $$-$$0.149 $$-$$0.021 0.000 0.021 0.149 2.542 Volume (1,000 shares) 289.5 0.0 2.4 55.4 147.6 350.7 2005 31153 Notes: The table reports summary statistics of the one-minute returns and one-minute trading volumes for the SPY ETF during regular trading hours from 10, April 2001 to 30, September 2014. Table 1 Summary statistics of high-frequency price returns and volume data Mean Min 1% 25% 50% 75% 99% Max Return (percent) 0.000 $$-$$2.014 $$-$$0.149 $$-$$0.021 0.000 0.021 0.149 2.542 Volume (1,000 shares) 289.5 0.0 2.4 55.4 147.6 350.7 2005 31153 Mean Min 1% 25% 50% 75% 99% Max Return (percent) 0.000 $$-$$2.014 $$-$$0.149 $$-$$0.021 0.000 0.021 0.149 2.542 Volume (1,000 shares) 289.5 0.0 2.4 55.4 147.6 350.7 2005 31153 Notes: The table reports summary statistics of the one-minute returns and one-minute trading volumes for the SPY ETF during regular trading hours from 10, April 2001 to 30, September 2014. To gain further insight regarding this (expected) volume–volatility relationship, Figure 1 illustrates how the $$m(\sigma )$$ function varies with the “disagreement component” $$\beta _{0}$$. Locally, when the volatility $$\sigma $$ is close to zero, the expected volume is positive if and only if the opinions of the traders differ (i.e.$$\mu _{1}\neq \mu _{2}$$). Globally, as $$\beta _{0}$$ increases (from the bottom to the top curves in the figure), the equilibrium relationship between the expected volume and volatility “ flattens out”. This pattern is consistent with the aforementioned intuition that disagreement among traders provide an additional trading motive which loosens the relationship between volume and volatility. Figure 1 View largeDownload slide Equilibrium volume–volatility relations Notes: The figure shows the equilibrium relationship between expected trading volume $$m$$ and price volatility $$\sigma$$ in the Kandel–Pearson model for various levels of disagreement, ranging from $$\beta_0 = 0$$ (bottom) to $$0.5, 1$$ and $$1.5$$ (top). $$\beta_1$$ is fixed at one in all of the graphs. Figure 1 View largeDownload slide Equilibrium volume–volatility relations Notes: The figure shows the equilibrium relationship between expected trading volume $$m$$ and price volatility $$\sigma$$ in the Kandel–Pearson model for various levels of disagreement, ranging from $$\beta_0 = 0$$ (bottom) to $$0.5, 1$$ and $$1.5$$ (top). $$\beta_1$$ is fixed at one in all of the graphs. The exact non-linear expression in equation (2.3) inevitably depends on the specific assumptions and setup underlying the KP model. Although this expression helps formalize the intuition about the way in which disagreement affects the volume–volatility relationship, the KP model is obviously too stylized to allow for a direct structural estimation. In our empirical analysis below, we therefore rely on a reduced-form approach. In so doing, we aim to test the basic economic intuition and empirical implications stemming from the differences-of-opinion class of models more generally. From an empirical perspective this is naturally accomplished by focusing on the volume–volatility relationship expressed in terms of the elasticity of $$m(\sigma )$$ with respect to $$\sigma $$. Not only does this volume–volatility elasticity provide a convenient “reduced-form” summary statistic, it also admits a clear economic interpretation within the KP model. Let $$\mathcal{E}$$ denote the volume–volatility elasticity. A straightforward calculation then yields \begin{equation} \mathcal{E}\equiv \frac{\partial m(\sigma )/m(\sigma )}{\partial \sigma /\sigma }=\frac{1}{1+\psi \left( \gamma /\sigma \right) }, \label{kpe} \end{equation} (2.4) where \begin{equation} \gamma \equiv \frac{|\beta _{0}|}{|\beta _{1}|}=\frac{h\left\vert \mu _{1}-\mu _{2}\right\vert }{\left\vert s_{1}-s_{2}\right\vert }, \label{kpe2} \end{equation} (2.5) and the function $$\psi $$ is defined by $$\psi \left( x\right) \equiv x\left(\Phi \left( x\right) -1/2\right) /\phi \left( x\right) $$, with $$\phi $$ being the density function of the standard normal distribution. The function $$\psi$$ is strictly increasing on $$[0,\infty )$$, with $$\psi (0)=0$$ and $$\lim_{x\rightarrow \infty }\psi (x)=\infty $$. The expressions in (2.4) and (2.5) embody two key features in regards to the volume–volatility elasticity. First, $$\mathcal{E}\leq 1$$ with the equality and an elasticity of unity obtaining if and only if $$\gamma =0$$. Secondly, $$\mathcal{E}$$ only depends on and is decreasing in $$\gamma /\sigma $$. This second feature provides a clear economic interpretation of the volume–volatility elasticity $$\mathcal{E}$$: it is low when differences-in-opinion is relatively high, and vice versa, with $$\gamma /\sigma $$ serving as the relative measure of the differences-of-opinion. This relative measure is higher when traders disagree more on how to interpret the public signal (i.e. larger $$\left\vert \mu _{1}-\mu _{2}\right\vert $$) and with more confidence (i.e. larger $$h$$), relative to the degree of asymmetric private information (i.e.$$\left\vert s_{1}-s_{2}\right\vert $$), and the overall price volatility (i.e.$$\sigma$$). These two features in turn translate into directly testable implications that we use to guide our empirical analysis. In particular, taking the KP model and the expressions in (2.4) and (2.5) at face value, it is possible to test for the presence of differences-of-opinion among traders by testing whether the volume–volatility elasticity around important news arrivals is less than or equal to unity. This strict implication, however, hinges on a number of specific parametric and distributional modeling assumptions.8 We therefore also investigate the second more qualitative implication arising from the model that the volume–volatility elasticity is decreasing with the overall level of disagreement. This implication reflects the more general economic intuition that disagreement among traders provides an extra trading motive, and as such this implication should hold true more broadly. To allow for a focused estimation of the elasticities, we base our empirical investigations on intraday high-frequency transaction data around well-defined public macroeconomic news announcements, along with various proxies for the heterogeneity in beliefs and economic uncertainty prevailing at the exact time of the announcements. We turn next to a discussion of the data that we use in doing so. 3. Data description and summary statistics Our empirical investigations are based on high-frequency intraday transaction prices and trading volume, together with precisely timed macroeconomic news announcements. We describe our data sets in turn. 3.1. High-frequency market prices and trading volume Our primary data are composed of intraday transaction prices and trading volume for the S&P 500 index ETF (ticker: SPY). All of the data are obtained from the TAQ database. The sample covers all regular trading days from April 10th, 2001 through September 30th, 2014. The raw data are cleaned following the procedures detailed in Brownlees and Gallo (2006) and Barndorff-Nielsen et al. (2009). Further, to mitigate the effect of market microstructure noise, we follow standard practice in the literature to sparsely sample the data at a one-minute sampling interval, resulting in a total of 1,315,470 one-minute return and volume observations. Summary statistics for the SPY returns and trading volumes (number of shares) are reported in Table 1. Consistent with prior empirical evidence (see, e.g.Bollerslev and Todorov, 2011), the high-frequency one-minute returns appear close to be symmetrically distributed. The one-minute volume series, on the other hand, is highly skewed to the right, with occasionally very large values. To highlight the general dynamic dependencies inherent in the data, Figure 2 plots the daily logarithmic trading volume (constructed by summing the one-minute trading volumes over each of the different days) and the logarithmic daily realized volatilities (constructed as the sum of squared one-minute returns over each of the days in the sample). As the figure shows, both of the daily series vary in a highly predictable fashion. The volume series, in particular, seem to exhibit an upward trend over the first half of the sample, but then levels off over the second half. Meanwhile, consistent with the extensive prior empirical evidence discussed above, there are strong dynamic commonalities evident in the two series. Figure 2 View largeDownload slide Time series of volume and volatility Notes: The figure shows the daily logarithmic trading volume (top panel) and logarithmic realized volatility (bottom panel) for the SPY ETF. The daily volume (in millions) is constructed by accumulating the intraday volume. The daily realized volatility (annualized in percentage) is constructed as the sum of one-minute squared returns over the day. Figure 2 View largeDownload slide Time series of volume and volatility Notes: The figure shows the daily logarithmic trading volume (top panel) and logarithmic realized volatility (bottom panel) for the SPY ETF. The daily volume (in millions) is constructed by accumulating the intraday volume. The daily realized volatility (annualized in percentage) is constructed as the sum of one-minute squared returns over the day. The volume and volatility series also both exhibit strong intraday patterns. To illustrate this, Figure 3 plots the square-root of the one-minute squared returns averaged across each minute-of-the-day (as an estimate for the volatility over that particular minute) and the average trading volume over each corresponding minute. To prevent abnormally large returns and volumes from distorting the picture, we only include non-announcement days that are discussed in Section 3.2. Consistent with the evidence in the extant literature, there is a clear U-shaped pattern in the average volatility and trading activity over the active part of the trading day.9 Figure 3 View largeDownload slide Intraday patterns of volatility and volume Notes: The figure shows the intraday volatility (annualized in percentage) for the SPY ETF (left panel) constructed as the square root of the one-minute squared returns averaged across all non-announcement days, along with the best quadratic fit. The intraday trading volume (in shares) for the SPY ETF (right panel) is similarly averaged across all non-announcement days. Figure 3 View largeDownload slide Intraday patterns of volatility and volume Notes: The figure shows the intraday volatility (annualized in percentage) for the SPY ETF (left panel) constructed as the square root of the one-minute squared returns averaged across all non-announcement days, along with the best quadratic fit. The intraday trading volume (in shares) for the SPY ETF (right panel) is similarly averaged across all non-announcement days. In addition to our main empirical results based on the high-frequency intraday data for the S&P 500 aggregate equity portfolio discussed above, we also report complementary empirical evidence for the U.S. Treasury bond market. Our intraday price and volume data for the ten-year U.S. T-bond futures are obtained from TickData, and spans the slightly shorter sample period from 1, July 2003 to 30, September 2014. The intraday T-bond volume and volatility series naturally exhibit their own distinct dynamic dependencies and intraday patterns. However, as discussed further below, the key features pertaining to the volume and volatility “ jumps” observed around important macroeconomic news announcements closely mirror those for the aggregate equity market. 3.2. Macroeconomic news announcements The Economic Calendar Economic Release section in Bloomberg includes the date and exact within day release time for over one hundred regularly scheduled macroeconomic news announcements. Most of these announcements occur before the market opens or after it closes. We purposely focus on announcements that occur during regular trading hours only.10 While this leaves out some key announcements (most notably the monthly employment report), importantly it allows us to accurately estimate volume and volatility “jumps” by harnessing the rich information inherent in intraday high-frequency data about the way in which markets process new information. In addition to the intraday announcements pertaining to specific macroeconomic variables and indicators, we also consider FOMC rate decisions. FOMC announcements have been analysed extensively in the existing literature, and we also pay special attention to those announcements in our empirical analysis discussed below. Based on prior empirical evidence (see, e.g.Andersen et al., 2003; Boudt and Petitjean, 2014; Jiang et al., 2011; Lee, 2012), we identify four announcements as being the most important overall in the sense of having the on-average largest price impact among all of the regularly scheduled intraday news announcements. In addition to the aforementioned FOMC rate decisions, this includes announcements pertaining to the ISM Manufacturing Index (ISMM), the ISM Non-Manufacturing Index (ISMNM), and the Consumer Confidence (CC) Index. Table 2 provides the typical release times and the number of releases over the April 10, 2001 to September 30, 2014 sample period for each of these important announcements.11 The remaining announcements are categorized as “Others”, a full list of which is provided in Table B.1 in Appendix B. Some of the announcement times for these other indicators invariably coincide, so that all-in-all our sample is composed of a total of 2,130 unique intraday public news announcement times. Table 2 Macroeconomic news announcements. No. of Obs. Time Source FOMC 109 Varies Federal Reserve Board ISM Manufacturing (ISMM) 160 10:00 Institute of Supply Management ISM Non-Manufacturing (ISMNM) 158 10:00 Institute of Supply Management Consumer Confidence (CC) 160 10:00 Conference Board Other Indicators (Others) 1682 Varies See Appendix B.1 No. of Obs. Time Source FOMC 109 Varies Federal Reserve Board ISM Manufacturing (ISMM) 160 10:00 Institute of Supply Management ISM Non-Manufacturing (ISMNM) 158 10:00 Institute of Supply Management Consumer Confidence (CC) 160 10:00 Conference Board Other Indicators (Others) 1682 Varies See Appendix B.1 Notes: The table reports the total number of observations, release time, and data source for each of the news announcements over the 10, April 2001 to 30, September 2014 sample. Table 2 Macroeconomic news announcements. No. of Obs. Time Source FOMC 109 Varies Federal Reserve Board ISM Manufacturing (ISMM) 160 10:00 Institute of Supply Management ISM Non-Manufacturing (ISMNM) 158 10:00 Institute of Supply Management Consumer Confidence (CC) 160 10:00 Conference Board Other Indicators (Others) 1682 Varies See Appendix B.1 No. of Obs. Time Source FOMC 109 Varies Federal Reserve Board ISM Manufacturing (ISMM) 160 10:00 Institute of Supply Management ISM Non-Manufacturing (ISMNM) 158 10:00 Institute of Supply Management Consumer Confidence (CC) 160 10:00 Conference Board Other Indicators (Others) 1682 Varies See Appendix B.1 Notes: The table reports the total number of observations, release time, and data source for each of the news announcements over the 10, April 2001 to 30, September 2014 sample. 4. A preliminary analysis of FOMC announcements To set the stage for our more in-depth subsequent empirical investigations, we begin by presenting a set of simple summary statistics and illustrative figures related to the volume–volatility relationship around FOMC announcements. We focus our preliminary analysis on FOMC announcements, because these are arguably among the most important public news announcements that occur during regular trading hours.12 For each announcement, let $$\tau $$ denote the pre-scheduled announcement time. The time $$\tau $$ is naturally associated with the integer $$i(\tau )$$ such that $$\tau =(i(\tau )-1)\Delta _{n}$$, where $$\Delta _{n}=1$$ minute is the sampling interval of our intraday data. We define the event window as $$(\left( i(\tau )-1\right) \Delta _{n},i(\tau )\Delta _{n}]$$. Further, we define the pre-event (resp. post-event) window to be the $$k_{n}$$-minute period immediately before (resp. after) the event. We denote the return and trading volume over the $$j$$th intraday time-interval $$((j-1)\Delta_{n},j\Delta _{n}]$$ by $$r_{j}$$ and $$V_{j\Delta _{n}}$$, respectively. The volume intensity $$m$$ (i.e., the instantaneous mean volume) and the spot volatility $$\sigma $$ before and after the announcement, denoted by $$m_{\tau-}$$, $$m_{\tau }$$, $$\sigma _{\tau -}$$ and $$\sigma _{\tau }$$, respectively, can then be estimated by \begin{equation} \begin{array}{ll} \hat{m}_{\tau -}\equiv \frac{1}{k_{n}}\sum_{j=1}^{k_{n}}V_{\left( i(\tau )-j\right) \Delta _{n}}, & \hat{m}_{\tau }\equiv \frac{1}{k_{n}} \sum_{j=1}^{k_{n}}V_{\left( i(\tau )+j\right) \Delta _{n}}, \\ & \\ \hat{\sigma}_{\tau -}\equiv \sqrt{\frac{1}{k_{n}\Delta _{n}} \sum_{j=1}^{k_{n}}r_{i(\tau )-j}^{2}}, & \hat{\sigma}_{\tau }\equiv \sqrt{\frac{1}{k_{n}\Delta _{n}}\sum_{j=1}^{k_{n}}r_{i(\tau )+j}^{2}}. \end{array} \label{EST} \end{equation} (4.1) These estimators are entirely non-parametric, in the sense that they only rely on simple averages of the data in local windows around the event time. The window size $$k_{n}$$ plays the same role as the bandwidth parameter in conventional non-parametric analysis and regressions; the standard technical assumptions needed to formally justify the estimators are further detailed in Appendix A.13 In the empirical analysis reported below we set $$k_{n}=30$$, corresponding to a 30-minute pre-event (resp. post-event) window.14 As an initial illustration of the volume and volatility jumps observed around FOMC announcements, Figure 4 plots the average (across all FOMC announcements in the sample) estimated volume intensity (top panel) and spot volatility (bottom panel) processes for the 15 minutes before and after the announcements. As the figure clearly shows, both volume and volatility sharply increase at the time of the announcement. Consistent with the recent work of Bernile et al. (2016) and Kurov et al. (2016) and the finding that some price-adjustment seemingly occurs in anticipation of the actual news release, there is also a slight increase in both series leading up to the news announcement time. However, this increase is clearly small compared to the “jumps” that manifest at the time of the announcement, and as such will not materially affect any of our subsequent empirical analysis.15 Figure 4 View largeDownload slide Average volume intensity and volatility jumps around FOMC announcements Notes: The figure plots the non-parametric estimates of the volume intensity (in shares) and volatility (annualized in percentage) before and after an “average” FOMC announcement. We compute $$\hat{m}_{\tau \pm s}$$ (resp. $$\hat{\sigma}_{\tau \pm s}$$) for $$s$$ ranging from 15 minutes before to 15 minutes after each FOMC announcement, and plot the averages across all of the announcements in the top (resp. bottom) panel. The local window is set to $$k_n = 30$$. Figure 4 View largeDownload slide Average volume intensity and volatility jumps around FOMC announcements Notes: The figure plots the non-parametric estimates of the volume intensity (in shares) and volatility (annualized in percentage) before and after an “average” FOMC announcement. We compute $$\hat{m}_{\tau \pm s}$$ (resp. $$\hat{\sigma}_{\tau \pm s}$$) for $$s$$ ranging from 15 minutes before to 15 minutes after each FOMC announcement, and plot the averages across all of the announcements in the top (resp. bottom) panel. The local window is set to $$k_n = 30$$. To examine whether the “average jumps” evident in Figure 4 are representative, Figure 5 plots the time series of individually estimated logarithmic volume intensities (top panel) and logarithmic spot volatilities (bottom panel) before and after each of the FOMC announcements.16 Consistent with the on-average estimates depicted in Figure 4, the figure shows marked bursts in the trading volume following each of the FOMC announcements, accompanied by positive jumps in the volatility. These jumps in the volume intensity and volatility are economically large, with average jump sizes (in log) of 1.41 and 1.09, respectively. As discussed in Section 6, they are also highly statistically significant. This therefore suggests that traders revise their beliefs about the stock market differently upon seeing the FOMC announcement. It is, of course, possible that traders have asymmetric private information about the overall market. However, in line with the reasoning of Hong and Stein (2007) regarding the large observed burst in trading volume, it seems much more likely that the differences are attributed to the traders’ disagreement regarding the news.17 Figure 5 View largeDownload slide Volume and volatility around FOMC announcements Notes: The figure plots the log-volume intensity (top panel, in shares) and log-volatility (bottom panel, in percentage) around FOMC announcements. The volume intensity and volatility are calculated using equation (4.1) with $$k_n = 30$$. Figure 5 View largeDownload slide Volume and volatility around FOMC announcements Notes: The figure plots the log-volume intensity (top panel, in shares) and log-volatility (bottom panel, in percentage) around FOMC announcements. The volume intensity and volatility are calculated using equation (4.1) with $$k_n = 30$$. To further buttress the importance of differences-of-opinion among investors, we consider two additional empirical approaches. First, as discussed in Section 2.1, if all investors agreed on the interpretation of the FOMC announcements, the trading volumes observed around the news releases should be approximately proportional to the price changes. Consequently, if there were no disagreement we would expect to see large trading volume accompanied by large changes in prices and vice versa. To investigate this hypothesis, we sort all of the FOMC announcements by the normalized one-minute event returns $$r_{i}/\hat{\sigma}_{\tau -}\sqrt{\Delta _{n}}$$, and plot the resulting time series of pre- and post-event volume intensity estimates in Figure 6.18 Consistent with the findings of KP (based on daily data), Figure 6 shows no systematic association between trading volumes and returns. Instead, we observe many sizable jumps in the volume intensity for events with absolute returns “close” to zero, that is, when they are less than one instantaneous standard deviation (highlighted by the shaded area). Figure 6 View largeDownload slide Sorted volume around FOMC announcements Notes: The figure shows the pre- and post-event log volume intensities (in shares) sorted on the basis of the one-minute normalized returns $$r_{i(\tau)}/\hat{\sigma}_{\tau-}\sqrt{\Delta_n}$$ around FOMC announcements (dots). The normalized return increases from left to right. Announcements with normalized returns less than 1 are highlighted by the shaded area. Figure 6 View largeDownload slide Sorted volume around FOMC announcements Notes: The figure shows the pre- and post-event log volume intensities (in shares) sorted on the basis of the one-minute normalized returns $$r_{i(\tau)}/\hat{\sigma}_{\tau-}\sqrt{\Delta_n}$$ around FOMC announcements (dots). The normalized return increases from left to right. Announcements with normalized returns less than 1 are highlighted by the shaded area. The empirical approach underlying Figure 6 mainly focuses on events with price changes close to zero and, hence, is local in nature. Our second empirical approach seeks to exploit a more global feature of the differences-of-opinion type models, namely that the volume–volatility elasticity should be below unity. While this prediction was derived from the explicit solution for the KP model in equation (2.3), the underlying economic intuition holds more generally: differences-of-opinion provides an additional trading motive that is not tied to the traders’ average valuation of the asset and, hence, serves to “loosen” the relationship between trading volume and volatility. To robustly examine this prediction for the volume–volatility elasticity, without relying on the specific functional form in (2.4), we adopt a less restrictive reduced-form estimation strategy. Further along those lines, the theoretical models discussed in Section 2 are inevitably stylized in nature, abstracting from other factors that might affect actual trading activity (e.g., liquidity or life-cycle trading, reduction in trading costs, advances in trading technology, to name but a few). As such, the theoretical predictions are more appropriately thought of as predictions about “abnormal” variations in the volume intensity and volatility. In the high-frequency data setting, abnormal movements conceptually translate into “jumps”. Below, we denote the corresponding log-volatility and log-volume intensity jumps by $$\Delta \log\left( \sigma _{\tau }\right) \equiv \log (\sigma _{\tau })-\log (\sigma_{\tau -})$$ and $$\Delta \log \left( m_{\tau }\right) \equiv \log (m_{\tau})-\log (m_{\tau -})$$, respectively. Figure 7 shows a scatter plot of the estimated $$\Delta\log \left( \sigma _{\tau }\right)$$ and $$\Delta \log \left(m_{\tau }\right)$$ jumps around FOMC announcement times. As expected, there is a clear positive association between the two series, with a correlation coefficient of 0.57. Moreover, consistent with the theoretical predictions and the idea that traders interpret FOMC announcements differently, the estimate for the volume–volatility elasticity implied by the slope coefficient from a simple linear fit equals $$0.66$$, which is much less than unity. Figure 7 View largeDownload slide Volume and volatility jumps around FOMC announcements Notes: The figure shows the scatter of the jumps in the log-volume intensity (in shares) versus the jumps in the log-volatility (in percentage) around FOMC announcements. The line represents the least-square fit. Figure 7 View largeDownload slide Volume and volatility jumps around FOMC announcements Notes: The figure shows the scatter of the jumps in the log-volume intensity (in shares) versus the jumps in the log-volatility (in percentage) around FOMC announcements. The line represents the least-square fit. The summary statistics and figures discussed above all corroborate the conjecture that differences-of-opinion among investors play an important role in the way in which the market responds to FOMC announcements. To proceed with a more formal empirical analysis involving other announcements and explanatory variables, we need econometric tools for conducting valid inference, to which we now turn. 5. High-frequency econometric procedures The econometrics in the high-frequency setting is notably different from more conventional settings, necessitating the development of new econometric tools properly tailored to the data and the questions of interest.19 To streamline the discussion, we focus on the practical implementation and heuristics of the underlying econometric theory, deferring the technical details to Appendix A. Following the discussion and the theoretical implications developed in Section 2.2, our primary interest centers on estimation and inference concerning the volume–volatility elasticity, and in particular, whether the elasticity decreases with the level of disagreement. To this end, we estimate the elasticity between announcement-induced “abnormal” volume and volatility variations by regressing jumps in the log-volume intensity on jumps in the log-volatility. Further, to help address how the elasticity is affected by disagreement, we parameterize the regression coefficient as a function of various measures of disagreement prevailing at the time of the announcement (e.g., the dispersion among professional forecasters). The resulting econometric model may be succinctly expressed as \begin{equation} \Delta \log \left( m_{\tau }\right) =(a_{0}+b_{0}^{\top }X_{0,\tau })+(a_{1}+b_{1}^{\top }X_{1,\tau })\cdot \Delta \log \left( \sigma _{\tau }\right) , \label{LL2} \end{equation} (5.1) where $$X_{\tau }\equiv (X_{0,\tau },X_{1,\tau })$$ is composed of the different explanatory/control variables employed in the estimation. In particular, by including measures of disagreement in $$X_{1,\tau}$$ it is possible to directly assess the aforementioned theoretical predictions based on the statistical significance of the estimated $$b_{1}$$ coefficients. From an econometric perspective, equation (5.1) is best understood as an instantaneous moment condition, in which the volume intensity process $$m_{t}$$ (resp. the spot volatility process $$\sigma _{t}$$) represents the latent local first (resp. second) moment of the volume (resp. price return) process.20 The sample analogue of (5.1) therefore takes the form \begin{equation} \widehat{\Delta \log \left( m_{\tau }\right) }=(a_{0}+b_{0}^{\top }X_{0,\tau })+(a_{1}+b_{1}^{\top }X_{1,\tau })\cdot \widehat{\Delta \log \left( \sigma _{\tau }\right) }+e_{\tau }, \label{LL3} \end{equation} (5.2) where the error term $$e_{\tau }$$ arises from the estimation errors associated with the local moments (i.e., $$m$$ and $$\sigma $$). Our goal is to conduct valid inference about the parameter vector $$\theta\equiv (a_{0},b_{0},a_{1},b_{1})$$, especially the components $$a_{1}$$ and $$b_{1}$$ that determine the volume–volatility elasticity. Consider the group $$\mathcal{A}$$ composed of a total of $$M$$ announcement times.21 Further, define $$S_{\tau }\equiv (m_{\tau -},m_{\tau },\sigma_{\tau -},\sigma _{\tau },X_{\tau })$$ and $$\mathbf{S}\equiv (S_{\tau})_{\tau \in \mathcal{A}}$$, where the latter collects the information on all announcements. Our estimator of $$\mathbf{S}$$ may then be expressed as $$\widehat{\mathbf{S}}_{n}\equiv (\widehat{S}_{\tau })_{\tau \in \mathcal{A}}$$, where $$\widehat{S}_{\tau }\equiv (\hat{m}_{\tau -},\hat{m}_{\tau },\hat{\sigma}_{\tau -},\hat{\sigma}_{\tau },X_{\tau })$$ is formed using the non-parametric pre- and post-event volume intensity and volatility estimators previously defined in (4.1). Correspondingly, summary statistics pertaining to the jumps in the volume intensity and volatility for the group of announcement times $$\mathcal{A}$$ may be succinctly expressed as $$f(\widehat{\mathbf{S}})$$ for some smooth function $$f(\cdot )$$.22 Moreover, we may estimate the parameter vector $$\theta \equiv(a_{0},b_{0},a_{1},b_{1})$$ in (5.1) for the group $$\mathcal{A}$$ using the following least-square estimator \begin{equation} \hat{\theta}\equiv \underset{\theta }{\text{ argmin}}\sum_{\tau \in \mathcal{ A}}\left( \widehat{\Delta \log \left( m_{\tau }\right) }-(a_{0}+b_{0}^{\top }X_{0,\tau })-(a_{1}+b_{1}^{\top }X_{1,\tau })\cdot \widehat{\Delta \log \left( \sigma _{\tau }\right) }\right) ^{2}. \label{basic_ols} \end{equation} (5.3) This estimator may similarly be expressed as $$\hat{\theta}=f(\widehat{\mathbf{S}})$$, albeit for a more complicated transform $$f(\cdot )$$. It can be shown that $$\widehat{\mathbf{S}}$$ is a consistent estimator of $$\mathbf{S}$$, which in turn implies that $$f(\widehat{\mathbf{S}})$$ consistently estimates $$f(\mathbf{S})$$, provided $$f(\cdot )$$ is a smooth function of the estimated quantities. The estimates that we reported in our preliminary analysis in Section 4 may be formally justified this way.23 The “raw” estimator defined above is asymptotically valid under general regularity conditions (again, we refer to Appendix A for the specific details). However, the non-parametric estimators $$\widehat{\Delta \log \left( \sigma _{\tau }\right) }$$ and $$\widehat{\Delta\log \left( m_{\tau }\right) }$$ underlying the simple estimator in (5.3) do not take into account the strong intraday U-shaped patterns in trading volume and volatility documented in Figure 3. While the influence of the intraday patterns vanishes asymptotically, they invariably contaminate our estimates of the jumps in finite samples, and thus affect our use of the jump estimates as measures of “abnormal” volume and volatility movements, to which the economic theory speaks. A failure to adjust for this may therefore result in a mismatch between the empirical strategy and the economic theory.24 To remedy this, we correct for the influence of the intraday pattern by differencing it out with respect to a control group. Since this differencing step is applied to the jumps, which are themselves differences between the post- and pre-event quantities, our empirical strategy for jump estimation may be naturally thought of as a high-frequency DID type approach, in which we consider the event-control difference of the jump estimates as our measure for the abnormal movements in the volume intensity and volatility. We then regress these DID jump estimates to obtain the volume–volatility elasticity. Following the terminology of Li et al. (2017), who considered “jump regressions” involving discontinuous price increments, we refer to our new procedure as a DID jump regression estimator.25 Formally, with each announcement time $$\tau $$, we associate a control group $$\mathcal{C}\left( \tau \right) $$ of non-announcement times. Based on this control group, we then correct for the intraday patterns in the “raw” jump estimators by differencing out the corresponding estimates averaged within the control group, resulting in the adjusted jump estimators \begin{equation} \begin{array}{l} \widetilde{\Delta \log \left( m_{\tau }\right) }\equiv \widehat{\Delta \log \left( m_{\tau }\right) }-\frac{1}{N_{C}}\sum_{\tau ^{\prime }\in C\left( \tau \right) }\widehat{\Delta \log \left( m_{\tau ^{\prime }}\right) }, \\ \widetilde{\Delta \log \left( \sigma _{\tau }\right) }\equiv \widehat{\Delta \log \left( \sigma _{\tau }\right) }-\frac{1}{N_{C}}\sum_{\tau ^{\prime }\in C\left( \tau \right) }\widehat{\Delta \log \left( \sigma _{\tau ^{\prime }}\right) }, \end{array} \label{mtil} \end{equation} (5.4) where $$N_{C}$$ refers to the number of times in the control group.26 Analogously to (5.3), we then estimate the parameters of interest by regressing the DID jump estimates as \begin{equation} \tilde{\theta}\equiv \underset{\theta }{\text{ argmin}}\sum_{\tau \in \mathcal{A}}\left( \widetilde{\Delta \log \left( m_{\tau }\right) } -(a_{0}+b_{0}^{\top }X_{0,\tau })-(a_{1}+b_{1}^{\top }X_{1,\tau })\cdot \widetilde{\Delta \log \left( \sigma _{\tau }\right) }\right) ^{2}. \label{did-reg} \end{equation} (5.5) Note that $$\tilde{\theta}$$ depends not only on $$(\widehat{S}_{\tau })_{\tau\in \mathcal{A}}$$ but also on $$(\widehat{S}_{\tau ^{\prime }})_{\tau^{\prime }\in \mathcal{C}}$$, where $$\mathcal{C\equiv \cup }_{\tau \in\mathcal{A}}\mathcal{C}\left( \mathcal{\tau }\right) $$ contains the times of all control groups. This estimator can be expressed as $$\tilde{\theta}$$$$=f(\widetilde{\mathbf{S}})$$ where $$\widetilde{\mathbf{S}}\equiv (\widehat{S}_{\tau })_{\tau \in \mathcal{T}}$$ for $$\mathcal{T}\equiv \mathcal{A\cup C}$$. In summary, our estimation procedure consists of two steps. The first step is to estimate the jumps in the volume intensity and volatility processes via DID. The second step consists in estimating the parameters that describe the relationship between the two via a least-square regression. Since jumps are discontinuities, the resulting DID jump regression estimator formally bears some resemblance to that in RDD commonly used in empirical microeconomics for the estimation of treatment effects (see, e.g., Lee and Lemieux, 2010). Importantly, however, our new econometric inference procedures (including the computation of standard errors) is very different and non-standard. Explicitly allowing for heterogeneity and dependence in the return and volume data, the sampling variability in $$\tilde{\theta}$$ arises exclusively from the non-parametric estimation errors in the pre- and post-event high-frequency-based volume intensity and volatility estimators, $$\hat{m}_{\tau \pm }$$ and $$\hat{\sigma}_{\pm }$$, respectively. While in theory we can characterize the resulting asymptotic covariance matrix and it would be possible to use it in the design of “plug-in” type standard errors, the control groups $$\mathcal{C}\left( \tau \right) $$ used for the different announcement times often partially overlap, which would severely complicate the formal derivation and implementation of the requisite formulas. To facilitate the practical implementation, we instead propose a novel easy-to-implement local i.i.d. bootstrap procedure for computing the standard errors, which is a localized version of the i.i.d. bootstrap of Gonçalves and Meddahi (2009). This procedure does not require the exact dependence of $$\tilde{\theta}$$ on $$\widetilde{\mathbf{S}}$$ to be fully specified. Instead, it merely requires repeated estimation over a large number of locally i.i.d. bootstrap samples for the pre-event and post-event windows around each of the announcement and control times. The “localization” is important, as it allows us to treat the conditional distributions as (nearly) constant, in turn permitting the use of an i.i.d. re-sampling scheme. The actual procedure is summarized by the following algorithm, for which the formal theoretical justification is given in the technical Appendix A. 5.1. Bootstrap Algorithm Step 1: For each $$\tau \in \mathcal{T}$$, generate i.i.d. draws $$(V_{i(\tau )-j}^{\ast },r_{i(\tau )-j}^{\ast })_{1\leq j\leq k_{n}}$$ and $$(V_{i(\tau )+j}^{\ast },r_{i(\tau )+j}^{\ast })_{1\leq j\leq k_{n}}$$ from $$(V_{i(\tau )-j},r_{i(\tau )-j})_{1\leq j\leq k_{n}}$$ and $$(V_{i(\tau)+j},r_{i(\tau )+j})_{1\leq j\leq k_{n}}$$, respectively. Step 2: Compute $$\widetilde{\Delta \log \left( m_{\tau }\right) }^{\ast }$$ and $$\widetilde{\Delta \log \left( \sigma _{\tau }\right) }^{\ast }$$ the same way as $$\widetilde{\Delta \log \left( m_{\tau }\right) }$$ and $$\widetilde{\Delta \log \left( \sigma _{\tau }\right) }$$, respectively, except that the original data $$(V_{i(\tau )-j},r_{i(\tau )-j})_{1\leq\left\vert j\right\vert \leq k_{n}}$$ is replaced with $$(V_{i(\tau )-j}^{\ast},r_{i(\tau )-j}^{\ast })_{1\leq \left\vert j\right\vert \leq k_{n}}$$. Similarly, compute $$\tilde{\theta}^{\ast }$$