Combination Return Forecasts and Portfolio Allocation with the Cross-Section of Book-to-Market Ratios

Combination Return Forecasts and Portfolio Allocation with the Cross-Section of Book-to-Market... Abstract * We thank Burton Hollifield (Editor), Mihail Velikov, and an anonymous referee for helpful comments. In this paper, we forecast industry returns out-of-sample using the cross-section of book-to-market (BM) ratios and investigate whether investors can exploit this predictability in portfolio allocation. Cash-flow and return forecasting regressions show that cross-industry BM ratios contain significant predictive information beyond aggregate and industry-specific BM ratios. Forecast combination methods based on industry BM ratios generate significant out-of-sample predictability for many industries. Real-time portfolio-rotation strategies that buy industries with high predicted returns and short industries with low predicted returns based on combination forecasts earn significant alpha with respect to standard asset pricing models net of transaction costs. 1. Introduction There is much evidence that average returns of stocks are positively correlated with their book-to-market (BM) equity ratios in the cross-section (e.g., Rosenberg, Reid, and Lanstein, 1985; Fama and French, 1992, 2015). Investment professionals since at least Graham and Dodd (1934) have exploited this pattern and demonstrated that “value” strategies that buy high BM stocks earn higher average returns than the overall market. In the time-series, Kothari and Shanken (1997), Pontiff and Schall (1998), and Lewellen (1999) find that the BM ratio of the aggregate market forecasts the market return. Both the time-series and cross-sectional patterns are consistent with present value identities that show, all-else-equal, higher expected returns on an asset lower that asset’s value relative to accounting fundamentals (e.g., Campbell and Shiller, 1988; Vuolteenaho, 2002). Perhaps in part because of this theory, prior studies on BM ratios and returns generally ignore whether the return of any one stock or portfolio can be predicted by the BM ratios of other stocks. Kelly and Pruitt (2013) argue that a common set of dynamic state variables drives the entire panel of BM ratios and expected returns. Thus, the whole panel of BM ratios provides information about these state variables. An asset’s BM ratio is also a noisy proxy for the asset’s expected return because it is a joint function of the asset’s expected future return and cash flows (CFs). Given imperfect correlation of CFs across stocks, commonality in expected returns implies that the BM ratio of any one stock conveys a relevant but non-redundant signal about the expected returns on other stocks. In this paper, we use forecast combination methods to extract information from the cross-section of BM ratios to help predict the returns of individual stock portfolios. Combination forecasts are weighted averages of individual forecasts and benefit from a diversification-like effect. If the prediction errors of individual forecasts are imperfectly correlated, forecast combinations can be more accurate out-of-sample than even the best individual forecast (for a recent survey, see Timmermann, 2006). Such imperfect correlation of return-forecast errors can arise from time-varying data-generating processes and idiosyncratic variation of cash-flow growth and returns across stocks. Consistent with the diversification benefit, prior studies show that combination forecasts effectively improve out-of-sample predictability of economic time-series such as output or stock market returns (e.g., Stock and Watson, 2004; Rapach, Strauss, and Zhou, 2010). We apply combination forecasts to construct portfolio-rotation strategies that buy portfolios with the highest predicted returns and short portfolios with the lowest predicted returns. The extent to which these portfolios improve investment opportunities relative to those represented by common asset pricing factors is a measure of the economic significance of our combination forecasts (e.g., Pesaran and Timmermann, 1995). Since investors face transaction costs, we follow Novy-Marx and Velikov (2016) and estimate effective transaction costs for individual stocks and apply them to our trading strategies. This method precisely captures variation in transaction costs over time and across portfolios. Our analysis focuses on value-weighted Fama–French industries, though we demonstrate the robustness of our methods using common characteristic-sorted portfolios (characteristic portfolios) formed on size and BM ratio, investment, and profitability. In forecasting, some aggregation of individual stocks is necessary to obtain regular-frequency time series with less idiosyncratic noise. We focus on industries for the following reasons. Expected returns on the characteristic portfolios tend to decrease with size and vary monotonically with the other characteristic. Hence, our forecast-based trading strategies should select the portfolios with the highest and lowest expected returns, but these portfolios are consistently those with the extreme size and characteristic ranks. In contrast, predicting the industries with the highest and lowest returns in real time is more challenging. Industries also lack the tight factor structure of characteristic portfolios and possess three to four times the variation in returns not explained by the Fama–French factors compared with characteristic portfolios. Hence, there is effectively more opportunity to demonstrate alpha with industry strategies (e.g., Lewellen, Nagel, and Shanken, 2010). Further, value-weighted industries have relatively low transaction costs as they put little weight on small-cap stocks. Conversely, the characteristic portfolios with the highest and lowest expected returns tend to consist of small-cap stocks that are expensive to trade. We summarize our findings as follows. Six principal components of industry BM ratios predict 1-year CFs and quarterly returns of the average industry with in-sample adjusted R2 of 46.5% and 6.1%, respectively. Principal components 2–6 explain most of this predictability. Conversely, industry-own BM ratios predict CFs and returns with a fraction of the R2 on average. Thus, the cross-section of BM ratios possesses significant information for predicting industry returns and CFs beyond aggregate and individual own-industry BM ratios. Almost all of the combination forecasts of industry returns using each industry’s BM ratio have positive out-of-sample R2 ( ROS2), and about half are individually significant. On average, the combination forecasts of industry returns have higher ROS2 than those based on the market and industry-own BM ratio by 3.5–4.0% and 4.9–5.4%, respectively. In contrast, BM ratios of each industry predict those industry’s returns with mostly negative ROS2. Similarly, the combination forecasts predict returns on the twenty-five size and BM portfolios with mostly positive ROS2. Depending on method, seven–seventeen of these ROS2 are individually significant. Unlike the combination forecasts, the market and portfolio-own BM ratios predict returns on the twenty-five portfolios with mostly negative ROS2. Thus, combining the forecasts from the cross-section of BM ratios improves real-time forecasts of most industry and size and BM portfolio returns relative to forecasts based only on those portfolio’s BM ratios or the market BM ratio. Long–short industry-rotation strategies based on the combination forecasts (hereafter “combo strategies”) earn large average returns, ranging from 7.7% to 11.6% per year. After transaction costs, the long–short industry-rotation portfolios significantly outperform the standard four-factor performance attribution model of Fama and French (1993) and Carhart (1997) by 5.8–8.9% per year. The long–short industry combo strategies have large positive loadings on the momentum factor, though much lower turnover and transaction costs than the momentum strategy. Thus, the gross four-factor α understates the net-of-costs performance of the industry combo strategies. Interestingly, the momentum exposure is not explained by lagged own-industry returns, but from the signals of other industries. The momentum loadings of the long–short strategies based only on industry-own BM ratios are negative. Adding the industry long–short combo strategies to the Fama–French–Carhart factors also increases the maximum net-of-costs Sharpe ratio by as much as adding two or three of the Fama–French–Carhart factors to the market return. The long–short combo strategies using the characteristic portfolios also produce large spreads in gross returns of 8.7–15.3% per year, and significant spreads in gross alphas of 4.7–8.0% per year. However, transaction costs of these strategies are higher than the industries as portfolios with the highest and lowest returns tend to consist of small-cap stocks. For example, among the size and BM portfolios, the highest and lowest BM stocks in the smallest size quintile tend to have the highest and lowest returns on average, but also the highest transaction costs. Transaction costs reduce returns on strategies made from characteristic-sorted portfolios by approximately 4% per year and eliminate the significance of alphas earned by several strategies. This paper relates to several recent studies. We have similar motivation as Kelly and Pruitt (2013) who incorporate the cross-section of BM ratios in forecasting the return on the aggregate US stock market. In contrast, we use the cross-section of BM ratios to forecast returns of industry and characteristic portfolios, different forecast combination methods, more timely accounting data, and apply these forecasts to trading strategies. Lewellen (2015) also predicts the cross-section of returns in real time with BM ratios and other characteristics. However, Lewellen focuses on the returns of individual stocks and those stock’s own characteristics, not the characteristics of other stocks. Rapach et al. (2015) find evidence of cross-industry predictability for a subset of industries, but they use industry returns as predictors. The remainder of this paper proceeds as follows. Section 2 motivates why BM ratios of different stocks are useful for predicting the returns of any given stock. Section 3 describes our data and combination forecast methods. Section 4 presents our combination forecasts of industry returns and CFs. Section 5 describes the construction of our forecast-based trading strategies and analyzes their performance. Section 6 concludes. 2. Information in BM Ratios Cohen, Polk, and Vuolteenaho (2003) posit a log-linear approximation to decompose the BM ratio for a stock i into log stock return (rt), log return-on-equity (et), and an approximation error (kt): BMi,t≈kt+∑j=1∞ρjri,t+j+∑j=1∞ρj(−ei,t+j), (1) where ρ<1 is a constant arising from the approximation. For ease of illustration, assume that μit=Et(ri,t+1) and git=Et(ei,t+1) follow AR(1) processes (e.g., van Binsbergen and Koijen, 2010; Golez, 2014). Then, taking expectations of Equation (1) yields: BMi,t≈kt+φi,rμit−φi,ggit, (2) where φi,r and φi,g are constants. To understand how the BM ratio of one stock or portfolio of stocks can help predict the returns (or CFs) of another, suppose that a set of stocks has a non-zero common component of expected returns and profitability. That is, assume μi,t and git can be decomposed into common and idiosyncratic components (denoted by *): μi,t=μt+μi,t*, and (3) gi,t=gt+gi,t*. (4) This commonality can be motivated by the factor structure in returns, inter-industry correlation, or the positive betas of most stocks with the market return. By Equation (2) regressions of ri,t+1 on the BM ratio of stock i, and that of another stock j take the form: E(ri,t+1|BMi,t)=α^ii+β^ii(kt+φi,r(μt+μi,t*)−φi,g(gt+gi,t*)), and (5) E(ri,t+1|BMj,t)=α^ij+β^ij(kt+φj,r(μt+μj,t*)−φj,g(gt+gj,t*)), (6) where μit and git are latent, and must be inferred from observables like BMit. Assuming μt≠0, both stock’s BM predict ri,t+1 ( β^ii>0 and β^ij>0). If gi,t is constant, then BMi,t is a perfect proxy for expected returns (μit) and BMj,t is redundant for j≠i. However, assuming gi,t is time-varying, as evidence suggests, BMi,t is not a perfect proxy for μi,t (e.g., Kelly and Pruitt, 2013). Barring a knife-edge correlation between git and gjt, BMj,t provides a non-redundant signal about the common μt, and therefore ri,t+1. Below, we combine these signals to forecast returns. 3. Data and Methodology 3.1 Return, CF, and Predictor Data We use monthly frequency value-weighted portfolio return data from Kenneth French’s website for the thirty-eight and forty-eight industries as well as each of the twenty-five portfolios formed on size and BM ratio, investment, or operating profitability. We refer to the latter three sets of portfolios as size/BM, size/investment, and size/profitability, respectively. We form quarterly returns by compounding monthly returns. We exclude industries that have ten or fewer firms at the start of the sample as they have long periods with missing returns and frequently cannot be matched with any accounting data. Our sample thus consists of thirty-two and forty-four of the thirty-eight and forty-eight industries, respectively.1 We also construct quarterly Fama and French (2015) and Carhart (1997) size, value, momentum, investment, and profitability factors (SMB, HML, MOM, CMA, and RMW, respectively) from the monthly returns on the corresponding 2× 3 base assets. We take the quarterly risk-free rate rft to be the compounded one-month bill rate. For each of the portfolios listed above, we estimate quarterly portfolio-level accounting and predictor variables as value-weighted averages of the stock-level counterparts within the portfolios. We obtain quarterly accounting data from COMPUSTAT Quarterly. We follow Hou, Xue, and Zhang (2015) and define book value of equity as shareholders’ equity, plus balance-sheet deferred taxes and investment tax credit (item TXDITCQ) if available, minus the book value of preferred stock. Depending on availability, we use stockholders’ equity (item SEQQ), common equity (item CEQQ) plus the carrying value of preferred stock (item PSTKQ), or total assets (item ATQ) minus total liabilities (item LTQ) in that order as shareholders equity. We use redemption value (item PSTKRQ) if available, or carrying value for the book value of preferred stock. Hou, Xue, and Zhang (2015) argue this definition of book equity provides the broadest coverage, particularly prior to 1980. To compute BM, we divide the book value of equity by the end-of-quarter market value of equity from CRSP. We compute operating CF following Hirshleifer, Hou, and Teoh (2009). Due to availability of COMPUSTAT quarterly data, we begin our analysis in the first quarter of 1972 following Hou, Xue, and Zhang (2015). To have a reasonable number of in-sample observations for initial estimation, while maintaining a long out-of-sample period, we begin out-of-sample analysis in 1980:1. This out-of-sample period includes a variety of market environments including the bull market of the 1990s, the dot-com collapse in 2000, a period of steady gains in the mid-2000s, the recent financial crisis of the late 2000s, and subsequent era of unconventional monetary policy. To evaluate the ability of forecast methods to generate real-time investment opportunities, it is important for accounting data to be available at the time of portfolio adjustments. Hence, we follow Hirshleifer, Hou, and Teoh (2009); Lewellen (2015); and Hou, Xue, and Zhang (2015), and lag accounting data by 4 months in forecasting returns. In spite of this lag, our quarterly accounting data are more timely than commonly used annual observations (e.g., Fama and French, 1993), which can easily be more than a year old. Using timely data allows us to construct forecasts that best approximate the market’s conditional expectations. Table I reports select summary statistics for the BM ratios, CFs, and returns of each of the forty-four industries and the value-weighted average across stocks (MKT). Average BM ratios range from 0.33 for DRUGS to 1.22 for AUTO, and 0.68 for MKT. The standard deviations display considerable dispersion from 0.12 for BUSSV to 0.61 for MINES, and 0.19 for MKT. The average industry’s BM is persistent with an AR1 of 0.88 and all but one industry has a BM with an AR1 of at least 0.75.2 The market BM is even more persistent with a one-quarter autocorrelation of 0.98. In predicting returns, Stambaugh (1999) and Ferson, Sarkissian, and Simin (2003) find that predictors with high persistence like BM ratios can produce inflated in-sample forecasting slopes and R2. This persistence bias further motivates examining the out-of-sample performance of returns forecasts with BM ratios. Table I. Summary statistics of industry BM ratios For each of the forty-four out of forty-eight Fama–French industries we use in the paper and the market portfolio, we present: time-series averages (Mean), standard deviations (SD), and first autocorrelations ( AR1) of the value-weighted industry BM ratios, as well as the average correlation between each industry’s BM, annual cash flow growth, and quarterly returns with those of all other industries ( ρ¯BM, ρ¯CF, and ρ¯ret, respectively). The ρ¯CF reports the average absolute value of the pairwise correlation as industry CFs can be negatively correlated. The sample is 1980:1–2015:4. Mean SD AR1 ρ¯BM ρ¯CF ρ¯ret Mean SD AR1 ρ¯BM ρ¯CF ρ¯ret AERO 0.71 0.44 0.96 0.55 0.14 0.61 INSUR 0.89 0.25 0.90 0.56 0.21 0.45 AGRIC 0.73 0.53 0.93 0.46 0.28 0.66 LABEQ 0.56 0.26 0.92 0.65 0.30 0.63 AUTOS 1.23 0.53 0.86 0.26 0.35 0.57 MACH 0.62 0.24 0.93 0.62 0.27 0.64 BANKS 0.89 0.31 0.91 0.53 0.28 0.63 MEALS 0.45 0.18 0.91 0.55 0.24 0.61 BEER 0.49 0.21 0.96 0.57 0.22 0.63 MEDEQ 0.42 0.19 0.95 0.59 0.15 0.61 BLDMT 0.70 0.24 0.84 0.59 0.31 0.52 MINES 1.22 0.61 0.06 0.00 0.21 0.63 BOOKS 0.60 0.31 0.82 0.27 0.13 0.52 OIL 0.89 0.22 0.82 0.40 0.20 0.65 BOXES 0.78 0.35 0.94 0.55 0.21 0.67 OTHER 0.62 0.27 0.94 0.09 0.25 0.51 BUSSV 0.36 0.12 0.95 0.46 0.15 0.57 PAPER 0.63 0.17 0.94 0.52 0.23 0.58 CHEM 0.63 0.23 0.93 0.55 0.18 0.65 PERSV 0.58 0.25 0.86 0.59 0.25 0.52 CHIPS 0.48 0.13 0.82 0.50 0.24 0.53 RLEST 0.96 0.43 0.86 0.52 0.14 0.63 CLTHS 0.69 0.45 0.94 0.59 0.18 0.49 RTAIL 0.50 0.22 0.96 0.61 0.22 0.61 CNSTR 0.77 0.21 0.79 0.56 0.34 0.61 RUBBR 0.69 0.31 0.96 0.62 0.25 0.67 COMPS 0.47 0.16 0.88 0.51 0.33 0.15 SHIPS 0.79 0.35 0.75 0.54 0.11 0.63 DRUGS 0.33 0.13 0.94 0.46 0.24 0.53 STEEL 1.14 0.36 0.87 0.47 0.30 0.44 ELCEQ 0.81 0.30 0.85 0.18 0.31 0.58 TELCM 0.97 0.34 0.91 0.41 0.21 0.67 FABPR 0.86 0.34 0.81 0.39 0.25 0.65 TOYS 0.60 0.24 0.86 0.59 0.23 0.53 FIN 0.75 0.29 0.85 0.62 0.23 0.63 TRANS 0.98 0.37 0.93 0.62 0.30 0.69 FOOD 0.56 0.20 0.96 0.59 0.25 0.66 TXTLS 0.98 0.45 0.92 0.61 0.30 0.67 FUN 0.65 0.30 0.85 0.55 0.22 0.62 UTIL 1.11 0.22 0.92 0.54 0.31 0.59 GOLD 0.58 0.19 0.92 0.19 0.33 0.61 WHLSL 0.68 0.27 0.95 0.63 0.26 0.63 HLTH 0.59 0.35 0.82 0.40 0.20 0.52 AVG 0.71 0.29 0.88 0.51 0.24 0.59 HSHLD 0.40 0.18 0.97 0.46 0.21 0.60 AGG 0.68 0.19 0.98 0.60 0.10 0.69 Mean SD AR1 ρ¯BM ρ¯CF ρ¯ret Mean SD AR1 ρ¯BM ρ¯CF ρ¯ret AERO 0.71 0.44 0.96 0.55 0.14 0.61 INSUR 0.89 0.25 0.90 0.56 0.21 0.45 AGRIC 0.73 0.53 0.93 0.46 0.28 0.66 LABEQ 0.56 0.26 0.92 0.65 0.30 0.63 AUTOS 1.23 0.53 0.86 0.26 0.35 0.57 MACH 0.62 0.24 0.93 0.62 0.27 0.64 BANKS 0.89 0.31 0.91 0.53 0.28 0.63 MEALS 0.45 0.18 0.91 0.55 0.24 0.61 BEER 0.49 0.21 0.96 0.57 0.22 0.63 MEDEQ 0.42 0.19 0.95 0.59 0.15 0.61 BLDMT 0.70 0.24 0.84 0.59 0.31 0.52 MINES 1.22 0.61 0.06 0.00 0.21 0.63 BOOKS 0.60 0.31 0.82 0.27 0.13 0.52 OIL 0.89 0.22 0.82 0.40 0.20 0.65 BOXES 0.78 0.35 0.94 0.55 0.21 0.67 OTHER 0.62 0.27 0.94 0.09 0.25 0.51 BUSSV 0.36 0.12 0.95 0.46 0.15 0.57 PAPER 0.63 0.17 0.94 0.52 0.23 0.58 CHEM 0.63 0.23 0.93 0.55 0.18 0.65 PERSV 0.58 0.25 0.86 0.59 0.25 0.52 CHIPS 0.48 0.13 0.82 0.50 0.24 0.53 RLEST 0.96 0.43 0.86 0.52 0.14 0.63 CLTHS 0.69 0.45 0.94 0.59 0.18 0.49 RTAIL 0.50 0.22 0.96 0.61 0.22 0.61 CNSTR 0.77 0.21 0.79 0.56 0.34 0.61 RUBBR 0.69 0.31 0.96 0.62 0.25 0.67 COMPS 0.47 0.16 0.88 0.51 0.33 0.15 SHIPS 0.79 0.35 0.75 0.54 0.11 0.63 DRUGS 0.33 0.13 0.94 0.46 0.24 0.53 STEEL 1.14 0.36 0.87 0.47 0.30 0.44 ELCEQ 0.81 0.30 0.85 0.18 0.31 0.58 TELCM 0.97 0.34 0.91 0.41 0.21 0.67 FABPR 0.86 0.34 0.81 0.39 0.25 0.65 TOYS 0.60 0.24 0.86 0.59 0.23 0.53 FIN 0.75 0.29 0.85 0.62 0.23 0.63 TRANS 0.98 0.37 0.93 0.62 0.30 0.69 FOOD 0.56 0.20 0.96 0.59 0.25 0.66 TXTLS 0.98 0.45 0.92 0.61 0.30 0.67 FUN 0.65 0.30 0.85 0.55 0.22 0.62 UTIL 1.11 0.22 0.92 0.54 0.31 0.59 GOLD 0.58 0.19 0.92 0.19 0.33 0.61 WHLSL 0.68 0.27 0.95 0.63 0.26 0.63 HLTH 0.59 0.35 0.82 0.40 0.20 0.52 AVG 0.71 0.29 0.88 0.51 0.24 0.59 HSHLD 0.40 0.18 0.97 0.46 0.21 0.60 AGG 0.68 0.19 0.98 0.60 0.10 0.69 Table I. Summary statistics of industry BM ratios For each of the forty-four out of forty-eight Fama–French industries we use in the paper and the market portfolio, we present: time-series averages (Mean), standard deviations (SD), and first autocorrelations ( AR1) of the value-weighted industry BM ratios, as well as the average correlation between each industry’s BM, annual cash flow growth, and quarterly returns with those of all other industries ( ρ¯BM, ρ¯CF, and ρ¯ret, respectively). The ρ¯CF reports the average absolute value of the pairwise correlation as industry CFs can be negatively correlated. The sample is 1980:1–2015:4. Mean SD AR1 ρ¯BM ρ¯CF ρ¯ret Mean SD AR1 ρ¯BM ρ¯CF ρ¯ret AERO 0.71 0.44 0.96 0.55 0.14 0.61 INSUR 0.89 0.25 0.90 0.56 0.21 0.45 AGRIC 0.73 0.53 0.93 0.46 0.28 0.66 LABEQ 0.56 0.26 0.92 0.65 0.30 0.63 AUTOS 1.23 0.53 0.86 0.26 0.35 0.57 MACH 0.62 0.24 0.93 0.62 0.27 0.64 BANKS 0.89 0.31 0.91 0.53 0.28 0.63 MEALS 0.45 0.18 0.91 0.55 0.24 0.61 BEER 0.49 0.21 0.96 0.57 0.22 0.63 MEDEQ 0.42 0.19 0.95 0.59 0.15 0.61 BLDMT 0.70 0.24 0.84 0.59 0.31 0.52 MINES 1.22 0.61 0.06 0.00 0.21 0.63 BOOKS 0.60 0.31 0.82 0.27 0.13 0.52 OIL 0.89 0.22 0.82 0.40 0.20 0.65 BOXES 0.78 0.35 0.94 0.55 0.21 0.67 OTHER 0.62 0.27 0.94 0.09 0.25 0.51 BUSSV 0.36 0.12 0.95 0.46 0.15 0.57 PAPER 0.63 0.17 0.94 0.52 0.23 0.58 CHEM 0.63 0.23 0.93 0.55 0.18 0.65 PERSV 0.58 0.25 0.86 0.59 0.25 0.52 CHIPS 0.48 0.13 0.82 0.50 0.24 0.53 RLEST 0.96 0.43 0.86 0.52 0.14 0.63 CLTHS 0.69 0.45 0.94 0.59 0.18 0.49 RTAIL 0.50 0.22 0.96 0.61 0.22 0.61 CNSTR 0.77 0.21 0.79 0.56 0.34 0.61 RUBBR 0.69 0.31 0.96 0.62 0.25 0.67 COMPS 0.47 0.16 0.88 0.51 0.33 0.15 SHIPS 0.79 0.35 0.75 0.54 0.11 0.63 DRUGS 0.33 0.13 0.94 0.46 0.24 0.53 STEEL 1.14 0.36 0.87 0.47 0.30 0.44 ELCEQ 0.81 0.30 0.85 0.18 0.31 0.58 TELCM 0.97 0.34 0.91 0.41 0.21 0.67 FABPR 0.86 0.34 0.81 0.39 0.25 0.65 TOYS 0.60 0.24 0.86 0.59 0.23 0.53 FIN 0.75 0.29 0.85 0.62 0.23 0.63 TRANS 0.98 0.37 0.93 0.62 0.30 0.69 FOOD 0.56 0.20 0.96 0.59 0.25 0.66 TXTLS 0.98 0.45 0.92 0.61 0.30 0.67 FUN 0.65 0.30 0.85 0.55 0.22 0.62 UTIL 1.11 0.22 0.92 0.54 0.31 0.59 GOLD 0.58 0.19 0.92 0.19 0.33 0.61 WHLSL 0.68 0.27 0.95 0.63 0.26 0.63 HLTH 0.59 0.35 0.82 0.40 0.20 0.52 AVG 0.71 0.29 0.88 0.51 0.24 0.59 HSHLD 0.40 0.18 0.97 0.46 0.21 0.60 AGG 0.68 0.19 0.98 0.60 0.10 0.69 Mean SD AR1 ρ¯BM ρ¯CF ρ¯ret Mean SD AR1 ρ¯BM ρ¯CF ρ¯ret AERO 0.71 0.44 0.96 0.55 0.14 0.61 INSUR 0.89 0.25 0.90 0.56 0.21 0.45 AGRIC 0.73 0.53 0.93 0.46 0.28 0.66 LABEQ 0.56 0.26 0.92 0.65 0.30 0.63 AUTOS 1.23 0.53 0.86 0.26 0.35 0.57 MACH 0.62 0.24 0.93 0.62 0.27 0.64 BANKS 0.89 0.31 0.91 0.53 0.28 0.63 MEALS 0.45 0.18 0.91 0.55 0.24 0.61 BEER 0.49 0.21 0.96 0.57 0.22 0.63 MEDEQ 0.42 0.19 0.95 0.59 0.15 0.61 BLDMT 0.70 0.24 0.84 0.59 0.31 0.52 MINES 1.22 0.61 0.06 0.00 0.21 0.63 BOOKS 0.60 0.31 0.82 0.27 0.13 0.52 OIL 0.89 0.22 0.82 0.40 0.20 0.65 BOXES 0.78 0.35 0.94 0.55 0.21 0.67 OTHER 0.62 0.27 0.94 0.09 0.25 0.51 BUSSV 0.36 0.12 0.95 0.46 0.15 0.57 PAPER 0.63 0.17 0.94 0.52 0.23 0.58 CHEM 0.63 0.23 0.93 0.55 0.18 0.65 PERSV 0.58 0.25 0.86 0.59 0.25 0.52 CHIPS 0.48 0.13 0.82 0.50 0.24 0.53 RLEST 0.96 0.43 0.86 0.52 0.14 0.63 CLTHS 0.69 0.45 0.94 0.59 0.18 0.49 RTAIL 0.50 0.22 0.96 0.61 0.22 0.61 CNSTR 0.77 0.21 0.79 0.56 0.34 0.61 RUBBR 0.69 0.31 0.96 0.62 0.25 0.67 COMPS 0.47 0.16 0.88 0.51 0.33 0.15 SHIPS 0.79 0.35 0.75 0.54 0.11 0.63 DRUGS 0.33 0.13 0.94 0.46 0.24 0.53 STEEL 1.14 0.36 0.87 0.47 0.30 0.44 ELCEQ 0.81 0.30 0.85 0.18 0.31 0.58 TELCM 0.97 0.34 0.91 0.41 0.21 0.67 FABPR 0.86 0.34 0.81 0.39 0.25 0.65 TOYS 0.60 0.24 0.86 0.59 0.23 0.53 FIN 0.75 0.29 0.85 0.62 0.23 0.63 TRANS 0.98 0.37 0.93 0.62 0.30 0.69 FOOD 0.56 0.20 0.96 0.59 0.25 0.66 TXTLS 0.98 0.45 0.92 0.61 0.30 0.67 FUN 0.65 0.30 0.85 0.55 0.22 0.62 UTIL 1.11 0.22 0.92 0.54 0.31 0.59 GOLD 0.58 0.19 0.92 0.19 0.33 0.61 WHLSL 0.68 0.27 0.95 0.63 0.26 0.63 HLTH 0.59 0.35 0.82 0.40 0.20 0.52 AVG 0.71 0.29 0.88 0.51 0.24 0.59 HSHLD 0.40 0.18 0.97 0.46 0.21 0.60 AGG 0.68 0.19 0.98 0.60 0.10 0.69 The average pairwise correlation across the industry BM is less than 0.5. The average correlation between industry returns is 0.59 and the average correlation between industry CFs is 0.24. The substantial commonality in returns and imperfect correlation in CFs illustrates how BM ratios should contain useful signals for the expected returns on different industries as argued in Section 2. 3.2 Transaction Cost Data In this paper, we evaluate the performance of trading strategies net of transaction costs. We closely follow Novy-Marx and Velikov (2016) and compute strategy trading costs in two steps. First, we estimate effective one-way stock-level transaction costs (“spreads” or “bid–ask spreads”) using daily CRSP return data following Hasbrouck (2009).3 Second, we use these stock-level spread estimates to compute portfolio-level costs. Hasbrouck (2009) shows this is an accurate measure of transaction costs as it has a 96.5% correlation with effective spreads based directly on Trade and Quote (TAQ) data. The primary limitation of the Hasbrouck spread measure is that it does not consider the price impact of very large trades. However, it is an upper bound of trading costs for small trades because it assumes market orders. Overall, this measure captures the marginal cost of a strategy for small traders. Additionally, since we use value-weighted industries, our portfolios are dominated by large stocks that are less subject to large price impacts. Panel A of Figure 1 depicts the value-weighted averages of our estimated one-way transaction costs of NYSE, AMEX, and NASDAQ stocks by NYSE size quintile each quarter from 1980:1 to 2015:4. Overall the figure looks similar to analogous plots in Hasbrouck (2009). Size negatively varies with transaction costs and the smallest size quintile has several times the transaction costs of the largest quintile. Transaction costs are known to decrease with size for several reasons (e.g., Stoll and Whaley, 1983; Bhushan, 1989; Amihud, 2002). Small-cap stocks are relatively costly for market makers to hold because they trade infrequently and have relatively high risk. Moreover, analyst coverage of small-cap stocks is lower and the risk of adverse selection higher. Search costs of small-cap stocks are also relatively high because they have a lower dollar volume of actively traded shares. Panel B of Figure 1 shows the distribution of time-series means of the estimated effective transaction costs of the value-weighted Fama–French forty-four industry portfolios. The industry transaction costs range from 20 to 69 basis points and average 35 basis points. The relatively low cost estimates indicate that value-weighted industries are relatively inexpensive to trade because they are dominated by large-cap stocks. Figure 1. View largeDownload slide Relevant patterns in transaction costs. Panel A depicts the time-series of value-weighted averages of effective one-way transaction costs for NYSE, AMEX, and NASDAQ stocks by NYSE size quintile. Panel B shows the histogram of time-series means of the value-weighted averages of estimated one-way transaction costs of stocks in each of the Fama–French forty-four industry portfolios (in units of %). Panel C depicts the four-quarter moving average of the transaction costs associated with the Fama–French size, value, momentum, investment, and profitability factors (SMB, HML, MOM, CMA, and RMW). The sample for all Panels is 1980:1–2015:4. Figure 1. View largeDownload slide Relevant patterns in transaction costs. Panel A depicts the time-series of value-weighted averages of effective one-way transaction costs for NYSE, AMEX, and NASDAQ stocks by NYSE size quintile. Panel B shows the histogram of time-series means of the value-weighted averages of estimated one-way transaction costs of stocks in each of the Fama–French forty-four industry portfolios (in units of %). Panel C depicts the four-quarter moving average of the transaction costs associated with the Fama–French size, value, momentum, investment, and profitability factors (SMB, HML, MOM, CMA, and RMW). The sample for all Panels is 1980:1–2015:4. Panel A of Figure 1 also shows that transaction costs decline over time along with the difference in transaction costs between large and small stocks. Common explanations for this decline include improvements in trading technology, decimalization in 2001, and the relatively aggressive delisting of small and illiquid NASDAQ stocks in the 1990s (e.g., Chakravarty, Panchapagesan, and Wood, 2005; Hasbrouck, 2009; Chordia, Subrahmanyam, and Tong, 2014). Panel C of Figure 1 depicts the four-quarter moving average trading costs of the Fama–French size, value, momentum, investment, and profitability factors (SMB, HML, MOM, CMA, and RMW) over our out-of-sample period 1980:1–2015:4.4 All of the factors except MOM are rebalanced only at the end of June each year, so on average their transaction costs are relatively modest at about 25 basis points per quarter. In contrast, MOM has significant monthly turnover leading to transaction costs that are several times higher than those of the other factors. 3.3 Forecasting Methodology The goal of this paper is to forecast industry returns using the whole cross-section of BM ratios, and then exploit this predictability with trading strategies. Regression-based return forecasts often exhibit structural breaks that result in poor out-of-sample performance (e.g., Goyal and Welch, 2008; Rapach, Strauss, and Zhou, 2010). In contrast, combination forecast methods tend to perform well out-of-sample in the presence of model uncertainty and structural breaks when forecasting market returns and other economic time series (e.g., Stock and Watson, 2004; Timmermann, 2006; Rapach, Strauss, and Zhou, 2010). The basic building block for our combination forecasts are standard univariate predictive regressions estimated recursively in real time for each industry i and j: ri,t+1=aij+bijBMtj+ei,t+1j, (7) where ri,t+1 is the excess return for industry i and BMtj is the BM of industry j. A combination forecast for industry i is simply a weighted average of the out-of-sample forecasts from Equation (7): r^i,t+1c=∑j=1Nwj,tc(a^i,tj+b^j,tjBMtj), (8) where c denotes a weighting method, ( a^i,t, b^i,t) are from estimates of Equation (7) based on data through time t – 1, and N is the number of industries. In the main body of the paper, we form univariate forecasts by estimating Equation (7) at each time t using “expanding windows”, that is data from time 1 to time t – 1. In the Online Appendix, we show results for forecasts that use “rolling windows” of data. Different combination forecasts are defined by the choice of weighting schemes {wj,tc}. The different combination forecast weights can be simple functions such as an equal-weighted mean (MEAN, wj,tMEAN≡1/N), or functions of prior forecast performance that give low weight to forecasts that have large past errors, and vice versa. If the forecast errors of the individual forecasts have equal variance and equal pairwise correlation, the MEAN combination method is optimal in that it produces the combination forecast with the minimum mean-squared forecast error. Further, MEAN involves no estimation error and therefore often empirically outperforms estimates of theoretically “optimal” weights in finite samples (e.g., Timmermann, 2008). There is generally no ex ante optimal combination method for a given time series, it is an empirical question (e.g., Timmermann, 2008). We therefore compare several methods, the MEAN method and two performance-based combination forecasts. The first performance-based method, Approximate Bayesian Model Averaging (ABMA), follows Garratt et al. (2003) and chooses: wi,tABMA= exp ⁡(Δi,t)∑j=1n exp ⁡(Δj,t), (9) where Δi,t=AICi,t−max⁡j(AICj,t), and AICi,t is the Akaike Information Criterion of model i. The ABMA thus gives higher weight to models with better historical fit measured by AIC. Garratt et al. (2003) argue the AIC is preferred when the “correct” model is possibly not in the set under consideration. The second performance-based method, the discounted mean-squared forecast error (DMSFE) follows Bates and Granger (1969) and Stock and Watson (2004) and chooses: wi,tDMSFE=φi,t−1∑j=1nφi,t−1, (10) where φi,t=∑s=1t−1θt−1−s(ri,s+1−r^i,s+1)2. (11) If θ = 1, DMSFE does not discount forecast errors further in the past. Because our individual forecasts have the same number of parameters, this generates nearly identical results to the ABMA, because the likelihood functions are driven by non-discounted MSFE. Hence, we choose θ=0.7 to examine the impact of discounting forecast errors further back in time. By discounting past observations more heavily, DMSFE works relatively well if the data-generating process is time-varying. However, the cost of discounting is a lower effective sample size and therefore higher volatility of estimated weights, which reduces forecast accuracy all else equal. 3.4 Forecast Evaluation Following Campbell and Thompson (2008), we constrain all predicted excess returns to be non-negative as (i) the equity risk premium should be positive, and (ii) in practice, these restrictions improve the out-of-sample performance of predictive regressions.5 We use the standard out-of-sample R2 statistic, ROS2, to evaluate the out-of-sample accuracy of our forecasts relative to the historical-average forecast r¯i,t+1, the natural benchmark under the null of no predictability. The ROS2 statistic is defined by: ROS2≡1−∑t=q+1T(ri,t−r^i,tc)2∑t=q+1T(ri,t−r¯i,t)2, (12) where q denotes the end of an initial in-sample period used to generate the first out-of-sample forecast from Equation (7). We select q=1979:4, yielding thirty-two initial in-sample quarters. When ROS2>0, r^i,tc outperforms the historical average forecast by earning a lower mean-squared-forecast error. We assess the significance of the ROS2 with the Clark and West (2007) MSFE-adjusted statistic. 4. Forecasting Results 4.1 In-Sample Results We argue above that the cross-section of BM should be useful for forecasting returns and CFs of stock portfolios. In this section, we provide evidence of this for the forty-four industry portfolios. Specifically, we estimate predictive regressions of CFs and excess log returns (r) for each industry i over the following four and one quarters, respectively, ∑j=14CFi,t+j=αiCF+βiCFXt, (13) ri,t+1=αir+βirXt. (14) Using four quarters in Equation (13) mitigates the effects of earnings seasonality. We consider four different sets of predictors X: the log BM for industry i (BMi), the log BM for the aggregate market (BMM), the first six principal components of the forty-four industry log BM, and principal components 2–6. Principal components 2–6 exclude the largest common trend in BM ratios but contain information across industries.6 Using timely market values in BM when predicting returns could pick up a momentum effect. Hence, we also consider the return over the prior four quarters ( Xt=rt−3,t) as a predictor in Equation (14). Table II contains in-sample adjusted R2 from the models given by Equations (13) and (14). Table II. In-sample R2 statistics from predictive regressions of industry CFs and returns on BM ratios For each of the forty-four Fama–French industries, this table presents in-sample adjusted-R2 statistics for predictive regressions of industry CFs over quarters t + 1 to t + 4 and returns (Returns) over quarter t + 1. In columns denoted BMi, the only predictor is the industry’s own BM in quarter t. In columns denoted BMM, the predictor is the market BM. In columns denoted PC and PCI, the predictors are the first six and second–sixth principal components of the forty-four industry BM ratios, respectively. In the column denoted rt−3,t, the predictor is the return over four quarters prior to t + 1. Avg denotes the average of the R2 across industries, and #Sig denotes the number of R2 that are significant at the 5% level. The sample period is 1980:1–2015:4. CF Returns IND BMM PC PCI IND BMM PC PCI rt−3,t AVG 8.16 10.95 46.45 33.52 1.72 3.30 6.11 4.03 −0.07 AERO −0.06 6.65 25.78 25.19 1.31 2.71 5.49 3.95 1.54 AGRIC 2.74 −0.70 33.95 32.35 1.30 6.96 8.15 2.91 0.57 AUTOS 5.04 4.52 28.26 20.48 −0.14 2.39 1.41 0.92 0.60 BANKS 1.69 6.68 22.14 10.10 4.02 5.67 13.34 8.41 −0.48 BEER 9.99 2.14 36.95 37.06 3.24 11.09 14.65 5.81 −0.62 BLDMT 28.98 20.58 78.72 77.47 3.86 4.67 7.90 5.81 1.78 BOOKS 11.74 12.46 81.49 46.47 2.11 7.89 14.22 9.86 −0.67 BOXES 23.25 23.25 51.18 5.20 −0.27 1.21 1.35 1.27 −0.36 BUSSV 0.29 8.18 42.64 29.53 3.25 7.51 12.45 5.40 −0.35 CHEM 2.27 15.76 72.20 56.47 1.90 0.99 0.56 1.08 −0.25 CHIPS 6.41 19.53 70.27 32.67 4.18 4.96 11.36 8.81 −0.54 CLTHS 0.14 3.66 29.23 29.65 1.82 3.88 9.44 6.20 −0.68 CNSTR −0.66 −0.23 22.11 22.04 8.87 4.47 6.21 3.97 −0.65 COMPS 16.94 12.67 48.65 38.36 3.90 3.80 7.07 4.98 −0.10 DRUGS 20.44 4.95 36.23 36.67 0.62 7.62 16.73 8.96 −0.49 ELCEQ 2.30 17.00 35.19 13.09 −0.40 4.19 5.78 5.55 0.41 FABPR −0.71 5.84 26.53 23.10 −0.72 1.27 1.91 1.07 −0.47 FIN 43.47 29.80 52.62 22.06 2.79 2.89 5.94 5.39 0.93 FOOD 1.61 −0.49 57.45 52.27 −0.62 −0.09 5.45 5.42 −0.72 FUN 1.29 15.50 41.18 30.77 0.17 1.00 5.57 4.58 −0.10 GOLD −0.16 29.97 47.15 4.05 −0.70 −0.19 −1.32 −0.89 −0.72 HLTH 3.07 −0.63 32.89 29.84 0.50 0.66 4.56 3.58 0.22 HSHLD 24.83 2.51 57.94 53.71 −0.30 1.54 2.98 2.34 −0.49 INSUR 8.00 −0.50 67.70 66.87 1.60 1.64 0.86 0.26 2.32 LABEQ 1.02 −0.59 31.98 28.57 0.80 0.81 4.56 3.81 −0.43 MACH −0.30 1.53 51.39 46.11 6.34 7.09 7.67 4.11 −0.40 MEALS 0.25 18.45 54.28 47.16 0.78 3.97 6.87 3.75 −0.33 MEDEQ 16.91 4.67 69.89 68.38 −0.67 0.76 1.02 3.25 −0.69 MINES 0.78 2.44 20.55 12.47 1.08 5.89 10.43 1.17 −0.72 OIL −0.39 10.85 63.82 48.83 2.10 3.59 7.53 4.36 −0.72 OTHER 1.26 39.09 63.79 18.14 0.71 1.71 5.46 3.50 0.22 PAPER 13.14 19.59 44.25 33.15 −0.68 0.28 2.46 2.74 2.47 PERSV 27.12 40.02 55.75 4.14 1.61 5.70 4.67 1.45 1.72 RLEST 0.16 1.77 25.88 21.22 2.58 2.06 2.73 0.13 −0.72 RTAIL −0.50 3.34 53.34 50.16 −0.68 −0.39 2.68 3.41 0.19 RUBBR 4.70 5.10 39.65 38.97 −0.25 −0.59 −3.05 −1.35 −0.43 SHIPS −0.98 2.53 39.20 38.71 2.58 5.09 6.69 4.08 −0.70 STEEL 24.12 7.43 30.78 27.88 7.99 5.32 10.18 7.02 0.88 TELCM 0.22 41.94 73.94 31.23 0.41 4.55 8.78 7.53 −0.12 TOYS 3.81 4.03 39.01 29.46 1.90 4.36 4.81 1.55 −0.57 TRANS −0.65 0.20 27.88 28.35 2.30 4.10 7.93 5.76 0.92 TXTLS 15.13 −0.69 32.50 26.17 0.27 0.25 7.81 7.54 −0.56 UTIL 31.22 23.00 62.49 35.78 0.93 0.02 2.35 3.01 −0.24 WHLSL 9.12 17.87 65.11 44.60 3.48 1.77 5.19 5.04 1.21 #SIG 24 33 44 44 14 25 33 32 4 CF Returns IND BMM PC PCI IND BMM PC PCI rt−3,t AVG 8.16 10.95 46.45 33.52 1.72 3.30 6.11 4.03 −0.07 AERO −0.06 6.65 25.78 25.19 1.31 2.71 5.49 3.95 1.54 AGRIC 2.74 −0.70 33.95 32.35 1.30 6.96 8.15 2.91 0.57 AUTOS 5.04 4.52 28.26 20.48 −0.14 2.39 1.41 0.92 0.60 BANKS 1.69 6.68 22.14 10.10 4.02 5.67 13.34 8.41 −0.48 BEER 9.99 2.14 36.95 37.06 3.24 11.09 14.65 5.81 −0.62 BLDMT 28.98 20.58 78.72 77.47 3.86 4.67 7.90 5.81 1.78 BOOKS 11.74 12.46 81.49 46.47 2.11 7.89 14.22 9.86 −0.67 BOXES 23.25 23.25 51.18 5.20 −0.27 1.21 1.35 1.27 −0.36 BUSSV 0.29 8.18 42.64 29.53 3.25 7.51 12.45 5.40 −0.35 CHEM 2.27 15.76 72.20 56.47 1.90 0.99 0.56 1.08 −0.25 CHIPS 6.41 19.53 70.27 32.67 4.18 4.96 11.36 8.81 −0.54 CLTHS 0.14 3.66 29.23 29.65 1.82 3.88 9.44 6.20 −0.68 CNSTR −0.66 −0.23 22.11 22.04 8.87 4.47 6.21 3.97 −0.65 COMPS 16.94 12.67 48.65 38.36 3.90 3.80 7.07 4.98 −0.10 DRUGS 20.44 4.95 36.23 36.67 0.62 7.62 16.73 8.96 −0.49 ELCEQ 2.30 17.00 35.19 13.09 −0.40 4.19 5.78 5.55 0.41 FABPR −0.71 5.84 26.53 23.10 −0.72 1.27 1.91 1.07 −0.47 FIN 43.47 29.80 52.62 22.06 2.79 2.89 5.94 5.39 0.93 FOOD 1.61 −0.49 57.45 52.27 −0.62 −0.09 5.45 5.42 −0.72 FUN 1.29 15.50 41.18 30.77 0.17 1.00 5.57 4.58 −0.10 GOLD −0.16 29.97 47.15 4.05 −0.70 −0.19 −1.32 −0.89 −0.72 HLTH 3.07 −0.63 32.89 29.84 0.50 0.66 4.56 3.58 0.22 HSHLD 24.83 2.51 57.94 53.71 −0.30 1.54 2.98 2.34 −0.49 INSUR 8.00 −0.50 67.70 66.87 1.60 1.64 0.86 0.26 2.32 LABEQ 1.02 −0.59 31.98 28.57 0.80 0.81 4.56 3.81 −0.43 MACH −0.30 1.53 51.39 46.11 6.34 7.09 7.67 4.11 −0.40 MEALS 0.25 18.45 54.28 47.16 0.78 3.97 6.87 3.75 −0.33 MEDEQ 16.91 4.67 69.89 68.38 −0.67 0.76 1.02 3.25 −0.69 MINES 0.78 2.44 20.55 12.47 1.08 5.89 10.43 1.17 −0.72 OIL −0.39 10.85 63.82 48.83 2.10 3.59 7.53 4.36 −0.72 OTHER 1.26 39.09 63.79 18.14 0.71 1.71 5.46 3.50 0.22 PAPER 13.14 19.59 44.25 33.15 −0.68 0.28 2.46 2.74 2.47 PERSV 27.12 40.02 55.75 4.14 1.61 5.70 4.67 1.45 1.72 RLEST 0.16 1.77 25.88 21.22 2.58 2.06 2.73 0.13 −0.72 RTAIL −0.50 3.34 53.34 50.16 −0.68 −0.39 2.68 3.41 0.19 RUBBR 4.70 5.10 39.65 38.97 −0.25 −0.59 −3.05 −1.35 −0.43 SHIPS −0.98 2.53 39.20 38.71 2.58 5.09 6.69 4.08 −0.70 STEEL 24.12 7.43 30.78 27.88 7.99 5.32 10.18 7.02 0.88 TELCM 0.22 41.94 73.94 31.23 0.41 4.55 8.78 7.53 −0.12 TOYS 3.81 4.03 39.01 29.46 1.90 4.36 4.81 1.55 −0.57 TRANS −0.65 0.20 27.88 28.35 2.30 4.10 7.93 5.76 0.92 TXTLS 15.13 −0.69 32.50 26.17 0.27 0.25 7.81 7.54 −0.56 UTIL 31.22 23.00 62.49 35.78 0.93 0.02 2.35 3.01 −0.24 WHLSL 9.12 17.87 65.11 44.60 3.48 1.77 5.19 5.04 1.21 #SIG 24 33 44 44 14 25 33 32 4 Table II. In-sample R2 statistics from predictive regressions of industry CFs and returns on BM ratios For each of the forty-four Fama–French industries, this table presents in-sample adjusted-R2 statistics for predictive regressions of industry CFs over quarters t + 1 to t + 4 and returns (Returns) over quarter t + 1. In columns denoted BMi, the only predictor is the industry’s own BM in quarter t. In columns denoted BMM, the predictor is the market BM. In columns denoted PC and PCI, the predictors are the first six and second–sixth principal components of the forty-four industry BM ratios, respectively. In the column denoted rt−3,t, the predictor is the return over four quarters prior to t + 1. Avg denotes the average of the R2 across industries, and #Sig denotes the number of R2 that are significant at the 5% level. The sample period is 1980:1–2015:4. CF Returns IND BMM PC PCI IND BMM PC PCI rt−3,t AVG 8.16 10.95 46.45 33.52 1.72 3.30 6.11 4.03 −0.07 AERO −0.06 6.65 25.78 25.19 1.31 2.71 5.49 3.95 1.54 AGRIC 2.74 −0.70 33.95 32.35 1.30 6.96 8.15 2.91 0.57 AUTOS 5.04 4.52 28.26 20.48 −0.14 2.39 1.41 0.92 0.60 BANKS 1.69 6.68 22.14 10.10 4.02 5.67 13.34 8.41 −0.48 BEER 9.99 2.14 36.95 37.06 3.24 11.09 14.65 5.81 −0.62 BLDMT 28.98 20.58 78.72 77.47 3.86 4.67 7.90 5.81 1.78 BOOKS 11.74 12.46 81.49 46.47 2.11 7.89 14.22 9.86 −0.67 BOXES 23.25 23.25 51.18 5.20 −0.27 1.21 1.35 1.27 −0.36 BUSSV 0.29 8.18 42.64 29.53 3.25 7.51 12.45 5.40 −0.35 CHEM 2.27 15.76 72.20 56.47 1.90 0.99 0.56 1.08 −0.25 CHIPS 6.41 19.53 70.27 32.67 4.18 4.96 11.36 8.81 −0.54 CLTHS 0.14 3.66 29.23 29.65 1.82 3.88 9.44 6.20 −0.68 CNSTR −0.66 −0.23 22.11 22.04 8.87 4.47 6.21 3.97 −0.65 COMPS 16.94 12.67 48.65 38.36 3.90 3.80 7.07 4.98 −0.10 DRUGS 20.44 4.95 36.23 36.67 0.62 7.62 16.73 8.96 −0.49 ELCEQ 2.30 17.00 35.19 13.09 −0.40 4.19 5.78 5.55 0.41 FABPR −0.71 5.84 26.53 23.10 −0.72 1.27 1.91 1.07 −0.47 FIN 43.47 29.80 52.62 22.06 2.79 2.89 5.94 5.39 0.93 FOOD 1.61 −0.49 57.45 52.27 −0.62 −0.09 5.45 5.42 −0.72 FUN 1.29 15.50 41.18 30.77 0.17 1.00 5.57 4.58 −0.10 GOLD −0.16 29.97 47.15 4.05 −0.70 −0.19 −1.32 −0.89 −0.72 HLTH 3.07 −0.63 32.89 29.84 0.50 0.66 4.56 3.58 0.22 HSHLD 24.83 2.51 57.94 53.71 −0.30 1.54 2.98 2.34 −0.49 INSUR 8.00 −0.50 67.70 66.87 1.60 1.64 0.86 0.26 2.32 LABEQ 1.02 −0.59 31.98 28.57 0.80 0.81 4.56 3.81 −0.43 MACH −0.30 1.53 51.39 46.11 6.34 7.09 7.67 4.11 −0.40 MEALS 0.25 18.45 54.28 47.16 0.78 3.97 6.87 3.75 −0.33 MEDEQ 16.91 4.67 69.89 68.38 −0.67 0.76 1.02 3.25 −0.69 MINES 0.78 2.44 20.55 12.47 1.08 5.89 10.43 1.17 −0.72 OIL −0.39 10.85 63.82 48.83 2.10 3.59 7.53 4.36 −0.72 OTHER 1.26 39.09 63.79 18.14 0.71 1.71 5.46 3.50 0.22 PAPER 13.14 19.59 44.25 33.15 −0.68 0.28 2.46 2.74 2.47 PERSV 27.12 40.02 55.75 4.14 1.61 5.70 4.67 1.45 1.72 RLEST 0.16 1.77 25.88 21.22 2.58 2.06 2.73 0.13 −0.72 RTAIL −0.50 3.34 53.34 50.16 −0.68 −0.39 2.68 3.41 0.19 RUBBR 4.70 5.10 39.65 38.97 −0.25 −0.59 −3.05 −1.35 −0.43 SHIPS −0.98 2.53 39.20 38.71 2.58 5.09 6.69 4.08 −0.70 STEEL 24.12 7.43 30.78 27.88 7.99 5.32 10.18 7.02 0.88 TELCM 0.22 41.94 73.94 31.23 0.41 4.55 8.78 7.53 −0.12 TOYS 3.81 4.03 39.01 29.46 1.90 4.36 4.81 1.55 −0.57 TRANS −0.65 0.20 27.88 28.35 2.30 4.10 7.93 5.76 0.92 TXTLS 15.13 −0.69 32.50 26.17 0.27 0.25 7.81 7.54 −0.56 UTIL 31.22 23.00 62.49 35.78 0.93 0.02 2.35 3.01 −0.24 WHLSL 9.12 17.87 65.11 44.60 3.48 1.77 5.19 5.04 1.21 #SIG 24 33 44 44 14 25 33 32 4 CF Returns IND BMM PC PCI IND BMM PC PCI rt−3,t AVG 8.16 10.95 46.45 33.52 1.72 3.30 6.11 4.03 −0.07 AERO −0.06 6.65 25.78 25.19 1.31 2.71 5.49 3.95 1.54 AGRIC 2.74 −0.70 33.95 32.35 1.30 6.96 8.15 2.91 0.57 AUTOS 5.04 4.52 28.26 20.48 −0.14 2.39 1.41 0.92 0.60 BANKS 1.69 6.68 22.14 10.10 4.02 5.67 13.34 8.41 −0.48 BEER 9.99 2.14 36.95 37.06 3.24 11.09 14.65 5.81 −0.62 BLDMT 28.98 20.58 78.72 77.47 3.86 4.67 7.90 5.81 1.78 BOOKS 11.74 12.46 81.49 46.47 2.11 7.89 14.22 9.86 −0.67 BOXES 23.25 23.25 51.18 5.20 −0.27 1.21 1.35 1.27 −0.36 BUSSV 0.29 8.18 42.64 29.53 3.25 7.51 12.45 5.40 −0.35 CHEM 2.27 15.76 72.20 56.47 1.90 0.99 0.56 1.08 −0.25 CHIPS 6.41 19.53 70.27 32.67 4.18 4.96 11.36 8.81 −0.54 CLTHS 0.14 3.66 29.23 29.65 1.82 3.88 9.44 6.20 −0.68 CNSTR −0.66 −0.23 22.11 22.04 8.87 4.47 6.21 3.97 −0.65 COMPS 16.94 12.67 48.65 38.36 3.90 3.80 7.07 4.98 −0.10 DRUGS 20.44 4.95 36.23 36.67 0.62 7.62 16.73 8.96 −0.49 ELCEQ 2.30 17.00 35.19 13.09 −0.40 4.19 5.78 5.55 0.41 FABPR −0.71 5.84 26.53 23.10 −0.72 1.27 1.91 1.07 −0.47 FIN 43.47 29.80 52.62 22.06 2.79 2.89 5.94 5.39 0.93 FOOD 1.61 −0.49 57.45 52.27 −0.62 −0.09 5.45 5.42 −0.72 FUN 1.29 15.50 41.18 30.77 0.17 1.00 5.57 4.58 −0.10 GOLD −0.16 29.97 47.15 4.05 −0.70 −0.19 −1.32 −0.89 −0.72 HLTH 3.07 −0.63 32.89 29.84 0.50 0.66 4.56 3.58 0.22 HSHLD 24.83 2.51 57.94 53.71 −0.30 1.54 2.98 2.34 −0.49 INSUR 8.00 −0.50 67.70 66.87 1.60 1.64 0.86 0.26 2.32 LABEQ 1.02 −0.59 31.98 28.57 0.80 0.81 4.56 3.81 −0.43 MACH −0.30 1.53 51.39 46.11 6.34 7.09 7.67 4.11 −0.40 MEALS 0.25 18.45 54.28 47.16 0.78 3.97 6.87 3.75 −0.33 MEDEQ 16.91 4.67 69.89 68.38 −0.67 0.76 1.02 3.25 −0.69 MINES 0.78 2.44 20.55 12.47 1.08 5.89 10.43 1.17 −0.72 OIL −0.39 10.85 63.82 48.83 2.10 3.59 7.53 4.36 −0.72 OTHER 1.26 39.09 63.79 18.14 0.71 1.71 5.46 3.50 0.22 PAPER 13.14 19.59 44.25 33.15 −0.68 0.28 2.46 2.74 2.47 PERSV 27.12 40.02 55.75 4.14 1.61 5.70 4.67 1.45 1.72 RLEST 0.16 1.77 25.88 21.22 2.58 2.06 2.73 0.13 −0.72 RTAIL −0.50 3.34 53.34 50.16 −0.68 −0.39 2.68 3.41 0.19 RUBBR 4.70 5.10 39.65 38.97 −0.25 −0.59 −3.05 −1.35 −0.43 SHIPS −0.98 2.53 39.20 38.71 2.58 5.09 6.69 4.08 −0.70 STEEL 24.12 7.43 30.78 27.88 7.99 5.32 10.18 7.02 0.88 TELCM 0.22 41.94 73.94 31.23 0.41 4.55 8.78 7.53 −0.12 TOYS 3.81 4.03 39.01 29.46 1.90 4.36 4.81 1.55 −0.57 TRANS −0.65 0.20 27.88 28.35 2.30 4.10 7.93 5.76 0.92 TXTLS 15.13 −0.69 32.50 26.17 0.27 0.25 7.81 7.54 −0.56 UTIL 31.22 23.00 62.49 35.78 0.93 0.02 2.35 3.01 −0.24 WHLSL 9.12 17.87 65.11 44.60 3.48 1.77 5.19 5.04 1.21 #SIG 24 33 44 44 14 25 33 32 4 On average, forecasts based only on BMi and BMM have smaller R2 than the forecasts based on the principal components. For both CFs and returns, the average R2 for the forecasts based on principal components 2–6 is about two-thirds of that based on all six principal components and greater than that based on either BMi or BMM. Moreover, principal components 2–6 predict the CFs and returns of at least thirty-two out of forty-four industries individually with statistically significant R2. In contrast to predictability by BM, the prior return has an average R2 of approximately zero in predicting returns. In sum, the evidence in Table II indicates that the cross-section of BM contains relevant information for predicting returns and CFs of individual industries. We report and discuss analogous in-sample results for the characteristic portfolios in the Online Appendix for brevity. Inferences are similar. 4.2 Out-of-Sample Results In-sample return predictability in regressions such as Equation (14) can break down in real time because of structural breaks or be overstated due to persistent-predictor biases (e.g., Stambaugh, 1999; Ferson, Sarkissian, and Simin, 2003; Goyal and Welch, 2008). Table III investigates the out-of-sample performance of combination forecasts based on the cross-section of BM. Panels A and B present ROS2 for the returns on the forty-four industry portfolios and twenty-five size/BM portfolios, respectively. Table III. Out-of-sample forecasting performance of combination forecasts of excess returns on industry and size/BM portfolios This table presents ROS2 from predictive regressions of one-quarter ahead excess returns using each of the combination forecast methods described in the paper (ABMA, DMSFE, and MEAN) as well as our market BM (BMM) and each portfolio’s own BM ratio (BMi). Panels A and B, respectively, present results for the forty-four industry and twenty-five size/BM portfolios. The out-of-sample period is 1980:1–2015:4. *, **, and *** denote significance based on the Clark and West (2007) MSFE-adjusted statistic at the 10%, 5%, and 1% levels, respectively. Panel A: Forty-four industries ABMA DMSFE MEAN BMM BMi AVG 2.10 1.76 1.59 −1.87 −3.27 AERO 3.72*** 2.92** 2.86** −2.59 −1.58 AGRIC 0.40 0.16 0.04 −2.07 −2.65 AUTOS 2.35** 1.91* 1.78* 0.03 −1.86 BANKS 1.31 1.05 1.03 −1.03 −4.50 BEER 3.87*** 3.39*** 3.32*** −2.90 2.55** BLDMT 2.20** 2.09* 1.42* 0.01 −6.03 BOOKS 4.44*** 4.02*** 4.00*** 1.82** −6.05 BOXES 1.28 0.71 0.67 −0.91 −3.39 BUSSV 1.73* 0.91 0.65 −5.98 −3.24 CHEM 0.74 0.23 0.20 0.06 −3.19 CHIPS 0.61 0.32 0.19 −4.02 0.30 CLTHS 4.66*** 3.51** 3.41*** −2.09 −4.40 CNSTR 2.08* 1.98* 1.52* −0.12 0.83 COMPS 0.62 0.77 0.69 −1.05 −7.55 DRUGS 1.24 1.24 1.09 −2.22 −4.78 ELCEQ 1.17 0.44 0.36 −6.15 −1.50 FABPR 0.37 1.01 0.87 0.02 −2.63 FIN 0.57 0.15 0.08 −3.07 −3.63 FOOD 1.75* 1.45 1.47* −0.02 −3.95 FUN 0.12 −0.30 −0.36 −4.28 0.10 GOLD 0.85 0.97 0.73 0.29 −5.12 HLTH 6.87*** 6.14** 5.76*** −2.11 −4.78 HSHLD 2.95** 2.36** 2.34*** −5.05 −5.79 INSUR 1.31 0.97 0.79 −4.10 −5.54 LABEQ 1.64 1.29 1.21 −1.60 −0.52 MACH −0.07 −0.23 −0.22 −2.02 2.63** MEALS 4.05*** 2.56** 2.52** −5.81 −6.58 MEDEQ 1.39 0.78 0.74 −2.87 −4.92 MINES 0.25 0.18 0.21 0.90 −3.51 OIL −0.99 −1.25 −1.45 −1.15 −3.04 OTHER 3.97*** 3.55*** 3.12** −5.19 −4.29 PAPER 1.19 0.79 0.90 3.30*** −4.40 PERSV 5.11*** 4.35*** 4.30** −1.55 −5.90 RLEST 4.25*** 4.79*** 4.40** 1.47 2.79** RTAIL 3.55* 2.53** 2.66** −3.32 −6.11 RUBBR 2.88** 2.88** 2.71** −0.88 −12.92 SHIPS 1.40* 1.43* 1.23 0.91 −1.17 STEEL −0.04 −0.08 −0.09 −0.66 −0.94 TELCM 0.09 0.40 0.40 −0.66 −2.69 TOYS 3.97*** 3.34** 3.19** 3.82*** −4.82 TRANS 2.89** 2.34* 1.98** −1.35 −6.65 TXTLS 2.96** 2.46** 2.47** 3.42*** −4.12 UTIL 1.06 0.70 0.66 1.65 4.23*** WHLSL 5.51*** 4.41*** 4.19** −3.12 −2.64 Panel A: Forty-four industries ABMA DMSFE MEAN BMM BMi AVG 2.10 1.76 1.59 −1.87 −3.27 AERO 3.72*** 2.92** 2.86** −2.59 −1.58 AGRIC 0.40 0.16 0.04 −2.07 −2.65 AUTOS 2.35** 1.91* 1.78* 0.03 −1.86 BANKS 1.31 1.05 1.03 −1.03 −4.50 BEER 3.87*** 3.39*** 3.32*** −2.90 2.55** BLDMT 2.20** 2.09* 1.42* 0.01 −6.03 BOOKS 4.44*** 4.02*** 4.00*** 1.82** −6.05 BOXES 1.28 0.71 0.67 −0.91 −3.39 BUSSV 1.73* 0.91 0.65 −5.98 −3.24 CHEM 0.74 0.23 0.20 0.06 −3.19 CHIPS 0.61 0.32 0.19 −4.02 0.30 CLTHS 4.66*** 3.51** 3.41*** −2.09 −4.40 CNSTR 2.08* 1.98* 1.52* −0.12 0.83 COMPS 0.62 0.77 0.69 −1.05 −7.55 DRUGS 1.24 1.24 1.09 −2.22 −4.78 ELCEQ 1.17 0.44 0.36 −6.15 −1.50 FABPR 0.37 1.01 0.87 0.02 −2.63 FIN 0.57 0.15 0.08 −3.07 −3.63 FOOD 1.75* 1.45 1.47* −0.02 −3.95 FUN 0.12 −0.30 −0.36 −4.28 0.10 GOLD 0.85 0.97 0.73 0.29 −5.12 HLTH 6.87*** 6.14** 5.76*** −2.11 −4.78 HSHLD 2.95** 2.36** 2.34*** −5.05 −5.79 INSUR 1.31 0.97 0.79 −4.10 −5.54 LABEQ 1.64 1.29 1.21 −1.60 −0.52 MACH −0.07 −0.23 −0.22 −2.02 2.63** MEALS 4.05*** 2.56** 2.52** −5.81 −6.58 MEDEQ 1.39 0.78 0.74 −2.87 −4.92 MINES 0.25 0.18 0.21 0.90 −3.51 OIL −0.99 −1.25 −1.45 −1.15 −3.04 OTHER 3.97*** 3.55*** 3.12** −5.19 −4.29 PAPER 1.19 0.79 0.90 3.30*** −4.40 PERSV 5.11*** 4.35*** 4.30** −1.55 −5.90 RLEST 4.25*** 4.79*** 4.40** 1.47 2.79** RTAIL 3.55* 2.53** 2.66** −3.32 −6.11 RUBBR 2.88** 2.88** 2.71** −0.88 −12.92 SHIPS 1.40* 1.43* 1.23 0.91 −1.17 STEEL −0.04 −0.08 −0.09 −0.66 −0.94 TELCM 0.09 0.40 0.40 −0.66 −2.69 TOYS 3.97*** 3.34** 3.19** 3.82*** −4.82 TRANS 2.89** 2.34* 1.98** −1.35 −6.65 TXTLS 2.96** 2.46** 2.47** 3.42*** −4.12 UTIL 1.06 0.70 0.66 1.65 4.23*** WHLSL 5.51*** 4.41*** 4.19** −3.12 −2.64 Panel B: Twenty-five size and BM portfolios Size BM ABMA DMSFE MEAN BMM BMi Avg 2.32 1.81 0.85 −2.06 −0.55 Small Low 2.31** 1.70 2.06** −0.32 0.09 2 1.77* 3.50*** 0.82 −2.68 −1.53 3 4.42*** 3.45*** 2.81** −2.49 0.91 4 3.80*** 5.27*** 2.82** −3.49 −0.02 High 5.72*** 2.98** 3.22*** −2.12 −0.27 2 Low 3.11*** 1.04 1.38 0.28 1.24 2 1.08 1.47 0.10 −0.54 0.90 3 1.63 1.13 0.48 −2.95 −1.30 4 0.93 0.90 −0.17 −3.82 −3.04 High 3.30*** 0.37 1.36 −2.48 −2.16 3 Low 1.84* 0.33 0.36 0.35 −1.04 2 2.70** 2.05** 0.65 −0.95 2.16 3 1.83* 0.22 0.60 −2.56 0.41 4 3.76*** 0.12 1.43* −2.79 1.30 High 3.25** 0.57 1.67* −2.68 −3.37 4 Low 2.52** 0.79 0.68 0.32 −0.82 2 1.50 1.49 −0.04 −2.12 1.55 3 2.11** 2.73** 0.05 −2.75 −1.24 4 2.28** 1.40 0.06 −5.49 −0.36 High 3.34** 1.58 1.87 −1.68 −3.07 Big Low 2.23* 1.48 2.85** 1.46 0.33 2 −0.11 2.66** −1.22 −2.59 −0.93 3 1.01 2.69** 0.01 −3.03 −0.18 4 0.93 0.53 −0.57 −2.31 −1.23 High 0.83 4.80*** −2.07 −3.99 −2.20 Panel B: Twenty-five size and BM portfolios Size BM ABMA DMSFE MEAN BMM BMi Avg 2.32 1.81 0.85 −2.06 −0.55 Small Low 2.31** 1.70 2.06** −0.32 0.09 2 1.77* 3.50*** 0.82 −2.68 −1.53 3 4.42*** 3.45*** 2.81** −2.49 0.91 4 3.80*** 5.27*** 2.82** −3.49 −0.02 High 5.72*** 2.98** 3.22*** −2.12 −0.27 2 Low 3.11*** 1.04 1.38 0.28 1.24 2 1.08 1.47 0.10 −0.54 0.90 3 1.63 1.13 0.48 −2.95 −1.30 4 0.93 0.90 −0.17 −3.82 −3.04 High 3.30*** 0.37 1.36 −2.48 −2.16 3 Low 1.84* 0.33 0.36 0.35 −1.04 2 2.70** 2.05** 0.65 −0.95 2.16 3 1.83* 0.22 0.60 −2.56 0.41 4 3.76*** 0.12 1.43* −2.79 1.30 High 3.25** 0.57 1.67* −2.68 −3.37 4 Low 2.52** 0.79 0.68 0.32 −0.82 2 1.50 1.49 −0.04 −2.12 1.55 3 2.11** 2.73** 0.05 −2.75 −1.24 4 2.28** 1.40 0.06 −5.49 −0.36 High 3.34** 1.58 1.87 −1.68 −3.07 Big Low 2.23* 1.48 2.85** 1.46 0.33 2 −0.11 2.66** −1.22 −2.59 −0.93 3 1.01 2.69** 0.01 −3.03 −0.18 4 0.93 0.53 −0.57 −2.31 −1.23 High 0.83 4.80*** −2.07 −3.99 −2.20 Table III. Out-of-sample forecasting performance of combination forecasts of excess returns on industry and size/BM portfolios This table presents ROS2 from predictive regressions of one-quarter ahead excess returns using each of the combination forecast methods described in the paper (ABMA, DMSFE, and MEAN) as well as our market BM (BMM) and each portfolio’s own BM ratio (BMi). Panels A and B, respectively, present results for the forty-four industry and twenty-five size/BM portfolios. The out-of-sample period is 1980:1–2015:4. *, **, and *** denote significance based on the Clark and West (2007) MSFE-adjusted statistic at the 10%, 5%, and 1% levels, respectively. Panel A: Forty-four industries ABMA DMSFE MEAN BMM BMi AVG 2.10 1.76 1.59 −1.87 −3.27 AERO 3.72*** 2.92** 2.86** −2.59 −1.58 AGRIC 0.40 0.16 0.04 −2.07 −2.65 AUTOS 2.35** 1.91* 1.78* 0.03 −1.86 BANKS 1.31 1.05 1.03 −1.03 −4.50 BEER 3.87*** 3.39*** 3.32*** −2.90 2.55** BLDMT 2.20** 2.09* 1.42* 0.01 −6.03 BOOKS 4.44*** 4.02*** 4.00*** 1.82** −6.05 BOXES 1.28 0.71 0.67 −0.91 −3.39 BUSSV 1.73* 0.91 0.65 −5.98 −3.24 CHEM 0.74 0.23 0.20 0.06 −3.19 CHIPS 0.61 0.32 0.19 −4.02 0.30 CLTHS 4.66*** 3.51** 3.41*** −2.09 −4.40 CNSTR 2.08* 1.98* 1.52* −0.12 0.83 COMPS 0.62 0.77 0.69 −1.05 −7.55 DRUGS 1.24 1.24 1.09 −2.22 −4.78 ELCEQ 1.17 0.44 0.36 −6.15 −1.50 FABPR 0.37 1.01 0.87 0.02 −2.63 FIN 0.57 0.15 0.08 −3.07 −3.63 FOOD 1.75* 1.45 1.47* −0.02 −3.95 FUN 0.12 −0.30 −0.36 −4.28 0.10 GOLD 0.85 0.97 0.73 0.29 −5.12 HLTH 6.87*** 6.14** 5.76*** −2.11 −4.78 HSHLD 2.95** 2.36** 2.34*** −5.05 −5.79 INSUR 1.31 0.97 0.79 −4.10 −5.54 LABEQ 1.64 1.29 1.21 −1.60 −0.52 MACH −0.07 −0.23 −0.22 −2.02 2.63** MEALS 4.05*** 2.56** 2.52** −5.81 −6.58 MEDEQ 1.39 0.78 0.74 −2.87 −4.92 MINES 0.25 0.18 0.21 0.90 −3.51 OIL −0.99 −1.25 −1.45 −1.15 −3.04 OTHER 3.97*** 3.55*** 3.12** −5.19 −4.29 PAPER 1.19 0.79 0.90 3.30*** −4.40 PERSV 5.11*** 4.35*** 4.30** −1.55 −5.90 RLEST 4.25*** 4.79*** 4.40** 1.47 2.79** RTAIL 3.55* 2.53** 2.66** −3.32 −6.11 RUBBR 2.88** 2.88** 2.71** −0.88 −12.92 SHIPS 1.40* 1.43* 1.23 0.91 −1.17 STEEL −0.04 −0.08 −0.09 −0.66 −0.94 TELCM 0.09 0.40 0.40 −0.66 −2.69 TOYS 3.97*** 3.34** 3.19** 3.82*** −4.82 TRANS 2.89** 2.34* 1.98** −1.35 −6.65 TXTLS 2.96** 2.46** 2.47** 3.42*** −4.12 UTIL 1.06 0.70 0.66 1.65 4.23*** WHLSL 5.51*** 4.41*** 4.19** −3.12 −2.64 Panel A: Forty-four industries ABMA DMSFE MEAN BMM BMi AVG 2.10 1.76 1.59 −1.87 −3.27 AERO 3.72*** 2.92** 2.86** −2.59 −1.58 AGRIC 0.40 0.16 0.04 −2.07 −2.65 AUTOS 2.35** 1.91* 1.78* 0.03 −1.86 BANKS 1.31 1.05 1.03 −1.03 −4.50 BEER 3.87*** 3.39*** 3.32*** −2.90 2.55** BLDMT 2.20** 2.09* 1.42* 0.01 −6.03 BOOKS 4.44*** 4.02*** 4.00*** 1.82** −6.05 BOXES 1.28 0.71 0.67 −0.91 −3.39 BUSSV 1.73* 0.91 0.65 −5.98 −3.24 CHEM 0.74 0.23 0.20 0.06 −3.19 CHIPS 0.61 0.32 0.19 −4.02 0.30 CLTHS 4.66*** 3.51** 3.41*** −2.09 −4.40 CNSTR 2.08* 1.98* 1.52* −0.12 0.83 COMPS 0.62 0.77 0.69 −1.05 −7.55 DRUGS 1.24 1.24 1.09 −2.22 −4.78 ELCEQ 1.17 0.44 0.36 −6.15 −1.50 FABPR 0.37 1.01 0.87 0.02 −2.63 FIN 0.57 0.15 0.08 −3.07 −3.63 FOOD 1.75* 1.45 1.47* −0.02 −3.95 FUN 0.12 −0.30 −0.36 −4.28 0.10 GOLD 0.85 0.97 0.73 0.29 −5.12 HLTH 6.87*** 6.14** 5.76*** −2.11 −4.78 HSHLD 2.95** 2.36** 2.34*** −5.05 −5.79 INSUR 1.31 0.97 0.79 −4.10 −5.54 LABEQ 1.64 1.29 1.21 −1.60 −0.52 MACH −0.07 −0.23 −0.22 −2.02 2.63** MEALS 4.05*** 2.56** 2.52** −5.81 −6.58 MEDEQ 1.39 0.78 0.74 −2.87 −4.92 MINES 0.25 0.18 0.21 0.90 −3.51 OIL −0.99 −1.25 −1.45 −1.15 −3.04 OTHER 3.97*** 3.55*** 3.12** −5.19 −4.29 PAPER 1.19 0.79 0.90 3.30*** −4.40 PERSV 5.11*** 4.35*** 4.30** −1.55 −5.90 RLEST 4.25*** 4.79*** 4.40** 1.47 2.79** RTAIL 3.55* 2.53** 2.66** −3.32 −6.11 RUBBR 2.88** 2.88** 2.71** −0.88 −12.92 SHIPS 1.40* 1.43* 1.23 0.91 −1.17 STEEL −0.04 −0.08 −0.09 −0.66 −0.94 TELCM 0.09 0.40 0.40 −0.66 −2.69 TOYS 3.97*** 3.34** 3.19** 3.82*** −4.82 TRANS 2.89** 2.34* 1.98** −1.35 −6.65 TXTLS 2.96** 2.46** 2.47** 3.42*** −4.12 UTIL 1.06 0.70 0.66 1.65 4.23*** WHLSL 5.51*** 4.41*** 4.19** −3.12 −2.64 Panel B: Twenty-five size and BM portfolios Size BM ABMA DMSFE MEAN BMM BMi Avg 2.32 1.81 0.85 −2.06 −0.55 Small Low 2.31** 1.70 2.06** −0.32 0.09 2 1.77* 3.50*** 0.82 −2.68 −1.53 3 4.42*** 3.45*** 2.81** −2.49 0.91 4 3.80*** 5.27*** 2.82** −3.49 −0.02 High 5.72*** 2.98** 3.22*** −2.12 −0.27 2 Low 3.11*** 1.04 1.38 0.28 1.24 2 1.08 1.47 0.10 −0.54 0.90 3 1.63 1.13 0.48 −2.95 −1.30 4 0.93 0.90 −0.17 −3.82 −3.04 High 3.30*** 0.37 1.36 −2.48 −2.16 3 Low 1.84* 0.33 0.36 0.35 −1.04 2 2.70** 2.05** 0.65 −0.95 2.16 3 1.83* 0.22 0.60 −2.56 0.41 4 3.76*** 0.12 1.43* −2.79 1.30 High 3.25** 0.57 1.67* −2.68 −3.37 4 Low 2.52** 0.79 0.68 0.32 −0.82 2 1.50 1.49 −0.04 −2.12 1.55 3 2.11** 2.73** 0.05 −2.75 −1.24 4 2.28** 1.40 0.06 −5.49 −0.36 High 3.34** 1.58 1.87 −1.68 −3.07 Big Low 2.23* 1.48 2.85** 1.46 0.33 2 −0.11 2.66** −1.22 −2.59 −0.93 3 1.01 2.69** 0.01 −3.03 −0.18 4 0.93 0.53 −0.57 −2.31 −1.23 High 0.83 4.80*** −2.07 −3.99 −2.20 Panel B: Twenty-five size and BM portfolios Size BM ABMA DMSFE MEAN BMM BMi Avg 2.32 1.81 0.85 −2.06 −0.55 Small Low 2.31** 1.70 2.06** −0.32 0.09 2 1.77* 3.50*** 0.82 −2.68 −1.53 3 4.42*** 3.45*** 2.81** −2.49 0.91 4 3.80*** 5.27*** 2.82** −3.49 −0.02 High 5.72*** 2.98** 3.22*** −2.12 −0.27 2 Low 3.11*** 1.04 1.38 0.28 1.24 2 1.08 1.47 0.10 −0.54 0.90 3 1.63 1.13 0.48 −2.95 −1.30 4 0.93 0.90 −0.17 −3.82 −3.04 High 3.30*** 0.37 1.36 −2.48 −2.16 3 Low 1.84* 0.33 0.36 0.35 −1.04 2 2.70** 2.05** 0.65 −0.95 2.16 3 1.83* 0.22 0.60 −2.56 0.41 4 3.76*** 0.12 1.43* −2.79 1.30 High 3.25** 0.57 1.67* −2.68 −3.37 4 Low 2.52** 0.79 0.68 0.32 −0.82 2 1.50 1.49 −0.04 −2.12 1.55 3 2.11** 2.73** 0.05 −2.75 −1.24 4 2.28** 1.40 0.06 −5.49 −0.36 High 3.34** 1.58 1.87 −1.68 −3.07 Big Low 2.23* 1.48 2.85** 1.46 0.33 2 −0.11 2.66** −1.22 −2.59 −0.93 3 1.01 2.69** 0.01 −3.03 −0.18 4 0.93 0.53 −0.57 −2.31 −1.23 High 0.83 4.80*** −2.07 −3.99 −2.20 Panel A shows that all three combination forecast methods generate statistically significant predictability for many industries. On average, the combination forecasts have ROS2 of 3.5–4.0% and 4.9–5.4% higher than those based on BMM and BMi, respectively. Depending on combination method, twenty to twenty-two of the individual ROS2 are statistically significant at the 10% significance level. In contrast, the ROS2 of the forecasts based on BMM and BMi are negative on average and all but four are insignificant or negative. Panel B shows that on average combination forecast methods also predict returns on the twenty-five size/BM portfolios better than BMM or BMi (by 2.9–4.4% and 1.4–2.9%, respectively). However, consistency of performance across combination methods is less than with the industry portfolios. Seventeen out of twenty-five of the ABMA forecasts have significant ROS2, compared with only nine and seven of the DMSFE and MEAN forecasts, respectively. All three methods predict returns in the smallest size quintile with generally significant ROS2, but there is no other clear pattern between size, BM, and ROS2. The ROS2 of the forecasts based on BMM or BMi are negative on average, and none are significantly positive. For both sets of portfolios, the average ROS2 decline from ABMA to DMSFE to MEAN, although the differences across methods are smaller for the industry portfolios. As Section 3 discusses, this finding has implications for the data-generating processes of the two sets of portfolio returns. First, the stronger performance of the ABMA and DMSFE relative to the MEAN method indicates that assigning higher time-varying weights to individual forecasts with lower MSFE produces a more accurate combination forecast than equal-weighting in spite of higher estimation error. Second, the weaker performance of DMSFE relative to ABMA shows that discounting past observations more heavily, which cuts the effective sample size, is not optimal. The benefit of the lower volatility in the estimated ABMA weights exceeds the cost of DMSFE method more quickly identifying time-variation in the data-generating process. Overall, the evidence from Table III indicates that combination forecast methods extract relevant information in the cross-section of BM ratios for predicting the returns of many industry and size/BM portfolios with statistically significant ROS2. Results in the Online Appendix show that similar inferences hold when using rolling estimation windows or size/investment and size/profitability portfolios. In the next section, we examine the joint and economic significance of the combination forecasts. 5. Trading Strategies Based on Return Forecasts We assess the performance of real-time portfolio-rotation strategies based on out-of-sample predicted returns from the combination forecasts. This exercise measures the economic significance of the combination forecasts to investors (e.g., Pesaran and Timmermann, 1995; Cochrane, 2008). 5.1 Industry-Rotation Portfolios Each quarter from 1980:1 to 2015:4, we rank industry portfolios based on their predicted returns. Our industry-rotation strategies (hereafter “combo strategies”) take an equal-weighted long position in the top four (roughly one decile) of the forty-four industry portfolios, and a short position in the bottom four.7 We compare these strategies to those based on univariate predictive regressions using BMi: ri,t+1=ai+bi·BMi,t+ϵt+1. (15) We evaluate the performance of strategies using the Fama–French–Carhart four-factor model. In the presence of transaction costs, however, four-factor intercepts do not in general represent obtainable returns. Hence, we measure abnormal net-of-costs returns with the generalized αnet of Novy-Marx and Velikov (2016). The αnet has the same units as the intercept-based α but properly accounts for transaction costs in measuring how access to a given asset expands the investment opportunity set relative to the Fama–French–Carhart factors (see Online Appendix for details). Table IV presents average returns, Fama–French–Carhart four-factor αs and factor loadings of the returns on our strategies ignoring transaction costs. Below these statistics are average net returns, αnet, turnover, and transaction costs. Reliability or consistency of performance is also an important concern to investors. The bottom four rows report the percentage of quarters in which our strategies “beat the market” after transaction costs over the whole sample (ALL) and subsamples defined by the row heading. For the long positions, beating the market benchmark means earning higher net-of-transaction costs returns than the market. A short position “beats the market” if after transaction costs, the short earns higher returns than short-selling the market. A long–short strategy is zero-investment and therefore “beats the market” if its net-of-costs return is positive. Table IV. Performance of strategies using forty-four industry portfolios Each column presents statistics for the long, short, and long-minus-short (L−S) industry-rotation strategies based on the three different combination forecast methods described in Section 5 (ABMA, DMSFE, and MEAN) or the benchmark real-time univariate predictive regression that uses industry-own BM (BMi). E(rgrosse) denotes annualized gross average excess return and αgross, β, s, and h and m denote estimates from the Fama–French–Carhart four-factor model: rgross,te=αgross+β*MKTt+s*SMBt+h*HMLt+m*MOMt+ϵt. (16) E(rnet) denotes the annualized average returns net of transaction costs and αnet denotes the generalized net-of-costs α of Novy-Marx and Velikov (2016) (which is non-negative by construction). The bottom four rows present the percentage of quarters in which the strategy beat the market benchmark after transaction costs over the whole sample (ALL) and subsamples defined in by the row heading. TO and T-Costs denote average turnover and transaction costs (%/quarter). The sample is 1980:1–2015:4 (N = 144). *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. ABMA DMSFE MEAN BMi Long Short L−S Long Short L−S Long Short L−S Long Short L−S E(rgrosse) 11.72 2.43 9.29 13.65 2.06 11.59 14.05 3.20 10.85 9.35 9.99 −0.64 αgross 2.22 −4.38** 6.60** 2.67 −5.47*** 8.15*** 4.02** −2.56 6.59** 1.78 −1.01 2.79 (1.38) (−2.05) (2.39) (1.59) (−2.75) (3.11) (2.38) (−1.18) (2.35) (0.84) (−0.65) (1.06) β 1.16*** 1.06*** 0.10 1.23*** 1.08*** 0.15* 1.22*** 1.03*** 0.19** 1.01*** 1.18*** −0.17* (21.52) (14.79) (1.14) (21.92) (16.28) (1.71) (21.59) (14.12) (2.06) (14.20) (22.51) (−1.95) s 0.14* 0.43*** −0.29** 0.10 0.46*** −0.36*** 0.00 0.44*** −0.43*** 0.24** 0.34*** −0.10 (1.79) (4.21) (−2.21) (1.18) (4.78) (−2.87) (0.05) (4.17) (−3.21) (2.37) (4.50) (−0.76) h −0.20*** 0.34*** −0.54*** −0.14* 0.43*** −0.57*** −0.27*** 0.26*** −0.54*** 0.20** 0.09 0.11 (−2.76) (3.46) (−4.29) (−1.76) (4.78) (−4.75) (−3.55) (2.64) (−4.20) (2.04) (1.23) (0.91) m 0.19*** −0.39*** 0.58*** 0.24*** −0.37*** 0.62*** 0.23*** −0.47*** 0.70*** −0.15** 0.16*** −0.31*** (3.69) (−5.84) (6.68) (4.59) (−5.94) (7.44) (4.32) (−6.86) (7.93) (−2.21) (3.29) (−3.72) Adjusted R2 0.82 0.73 0.35 0.82 0.76 0.41 0.82 0.72 0.42 0.66 0.83 0.12 E(rnete) 10.75 −3.64 7.12 12.56 −3.45 9.11 13.13 −4.34 8.79 8.08 −11.16 −3.08 αnet 2.03 4.97** 7.08*** 3.13** 5.68*** 8.87*** 4.34*** 3.67* 7.99*** 0.00 0.00 0.00 (1.33) (2.44) (2.70) (1.99) (3.00) (3.54) (2.71) (1.74) (2.97) TO 31.5 34.6 33.0 36.6 39.0 37.8 30.3 31.5 30.9 39.2 34.3 36.7 T-Costs 0.24 0.30 0.54 0.27 0.35 0.62 0.23 0.28 0.52 0.32 0.29 0.61 Consistency ALL 53 58 59 55 59 63 57 51 60 51 45 45 1980s 42 60 52 48 60 62 55 45 55 50 45 40 1990s 62 70 70 55 65 65 55 65 70 48 50 55 2000s 54 49 56 59 54 62 60 46 57 54 41 43 ABMA DMSFE MEAN BMi Long Short L−S Long Short L−S Long Short L−S Long Short L−S E(rgrosse) 11.72 2.43 9.29 13.65 2.06 11.59 14.05 3.20 10.85 9.35 9.99 −0.64 αgross 2.22 −4.38** 6.60** 2.67 −5.47*** 8.15*** 4.02** −2.56 6.59** 1.78 −1.01 2.79 (1.38) (−2.05) (2.39) (1.59) (−2.75) (3.11) (2.38) (−1.18) (2.35) (0.84) (−0.65) (1.06) β 1.16*** 1.06*** 0.10 1.23*** 1.08*** 0.15* 1.22*** 1.03*** 0.19** 1.01*** 1.18*** −0.17* (21.52) (14.79) (1.14) (21.92) (16.28) (1.71) (21.59) (14.12) (2.06) (14.20) (22.51) (−1.95) s 0.14* 0.43*** −0.29** 0.10 0.46*** −0.36*** 0.00 0.44*** −0.43*** 0.24** 0.34*** −0.10 (1.79) (4.21) (−2.21) (1.18) (4.78) (−2.87) (0.05) (4.17) (−3.21) (2.37) (4.50) (−0.76) h −0.20*** 0.34*** −0.54*** −0.14* 0.43*** −0.57*** −0.27*** 0.26*** −0.54*** 0.20** 0.09 0.11 (−2.76) (3.46) (−4.29) (−1.76) (4.78) (−4.75) (−3.55) (2.64) (−4.20) (2.04) (1.23) (0.91) m 0.19*** −0.39*** 0.58*** 0.24*** −0.37*** 0.62*** 0.23*** −0.47*** 0.70*** −0.15** 0.16*** −0.31*** (3.69) (−5.84) (6.68) (4.59) (−5.94) (7.44) (4.32) (−6.86) (7.93) (−2.21) (3.29) (−3.72) Adjusted R2 0.82 0.73 0.35 0.82 0.76 0.41 0.82 0.72 0.42 0.66 0.83 0.12 E(rnete) 10.75 −3.64 7.12 12.56 −3.45 9.11 13.13 −4.34 8.79 8.08 −11.16 −3.08 αnet 2.03 4.97** 7.08*** 3.13** 5.68*** 8.87*** 4.34*** 3.67* 7.99*** 0.00 0.00 0.00 (1.33) (2.44) (2.70) (1.99) (3.00) (3.54) (2.71) (1.74) (2.97) TO 31.5 34.6 33.0 36.6 39.0 37.8 30.3 31.5 30.9 39.2 34.3 36.7 T-Costs 0.24 0.30 0.54 0.27 0.35 0.62 0.23 0.28 0.52 0.32 0.29 0.61 Consistency ALL 53 58 59 55 59 63 57 51 60 51 45 45 1980s 42 60 52 48 60 62 55 45 55 50 45 40 1990s 62 70 70 55 65 65 55 65 70 48 50 55 2000s 54 49 56 59 54 62 60 46 57 54 41 43 Table IV. Performance of strategies using forty-four industry portfolios Each column presents statistics for the long, short, and long-minus-short (L−S) industry-rotation strategies based on the three different combination forecast methods described in Section 5 (ABMA, DMSFE, and MEAN) or the benchmark real-time univariate predictive regression that uses industry-own BM (BMi). E(rgrosse) denotes annualized gross average excess return and αgross, β, s, and h and m denote estimates from the Fama–French–Carhart four-factor model: rgross,te=αgross+β*MKTt+s*SMBt+h*HMLt+m*MOMt+ϵt. (16) E(rnet) denotes the annualized average returns net of transaction costs and αnet denotes the generalized net-of-costs α of Novy-Marx and Velikov (2016) (which is non-negative by construction). The bottom four rows present the percentage of quarters in which the strategy beat the market benchmark after transaction costs over the whole sample (ALL) and subsamples defined in by the row heading. TO and T-Costs denote average turnover and transaction costs (%/quarter). The sample is 1980:1–2015:4 (N = 144). *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. ABMA DMSFE MEAN BMi Long Short L−S Long Short L−S Long Short L−S Long Short L−S E(rgrosse) 11.72 2.43 9.29 13.65 2.06 11.59 14.05 3.20 10.85 9.35 9.99 −0.64 αgross 2.22 −4.38** 6.60** 2.67 −5.47*** 8.15*** 4.02** −2.56 6.59** 1.78 −1.01 2.79 (1.38) (−2.05) (2.39) (1.59) (−2.75) (3.11) (2.38) (−1.18) (2.35) (0.84) (−0.65) (1.06) β 1.16*** 1.06*** 0.10 1.23*** 1.08*** 0.15* 1.22*** 1.03*** 0.19** 1.01*** 1.18*** −0.17* (21.52) (14.79) (1.14) (21.92) (16.28) (1.71) (21.59) (14.12) (2.06) (14.20) (22.51) (−1.95) s 0.14* 0.43*** −0.29** 0.10 0.46*** −0.36*** 0.00 0.44*** −0.43*** 0.24** 0.34*** −0.10 (1.79) (4.21) (−2.21) (1.18) (4.78) (−2.87) (0.05) (4.17) (−3.21) (2.37) (4.50) (−0.76) h −0.20*** 0.34*** −0.54*** −0.14* 0.43*** −0.57*** −0.27*** 0.26*** −0.54*** 0.20** 0.09 0.11 (−2.76) (3.46) (−4.29) (−1.76) (4.78) (−4.75) (−3.55) (2.64) (−4.20) (2.04) (1.23) (0.91) m 0.19*** −0.39*** 0.58*** 0.24*** −0.37*** 0.62*** 0.23*** −0.47*** 0.70*** −0.15** 0.16*** −0.31*** (3.69) (−5.84) (6.68) (4.59) (−5.94) (7.44) (4.32) (−6.86) (7.93) (−2.21) (3.29) (−3.72) Adjusted R2 0.82 0.73 0.35 0.82 0.76 0.41 0.82 0.72 0.42 0.66 0.83 0.12 E(rnete) 10.75 −3.64 7.12 12.56 −3.45 9.11 13.13 −4.34 8.79 8.08 −11.16 −3.08 αnet 2.03 4.97** 7.08*** 3.13** 5.68*** 8.87*** 4.34*** 3.67* 7.99*** 0.00 0.00 0.00 (1.33) (2.44) (2.70) (1.99) (3.00) (3.54) (2.71) (1.74) (2.97) TO 31.5 34.6 33.0 36.6 39.0 37.8 30.3 31.5 30.9 39.2 34.3 36.7 T-Costs 0.24 0.30 0.54 0.27 0.35 0.62 0.23 0.28 0.52 0.32 0.29 0.61 Consistency ALL 53 58 59 55 59 63 57 51 60 51 45 45 1980s 42 60 52 48 60 62 55 45 55 50 45 40 1990s 62 70 70 55 65 65 55 65 70 48 50 55 2000s 54 49 56 59 54 62 60 46 57 54 41 43 ABMA DMSFE MEAN BMi Long Short L−S Long Short L−S Long Short L−S Long Short L−S E(rgrosse) 11.72 2.43 9.29 13.65 2.06 11.59 14.05 3.20 10.85 9.35 9.99 −0.64 αgross 2.22 −4.38** 6.60** 2.67 −5.47*** 8.15*** 4.02** −2.56 6.59** 1.78 −1.01 2.79 (1.38) (−2.05) (2.39) (1.59) (−2.75) (3.11) (2.38) (−1.18) (2.35) (0.84) (−0.65) (1.06) β 1.16*** 1.06*** 0.10 1.23*** 1.08*** 0.15* 1.22*** 1.03*** 0.19** 1.01*** 1.18*** −0.17* (21.52) (14.79) (1.14) (21.92) (16.28) (1.71) (21.59) (14.12) (2.06) (14.20) (22.51) (−1.95) s 0.14* 0.43*** −0.29** 0.10 0.46*** −0.36*** 0.00 0.44*** −0.43*** 0.24** 0.34*** −0.10 (1.79) (4.21) (−2.21) (1.18) (4.78) (−2.87) (0.05) (4.17) (−3.21) (2.37) (4.50) (−0.76) h −0.20*** 0.34*** −0.54*** −0.14* 0.43*** −0.57*** −0.27*** 0.26*** −0.54*** 0.20** 0.09 0.11 (−2.76) (3.46) (−4.29) (−1.76) (4.78) (−4.75) (−3.55) (2.64) (−4.20) (2.04) (1.23) (0.91) m 0.19*** −0.39*** 0.58*** 0.24*** −0.37*** 0.62*** 0.23*** −0.47*** 0.70*** −0.15** 0.16*** −0.31*** (3.69) (−5.84) (6.68) (4.59) (−5.94) (7.44) (4.32) (−6.86) (7.93) (−2.21) (3.29) (−3.72) Adjusted R2 0.82 0.73 0.35 0.82 0.76 0.41 0.82 0.72 0.42 0.66 0.83 0.12 E(rnete) 10.75 −3.64 7.12 12.56 −3.45 9.11 13.13 −4.34 8.79 8.08 −11.16 −3.08 αnet 2.03 4.97** 7.08*** 3.13** 5.68*** 8.87*** 4.34*** 3.67* 7.99*** 0.00 0.00 0.00 (1.33) (2.44) (2.70) (1.99) (3.00) (3.54) (2.71) (1.74) (2.97) TO 31.5 34.6 33.0 36.6 39.0 37.8 30.3 31.5 30.9 39.2 34.3 36.7 T-Costs 0.24 0.30 0.54 0.27 0.35 0.62 0.23 0.28 0.52 0.32 0.29 0.61 Consistency ALL 53 58 59 55 59 63 57 51 60 51 45 45 1980s 42 60 52 48 60 62 55 45 55 50 45 40 1990s 62 70 70 55 65 65 55 65 70 48 50 55 2000s 54 49 56 59 54 62 60 46 57 54 41 43 Table IV shows that each of the combination methods generates substantial spreads in average returns of about 9.3–11.6% per annum between the long and short legs of the forty-four industry combo strategies. Further, the long–short strategies earn significant four-factor αgross of about 6.6–8.2% as exposure to the risk factors explains only a few percent of the spread in average returns.8 The BMi strategy does not generate a significant spread in returns or αgross. The combo strategy returns do not arise from a value effect. The HML loadings on the long leg of the combo strategies are lower than those of the short leg. In contrast to the HML loadings, the MOM loadings are significantly higher for the long legs of the combo strategies than for the short legs. Further, the MOM exposure of the long–short combo strategies actually lowers αgross relative to αnet by about 0.5–1.2% because the transaction costs on the combo strategies are significantly lower than those of MOM. While the momentum factor is correlated with our strategies returns, it does not explain the returns on our strategies as the α is not 0. In the Online Appendix, we also show that a factor model similar to the Fama–French–Carhart model that includes an industry-momentum factor explains even less of our strategies’ returns than the Fama–French–Carhart model. Further, the spread in MOM loadings is driven by other industries; the opposite pattern occurs with the BMi strategy. The more likely explanation of the MOM loadings is that our rotation strategies take long and short positions, respectively, in industries with high and low predicted returns. Thus, we should expect our long–short strategy returns to be positively correlated with characteristics that are themselves positively correlated with expected returns, such as the market beta and momentum. Hou, Xue, and Zhang (2015) also find that the momentum premium is explained by profitability and investment, so the momentum loadings in Table IV likely proxy for more fundamental characteristics that are correlated with expected returns. The bottom four rows of Table IV panel demonstrate that for both sets of industries, the long–short combo strategies earn positive net-of-costs returns over 59–63% of quarters over the whole sample. For most decades, the long, short, and long–short combo strategies outperform the market benchmarks as well. Thus, the combo strategies generally perform reliably over the sample, exhibiting no sign of deterioration over time. The high αnet relative to αgross in Table IV demonstrate how the value-weighting of the industries in our strategies economize on transaction costs. The value-weighting of the industry portfolios ensures that the short legs of our strategies are feasible to execute. D’Avolio (2002), for example, finds that stocks that cannot be shorted combine to less than 1% of the aggregate stock market. A related concern is that short-sale fees would eliminate the profitability of our strategies. However, D’Avolio (2002) also finds the short-sale costs on 91% of stocks are less than 1% per annum and the average shorting cost on the remaining 9% averages only 4.9%. Drechsler and Drechsler (2016) similarly show that stocks that are expensive to short have the smallest market capitalizations. While accounting for short-sale fees could create a modest drag on the absolute profitability of our industry-rotation strategies, it should improve the net-of-costs α for the same reason that accounting for effective spreads does. Long and short positions in HML and MOM both involve short-selling portfolios of expensive-to-short small-cap stocks.9 Similarly, our forty-four industry strategies are generally net short SMB, which would further involve short-selling small-cap stocks. Overall, investors facing transaction costs can earn positive abnormal returns relative to the Fama–French–Carhart four-factor model by trading industries based on all three combination forecasts. 5.1.a. Reliability and economic significance to investors Figure 2 depicts the time series of the natural log of the net-of-costs accumulated value of $1 invested in the different each leg of the forty-four industry combo strategies and the market (MKT) benchmark over the 36 year out-of-sample period. For ease of visualization, the short positions in Figure 2 are expressed as long positions. The long legs of the three combo strategies steadily gain relative to the market bench, and exhibit no obvious break in performance. Similarly, the short legs of the combo strategies generally under-perform the market benchmark throughout the entire 36-year period. Although transaction costs reduce final payoffs, the combo strategies steadily outperform the market over several decades. Figure 2. View largeDownload slide Cumulative log returns, 1980:1–2015:4. This figure depicts the natural log of the accumulated net-of-costs value of $1 invested in each of the long and short positions associated with the combination forecast-trading strategies described in Section 4 applied to the Fama–French forty-eight industries. BMi denotes the analogous trading strategy based on a real-time univariate predictive regression of returns on industry i on the lagged BM of industry i. Figure 2. View largeDownload slide Cumulative log returns, 1980:1–2015:4. This figure depicts the natural log of the accumulated net-of-costs value of $1 invested in each of the long and short positions associated with the combination forecast-trading strategies described in Section 4 applied to the Fama–French forty-eight industries. BMi denotes the analogous trading strategy based on a real-time univariate predictive regression of returns on industry i on the lagged BM of industry i. Figure 3 sorts industries in ascending order by average BMi, and then plots the percentage of quarters that each industry is chosen in the ABMA-based strategy (other strategies are not shown for brevity, but look qualitatively similar). The figure shows no clear relationship between BMi and industry selection in the ABMA strategy. In fact, the ABMA chooses the industry with the lowest average BMi (DRUGS) most often. Moreover, no industry is picked most of the time and most industries are chosen at least once. Hence, the industry selection of the ABMA strategy reflects meaningful time-variation in relative predicted returns across industries. Figure 3. View largeDownload slide Frequency each industry is picked in ABMA strategy BM ratios with industries sorted by B/M. Panel (A) depicts the percentage of times each industry is selected in the long leg of the rotation portfolio based on the ABMA forecasts. Panel (B) depicts the short leg. Industries are sorted by their time-series average BM along the horizontal axis from smallest on the left and highest on the right. Figure 3. View largeDownload slide Frequency each industry is picked in ABMA strategy BM ratios with industries sorted by B/M. Panel (A) depicts the percentage of times each industry is selected in the long leg of the rotation portfolio based on the ABMA forecasts. Panel (B) depicts the short leg. Industries are sorted by their time-series average BM along the horizontal axis from smallest on the left and highest on the right. Investors choose from many assets and their utility depends on the Sharpe ratio of their whole portfolio. Table V presents the ex post mean–variance frontier and maximum Sharpe ratio attainable by investing in the benchmark Fama–French–Carhart factors and each of our long–short industry-combo strategies in the presence of transaction costs. The forty-four industry combo strategies improve the maximum Sharpe ratio by 0.15–0.24 per year relative to the four-factor model, which is comparable to the gains from adding the Fama–French and Carhart factors to the market return. Moreover, given access to the forty-four industry combo strategies, the momentum factor has effectively zero weight in the ex post tangency portfolio. Hence, access to our industry-rotation strategies is an economically significant improvement in investment opportunities transaction costs. Table V. Ex post mean–variance efficient portfolios net of transaction costs This table reports ex post tangency portfolio weights and Sharpe ratios (SRs) on the net-of-costs returns on the Fama–French–Carhart factors and one of each of the different long–short strategies based on the four combination forecast methods described in Section 3 and constructed from the forty-four Fama–French industries. Panel A presents results for the Fama–French–Carhart factors. Panel B adds the long–short strategies. Panel A: Fama–French factors MKT SMB HML MOM L−S SR CAPM 1.00 0.46 FF3 0.51 0.00 0.49 0.61 FF4 0.38 0.00 0.41 0.21 0.71 Panel B: Forty-four industries ABMA 0.29 0.03 0.44 0.01 0.23 0.86 DMSFE 0.25 0.06 0.44 0.00 0.25 0.95 MEAN 0.27 0.06 0.45 0.00 0.23 0.89 Panel A: Fama–French factors MKT SMB HML MOM L−S SR CAPM 1.00 0.46 FF3 0.51 0.00 0.49 0.61 FF4 0.38 0.00 0.41 0.21 0.71 Panel B: Forty-four industries ABMA 0.29 0.03 0.44 0.01 0.23 0.86 DMSFE 0.25 0.06 0.44 0.00 0.25 0.95 MEAN 0.27 0.06 0.45 0.00 0.23 0.89 Table V. Ex post mean–variance efficient portfolios net of transaction costs This table reports ex post tangency portfolio weights and Sharpe ratios (SRs) on the net-of-costs returns on the Fama–French–Carhart factors and one of each of the different long–short strategies based on the four combination forecast methods described in Section 3 and constructed from the forty-four Fama–French industries. Panel A presents results for the Fama–French–Carhart factors. Panel B adds the long–short strategies. Panel A: Fama–French factors MKT SMB HML MOM L−S SR CAPM 1.00 0.46 FF3 0.51 0.00 0.49 0.61 FF4 0.38 0.00 0.41 0.21 0.71 Panel B: Forty-four industries ABMA 0.29 0.03 0.44 0.01 0.23 0.86 DMSFE 0.25 0.06 0.44 0.00 0.25 0.95 MEAN 0.27 0.06 0.45 0.00 0.23 0.89 Panel A: Fama–French factors MKT SMB HML MOM L−S SR CAPM 1.00 0.46 FF3 0.51 0.00 0.49 0.61 FF4 0.38 0.00 0.41 0.21 0.71 Panel B: Forty-four industries ABMA 0.29 0.03 0.44 0.01 0.23 0.86 DMSFE 0.25 0.06 0.44 0.00 0.25 0.95 MEAN 0.27 0.06 0.45 0.00 0.23 0.89 5.2 Alternative Test Assets We apply the portfolio-rotation strategies from the previous section to each of the twenty-five size/BM, size/investment, and size/profitability portfolios. However, we relegate the size/investment and size/profitability results and discussion to the Online Appendix for brevity. Characteristic portfolios are interesting for a variety of reasons. Many asset pricing studies use them (e.g., Fama and French, 1993, 2015, 2016). Further, the characteristic portfolios allow us to check the robustness of combination forecast methods to returns on test assets formed on economically different criterion than industry. The evidence from Section 4 also shows that return predictability varies with characteristics. Moreover, strategies based on characteristic sorts are potentially useful to investors who allocate funds across characteristic classes (e.g., Dimensional Fund Advisors, or Morningstar Equity Style Box10). The characteristic portfolios have several drawbacks, however, which lead us to advocate using industries in rotation strategies. First, multiple portfolios in the characteristic sorts consist entirely of small-cap stocks that are expensive to trade. Second, expected returns on characteristic portfolios have a well-known high correlation between characteristic ranks and expected returns. Our combo strategies should invest in the portfolios with the highest expected returns, but these portfolios tend to be easy to predict with their characteristic rank (e.g., small-cap and high BM). Verifying this pattern would confirm that the combination forecasts work properly, but is less informative than forming combo strategies with the industry returns whose expected return ranks are time-varying. Characteristic portfolios also have a tight factor structure (e.g., Lewellen, Nagel, and Shanken, 2010). The Fama–French factors in particular explain most of the variation in returns across the characteristic-sorted portfolios, which make it difficult to construct strategies with them that earn significant alpha. To illustrate this point, Figure 4 shows the distribution of the standard deviation ( σ(ϵt)) of the residuals from the Fama and French (2015) five-factor model: rte=α+β*MKTt+s*SMBt+h*HMLt+c*CMAt+r*RMWt+ϵt, (17) where rte denotes the gross excess returns one of the forty-four industry portfolios or the characteristic portfolios. Figure 4 shows that relative to the industry portfolios, the characteristic portfolios have a fraction of the variation in returns that is not explained by exposure to the Fama–French factors. Hence, the characteristic portfolios provide less opportunity than industry portfolios to demonstrate the effectiveness of the combination forecasts to improve the investment opportunity set spanned by these factors. Figure 4. View largeDownload slide Standard deviations of residuals from regressions of portfolio returns on Fama–French factors. This figure depicts histograms of the standard deviation of residuals from regressions of portfolio excess returns on the relevant Fama–French factors; MKT, SMB, HML, and MOM for the industry and size/BM portfolios, and the four factors plus CMA and RMW for the size/investment and size/profitability portfolios. The sample is 1980:1–2015:4. Figure 4. View largeDownload slide Standard deviations of residuals from regressions of portfolio returns on Fama–French factors. This figure depicts histograms of the standard deviation of residuals from regressions of portfolio excess returns on the relevant Fama–French factors; MKT, SMB, HML, and MOM for the industry and size/BM portfolios, and the four factors plus CMA and RMW for the size/investment and size/profitability portfolios. The sample is 1980:1–2015:4. We apply our long–short strategies to the characteristic-sorted portfolios following the industry strategies in Section 5. Because of the tight factor structure of these portfolios, we use fewer portfolios to long and short than with the industries to get the most extreme predicted returns. For the size and BM portfolios, we take a long position in the two portfolios with the highest predicted returns, and vice versa for the short position. Table VI presents average returns, Fama–French–Carhart four-factor αgross and factor loadings ignoring transaction costs, average net returns, four-factor αnet, turnover (%/quarter), and transaction costs (%/quarter) of the size and BM combo strategies as well as the benchmark BMi strategy. The combo strategies all generate a sizable spread in returns of about 8.7–10.7% per annum. Conversely, the BMi strategy yields less than half the spread in average returns, verifying the effectiveness of the combination return forecasts. The long–short size/BM combo strategies earn significant αgross in spite of the expected significant exposure to HML. The size and significance of the long–short α also parallel the patterns in ROS2 from Table III. ABMA generated the highest ROS2, followed by DMSFE and then MEAN. Similarly, the rotation strategies’ α is highest for ABMA, followed by DMSFE and then MEAN. Table VI. Performance of strategies using twenty-five size and BM portfolios Each column presents statistics for the long, short, and long-minus-short (L−S) rotation strategies described in Section 5 based on one of the three different combination forecast methods (ABMA, DMSFE, and MEAN) or the benchmark univariate real-time predictive regression that uses portfolio-own BM (BMi). The strategies use the Fama–French twenty-five size and BM portfolios. E(rgrosse) denotes annualized gross average excess return and αgross, β, s, and h and m denote estimates from the Fama–French–Carhart four-factor model: rgross,te=αgross+β*MKTt+s*SMBt+h*HMLt+m*MOMt+ϵt. (18) E(rnet) denotes the annualized average returns net of transaction costs and αnet denotes the generalized net-of-costs α of Novy-Marx and Velikov (2016). TO and T-Costs denote average turnover and transaction costs (%/quarter). The sample is 1980:1–2015:4 (N = 144). *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. ABMA DMSFE MEAN BMi Long Short L−S Long Short L−S Long Short L−S Long Short L−S E(rgrosse) 13.45 2.73 10.71 13.26 3.38 9.88 12.86 4.16 8.69 10.07 5.96 4.11 αgross 2.62** −4.54*** 7.16*** 2.67** −4.35*** 7.01*** 2.12 −3.05** 5.18** −0.13 −3.46** 3.33 (2.11) (−3.13) (3.29) (2.06) (−2.98) (2.87) (1.58) (−2.11) (2.19) (−0.13) (−2.24) (1.60) β 0.97*** 1.01*** −0.04 0.98*** 1.01*** −0.03 1.02*** 0.98*** 0.04 0.95*** 1.05*** −0.10 (23.22) (20.77) (−0.61) (22.52) (20.56) (−0.35) (22.55) (20.17) (0.51) (27.72) (20.32) (−1.51) s 0.74*** 0.54*** 0.20* 0.67*** 0.64*** 0.03 0.56*** 0.57*** −0.01 0.59*** 0.74*** −0.15 (12.48) (7.79) (1.93) (10.75) (9.08) (0.27) (8.67) (8.26) (−0.11) (12.00) (9.94) (−1.51) h 0.53*** −0.31*** 0.84*** 0.62*** −0.18*** 0.80*** 0.60*** −0.26*** 0.86*** 0.62*** −0.02 0.64*** (9.34) (−4.62) (8.43) (10.47) (−2.63) (7.13) (9.66) (−3.98) (7.93) (13.16) (−0.27) (6.68) m 0.07* 0.00 0.06 0.00 −0.04 0.04 0.01 0.00 0.01 −0.01 0.05 −0.05 (1.70) (0.06) (0.93) (0.07) (−0.78) (0.50) (0.34) (0.01) (0.19) (−0.25) (0.95) (−0.83) Adjusted R2 0.88 0.86 0.34 0.86 0.86 0.28 0.85 0.85 0.32 0.90 0.85 0.33 E(rnete) 11.68 −3.87 7.80 10.74 −5.22 5.52 11.46 −4.91 6.55 11.06 −5.90 5.16 αnet 1.49 3.44** 5.43** 1.44 2.67* 4.44* 1.00 1.42 2.56 0.00 0.00 0.00 (1.08) (2.44) (2.58) (0.91) (1.88) (1.88) (0.63) (0.95) (1.12) TO 38.7 24.9 31.8 41.4 31.7 36.6 35.5 37.8 36.7 49.6 45.3 47.5 T-Costs 0.44 0.28 0.73 0.45 0.38 0.83 0.45 0.43 0.88 0.52 0.54 1.06 ABMA DMSFE MEAN BMi Long Short L−S Long Short L−S Long Short L−S Long Short L−S E(rgrosse) 13.45 2.73 10.71 13.26 3.38 9.88 12.86 4.16 8.69 10.07 5.96 4.11 αgross 2.62** −4.54*** 7.16*** 2.67** −4.35*** 7.01*** 2.12 −3.05** 5.18** −0.13 −3.46** 3.33 (2.11) (−3.13) (3.29) (2.06) (−2.98) (2.87) (1.58) (−2.11) (2.19) (−0.13) (−2.24) (1.60) β 0.97*** 1.01*** −0.04 0.98*** 1.01*** −0.03 1.02*** 0.98*** 0.04 0.95*** 1.05*** −0.10 (23.22) (20.77) (−0.61) (22.52) (20.56) (−0.35) (22.55) (20.17) (0.51) (27.72) (20.32) (−1.51) s 0.74*** 0.54*** 0.20* 0.67*** 0.64*** 0.03 0.56*** 0.57*** −0.01 0.59*** 0.74*** −0.15 (12.48) (7.79) (1.93) (10.75) (9.08) (0.27) (8.67) (8.26) (−0.11) (12.00) (9.94) (−1.51) h 0.53*** −0.31*** 0.84*** 0.62*** −0.18*** 0.80*** 0.60*** −0.26*** 0.86*** 0.62*** −0.02 0.64*** (9.34) (−4.62) (8.43) (10.47) (−2.63) (7.13) (9.66) (−3.98) (7.93) (13.16) (−0.27) (6.68) m 0.07* 0.00 0.06 0.00 −0.04 0.04 0.01 0.00 0.01 −0.01 0.05 −0.05 (1.70) (0.06) (0.93) (0.07) (−0.78) (0.50) (0.34) (0.01) (0.19) (−0.25) (0.95) (−0.83) Adjusted R2 0.88 0.86 0.34 0.86 0.86 0.28 0.85 0.85 0.32 0.90 0.85 0.33 E(rnete) 11.68 −3.87 7.80 10.74 −5.22 5.52 11.46 −4.91 6.55 11.06 −5.90 5.16 αnet 1.49 3.44** 5.43** 1.44 2.67* 4.44* 1.00 1.42 2.56 0.00 0.00 0.00 (1.08) (2.44) (2.58) (0.91) (1.88) (1.88) (0.63) (0.95) (1.12) TO 38.7 24.9 31.8 41.4 31.7 36.6 35.5 37.8 36.7 49.6 45.3 47.5 T-Costs 0.44 0.28 0.73 0.45 0.38 0.83 0.45 0.43 0.88 0.52 0.54 1.06 Table VI. Performance of strategies using twenty-five size and BM portfolios Each column presents statistics for the long, short, and long-minus-short (L−S) rotation strategies described in Section 5 based on one of the three different combination forecast methods (ABMA, DMSFE, and MEAN) or the benchmark univariate real-time predictive regression that uses portfolio-own BM (BMi). The strategies use the Fama–French twenty-five size and BM portfolios. E(rgrosse) denotes annualized gross average excess return and αgross, β, s, and h and m denote estimates from the Fama–French–Carhart four-factor model: rgross,te=αgross+β*MKTt+s*SMBt+h*HMLt+m*MOMt+ϵt. (18) E(rnet) denotes the annualized average returns net of transaction costs and αnet denotes the generalized net-of-costs α of Novy-Marx and Velikov (2016). TO and T-Costs denote average turnover and transaction costs (%/quarter). The sample is 1980:1–2015:4 (N = 144). *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. ABMA DMSFE MEAN BMi Long Short L−S Long Short L−S Long Short L−S Long Short L−S E(rgrosse) 13.45 2.73 10.71 13.26 3.38 9.88 12.86 4.16 8.69 10.07 5.96 4.11 αgross 2.62** −4.54*** 7.16*** 2.67** −4.35*** 7.01*** 2.12 −3.05** 5.18** −0.13 −3.46** 3.33 (2.11) (−3.13) (3.29) (2.06) (−2.98) (2.87) (1.58) (−2.11) (2.19) (−0.13) (−2.24) (1.60) β 0.97*** 1.01*** −0.04 0.98*** 1.01*** −0.03 1.02*** 0.98*** 0.04 0.95*** 1.05*** −0.10 (23.22) (20.77) (−0.61) (22.52) (20.56) (−0.35) (22.55) (20.17) (0.51) (27.72) (20.32) (−1.51) s 0.74*** 0.54*** 0.20* 0.67*** 0.64*** 0.03 0.56*** 0.57*** −0.01 0.59*** 0.74*** −0.15 (12.48) (7.79) (1.93) (10.75) (9.08) (0.27) (8.67) (8.26) (−0.11) (12.00) (9.94) (−1.51) h 0.53*** −0.31*** 0.84*** 0.62*** −0.18*** 0.80*** 0.60*** −0.26*** 0.86*** 0.62*** −0.02 0.64*** (9.34) (−4.62) (8.43) (10.47) (−2.63) (7.13) (9.66) (−3.98) (7.93) (13.16) (−0.27) (6.68) m 0.07* 0.00 0.06 0.00 −0.04 0.04 0.01 0.00 0.01 −0.01 0.05 −0.05 (1.70) (0.06) (0.93) (0.07) (−0.78) (0.50) (0.34) (0.01) (0.19) (−0.25) (0.95) (−0.83) Adjusted R2 0.88 0.86 0.34 0.86 0.86 0.28 0.85 0.85 0.32 0.90 0.85 0.33 E(rnete) 11.68 −3.87 7.80 10.74 −5.22 5.52 11.46 −4.91 6.55 11.06 −5.90 5.16 αnet 1.49 3.44** 5.43** 1.44 2.67* 4.44* 1.00 1.42 2.56 0.00 0.00 0.00 (1.08) (2.44) (2.58) (0.91) (1.88) (1.88) (0.63) (0.95) (1.12) TO 38.7 24.9 31.8 41.4 31.7 36.6 35.5 37.8 36.7 49.6 45.3 47.5 T-Costs 0.44 0.28 0.73 0.45 0.38 0.83 0.45 0.43 0.88 0.52 0.54 1.06 ABMA DMSFE MEAN BMi Long Short L−S Long Short L−S Long Short L−S Long Short L−S E(rgrosse) 13.45 2.73 10.71 13.26 3.38 9.88 12.86 4.16 8.69 10.07 5.96 4.11 αgross 2.62** −4.54*** 7.16*** 2.67** −4.35*** 7.01*** 2.12 −3.05** 5.18** −0.13 −3.46** 3.33 (2.11) (−3.13) (3.29) (2.06) (−2.98) (2.87) (1.58) (−2.11) (2.19) (−0.13) (−2.24) (1.60) β 0.97*** 1.01*** −0.04 0.98*** 1.01*** −0.03 1.02*** 0.98*** 0.04 0.95*** 1.05*** −0.10 (23.22) (20.77) (−0.61) (22.52) (20.56) (−0.35) (22.55) (20.17) (0.51) (27.72) (20.32) (−1.51) s 0.74*** 0.54*** 0.20* 0.67*** 0.64*** 0.03 0.56*** 0.57*** −0.01 0.59*** 0.74*** −0.15 (12.48) (7.79) (1.93) (10.75) (9.08) (0.27) (8.67) (8.26) (−0.11) (12.00) (9.94) (−1.51) h 0.53*** −0.31*** 0.84*** 0.62*** −0.18*** 0.80*** 0.60*** −0.26*** 0.86*** 0.62*** −0.02 0.64*** (9.34) (−4.62) (8.43) (10.47) (−2.63) (7.13) (9.66) (−3.98) (7.93) (13.16) (−0.27) (6.68) m 0.07* 0.00 0.06 0.00 −0.04 0.04 0.01 0.00 0.01 −0.01 0.05 −0.05 (1.70) (0.06) (0.93) (0.07) (−0.78) (0.50) (0.34) (0.01) (0.19) (−0.25) (0.95) (−0.83) Adjusted R2 0.88 0.86 0.34 0.86 0.86 0.28 0.85 0.85 0.32 0.90 0.85 0.33 E(rnete) 11.68 −3.87 7.80 10.74 −5.22 5.52 11.46 −4.91 6.55 11.06 −5.90 5.16 αnet 1.49 3.44** 5.43** 1.44 2.67* 4.44* 1.00 1.42 2.56 0.00 0.00 0.00 (1.08) (2.44) (2.58) (0.91) (1.88) (1.88) (0.63) (0.95) (1.12) TO 38.7 24.9 31.8 41.4 31.7 36.6 35.5 37.8 36.7 49.6 45.3 47.5 T-Costs 0.44 0.28 0.73 0.45 0.38 0.83 0.45 0.43 0.88 0.52 0.54 1.06 The αnet on the size/BM combo strategies are lower than the αgross for each of the combo strategies, and also decrease going from the ABMA to DMSFE to MEAN. The αnet are significant for the ABMA strategy, marginally significant for the DMSFE strategy, and are insignificant for the MEAN strategy. The lower magnitude and significance of the αnet than αgross reflects the fact that the transaction costs are higher for smaller stocks that are selected relatively often because they have the most extreme returns on average. Results in the Online Appendix show the long–short size/investment combo strategies have similar average returns and α as the size/BM strategies. In contrast, the long–short size/profitability combo strategies have similar average returns, but only marginally significant α. Regardless of choice of test assets, the long–short spreads in average returns on the portfolio-rotation strategies are large, ranging from 7.7% to 14.2% depending on method and assets. These spreads in returns need not accompany Fama–French–Carhart α, but the long–short combo strategies do mostly earn significant α, even net of transaction costs. Similarly, adding the industry long–short combo strategies to the Fama–French–Carhart factors substantially increases the maximum attainable Sharpe ratio. Thus, combining information in the cross-section of BM ratios improves investment opportunities for investors net of transaction costs. 6. Conclusion Predictive regressions show that the cross-section of industry BM ratios has greater explanatory power for forecasting both individual industry returns and CFs than industry-own or aggregate BM ratios. We use combination forecast methods to capture this cross-industry predictive information in real time. Since 1980, the combination forecasts predict quarterly industry returns with out-of-sample R2 that average 1.6–2.1%. Trading strategies that buy and sell industries with the highest and lowest out-of-sample predicted returns, respectively, generate significant a Fama–French–Carhart alpha of 7.1–8.9% per year net of transaction costs. Combination forecasts based on the cross-section of characteristic portfolios’ BM ratios also generate significant out-of-sample R2 and their trading strategies frequently result in significant Fama–French–Carhart alpha. Overall, our results reveal that a stock’s BM ratio can provide useful predictive information for the expected returns on other stocks. This finding can be explained by reexamining present value models. An asset’s BM ratio is a noisy proxy for that asset’s expected return because it is a joint function of the asset’s expected future return and CFs. Given commonality in expected returns and imperfect correlation of CFs across stocks, BM ratios of any one stock portfolio can provide relevant but non-redundant signals about the expected returns on other stocks or portfolios. Future studies that predict returns on industry or characteristic portfolios should therefore consider the cross-section of valuation ratios. Supplementary Material Supplementary data are available at Review of Finance online. Footnotes 1 From the thirty-eight industries, we exclude Agric, Stone, Garbg, Steam, Water, and Govt. From the forty-eight industries, we exclude Coal, Guns, Smoke, and Soda. 2 The industry MINES has a very low first-autocorrelation coefficient of 0.06. This is caused by MINES having a relatively small number of firms, several of which have volatile market values and only report COMPUSTAT accounting data every other quarter. In the Online Appendix, we demonstrate that all results are robust to an alternative construction of BM that corrects for this problem by using the prior quarter’s book equity if the current quarter’s is missing. 3 Hasbrouck provides SAS code to estimate effective spreads at http://people.stern.nyu.edu/jhasbrou/. This code runs directly on the WRDS server after only changing relevant file paths. 4 Novy-Marx and Velikov (2016) present an analogous plot for all but CMA and RMW. 5 In the Online Appendix, we discuss how relaxing the Campbell and Thompson (2008) restriction impacts the combination forecasts. 6 The first principal component of industry BM has a 93% correlation with BMM, so principal components 2–6 are effectively common factors in the cross-section of BM beyond the market BM. 7 We present results for analogous strategies using the thirty-two industry portfolios in the Online Appendix. Inferences are unchanged. 8 We also construct combo strategies using BM ratios with annual accounting data. 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Published by Oxford University Press on behalf of the European Finance Association. All rights reserved. For Permissions, please email: journals.permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Review of Finance Oxford University Press

Combination Return Forecasts and Portfolio Allocation with the Cross-Section of Book-to-Market Ratios

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© The Authors 2017. Published by Oxford University Press on behalf of the European Finance Association. All rights reserved. For Permissions, please email: journals.permissions@oup.com
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Abstract

Abstract * We thank Burton Hollifield (Editor), Mihail Velikov, and an anonymous referee for helpful comments. In this paper, we forecast industry returns out-of-sample using the cross-section of book-to-market (BM) ratios and investigate whether investors can exploit this predictability in portfolio allocation. Cash-flow and return forecasting regressions show that cross-industry BM ratios contain significant predictive information beyond aggregate and industry-specific BM ratios. Forecast combination methods based on industry BM ratios generate significant out-of-sample predictability for many industries. Real-time portfolio-rotation strategies that buy industries with high predicted returns and short industries with low predicted returns based on combination forecasts earn significant alpha with respect to standard asset pricing models net of transaction costs. 1. Introduction There is much evidence that average returns of stocks are positively correlated with their book-to-market (BM) equity ratios in the cross-section (e.g., Rosenberg, Reid, and Lanstein, 1985; Fama and French, 1992, 2015). Investment professionals since at least Graham and Dodd (1934) have exploited this pattern and demonstrated that “value” strategies that buy high BM stocks earn higher average returns than the overall market. In the time-series, Kothari and Shanken (1997), Pontiff and Schall (1998), and Lewellen (1999) find that the BM ratio of the aggregate market forecasts the market return. Both the time-series and cross-sectional patterns are consistent with present value identities that show, all-else-equal, higher expected returns on an asset lower that asset’s value relative to accounting fundamentals (e.g., Campbell and Shiller, 1988; Vuolteenaho, 2002). Perhaps in part because of this theory, prior studies on BM ratios and returns generally ignore whether the return of any one stock or portfolio can be predicted by the BM ratios of other stocks. Kelly and Pruitt (2013) argue that a common set of dynamic state variables drives the entire panel of BM ratios and expected returns. Thus, the whole panel of BM ratios provides information about these state variables. An asset’s BM ratio is also a noisy proxy for the asset’s expected return because it is a joint function of the asset’s expected future return and cash flows (CFs). Given imperfect correlation of CFs across stocks, commonality in expected returns implies that the BM ratio of any one stock conveys a relevant but non-redundant signal about the expected returns on other stocks. In this paper, we use forecast combination methods to extract information from the cross-section of BM ratios to help predict the returns of individual stock portfolios. Combination forecasts are weighted averages of individual forecasts and benefit from a diversification-like effect. If the prediction errors of individual forecasts are imperfectly correlated, forecast combinations can be more accurate out-of-sample than even the best individual forecast (for a recent survey, see Timmermann, 2006). Such imperfect correlation of return-forecast errors can arise from time-varying data-generating processes and idiosyncratic variation of cash-flow growth and returns across stocks. Consistent with the diversification benefit, prior studies show that combination forecasts effectively improve out-of-sample predictability of economic time-series such as output or stock market returns (e.g., Stock and Watson, 2004; Rapach, Strauss, and Zhou, 2010). We apply combination forecasts to construct portfolio-rotation strategies that buy portfolios with the highest predicted returns and short portfolios with the lowest predicted returns. The extent to which these portfolios improve investment opportunities relative to those represented by common asset pricing factors is a measure of the economic significance of our combination forecasts (e.g., Pesaran and Timmermann, 1995). Since investors face transaction costs, we follow Novy-Marx and Velikov (2016) and estimate effective transaction costs for individual stocks and apply them to our trading strategies. This method precisely captures variation in transaction costs over time and across portfolios. Our analysis focuses on value-weighted Fama–French industries, though we demonstrate the robustness of our methods using common characteristic-sorted portfolios (characteristic portfolios) formed on size and BM ratio, investment, and profitability. In forecasting, some aggregation of individual stocks is necessary to obtain regular-frequency time series with less idiosyncratic noise. We focus on industries for the following reasons. Expected returns on the characteristic portfolios tend to decrease with size and vary monotonically with the other characteristic. Hence, our forecast-based trading strategies should select the portfolios with the highest and lowest expected returns, but these portfolios are consistently those with the extreme size and characteristic ranks. In contrast, predicting the industries with the highest and lowest returns in real time is more challenging. Industries also lack the tight factor structure of characteristic portfolios and possess three to four times the variation in returns not explained by the Fama–French factors compared with characteristic portfolios. Hence, there is effectively more opportunity to demonstrate alpha with industry strategies (e.g., Lewellen, Nagel, and Shanken, 2010). Further, value-weighted industries have relatively low transaction costs as they put little weight on small-cap stocks. Conversely, the characteristic portfolios with the highest and lowest expected returns tend to consist of small-cap stocks that are expensive to trade. We summarize our findings as follows. Six principal components of industry BM ratios predict 1-year CFs and quarterly returns of the average industry with in-sample adjusted R2 of 46.5% and 6.1%, respectively. Principal components 2–6 explain most of this predictability. Conversely, industry-own BM ratios predict CFs and returns with a fraction of the R2 on average. Thus, the cross-section of BM ratios possesses significant information for predicting industry returns and CFs beyond aggregate and individual own-industry BM ratios. Almost all of the combination forecasts of industry returns using each industry’s BM ratio have positive out-of-sample R2 ( ROS2), and about half are individually significant. On average, the combination forecasts of industry returns have higher ROS2 than those based on the market and industry-own BM ratio by 3.5–4.0% and 4.9–5.4%, respectively. In contrast, BM ratios of each industry predict those industry’s returns with mostly negative ROS2. Similarly, the combination forecasts predict returns on the twenty-five size and BM portfolios with mostly positive ROS2. Depending on method, seven–seventeen of these ROS2 are individually significant. Unlike the combination forecasts, the market and portfolio-own BM ratios predict returns on the twenty-five portfolios with mostly negative ROS2. Thus, combining the forecasts from the cross-section of BM ratios improves real-time forecasts of most industry and size and BM portfolio returns relative to forecasts based only on those portfolio’s BM ratios or the market BM ratio. Long–short industry-rotation strategies based on the combination forecasts (hereafter “combo strategies”) earn large average returns, ranging from 7.7% to 11.6% per year. After transaction costs, the long–short industry-rotation portfolios significantly outperform the standard four-factor performance attribution model of Fama and French (1993) and Carhart (1997) by 5.8–8.9% per year. The long–short industry combo strategies have large positive loadings on the momentum factor, though much lower turnover and transaction costs than the momentum strategy. Thus, the gross four-factor α understates the net-of-costs performance of the industry combo strategies. Interestingly, the momentum exposure is not explained by lagged own-industry returns, but from the signals of other industries. The momentum loadings of the long–short strategies based only on industry-own BM ratios are negative. Adding the industry long–short combo strategies to the Fama–French–Carhart factors also increases the maximum net-of-costs Sharpe ratio by as much as adding two or three of the Fama–French–Carhart factors to the market return. The long–short combo strategies using the characteristic portfolios also produce large spreads in gross returns of 8.7–15.3% per year, and significant spreads in gross alphas of 4.7–8.0% per year. However, transaction costs of these strategies are higher than the industries as portfolios with the highest and lowest returns tend to consist of small-cap stocks. For example, among the size and BM portfolios, the highest and lowest BM stocks in the smallest size quintile tend to have the highest and lowest returns on average, but also the highest transaction costs. Transaction costs reduce returns on strategies made from characteristic-sorted portfolios by approximately 4% per year and eliminate the significance of alphas earned by several strategies. This paper relates to several recent studies. We have similar motivation as Kelly and Pruitt (2013) who incorporate the cross-section of BM ratios in forecasting the return on the aggregate US stock market. In contrast, we use the cross-section of BM ratios to forecast returns of industry and characteristic portfolios, different forecast combination methods, more timely accounting data, and apply these forecasts to trading strategies. Lewellen (2015) also predicts the cross-section of returns in real time with BM ratios and other characteristics. However, Lewellen focuses on the returns of individual stocks and those stock’s own characteristics, not the characteristics of other stocks. Rapach et al. (2015) find evidence of cross-industry predictability for a subset of industries, but they use industry returns as predictors. The remainder of this paper proceeds as follows. Section 2 motivates why BM ratios of different stocks are useful for predicting the returns of any given stock. Section 3 describes our data and combination forecast methods. Section 4 presents our combination forecasts of industry returns and CFs. Section 5 describes the construction of our forecast-based trading strategies and analyzes their performance. Section 6 concludes. 2. Information in BM Ratios Cohen, Polk, and Vuolteenaho (2003) posit a log-linear approximation to decompose the BM ratio for a stock i into log stock return (rt), log return-on-equity (et), and an approximation error (kt): BMi,t≈kt+∑j=1∞ρjri,t+j+∑j=1∞ρj(−ei,t+j), (1) where ρ<1 is a constant arising from the approximation. For ease of illustration, assume that μit=Et(ri,t+1) and git=Et(ei,t+1) follow AR(1) processes (e.g., van Binsbergen and Koijen, 2010; Golez, 2014). Then, taking expectations of Equation (1) yields: BMi,t≈kt+φi,rμit−φi,ggit, (2) where φi,r and φi,g are constants. To understand how the BM ratio of one stock or portfolio of stocks can help predict the returns (or CFs) of another, suppose that a set of stocks has a non-zero common component of expected returns and profitability. That is, assume μi,t and git can be decomposed into common and idiosyncratic components (denoted by *): μi,t=μt+μi,t*, and (3) gi,t=gt+gi,t*. (4) This commonality can be motivated by the factor structure in returns, inter-industry correlation, or the positive betas of most stocks with the market return. By Equation (2) regressions of ri,t+1 on the BM ratio of stock i, and that of another stock j take the form: E(ri,t+1|BMi,t)=α^ii+β^ii(kt+φi,r(μt+μi,t*)−φi,g(gt+gi,t*)), and (5) E(ri,t+1|BMj,t)=α^ij+β^ij(kt+φj,r(μt+μj,t*)−φj,g(gt+gj,t*)), (6) where μit and git are latent, and must be inferred from observables like BMit. Assuming μt≠0, both stock’s BM predict ri,t+1 ( β^ii>0 and β^ij>0). If gi,t is constant, then BMi,t is a perfect proxy for expected returns (μit) and BMj,t is redundant for j≠i. However, assuming gi,t is time-varying, as evidence suggests, BMi,t is not a perfect proxy for μi,t (e.g., Kelly and Pruitt, 2013). Barring a knife-edge correlation between git and gjt, BMj,t provides a non-redundant signal about the common μt, and therefore ri,t+1. Below, we combine these signals to forecast returns. 3. Data and Methodology 3.1 Return, CF, and Predictor Data We use monthly frequency value-weighted portfolio return data from Kenneth French’s website for the thirty-eight and forty-eight industries as well as each of the twenty-five portfolios formed on size and BM ratio, investment, or operating profitability. We refer to the latter three sets of portfolios as size/BM, size/investment, and size/profitability, respectively. We form quarterly returns by compounding monthly returns. We exclude industries that have ten or fewer firms at the start of the sample as they have long periods with missing returns and frequently cannot be matched with any accounting data. Our sample thus consists of thirty-two and forty-four of the thirty-eight and forty-eight industries, respectively.1 We also construct quarterly Fama and French (2015) and Carhart (1997) size, value, momentum, investment, and profitability factors (SMB, HML, MOM, CMA, and RMW, respectively) from the monthly returns on the corresponding 2× 3 base assets. We take the quarterly risk-free rate rft to be the compounded one-month bill rate. For each of the portfolios listed above, we estimate quarterly portfolio-level accounting and predictor variables as value-weighted averages of the stock-level counterparts within the portfolios. We obtain quarterly accounting data from COMPUSTAT Quarterly. We follow Hou, Xue, and Zhang (2015) and define book value of equity as shareholders’ equity, plus balance-sheet deferred taxes and investment tax credit (item TXDITCQ) if available, minus the book value of preferred stock. Depending on availability, we use stockholders’ equity (item SEQQ), common equity (item CEQQ) plus the carrying value of preferred stock (item PSTKQ), or total assets (item ATQ) minus total liabilities (item LTQ) in that order as shareholders equity. We use redemption value (item PSTKRQ) if available, or carrying value for the book value of preferred stock. Hou, Xue, and Zhang (2015) argue this definition of book equity provides the broadest coverage, particularly prior to 1980. To compute BM, we divide the book value of equity by the end-of-quarter market value of equity from CRSP. We compute operating CF following Hirshleifer, Hou, and Teoh (2009). Due to availability of COMPUSTAT quarterly data, we begin our analysis in the first quarter of 1972 following Hou, Xue, and Zhang (2015). To have a reasonable number of in-sample observations for initial estimation, while maintaining a long out-of-sample period, we begin out-of-sample analysis in 1980:1. This out-of-sample period includes a variety of market environments including the bull market of the 1990s, the dot-com collapse in 2000, a period of steady gains in the mid-2000s, the recent financial crisis of the late 2000s, and subsequent era of unconventional monetary policy. To evaluate the ability of forecast methods to generate real-time investment opportunities, it is important for accounting data to be available at the time of portfolio adjustments. Hence, we follow Hirshleifer, Hou, and Teoh (2009); Lewellen (2015); and Hou, Xue, and Zhang (2015), and lag accounting data by 4 months in forecasting returns. In spite of this lag, our quarterly accounting data are more timely than commonly used annual observations (e.g., Fama and French, 1993), which can easily be more than a year old. Using timely data allows us to construct forecasts that best approximate the market’s conditional expectations. Table I reports select summary statistics for the BM ratios, CFs, and returns of each of the forty-four industries and the value-weighted average across stocks (MKT). Average BM ratios range from 0.33 for DRUGS to 1.22 for AUTO, and 0.68 for MKT. The standard deviations display considerable dispersion from 0.12 for BUSSV to 0.61 for MINES, and 0.19 for MKT. The average industry’s BM is persistent with an AR1 of 0.88 and all but one industry has a BM with an AR1 of at least 0.75.2 The market BM is even more persistent with a one-quarter autocorrelation of 0.98. In predicting returns, Stambaugh (1999) and Ferson, Sarkissian, and Simin (2003) find that predictors with high persistence like BM ratios can produce inflated in-sample forecasting slopes and R2. This persistence bias further motivates examining the out-of-sample performance of returns forecasts with BM ratios. Table I. Summary statistics of industry BM ratios For each of the forty-four out of forty-eight Fama–French industries we use in the paper and the market portfolio, we present: time-series averages (Mean), standard deviations (SD), and first autocorrelations ( AR1) of the value-weighted industry BM ratios, as well as the average correlation between each industry’s BM, annual cash flow growth, and quarterly returns with those of all other industries ( ρ¯BM, ρ¯CF, and ρ¯ret, respectively). The ρ¯CF reports the average absolute value of the pairwise correlation as industry CFs can be negatively correlated. The sample is 1980:1–2015:4. Mean SD AR1 ρ¯BM ρ¯CF ρ¯ret Mean SD AR1 ρ¯BM ρ¯CF ρ¯ret AERO 0.71 0.44 0.96 0.55 0.14 0.61 INSUR 0.89 0.25 0.90 0.56 0.21 0.45 AGRIC 0.73 0.53 0.93 0.46 0.28 0.66 LABEQ 0.56 0.26 0.92 0.65 0.30 0.63 AUTOS 1.23 0.53 0.86 0.26 0.35 0.57 MACH 0.62 0.24 0.93 0.62 0.27 0.64 BANKS 0.89 0.31 0.91 0.53 0.28 0.63 MEALS 0.45 0.18 0.91 0.55 0.24 0.61 BEER 0.49 0.21 0.96 0.57 0.22 0.63 MEDEQ 0.42 0.19 0.95 0.59 0.15 0.61 BLDMT 0.70 0.24 0.84 0.59 0.31 0.52 MINES 1.22 0.61 0.06 0.00 0.21 0.63 BOOKS 0.60 0.31 0.82 0.27 0.13 0.52 OIL 0.89 0.22 0.82 0.40 0.20 0.65 BOXES 0.78 0.35 0.94 0.55 0.21 0.67 OTHER 0.62 0.27 0.94 0.09 0.25 0.51 BUSSV 0.36 0.12 0.95 0.46 0.15 0.57 PAPER 0.63 0.17 0.94 0.52 0.23 0.58 CHEM 0.63 0.23 0.93 0.55 0.18 0.65 PERSV 0.58 0.25 0.86 0.59 0.25 0.52 CHIPS 0.48 0.13 0.82 0.50 0.24 0.53 RLEST 0.96 0.43 0.86 0.52 0.14 0.63 CLTHS 0.69 0.45 0.94 0.59 0.18 0.49 RTAIL 0.50 0.22 0.96 0.61 0.22 0.61 CNSTR 0.77 0.21 0.79 0.56 0.34 0.61 RUBBR 0.69 0.31 0.96 0.62 0.25 0.67 COMPS 0.47 0.16 0.88 0.51 0.33 0.15 SHIPS 0.79 0.35 0.75 0.54 0.11 0.63 DRUGS 0.33 0.13 0.94 0.46 0.24 0.53 STEEL 1.14 0.36 0.87 0.47 0.30 0.44 ELCEQ 0.81 0.30 0.85 0.18 0.31 0.58 TELCM 0.97 0.34 0.91 0.41 0.21 0.67 FABPR 0.86 0.34 0.81 0.39 0.25 0.65 TOYS 0.60 0.24 0.86 0.59 0.23 0.53 FIN 0.75 0.29 0.85 0.62 0.23 0.63 TRANS 0.98 0.37 0.93 0.62 0.30 0.69 FOOD 0.56 0.20 0.96 0.59 0.25 0.66 TXTLS 0.98 0.45 0.92 0.61 0.30 0.67 FUN 0.65 0.30 0.85 0.55 0.22 0.62 UTIL 1.11 0.22 0.92 0.54 0.31 0.59 GOLD 0.58 0.19 0.92 0.19 0.33 0.61 WHLSL 0.68 0.27 0.95 0.63 0.26 0.63 HLTH 0.59 0.35 0.82 0.40 0.20 0.52 AVG 0.71 0.29 0.88 0.51 0.24 0.59 HSHLD 0.40 0.18 0.97 0.46 0.21 0.60 AGG 0.68 0.19 0.98 0.60 0.10 0.69 Mean SD AR1 ρ¯BM ρ¯CF ρ¯ret Mean SD AR1 ρ¯BM ρ¯CF ρ¯ret AERO 0.71 0.44 0.96 0.55 0.14 0.61 INSUR 0.89 0.25 0.90 0.56 0.21 0.45 AGRIC 0.73 0.53 0.93 0.46 0.28 0.66 LABEQ 0.56 0.26 0.92 0.65 0.30 0.63 AUTOS 1.23 0.53 0.86 0.26 0.35 0.57 MACH 0.62 0.24 0.93 0.62 0.27 0.64 BANKS 0.89 0.31 0.91 0.53 0.28 0.63 MEALS 0.45 0.18 0.91 0.55 0.24 0.61 BEER 0.49 0.21 0.96 0.57 0.22 0.63 MEDEQ 0.42 0.19 0.95 0.59 0.15 0.61 BLDMT 0.70 0.24 0.84 0.59 0.31 0.52 MINES 1.22 0.61 0.06 0.00 0.21 0.63 BOOKS 0.60 0.31 0.82 0.27 0.13 0.52 OIL 0.89 0.22 0.82 0.40 0.20 0.65 BOXES 0.78 0.35 0.94 0.55 0.21 0.67 OTHER 0.62 0.27 0.94 0.09 0.25 0.51 BUSSV 0.36 0.12 0.95 0.46 0.15 0.57 PAPER 0.63 0.17 0.94 0.52 0.23 0.58 CHEM 0.63 0.23 0.93 0.55 0.18 0.65 PERSV 0.58 0.25 0.86 0.59 0.25 0.52 CHIPS 0.48 0.13 0.82 0.50 0.24 0.53 RLEST 0.96 0.43 0.86 0.52 0.14 0.63 CLTHS 0.69 0.45 0.94 0.59 0.18 0.49 RTAIL 0.50 0.22 0.96 0.61 0.22 0.61 CNSTR 0.77 0.21 0.79 0.56 0.34 0.61 RUBBR 0.69 0.31 0.96 0.62 0.25 0.67 COMPS 0.47 0.16 0.88 0.51 0.33 0.15 SHIPS 0.79 0.35 0.75 0.54 0.11 0.63 DRUGS 0.33 0.13 0.94 0.46 0.24 0.53 STEEL 1.14 0.36 0.87 0.47 0.30 0.44 ELCEQ 0.81 0.30 0.85 0.18 0.31 0.58 TELCM 0.97 0.34 0.91 0.41 0.21 0.67 FABPR 0.86 0.34 0.81 0.39 0.25 0.65 TOYS 0.60 0.24 0.86 0.59 0.23 0.53 FIN 0.75 0.29 0.85 0.62 0.23 0.63 TRANS 0.98 0.37 0.93 0.62 0.30 0.69 FOOD 0.56 0.20 0.96 0.59 0.25 0.66 TXTLS 0.98 0.45 0.92 0.61 0.30 0.67 FUN 0.65 0.30 0.85 0.55 0.22 0.62 UTIL 1.11 0.22 0.92 0.54 0.31 0.59 GOLD 0.58 0.19 0.92 0.19 0.33 0.61 WHLSL 0.68 0.27 0.95 0.63 0.26 0.63 HLTH 0.59 0.35 0.82 0.40 0.20 0.52 AVG 0.71 0.29 0.88 0.51 0.24 0.59 HSHLD 0.40 0.18 0.97 0.46 0.21 0.60 AGG 0.68 0.19 0.98 0.60 0.10 0.69 Table I. Summary statistics of industry BM ratios For each of the forty-four out of forty-eight Fama–French industries we use in the paper and the market portfolio, we present: time-series averages (Mean), standard deviations (SD), and first autocorrelations ( AR1) of the value-weighted industry BM ratios, as well as the average correlation between each industry’s BM, annual cash flow growth, and quarterly returns with those of all other industries ( ρ¯BM, ρ¯CF, and ρ¯ret, respectively). The ρ¯CF reports the average absolute value of the pairwise correlation as industry CFs can be negatively correlated. The sample is 1980:1–2015:4. Mean SD AR1 ρ¯BM ρ¯CF ρ¯ret Mean SD AR1 ρ¯BM ρ¯CF ρ¯ret AERO 0.71 0.44 0.96 0.55 0.14 0.61 INSUR 0.89 0.25 0.90 0.56 0.21 0.45 AGRIC 0.73 0.53 0.93 0.46 0.28 0.66 LABEQ 0.56 0.26 0.92 0.65 0.30 0.63 AUTOS 1.23 0.53 0.86 0.26 0.35 0.57 MACH 0.62 0.24 0.93 0.62 0.27 0.64 BANKS 0.89 0.31 0.91 0.53 0.28 0.63 MEALS 0.45 0.18 0.91 0.55 0.24 0.61 BEER 0.49 0.21 0.96 0.57 0.22 0.63 MEDEQ 0.42 0.19 0.95 0.59 0.15 0.61 BLDMT 0.70 0.24 0.84 0.59 0.31 0.52 MINES 1.22 0.61 0.06 0.00 0.21 0.63 BOOKS 0.60 0.31 0.82 0.27 0.13 0.52 OIL 0.89 0.22 0.82 0.40 0.20 0.65 BOXES 0.78 0.35 0.94 0.55 0.21 0.67 OTHER 0.62 0.27 0.94 0.09 0.25 0.51 BUSSV 0.36 0.12 0.95 0.46 0.15 0.57 PAPER 0.63 0.17 0.94 0.52 0.23 0.58 CHEM 0.63 0.23 0.93 0.55 0.18 0.65 PERSV 0.58 0.25 0.86 0.59 0.25 0.52 CHIPS 0.48 0.13 0.82 0.50 0.24 0.53 RLEST 0.96 0.43 0.86 0.52 0.14 0.63 CLTHS 0.69 0.45 0.94 0.59 0.18 0.49 RTAIL 0.50 0.22 0.96 0.61 0.22 0.61 CNSTR 0.77 0.21 0.79 0.56 0.34 0.61 RUBBR 0.69 0.31 0.96 0.62 0.25 0.67 COMPS 0.47 0.16 0.88 0.51 0.33 0.15 SHIPS 0.79 0.35 0.75 0.54 0.11 0.63 DRUGS 0.33 0.13 0.94 0.46 0.24 0.53 STEEL 1.14 0.36 0.87 0.47 0.30 0.44 ELCEQ 0.81 0.30 0.85 0.18 0.31 0.58 TELCM 0.97 0.34 0.91 0.41 0.21 0.67 FABPR 0.86 0.34 0.81 0.39 0.25 0.65 TOYS 0.60 0.24 0.86 0.59 0.23 0.53 FIN 0.75 0.29 0.85 0.62 0.23 0.63 TRANS 0.98 0.37 0.93 0.62 0.30 0.69 FOOD 0.56 0.20 0.96 0.59 0.25 0.66 TXTLS 0.98 0.45 0.92 0.61 0.30 0.67 FUN 0.65 0.30 0.85 0.55 0.22 0.62 UTIL 1.11 0.22 0.92 0.54 0.31 0.59 GOLD 0.58 0.19 0.92 0.19 0.33 0.61 WHLSL 0.68 0.27 0.95 0.63 0.26 0.63 HLTH 0.59 0.35 0.82 0.40 0.20 0.52 AVG 0.71 0.29 0.88 0.51 0.24 0.59 HSHLD 0.40 0.18 0.97 0.46 0.21 0.60 AGG 0.68 0.19 0.98 0.60 0.10 0.69 Mean SD AR1 ρ¯BM ρ¯CF ρ¯ret Mean SD AR1 ρ¯BM ρ¯CF ρ¯ret AERO 0.71 0.44 0.96 0.55 0.14 0.61 INSUR 0.89 0.25 0.90 0.56 0.21 0.45 AGRIC 0.73 0.53 0.93 0.46 0.28 0.66 LABEQ 0.56 0.26 0.92 0.65 0.30 0.63 AUTOS 1.23 0.53 0.86 0.26 0.35 0.57 MACH 0.62 0.24 0.93 0.62 0.27 0.64 BANKS 0.89 0.31 0.91 0.53 0.28 0.63 MEALS 0.45 0.18 0.91 0.55 0.24 0.61 BEER 0.49 0.21 0.96 0.57 0.22 0.63 MEDEQ 0.42 0.19 0.95 0.59 0.15 0.61 BLDMT 0.70 0.24 0.84 0.59 0.31 0.52 MINES 1.22 0.61 0.06 0.00 0.21 0.63 BOOKS 0.60 0.31 0.82 0.27 0.13 0.52 OIL 0.89 0.22 0.82 0.40 0.20 0.65 BOXES 0.78 0.35 0.94 0.55 0.21 0.67 OTHER 0.62 0.27 0.94 0.09 0.25 0.51 BUSSV 0.36 0.12 0.95 0.46 0.15 0.57 PAPER 0.63 0.17 0.94 0.52 0.23 0.58 CHEM 0.63 0.23 0.93 0.55 0.18 0.65 PERSV 0.58 0.25 0.86 0.59 0.25 0.52 CHIPS 0.48 0.13 0.82 0.50 0.24 0.53 RLEST 0.96 0.43 0.86 0.52 0.14 0.63 CLTHS 0.69 0.45 0.94 0.59 0.18 0.49 RTAIL 0.50 0.22 0.96 0.61 0.22 0.61 CNSTR 0.77 0.21 0.79 0.56 0.34 0.61 RUBBR 0.69 0.31 0.96 0.62 0.25 0.67 COMPS 0.47 0.16 0.88 0.51 0.33 0.15 SHIPS 0.79 0.35 0.75 0.54 0.11 0.63 DRUGS 0.33 0.13 0.94 0.46 0.24 0.53 STEEL 1.14 0.36 0.87 0.47 0.30 0.44 ELCEQ 0.81 0.30 0.85 0.18 0.31 0.58 TELCM 0.97 0.34 0.91 0.41 0.21 0.67 FABPR 0.86 0.34 0.81 0.39 0.25 0.65 TOYS 0.60 0.24 0.86 0.59 0.23 0.53 FIN 0.75 0.29 0.85 0.62 0.23 0.63 TRANS 0.98 0.37 0.93 0.62 0.30 0.69 FOOD 0.56 0.20 0.96 0.59 0.25 0.66 TXTLS 0.98 0.45 0.92 0.61 0.30 0.67 FUN 0.65 0.30 0.85 0.55 0.22 0.62 UTIL 1.11 0.22 0.92 0.54 0.31 0.59 GOLD 0.58 0.19 0.92 0.19 0.33 0.61 WHLSL 0.68 0.27 0.95 0.63 0.26 0.63 HLTH 0.59 0.35 0.82 0.40 0.20 0.52 AVG 0.71 0.29 0.88 0.51 0.24 0.59 HSHLD 0.40 0.18 0.97 0.46 0.21 0.60 AGG 0.68 0.19 0.98 0.60 0.10 0.69 The average pairwise correlation across the industry BM is less than 0.5. The average correlation between industry returns is 0.59 and the average correlation between industry CFs is 0.24. The substantial commonality in returns and imperfect correlation in CFs illustrates how BM ratios should contain useful signals for the expected returns on different industries as argued in Section 2. 3.2 Transaction Cost Data In this paper, we evaluate the performance of trading strategies net of transaction costs. We closely follow Novy-Marx and Velikov (2016) and compute strategy trading costs in two steps. First, we estimate effective one-way stock-level transaction costs (“spreads” or “bid–ask spreads”) using daily CRSP return data following Hasbrouck (2009).3 Second, we use these stock-level spread estimates to compute portfolio-level costs. Hasbrouck (2009) shows this is an accurate measure of transaction costs as it has a 96.5% correlation with effective spreads based directly on Trade and Quote (TAQ) data. The primary limitation of the Hasbrouck spread measure is that it does not consider the price impact of very large trades. However, it is an upper bound of trading costs for small trades because it assumes market orders. Overall, this measure captures the marginal cost of a strategy for small traders. Additionally, since we use value-weighted industries, our portfolios are dominated by large stocks that are less subject to large price impacts. Panel A of Figure 1 depicts the value-weighted averages of our estimated one-way transaction costs of NYSE, AMEX, and NASDAQ stocks by NYSE size quintile each quarter from 1980:1 to 2015:4. Overall the figure looks similar to analogous plots in Hasbrouck (2009). Size negatively varies with transaction costs and the smallest size quintile has several times the transaction costs of the largest quintile. Transaction costs are known to decrease with size for several reasons (e.g., Stoll and Whaley, 1983; Bhushan, 1989; Amihud, 2002). Small-cap stocks are relatively costly for market makers to hold because they trade infrequently and have relatively high risk. Moreover, analyst coverage of small-cap stocks is lower and the risk of adverse selection higher. Search costs of small-cap stocks are also relatively high because they have a lower dollar volume of actively traded shares. Panel B of Figure 1 shows the distribution of time-series means of the estimated effective transaction costs of the value-weighted Fama–French forty-four industry portfolios. The industry transaction costs range from 20 to 69 basis points and average 35 basis points. The relatively low cost estimates indicate that value-weighted industries are relatively inexpensive to trade because they are dominated by large-cap stocks. Figure 1. View largeDownload slide Relevant patterns in transaction costs. Panel A depicts the time-series of value-weighted averages of effective one-way transaction costs for NYSE, AMEX, and NASDAQ stocks by NYSE size quintile. Panel B shows the histogram of time-series means of the value-weighted averages of estimated one-way transaction costs of stocks in each of the Fama–French forty-four industry portfolios (in units of %). Panel C depicts the four-quarter moving average of the transaction costs associated with the Fama–French size, value, momentum, investment, and profitability factors (SMB, HML, MOM, CMA, and RMW). The sample for all Panels is 1980:1–2015:4. Figure 1. View largeDownload slide Relevant patterns in transaction costs. Panel A depicts the time-series of value-weighted averages of effective one-way transaction costs for NYSE, AMEX, and NASDAQ stocks by NYSE size quintile. Panel B shows the histogram of time-series means of the value-weighted averages of estimated one-way transaction costs of stocks in each of the Fama–French forty-four industry portfolios (in units of %). Panel C depicts the four-quarter moving average of the transaction costs associated with the Fama–French size, value, momentum, investment, and profitability factors (SMB, HML, MOM, CMA, and RMW). The sample for all Panels is 1980:1–2015:4. Panel A of Figure 1 also shows that transaction costs decline over time along with the difference in transaction costs between large and small stocks. Common explanations for this decline include improvements in trading technology, decimalization in 2001, and the relatively aggressive delisting of small and illiquid NASDAQ stocks in the 1990s (e.g., Chakravarty, Panchapagesan, and Wood, 2005; Hasbrouck, 2009; Chordia, Subrahmanyam, and Tong, 2014). Panel C of Figure 1 depicts the four-quarter moving average trading costs of the Fama–French size, value, momentum, investment, and profitability factors (SMB, HML, MOM, CMA, and RMW) over our out-of-sample period 1980:1–2015:4.4 All of the factors except MOM are rebalanced only at the end of June each year, so on average their transaction costs are relatively modest at about 25 basis points per quarter. In contrast, MOM has significant monthly turnover leading to transaction costs that are several times higher than those of the other factors. 3.3 Forecasting Methodology The goal of this paper is to forecast industry returns using the whole cross-section of BM ratios, and then exploit this predictability with trading strategies. Regression-based return forecasts often exhibit structural breaks that result in poor out-of-sample performance (e.g., Goyal and Welch, 2008; Rapach, Strauss, and Zhou, 2010). In contrast, combination forecast methods tend to perform well out-of-sample in the presence of model uncertainty and structural breaks when forecasting market returns and other economic time series (e.g., Stock and Watson, 2004; Timmermann, 2006; Rapach, Strauss, and Zhou, 2010). The basic building block for our combination forecasts are standard univariate predictive regressions estimated recursively in real time for each industry i and j: ri,t+1=aij+bijBMtj+ei,t+1j, (7) where ri,t+1 is the excess return for industry i and BMtj is the BM of industry j. A combination forecast for industry i is simply a weighted average of the out-of-sample forecasts from Equation (7): r^i,t+1c=∑j=1Nwj,tc(a^i,tj+b^j,tjBMtj), (8) where c denotes a weighting method, ( a^i,t, b^i,t) are from estimates of Equation (7) based on data through time t – 1, and N is the number of industries. In the main body of the paper, we form univariate forecasts by estimating Equation (7) at each time t using “expanding windows”, that is data from time 1 to time t – 1. In the Online Appendix, we show results for forecasts that use “rolling windows” of data. Different combination forecasts are defined by the choice of weighting schemes {wj,tc}. The different combination forecast weights can be simple functions such as an equal-weighted mean (MEAN, wj,tMEAN≡1/N), or functions of prior forecast performance that give low weight to forecasts that have large past errors, and vice versa. If the forecast errors of the individual forecasts have equal variance and equal pairwise correlation, the MEAN combination method is optimal in that it produces the combination forecast with the minimum mean-squared forecast error. Further, MEAN involves no estimation error and therefore often empirically outperforms estimates of theoretically “optimal” weights in finite samples (e.g., Timmermann, 2008). There is generally no ex ante optimal combination method for a given time series, it is an empirical question (e.g., Timmermann, 2008). We therefore compare several methods, the MEAN method and two performance-based combination forecasts. The first performance-based method, Approximate Bayesian Model Averaging (ABMA), follows Garratt et al. (2003) and chooses: wi,tABMA= exp ⁡(Δi,t)∑j=1n exp ⁡(Δj,t), (9) where Δi,t=AICi,t−max⁡j(AICj,t), and AICi,t is the Akaike Information Criterion of model i. The ABMA thus gives higher weight to models with better historical fit measured by AIC. Garratt et al. (2003) argue the AIC is preferred when the “correct” model is possibly not in the set under consideration. The second performance-based method, the discounted mean-squared forecast error (DMSFE) follows Bates and Granger (1969) and Stock and Watson (2004) and chooses: wi,tDMSFE=φi,t−1∑j=1nφi,t−1, (10) where φi,t=∑s=1t−1θt−1−s(ri,s+1−r^i,s+1)2. (11) If θ = 1, DMSFE does not discount forecast errors further in the past. Because our individual forecasts have the same number of parameters, this generates nearly identical results to the ABMA, because the likelihood functions are driven by non-discounted MSFE. Hence, we choose θ=0.7 to examine the impact of discounting forecast errors further back in time. By discounting past observations more heavily, DMSFE works relatively well if the data-generating process is time-varying. However, the cost of discounting is a lower effective sample size and therefore higher volatility of estimated weights, which reduces forecast accuracy all else equal. 3.4 Forecast Evaluation Following Campbell and Thompson (2008), we constrain all predicted excess returns to be non-negative as (i) the equity risk premium should be positive, and (ii) in practice, these restrictions improve the out-of-sample performance of predictive regressions.5 We use the standard out-of-sample R2 statistic, ROS2, to evaluate the out-of-sample accuracy of our forecasts relative to the historical-average forecast r¯i,t+1, the natural benchmark under the null of no predictability. The ROS2 statistic is defined by: ROS2≡1−∑t=q+1T(ri,t−r^i,tc)2∑t=q+1T(ri,t−r¯i,t)2, (12) where q denotes the end of an initial in-sample period used to generate the first out-of-sample forecast from Equation (7). We select q=1979:4, yielding thirty-two initial in-sample quarters. When ROS2>0, r^i,tc outperforms the historical average forecast by earning a lower mean-squared-forecast error. We assess the significance of the ROS2 with the Clark and West (2007) MSFE-adjusted statistic. 4. Forecasting Results 4.1 In-Sample Results We argue above that the cross-section of BM should be useful for forecasting returns and CFs of stock portfolios. In this section, we provide evidence of this for the forty-four industry portfolios. Specifically, we estimate predictive regressions of CFs and excess log returns (r) for each industry i over the following four and one quarters, respectively, ∑j=14CFi,t+j=αiCF+βiCFXt, (13) ri,t+1=αir+βirXt. (14) Using four quarters in Equation (13) mitigates the effects of earnings seasonality. We consider four different sets of predictors X: the log BM for industry i (BMi), the log BM for the aggregate market (BMM), the first six principal components of the forty-four industry log BM, and principal components 2–6. Principal components 2–6 exclude the largest common trend in BM ratios but contain information across industries.6 Using timely market values in BM when predicting returns could pick up a momentum effect. Hence, we also consider the return over the prior four quarters ( Xt=rt−3,t) as a predictor in Equation (14). Table II contains in-sample adjusted R2 from the models given by Equations (13) and (14). Table II. In-sample R2 statistics from predictive regressions of industry CFs and returns on BM ratios For each of the forty-four Fama–French industries, this table presents in-sample adjusted-R2 statistics for predictive regressions of industry CFs over quarters t + 1 to t + 4 and returns (Returns) over quarter t + 1. In columns denoted BMi, the only predictor is the industry’s own BM in quarter t. In columns denoted BMM, the predictor is the market BM. In columns denoted PC and PCI, the predictors are the first six and second–sixth principal components of the forty-four industry BM ratios, respectively. In the column denoted rt−3,t, the predictor is the return over four quarters prior to t + 1. Avg denotes the average of the R2 across industries, and #Sig denotes the number of R2 that are significant at the 5% level. The sample period is 1980:1–2015:4. CF Returns IND BMM PC PCI IND BMM PC PCI rt−3,t AVG 8.16 10.95 46.45 33.52 1.72 3.30 6.11 4.03 −0.07 AERO −0.06 6.65 25.78 25.19 1.31 2.71 5.49 3.95 1.54 AGRIC 2.74 −0.70 33.95 32.35 1.30 6.96 8.15 2.91 0.57 AUTOS 5.04 4.52 28.26 20.48 −0.14 2.39 1.41 0.92 0.60 BANKS 1.69 6.68 22.14 10.10 4.02 5.67 13.34 8.41 −0.48 BEER 9.99 2.14 36.95 37.06 3.24 11.09 14.65 5.81 −0.62 BLDMT 28.98 20.58 78.72 77.47 3.86 4.67 7.90 5.81 1.78 BOOKS 11.74 12.46 81.49 46.47 2.11 7.89 14.22 9.86 −0.67 BOXES 23.25 23.25 51.18 5.20 −0.27 1.21 1.35 1.27 −0.36 BUSSV 0.29 8.18 42.64 29.53 3.25 7.51 12.45 5.40 −0.35 CHEM 2.27 15.76 72.20 56.47 1.90 0.99 0.56 1.08 −0.25 CHIPS 6.41 19.53 70.27 32.67 4.18 4.96 11.36 8.81 −0.54 CLTHS 0.14 3.66 29.23 29.65 1.82 3.88 9.44 6.20 −0.68 CNSTR −0.66 −0.23 22.11 22.04 8.87 4.47 6.21 3.97 −0.65 COMPS 16.94 12.67 48.65 38.36 3.90 3.80 7.07 4.98 −0.10 DRUGS 20.44 4.95 36.23 36.67 0.62 7.62 16.73 8.96 −0.49 ELCEQ 2.30 17.00 35.19 13.09 −0.40 4.19 5.78 5.55 0.41 FABPR −0.71 5.84 26.53 23.10 −0.72 1.27 1.91 1.07 −0.47 FIN 43.47 29.80 52.62 22.06 2.79 2.89 5.94 5.39 0.93 FOOD 1.61 −0.49 57.45 52.27 −0.62 −0.09 5.45 5.42 −0.72 FUN 1.29 15.50 41.18 30.77 0.17 1.00 5.57 4.58 −0.10 GOLD −0.16 29.97 47.15 4.05 −0.70 −0.19 −1.32 −0.89 −0.72 HLTH 3.07 −0.63 32.89 29.84 0.50 0.66 4.56 3.58 0.22 HSHLD 24.83 2.51 57.94 53.71 −0.30 1.54 2.98 2.34 −0.49 INSUR 8.00 −0.50 67.70 66.87 1.60 1.64 0.86 0.26 2.32 LABEQ 1.02 −0.59 31.98 28.57 0.80 0.81 4.56 3.81 −0.43 MACH −0.30 1.53 51.39 46.11 6.34 7.09 7.67 4.11 −0.40 MEALS 0.25 18.45 54.28 47.16 0.78 3.97 6.87 3.75 −0.33 MEDEQ 16.91 4.67 69.89 68.38 −0.67 0.76 1.02 3.25 −0.69 MINES 0.78 2.44 20.55 12.47 1.08 5.89 10.43 1.17 −0.72 OIL −0.39 10.85 63.82 48.83 2.10 3.59 7.53 4.36 −0.72 OTHER 1.26 39.09 63.79 18.14 0.71 1.71 5.46 3.50 0.22 PAPER 13.14 19.59 44.25 33.15 −0.68 0.28 2.46 2.74 2.47 PERSV 27.12 40.02 55.75 4.14 1.61 5.70 4.67 1.45 1.72 RLEST 0.16 1.77 25.88 21.22 2.58 2.06 2.73 0.13 −0.72 RTAIL −0.50 3.34 53.34 50.16 −0.68 −0.39 2.68 3.41 0.19 RUBBR 4.70 5.10 39.65 38.97 −0.25 −0.59 −3.05 −1.35 −0.43 SHIPS −0.98 2.53 39.20 38.71 2.58 5.09 6.69 4.08 −0.70 STEEL 24.12 7.43 30.78 27.88 7.99 5.32 10.18 7.02 0.88 TELCM 0.22 41.94 73.94 31.23 0.41 4.55 8.78 7.53 −0.12 TOYS 3.81 4.03 39.01 29.46 1.90 4.36 4.81 1.55 −0.57 TRANS −0.65 0.20 27.88 28.35 2.30 4.10 7.93 5.76 0.92 TXTLS 15.13 −0.69 32.50 26.17 0.27 0.25 7.81 7.54 −0.56 UTIL 31.22 23.00 62.49 35.78 0.93 0.02 2.35 3.01 −0.24 WHLSL 9.12 17.87 65.11 44.60 3.48 1.77 5.19 5.04 1.21 #SIG 24 33 44 44 14 25 33 32 4 CF Returns IND BMM PC PCI IND BMM PC PCI rt−3,t AVG 8.16 10.95 46.45 33.52 1.72 3.30 6.11 4.03 −0.07 AERO −0.06 6.65 25.78 25.19 1.31 2.71 5.49 3.95 1.54 AGRIC 2.74 −0.70 33.95 32.35 1.30 6.96 8.15 2.91 0.57 AUTOS 5.04 4.52 28.26 20.48 −0.14 2.39 1.41 0.92 0.60 BANKS 1.69 6.68 22.14 10.10 4.02 5.67 13.34 8.41 −0.48 BEER 9.99 2.14 36.95 37.06 3.24 11.09 14.65 5.81 −0.62 BLDMT 28.98 20.58 78.72 77.47 3.86 4.67 7.90 5.81 1.78 BOOKS 11.74 12.46 81.49 46.47 2.11 7.89 14.22 9.86 −0.67 BOXES 23.25 23.25 51.18 5.20 −0.27 1.21 1.35 1.27 −0.36 BUSSV 0.29 8.18 42.64 29.53 3.25 7.51 12.45 5.40 −0.35 CHEM 2.27 15.76 72.20 56.47 1.90 0.99 0.56 1.08 −0.25 CHIPS 6.41 19.53 70.27 32.67 4.18 4.96 11.36 8.81 −0.54 CLTHS 0.14 3.66 29.23 29.65 1.82 3.88 9.44 6.20 −0.68 CNSTR −0.66 −0.23 22.11 22.04 8.87 4.47 6.21 3.97 −0.65 COMPS 16.94 12.67 48.65 38.36 3.90 3.80 7.07 4.98 −0.10 DRUGS 20.44 4.95 36.23 36.67 0.62 7.62 16.73 8.96 −0.49 ELCEQ 2.30 17.00 35.19 13.09 −0.40 4.19 5.78 5.55 0.41 FABPR −0.71 5.84 26.53 23.10 −0.72 1.27 1.91 1.07 −0.47 FIN 43.47 29.80 52.62 22.06 2.79 2.89 5.94 5.39 0.93 FOOD 1.61 −0.49 57.45 52.27 −0.62 −0.09 5.45 5.42 −0.72 FUN 1.29 15.50 41.18 30.77 0.17 1.00 5.57 4.58 −0.10 GOLD −0.16 29.97 47.15 4.05 −0.70 −0.19 −1.32 −0.89 −0.72 HLTH 3.07 −0.63 32.89 29.84 0.50 0.66 4.56 3.58 0.22 HSHLD 24.83 2.51 57.94 53.71 −0.30 1.54 2.98 2.34 −0.49 INSUR 8.00 −0.50 67.70 66.87 1.60 1.64 0.86 0.26 2.32 LABEQ 1.02 −0.59 31.98 28.57 0.80 0.81 4.56 3.81 −0.43 MACH −0.30 1.53 51.39 46.11 6.34 7.09 7.67 4.11 −0.40 MEALS 0.25 18.45 54.28 47.16 0.78 3.97 6.87 3.75 −0.33 MEDEQ 16.91 4.67 69.89 68.38 −0.67 0.76 1.02 3.25 −0.69 MINES 0.78 2.44 20.55 12.47 1.08 5.89 10.43 1.17 −0.72 OIL −0.39 10.85 63.82 48.83 2.10 3.59 7.53 4.36 −0.72 OTHER 1.26 39.09 63.79 18.14 0.71 1.71 5.46 3.50 0.22 PAPER 13.14 19.59 44.25 33.15 −0.68 0.28 2.46 2.74 2.47 PERSV 27.12 40.02 55.75 4.14 1.61 5.70 4.67 1.45 1.72 RLEST 0.16 1.77 25.88 21.22 2.58 2.06 2.73 0.13 −0.72 RTAIL −0.50 3.34 53.34 50.16 −0.68 −0.39 2.68 3.41 0.19 RUBBR 4.70 5.10 39.65 38.97 −0.25 −0.59 −3.05 −1.35 −0.43 SHIPS −0.98 2.53 39.20 38.71 2.58 5.09 6.69 4.08 −0.70 STEEL 24.12 7.43 30.78 27.88 7.99 5.32 10.18 7.02 0.88 TELCM 0.22 41.94 73.94 31.23 0.41 4.55 8.78 7.53 −0.12 TOYS 3.81 4.03 39.01 29.46 1.90 4.36 4.81 1.55 −0.57 TRANS −0.65 0.20 27.88 28.35 2.30 4.10 7.93 5.76 0.92 TXTLS 15.13 −0.69 32.50 26.17 0.27 0.25 7.81 7.54 −0.56 UTIL 31.22 23.00 62.49 35.78 0.93 0.02 2.35 3.01 −0.24 WHLSL 9.12 17.87 65.11 44.60 3.48 1.77 5.19 5.04 1.21 #SIG 24 33 44 44 14 25 33 32 4 Table II. In-sample R2 statistics from predictive regressions of industry CFs and returns on BM ratios For each of the forty-four Fama–French industries, this table presents in-sample adjusted-R2 statistics for predictive regressions of industry CFs over quarters t + 1 to t + 4 and returns (Returns) over quarter t + 1. In columns denoted BMi, the only predictor is the industry’s own BM in quarter t. In columns denoted BMM, the predictor is the market BM. In columns denoted PC and PCI, the predictors are the first six and second–sixth principal components of the forty-four industry BM ratios, respectively. In the column denoted rt−3,t, the predictor is the return over four quarters prior to t + 1. Avg denotes the average of the R2 across industries, and #Sig denotes the number of R2 that are significant at the 5% level. The sample period is 1980:1–2015:4. CF Returns IND BMM PC PCI IND BMM PC PCI rt−3,t AVG 8.16 10.95 46.45 33.52 1.72 3.30 6.11 4.03 −0.07 AERO −0.06 6.65 25.78 25.19 1.31 2.71 5.49 3.95 1.54 AGRIC 2.74 −0.70 33.95 32.35 1.30 6.96 8.15 2.91 0.57 AUTOS 5.04 4.52 28.26 20.48 −0.14 2.39 1.41 0.92 0.60 BANKS 1.69 6.68 22.14 10.10 4.02 5.67 13.34 8.41 −0.48 BEER 9.99 2.14 36.95 37.06 3.24 11.09 14.65 5.81 −0.62 BLDMT 28.98 20.58 78.72 77.47 3.86 4.67 7.90 5.81 1.78 BOOKS 11.74 12.46 81.49 46.47 2.11 7.89 14.22 9.86 −0.67 BOXES 23.25 23.25 51.18 5.20 −0.27 1.21 1.35 1.27 −0.36 BUSSV 0.29 8.18 42.64 29.53 3.25 7.51 12.45 5.40 −0.35 CHEM 2.27 15.76 72.20 56.47 1.90 0.99 0.56 1.08 −0.25 CHIPS 6.41 19.53 70.27 32.67 4.18 4.96 11.36 8.81 −0.54 CLTHS 0.14 3.66 29.23 29.65 1.82 3.88 9.44 6.20 −0.68 CNSTR −0.66 −0.23 22.11 22.04 8.87 4.47 6.21 3.97 −0.65 COMPS 16.94 12.67 48.65 38.36 3.90 3.80 7.07 4.98 −0.10 DRUGS 20.44 4.95 36.23 36.67 0.62 7.62 16.73 8.96 −0.49 ELCEQ 2.30 17.00 35.19 13.09 −0.40 4.19 5.78 5.55 0.41 FABPR −0.71 5.84 26.53 23.10 −0.72 1.27 1.91 1.07 −0.47 FIN 43.47 29.80 52.62 22.06 2.79 2.89 5.94 5.39 0.93 FOOD 1.61 −0.49 57.45 52.27 −0.62 −0.09 5.45 5.42 −0.72 FUN 1.29 15.50 41.18 30.77 0.17 1.00 5.57 4.58 −0.10 GOLD −0.16 29.97 47.15 4.05 −0.70 −0.19 −1.32 −0.89 −0.72 HLTH 3.07 −0.63 32.89 29.84 0.50 0.66 4.56 3.58 0.22 HSHLD 24.83 2.51 57.94 53.71 −0.30 1.54 2.98 2.34 −0.49 INSUR 8.00 −0.50 67.70 66.87 1.60 1.64 0.86 0.26 2.32 LABEQ 1.02 −0.59 31.98 28.57 0.80 0.81 4.56 3.81 −0.43 MACH −0.30 1.53 51.39 46.11 6.34 7.09 7.67 4.11 −0.40 MEALS 0.25 18.45 54.28 47.16 0.78 3.97 6.87 3.75 −0.33 MEDEQ 16.91 4.67 69.89 68.38 −0.67 0.76 1.02 3.25 −0.69 MINES 0.78 2.44 20.55 12.47 1.08 5.89 10.43 1.17 −0.72 OIL −0.39 10.85 63.82 48.83 2.10 3.59 7.53 4.36 −0.72 OTHER 1.26 39.09 63.79 18.14 0.71 1.71 5.46 3.50 0.22 PAPER 13.14 19.59 44.25 33.15 −0.68 0.28 2.46 2.74 2.47 PERSV 27.12 40.02 55.75 4.14 1.61 5.70 4.67 1.45 1.72 RLEST 0.16 1.77 25.88 21.22 2.58 2.06 2.73 0.13 −0.72 RTAIL −0.50 3.34 53.34 50.16 −0.68 −0.39 2.68 3.41 0.19 RUBBR 4.70 5.10 39.65 38.97 −0.25 −0.59 −3.05 −1.35 −0.43 SHIPS −0.98 2.53 39.20 38.71 2.58 5.09 6.69 4.08 −0.70 STEEL 24.12 7.43 30.78 27.88 7.99 5.32 10.18 7.02 0.88 TELCM 0.22 41.94 73.94 31.23 0.41 4.55 8.78 7.53 −0.12 TOYS 3.81 4.03 39.01 29.46 1.90 4.36 4.81 1.55 −0.57 TRANS −0.65 0.20 27.88 28.35 2.30 4.10 7.93 5.76 0.92 TXTLS 15.13 −0.69 32.50 26.17 0.27 0.25 7.81 7.54 −0.56 UTIL 31.22 23.00 62.49 35.78 0.93 0.02 2.35 3.01 −0.24 WHLSL 9.12 17.87 65.11 44.60 3.48 1.77 5.19 5.04 1.21 #SIG 24 33 44 44 14 25 33 32 4 CF Returns IND BMM PC PCI IND BMM PC PCI rt−3,t AVG 8.16 10.95 46.45 33.52 1.72 3.30 6.11 4.03 −0.07 AERO −0.06 6.65 25.78 25.19 1.31 2.71 5.49 3.95 1.54 AGRIC 2.74 −0.70 33.95 32.35 1.30 6.96 8.15 2.91 0.57 AUTOS 5.04 4.52 28.26 20.48 −0.14 2.39 1.41 0.92 0.60 BANKS 1.69 6.68 22.14 10.10 4.02 5.67 13.34 8.41 −0.48 BEER 9.99 2.14 36.95 37.06 3.24 11.09 14.65 5.81 −0.62 BLDMT 28.98 20.58 78.72 77.47 3.86 4.67 7.90 5.81 1.78 BOOKS 11.74 12.46 81.49 46.47 2.11 7.89 14.22 9.86 −0.67 BOXES 23.25 23.25 51.18 5.20 −0.27 1.21 1.35 1.27 −0.36 BUSSV 0.29 8.18 42.64 29.53 3.25 7.51 12.45 5.40 −0.35 CHEM 2.27 15.76 72.20 56.47 1.90 0.99 0.56 1.08 −0.25 CHIPS 6.41 19.53 70.27 32.67 4.18 4.96 11.36 8.81 −0.54 CLTHS 0.14 3.66 29.23 29.65 1.82 3.88 9.44 6.20 −0.68 CNSTR −0.66 −0.23 22.11 22.04 8.87 4.47 6.21 3.97 −0.65 COMPS 16.94 12.67 48.65 38.36 3.90 3.80 7.07 4.98 −0.10 DRUGS 20.44 4.95 36.23 36.67 0.62 7.62 16.73 8.96 −0.49 ELCEQ 2.30 17.00 35.19 13.09 −0.40 4.19 5.78 5.55 0.41 FABPR −0.71 5.84 26.53 23.10 −0.72 1.27 1.91 1.07 −0.47 FIN 43.47 29.80 52.62 22.06 2.79 2.89 5.94 5.39 0.93 FOOD 1.61 −0.49 57.45 52.27 −0.62 −0.09 5.45 5.42 −0.72 FUN 1.29 15.50 41.18 30.77 0.17 1.00 5.57 4.58 −0.10 GOLD −0.16 29.97 47.15 4.05 −0.70 −0.19 −1.32 −0.89 −0.72 HLTH 3.07 −0.63 32.89 29.84 0.50 0.66 4.56 3.58 0.22 HSHLD 24.83 2.51 57.94 53.71 −0.30 1.54 2.98 2.34 −0.49 INSUR 8.00 −0.50 67.70 66.87 1.60 1.64 0.86 0.26 2.32 LABEQ 1.02 −0.59 31.98 28.57 0.80 0.81 4.56 3.81 −0.43 MACH −0.30 1.53 51.39 46.11 6.34 7.09 7.67 4.11 −0.40 MEALS 0.25 18.45 54.28 47.16 0.78 3.97 6.87 3.75 −0.33 MEDEQ 16.91 4.67 69.89 68.38 −0.67 0.76 1.02 3.25 −0.69 MINES 0.78 2.44 20.55 12.47 1.08 5.89 10.43 1.17 −0.72 OIL −0.39 10.85 63.82 48.83 2.10 3.59 7.53 4.36 −0.72 OTHER 1.26 39.09 63.79 18.14 0.71 1.71 5.46 3.50 0.22 PAPER 13.14 19.59 44.25 33.15 −0.68 0.28 2.46 2.74 2.47 PERSV 27.12 40.02 55.75 4.14 1.61 5.70 4.67 1.45 1.72 RLEST 0.16 1.77 25.88 21.22 2.58 2.06 2.73 0.13 −0.72 RTAIL −0.50 3.34 53.34 50.16 −0.68 −0.39 2.68 3.41 0.19 RUBBR 4.70 5.10 39.65 38.97 −0.25 −0.59 −3.05 −1.35 −0.43 SHIPS −0.98 2.53 39.20 38.71 2.58 5.09 6.69 4.08 −0.70 STEEL 24.12 7.43 30.78 27.88 7.99 5.32 10.18 7.02 0.88 TELCM 0.22 41.94 73.94 31.23 0.41 4.55 8.78 7.53 −0.12 TOYS 3.81 4.03 39.01 29.46 1.90 4.36 4.81 1.55 −0.57 TRANS −0.65 0.20 27.88 28.35 2.30 4.10 7.93 5.76 0.92 TXTLS 15.13 −0.69 32.50 26.17 0.27 0.25 7.81 7.54 −0.56 UTIL 31.22 23.00 62.49 35.78 0.93 0.02 2.35 3.01 −0.24 WHLSL 9.12 17.87 65.11 44.60 3.48 1.77 5.19 5.04 1.21 #SIG 24 33 44 44 14 25 33 32 4 On average, forecasts based only on BMi and BMM have smaller R2 than the forecasts based on the principal components. For both CFs and returns, the average R2 for the forecasts based on principal components 2–6 is about two-thirds of that based on all six principal components and greater than that based on either BMi or BMM. Moreover, principal components 2–6 predict the CFs and returns of at least thirty-two out of forty-four industries individually with statistically significant R2. In contrast to predictability by BM, the prior return has an average R2 of approximately zero in predicting returns. In sum, the evidence in Table II indicates that the cross-section of BM contains relevant information for predicting returns and CFs of individual industries. We report and discuss analogous in-sample results for the characteristic portfolios in the Online Appendix for brevity. Inferences are similar. 4.2 Out-of-Sample Results In-sample return predictability in regressions such as Equation (14) can break down in real time because of structural breaks or be overstated due to persistent-predictor biases (e.g., Stambaugh, 1999; Ferson, Sarkissian, and Simin, 2003; Goyal and Welch, 2008). Table III investigates the out-of-sample performance of combination forecasts based on the cross-section of BM. Panels A and B present ROS2 for the returns on the forty-four industry portfolios and twenty-five size/BM portfolios, respectively. Table III. Out-of-sample forecasting performance of combination forecasts of excess returns on industry and size/BM portfolios This table presents ROS2 from predictive regressions of one-quarter ahead excess returns using each of the combination forecast methods described in the paper (ABMA, DMSFE, and MEAN) as well as our market BM (BMM) and each portfolio’s own BM ratio (BMi). Panels A and B, respectively, present results for the forty-four industry and twenty-five size/BM portfolios. The out-of-sample period is 1980:1–2015:4. *, **, and *** denote significance based on the Clark and West (2007) MSFE-adjusted statistic at the 10%, 5%, and 1% levels, respectively. Panel A: Forty-four industries ABMA DMSFE MEAN BMM BMi AVG 2.10 1.76 1.59 −1.87 −3.27 AERO 3.72*** 2.92** 2.86** −2.59 −1.58 AGRIC 0.40 0.16 0.04 −2.07 −2.65 AUTOS 2.35** 1.91* 1.78* 0.03 −1.86 BANKS 1.31 1.05 1.03 −1.03 −4.50 BEER 3.87*** 3.39*** 3.32*** −2.90 2.55** BLDMT 2.20** 2.09* 1.42* 0.01 −6.03 BOOKS 4.44*** 4.02*** 4.00*** 1.82** −6.05 BOXES 1.28 0.71 0.67 −0.91 −3.39 BUSSV 1.73* 0.91 0.65 −5.98 −3.24 CHEM 0.74 0.23 0.20 0.06 −3.19 CHIPS 0.61 0.32 0.19 −4.02 0.30 CLTHS 4.66*** 3.51** 3.41*** −2.09 −4.40 CNSTR 2.08* 1.98* 1.52* −0.12 0.83 COMPS 0.62 0.77 0.69 −1.05 −7.55 DRUGS 1.24 1.24 1.09 −2.22 −4.78 ELCEQ 1.17 0.44 0.36 −6.15 −1.50 FABPR 0.37 1.01 0.87 0.02 −2.63 FIN 0.57 0.15 0.08 −3.07 −3.63 FOOD 1.75* 1.45 1.47* −0.02 −3.95 FUN 0.12 −0.30 −0.36 −4.28 0.10 GOLD 0.85 0.97 0.73 0.29 −5.12 HLTH 6.87*** 6.14** 5.76*** −2.11 −4.78 HSHLD 2.95** 2.36** 2.34*** −5.05 −5.79 INSUR 1.31 0.97 0.79 −4.10 −5.54 LABEQ 1.64 1.29 1.21 −1.60 −0.52 MACH −0.07 −0.23 −0.22 −2.02 2.63** MEALS 4.05*** 2.56** 2.52** −5.81 −6.58 MEDEQ 1.39 0.78 0.74 −2.87 −4.92 MINES 0.25 0.18 0.21 0.90 −3.51 OIL −0.99 −1.25 −1.45 −1.15 −3.04 OTHER 3.97*** 3.55*** 3.12** −5.19 −4.29 PAPER 1.19 0.79 0.90 3.30*** −4.40 PERSV 5.11*** 4.35*** 4.30** −1.55 −5.90 RLEST 4.25*** 4.79*** 4.40** 1.47 2.79** RTAIL 3.55* 2.53** 2.66** −3.32 −6.11 RUBBR 2.88** 2.88** 2.71** −0.88 −12.92 SHIPS 1.40* 1.43* 1.23 0.91 −1.17 STEEL −0.04 −0.08 −0.09 −0.66 −0.94 TELCM 0.09 0.40 0.40 −0.66 −2.69 TOYS 3.97*** 3.34** 3.19** 3.82*** −4.82 TRANS 2.89** 2.34* 1.98** −1.35 −6.65 TXTLS 2.96** 2.46** 2.47** 3.42*** −4.12 UTIL 1.06 0.70 0.66 1.65 4.23*** WHLSL 5.51*** 4.41*** 4.19** −3.12 −2.64 Panel A: Forty-four industries ABMA DMSFE MEAN BMM BMi AVG 2.10 1.76 1.59 −1.87 −3.27 AERO 3.72*** 2.92** 2.86** −2.59 −1.58 AGRIC 0.40 0.16 0.04 −2.07 −2.65 AUTOS 2.35** 1.91* 1.78* 0.03 −1.86 BANKS 1.31 1.05 1.03 −1.03 −4.50 BEER 3.87*** 3.39*** 3.32*** −2.90 2.55** BLDMT 2.20** 2.09* 1.42* 0.01 −6.03 BOOKS 4.44*** 4.02*** 4.00*** 1.82** −6.05 BOXES 1.28 0.71 0.67 −0.91 −3.39 BUSSV 1.73* 0.91 0.65 −5.98 −3.24 CHEM 0.74 0.23 0.20 0.06 −3.19 CHIPS 0.61 0.32 0.19 −4.02 0.30 CLTHS 4.66*** 3.51** 3.41*** −2.09 −4.40 CNSTR 2.08* 1.98* 1.52* −0.12 0.83 COMPS 0.62 0.77 0.69 −1.05 −7.55 DRUGS 1.24 1.24 1.09 −2.22 −4.78 ELCEQ 1.17 0.44 0.36 −6.15 −1.50 FABPR 0.37 1.01 0.87 0.02 −2.63 FIN 0.57 0.15 0.08 −3.07 −3.63 FOOD 1.75* 1.45 1.47* −0.02 −3.95 FUN 0.12 −0.30 −0.36 −4.28 0.10 GOLD 0.85 0.97 0.73 0.29 −5.12 HLTH 6.87*** 6.14** 5.76*** −2.11 −4.78 HSHLD 2.95** 2.36** 2.34*** −5.05 −5.79 INSUR 1.31 0.97 0.79 −4.10 −5.54 LABEQ 1.64 1.29 1.21 −1.60 −0.52 MACH −0.07 −0.23 −0.22 −2.02 2.63** MEALS 4.05*** 2.56** 2.52** −5.81 −6.58 MEDEQ 1.39 0.78 0.74 −2.87 −4.92 MINES 0.25 0.18 0.21 0.90 −3.51 OIL −0.99 −1.25 −1.45 −1.15 −3.04 OTHER 3.97*** 3.55*** 3.12** −5.19 −4.29 PAPER 1.19 0.79 0.90 3.30*** −4.40 PERSV 5.11*** 4.35*** 4.30** −1.55 −5.90 RLEST 4.25*** 4.79*** 4.40** 1.47 2.79** RTAIL 3.55* 2.53** 2.66** −3.32 −6.11 RUBBR 2.88** 2.88** 2.71** −0.88 −12.92 SHIPS 1.40* 1.43* 1.23 0.91 −1.17 STEEL −0.04 −0.08 −0.09 −0.66 −0.94 TELCM 0.09 0.40 0.40 −0.66 −2.69 TOYS 3.97*** 3.34** 3.19** 3.82*** −4.82 TRANS 2.89** 2.34* 1.98** −1.35 −6.65 TXTLS 2.96** 2.46** 2.47** 3.42*** −4.12 UTIL 1.06 0.70 0.66 1.65 4.23*** WHLSL 5.51*** 4.41*** 4.19** −3.12 −2.64 Panel B: Twenty-five size and BM portfolios Size BM ABMA DMSFE MEAN BMM BMi Avg 2.32 1.81 0.85 −2.06 −0.55 Small Low 2.31** 1.70 2.06** −0.32 0.09 2 1.77* 3.50*** 0.82 −2.68 −1.53 3 4.42*** 3.45*** 2.81** −2.49 0.91 4 3.80*** 5.27*** 2.82** −3.49 −0.02 High 5.72*** 2.98** 3.22*** −2.12 −0.27 2 Low 3.11*** 1.04 1.38 0.28 1.24 2 1.08 1.47 0.10 −0.54 0.90 3 1.63 1.13 0.48 −2.95 −1.30 4 0.93 0.90 −0.17 −3.82 −3.04 High 3.30*** 0.37 1.36 −2.48 −2.16 3 Low 1.84* 0.33 0.36 0.35 −1.04 2 2.70** 2.05** 0.65 −0.95 2.16 3 1.83* 0.22 0.60 −2.56 0.41 4 3.76*** 0.12 1.43* −2.79 1.30 High 3.25** 0.57 1.67* −2.68 −3.37 4 Low 2.52** 0.79 0.68 0.32 −0.82 2 1.50 1.49 −0.04 −2.12 1.55 3 2.11** 2.73** 0.05 −2.75 −1.24 4 2.28** 1.40 0.06 −5.49 −0.36 High 3.34** 1.58 1.87 −1.68 −3.07 Big Low 2.23* 1.48 2.85** 1.46 0.33 2 −0.11 2.66** −1.22 −2.59 −0.93 3 1.01 2.69** 0.01 −3.03 −0.18 4 0.93 0.53 −0.57 −2.31 −1.23 High 0.83 4.80*** −2.07 −3.99 −2.20 Panel B: Twenty-five size and BM portfolios Size BM ABMA DMSFE MEAN BMM BMi Avg 2.32 1.81 0.85 −2.06 −0.55 Small Low 2.31** 1.70 2.06** −0.32 0.09 2 1.77* 3.50*** 0.82 −2.68 −1.53 3 4.42*** 3.45*** 2.81** −2.49 0.91 4 3.80*** 5.27*** 2.82** −3.49 −0.02 High 5.72*** 2.98** 3.22*** −2.12 −0.27 2 Low 3.11*** 1.04 1.38 0.28 1.24 2 1.08 1.47 0.10 −0.54 0.90 3 1.63 1.13 0.48 −2.95 −1.30 4 0.93 0.90 −0.17 −3.82 −3.04 High 3.30*** 0.37 1.36 −2.48 −2.16 3 Low 1.84* 0.33 0.36 0.35 −1.04 2 2.70** 2.05** 0.65 −0.95 2.16 3 1.83* 0.22 0.60 −2.56 0.41 4 3.76*** 0.12 1.43* −2.79 1.30 High 3.25** 0.57 1.67* −2.68 −3.37 4 Low 2.52** 0.79 0.68 0.32 −0.82 2 1.50 1.49 −0.04 −2.12 1.55 3 2.11** 2.73** 0.05 −2.75 −1.24 4 2.28** 1.40 0.06 −5.49 −0.36 High 3.34** 1.58 1.87 −1.68 −3.07 Big Low 2.23* 1.48 2.85** 1.46 0.33 2 −0.11 2.66** −1.22 −2.59 −0.93 3 1.01 2.69** 0.01 −3.03 −0.18 4 0.93 0.53 −0.57 −2.31 −1.23 High 0.83 4.80*** −2.07 −3.99 −2.20 Table III. Out-of-sample forecasting performance of combination forecasts of excess returns on industry and size/BM portfolios This table presents ROS2 from predictive regressions of one-quarter ahead excess returns using each of the combination forecast methods described in the paper (ABMA, DMSFE, and MEAN) as well as our market BM (BMM) and each portfolio’s own BM ratio (BMi). Panels A and B, respectively, present results for the forty-four industry and twenty-five size/BM portfolios. The out-of-sample period is 1980:1–2015:4. *, **, and *** denote significance based on the Clark and West (2007) MSFE-adjusted statistic at the 10%, 5%, and 1% levels, respectively. Panel A: Forty-four industries ABMA DMSFE MEAN BMM BMi AVG 2.10 1.76 1.59 −1.87 −3.27 AERO 3.72*** 2.92** 2.86** −2.59 −1.58 AGRIC 0.40 0.16 0.04 −2.07 −2.65 AUTOS 2.35** 1.91* 1.78* 0.03 −1.86 BANKS 1.31 1.05 1.03 −1.03 −4.50 BEER 3.87*** 3.39*** 3.32*** −2.90 2.55** BLDMT 2.20** 2.09* 1.42* 0.01 −6.03 BOOKS 4.44*** 4.02*** 4.00*** 1.82** −6.05 BOXES 1.28 0.71 0.67 −0.91 −3.39 BUSSV 1.73* 0.91 0.65 −5.98 −3.24 CHEM 0.74 0.23 0.20 0.06 −3.19 CHIPS 0.61 0.32 0.19 −4.02 0.30 CLTHS 4.66*** 3.51** 3.41*** −2.09 −4.40 CNSTR 2.08* 1.98* 1.52* −0.12 0.83 COMPS 0.62 0.77 0.69 −1.05 −7.55 DRUGS 1.24 1.24 1.09 −2.22 −4.78 ELCEQ 1.17 0.44 0.36 −6.15 −1.50 FABPR 0.37 1.01 0.87 0.02 −2.63 FIN 0.57 0.15 0.08 −3.07 −3.63 FOOD 1.75* 1.45 1.47* −0.02 −3.95 FUN 0.12 −0.30 −0.36 −4.28 0.10 GOLD 0.85 0.97 0.73 0.29 −5.12 HLTH 6.87*** 6.14** 5.76*** −2.11 −4.78 HSHLD 2.95** 2.36** 2.34*** −5.05 −5.79 INSUR 1.31 0.97 0.79 −4.10 −5.54 LABEQ 1.64 1.29 1.21 −1.60 −0.52 MACH −0.07 −0.23 −0.22 −2.02 2.63** MEALS 4.05*** 2.56** 2.52** −5.81 −6.58 MEDEQ 1.39 0.78 0.74 −2.87 −4.92 MINES 0.25 0.18 0.21 0.90 −3.51 OIL −0.99 −1.25 −1.45 −1.15 −3.04 OTHER 3.97*** 3.55*** 3.12** −5.19 −4.29 PAPER 1.19 0.79 0.90 3.30*** −4.40 PERSV 5.11*** 4.35*** 4.30** −1.55 −5.90 RLEST 4.25*** 4.79*** 4.40** 1.47 2.79** RTAIL 3.55* 2.53** 2.66** −3.32 −6.11 RUBBR 2.88** 2.88** 2.71** −0.88 −12.92 SHIPS 1.40* 1.43* 1.23 0.91 −1.17 STEEL −0.04 −0.08 −0.09 −0.66 −0.94 TELCM 0.09 0.40 0.40 −0.66 −2.69 TOYS 3.97*** 3.34** 3.19** 3.82*** −4.82 TRANS 2.89** 2.34* 1.98** −1.35 −6.65 TXTLS 2.96** 2.46** 2.47** 3.42*** −4.12 UTIL 1.06 0.70 0.66 1.65 4.23*** WHLSL 5.51*** 4.41*** 4.19** −3.12 −2.64 Panel A: Forty-four industries ABMA DMSFE MEAN BMM BMi AVG 2.10 1.76 1.59 −1.87 −3.27 AERO 3.72*** 2.92** 2.86** −2.59 −1.58 AGRIC 0.40 0.16 0.04 −2.07 −2.65 AUTOS 2.35** 1.91* 1.78* 0.03 −1.86 BANKS 1.31 1.05 1.03 −1.03 −4.50 BEER 3.87*** 3.39*** 3.32*** −2.90 2.55** BLDMT 2.20** 2.09* 1.42* 0.01 −6.03 BOOKS 4.44*** 4.02*** 4.00*** 1.82** −6.05 BOXES 1.28 0.71 0.67 −0.91 −3.39 BUSSV 1.73* 0.91 0.65 −5.98 −3.24 CHEM 0.74 0.23 0.20 0.06 −3.19 CHIPS 0.61 0.32 0.19 −4.02 0.30 CLTHS 4.66*** 3.51** 3.41*** −2.09 −4.40 CNSTR 2.08* 1.98* 1.52* −0.12 0.83 COMPS 0.62 0.77 0.69 −1.05 −7.55 DRUGS 1.24 1.24 1.09 −2.22 −4.78 ELCEQ 1.17 0.44 0.36 −6.15 −1.50 FABPR 0.37 1.01 0.87 0.02 −2.63 FIN 0.57 0.15 0.08 −3.07 −3.63 FOOD 1.75* 1.45 1.47* −0.02 −3.95 FUN 0.12 −0.30 −0.36 −4.28 0.10 GOLD 0.85 0.97 0.73 0.29 −5.12 HLTH 6.87*** 6.14** 5.76*** −2.11 −4.78 HSHLD 2.95** 2.36** 2.34*** −5.05 −5.79 INSUR 1.31 0.97 0.79 −4.10 −5.54 LABEQ 1.64 1.29 1.21 −1.60 −0.52 MACH −0.07 −0.23 −0.22 −2.02 2.63** MEALS 4.05*** 2.56** 2.52** −5.81 −6.58 MEDEQ 1.39 0.78 0.74 −2.87 −4.92 MINES 0.25 0.18 0.21 0.90 −3.51 OIL −0.99 −1.25 −1.45 −1.15 −3.04 OTHER 3.97*** 3.55*** 3.12** −5.19 −4.29 PAPER 1.19 0.79 0.90 3.30*** −4.40 PERSV 5.11*** 4.35*** 4.30** −1.55 −5.90 RLEST 4.25*** 4.79*** 4.40** 1.47 2.79** RTAIL 3.55* 2.53** 2.66** −3.32 −6.11 RUBBR 2.88** 2.88** 2.71** −0.88 −12.92 SHIPS 1.40* 1.43* 1.23 0.91 −1.17 STEEL −0.04 −0.08 −0.09 −0.66 −0.94 TELCM 0.09 0.40 0.40 −0.66 −2.69 TOYS 3.97*** 3.34** 3.19** 3.82*** −4.82 TRANS 2.89** 2.34* 1.98** −1.35 −6.65 TXTLS 2.96** 2.46** 2.47** 3.42*** −4.12 UTIL 1.06 0.70 0.66 1.65 4.23*** WHLSL 5.51*** 4.41*** 4.19** −3.12 −2.64 Panel B: Twenty-five size and BM portfolios Size BM ABMA DMSFE MEAN BMM BMi Avg 2.32 1.81 0.85 −2.06 −0.55 Small Low 2.31** 1.70 2.06** −0.32 0.09 2 1.77* 3.50*** 0.82 −2.68 −1.53 3 4.42*** 3.45*** 2.81** −2.49 0.91 4 3.80*** 5.27*** 2.82** −3.49 −0.02 High 5.72*** 2.98** 3.22*** −2.12 −0.27 2 Low 3.11*** 1.04 1.38 0.28 1.24 2 1.08 1.47 0.10 −0.54 0.90 3 1.63 1.13 0.48 −2.95 −1.30 4 0.93 0.90 −0.17 −3.82 −3.04 High 3.30*** 0.37 1.36 −2.48 −2.16 3 Low 1.84* 0.33 0.36 0.35 −1.04 2 2.70** 2.05** 0.65 −0.95 2.16 3 1.83* 0.22 0.60 −2.56 0.41 4 3.76*** 0.12 1.43* −2.79 1.30 High 3.25** 0.57 1.67* −2.68 −3.37 4 Low 2.52** 0.79 0.68 0.32 −0.82 2 1.50 1.49 −0.04 −2.12 1.55 3 2.11** 2.73** 0.05 −2.75 −1.24 4 2.28** 1.40 0.06 −5.49 −0.36 High 3.34** 1.58 1.87 −1.68 −3.07 Big Low 2.23* 1.48 2.85** 1.46 0.33 2 −0.11 2.66** −1.22 −2.59 −0.93 3 1.01 2.69** 0.01 −3.03 −0.18 4 0.93 0.53 −0.57 −2.31 −1.23 High 0.83 4.80*** −2.07 −3.99 −2.20 Panel B: Twenty-five size and BM portfolios Size BM ABMA DMSFE MEAN BMM BMi Avg 2.32 1.81 0.85 −2.06 −0.55 Small Low 2.31** 1.70 2.06** −0.32 0.09 2 1.77* 3.50*** 0.82 −2.68 −1.53 3 4.42*** 3.45*** 2.81** −2.49 0.91 4 3.80*** 5.27*** 2.82** −3.49 −0.02 High 5.72*** 2.98** 3.22*** −2.12 −0.27 2 Low 3.11*** 1.04 1.38 0.28 1.24 2 1.08 1.47 0.10 −0.54 0.90 3 1.63 1.13 0.48 −2.95 −1.30 4 0.93 0.90 −0.17 −3.82 −3.04 High 3.30*** 0.37 1.36 −2.48 −2.16 3 Low 1.84* 0.33 0.36 0.35 −1.04 2 2.70** 2.05** 0.65 −0.95 2.16 3 1.83* 0.22 0.60 −2.56 0.41 4 3.76*** 0.12 1.43* −2.79 1.30 High 3.25** 0.57 1.67* −2.68 −3.37 4 Low 2.52** 0.79 0.68 0.32 −0.82 2 1.50 1.49 −0.04 −2.12 1.55 3 2.11** 2.73** 0.05 −2.75 −1.24 4 2.28** 1.40 0.06 −5.49 −0.36 High 3.34** 1.58 1.87 −1.68 −3.07 Big Low 2.23* 1.48 2.85** 1.46 0.33 2 −0.11 2.66** −1.22 −2.59 −0.93 3 1.01 2.69** 0.01 −3.03 −0.18 4 0.93 0.53 −0.57 −2.31 −1.23 High 0.83 4.80*** −2.07 −3.99 −2.20 Panel A shows that all three combination forecast methods generate statistically significant predictability for many industries. On average, the combination forecasts have ROS2 of 3.5–4.0% and 4.9–5.4% higher than those based on BMM and BMi, respectively. Depending on combination method, twenty to twenty-two of the individual ROS2 are statistically significant at the 10% significance level. In contrast, the ROS2 of the forecasts based on BMM and BMi are negative on average and all but four are insignificant or negative. Panel B shows that on average combination forecast methods also predict returns on the twenty-five size/BM portfolios better than BMM or BMi (by 2.9–4.4% and 1.4–2.9%, respectively). However, consistency of performance across combination methods is less than with the industry portfolios. Seventeen out of twenty-five of the ABMA forecasts have significant ROS2, compared with only nine and seven of the DMSFE and MEAN forecasts, respectively. All three methods predict returns in the smallest size quintile with generally significant ROS2, but there is no other clear pattern between size, BM, and ROS2. The ROS2 of the forecasts based on BMM or BMi are negative on average, and none are significantly positive. For both sets of portfolios, the average ROS2 decline from ABMA to DMSFE to MEAN, although the differences across methods are smaller for the industry portfolios. As Section 3 discusses, this finding has implications for the data-generating processes of the two sets of portfolio returns. First, the stronger performance of the ABMA and DMSFE relative to the MEAN method indicates that assigning higher time-varying weights to individual forecasts with lower MSFE produces a more accurate combination forecast than equal-weighting in spite of higher estimation error. Second, the weaker performance of DMSFE relative to ABMA shows that discounting past observations more heavily, which cuts the effective sample size, is not optimal. The benefit of the lower volatility in the estimated ABMA weights exceeds the cost of DMSFE method more quickly identifying time-variation in the data-generating process. Overall, the evidence from Table III indicates that combination forecast methods extract relevant information in the cross-section of BM ratios for predicting the returns of many industry and size/BM portfolios with statistically significant ROS2. Results in the Online Appendix show that similar inferences hold when using rolling estimation windows or size/investment and size/profitability portfolios. In the next section, we examine the joint and economic significance of the combination forecasts. 5. Trading Strategies Based on Return Forecasts We assess the performance of real-time portfolio-rotation strategies based on out-of-sample predicted returns from the combination forecasts. This exercise measures the economic significance of the combination forecasts to investors (e.g., Pesaran and Timmermann, 1995; Cochrane, 2008). 5.1 Industry-Rotation Portfolios Each quarter from 1980:1 to 2015:4, we rank industry portfolios based on their predicted returns. Our industry-rotation strategies (hereafter “combo strategies”) take an equal-weighted long position in the top four (roughly one decile) of the forty-four industry portfolios, and a short position in the bottom four.7 We compare these strategies to those based on univariate predictive regressions using BMi: ri,t+1=ai+bi·BMi,t+ϵt+1. (15) We evaluate the performance of strategies using the Fama–French–Carhart four-factor model. In the presence of transaction costs, however, four-factor intercepts do not in general represent obtainable returns. Hence, we measure abnormal net-of-costs returns with the generalized αnet of Novy-Marx and Velikov (2016). The αnet has the same units as the intercept-based α but properly accounts for transaction costs in measuring how access to a given asset expands the investment opportunity set relative to the Fama–French–Carhart factors (see Online Appendix for details). Table IV presents average returns, Fama–French–Carhart four-factor αs and factor loadings of the returns on our strategies ignoring transaction costs. Below these statistics are average net returns, αnet, turnover, and transaction costs. Reliability or consistency of performance is also an important concern to investors. The bottom four rows report the percentage of quarters in which our strategies “beat the market” after transaction costs over the whole sample (ALL) and subsamples defined by the row heading. For the long positions, beating the market benchmark means earning higher net-of-transaction costs returns than the market. A short position “beats the market” if after transaction costs, the short earns higher returns than short-selling the market. A long–short strategy is zero-investment and therefore “beats the market” if its net-of-costs return is positive. Table IV. Performance of strategies using forty-four industry portfolios Each column presents statistics for the long, short, and long-minus-short (L−S) industry-rotation strategies based on the three different combination forecast methods described in Section 5 (ABMA, DMSFE, and MEAN) or the benchmark real-time univariate predictive regression that uses industry-own BM (BMi). E(rgrosse) denotes annualized gross average excess return and αgross, β, s, and h and m denote estimates from the Fama–French–Carhart four-factor model: rgross,te=αgross+β*MKTt+s*SMBt+h*HMLt+m*MOMt+ϵt. (16) E(rnet) denotes the annualized average returns net of transaction costs and αnet denotes the generalized net-of-costs α of Novy-Marx and Velikov (2016) (which is non-negative by construction). The bottom four rows present the percentage of quarters in which the strategy beat the market benchmark after transaction costs over the whole sample (ALL) and subsamples defined in by the row heading. TO and T-Costs denote average turnover and transaction costs (%/quarter). The sample is 1980:1–2015:4 (N = 144). *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. ABMA DMSFE MEAN BMi Long Short L−S Long Short L−S Long Short L−S Long Short L−S E(rgrosse) 11.72 2.43 9.29 13.65 2.06 11.59 14.05 3.20 10.85 9.35 9.99 −0.64 αgross 2.22 −4.38** 6.60** 2.67 −5.47*** 8.15*** 4.02** −2.56 6.59** 1.78 −1.01 2.79 (1.38) (−2.05) (2.39) (1.59) (−2.75) (3.11) (2.38) (−1.18) (2.35) (0.84) (−0.65) (1.06) β 1.16*** 1.06*** 0.10 1.23*** 1.08*** 0.15* 1.22*** 1.03*** 0.19** 1.01*** 1.18*** −0.17* (21.52) (14.79) (1.14) (21.92) (16.28) (1.71) (21.59) (14.12) (2.06) (14.20) (22.51) (−1.95) s 0.14* 0.43*** −0.29** 0.10 0.46*** −0.36*** 0.00 0.44*** −0.43*** 0.24** 0.34*** −0.10 (1.79) (4.21) (−2.21) (1.18) (4.78) (−2.87) (0.05) (4.17) (−3.21) (2.37) (4.50) (−0.76) h −0.20*** 0.34*** −0.54*** −0.14* 0.43*** −0.57*** −0.27*** 0.26*** −0.54*** 0.20** 0.09 0.11 (−2.76) (3.46) (−4.29) (−1.76) (4.78) (−4.75) (−3.55) (2.64) (−4.20) (2.04) (1.23) (0.91) m 0.19*** −0.39*** 0.58*** 0.24*** −0.37*** 0.62*** 0.23*** −0.47*** 0.70*** −0.15** 0.16*** −0.31*** (3.69) (−5.84) (6.68) (4.59) (−5.94) (7.44) (4.32) (−6.86) (7.93) (−2.21) (3.29) (−3.72) Adjusted R2 0.82 0.73 0.35 0.82 0.76 0.41 0.82 0.72 0.42 0.66 0.83 0.12 E(rnete) 10.75 −3.64 7.12 12.56 −3.45 9.11 13.13 −4.34 8.79 8.08 −11.16 −3.08 αnet 2.03 4.97** 7.08*** 3.13** 5.68*** 8.87*** 4.34*** 3.67* 7.99*** 0.00 0.00 0.00 (1.33) (2.44) (2.70) (1.99) (3.00) (3.54) (2.71) (1.74) (2.97) TO 31.5 34.6 33.0 36.6 39.0 37.8 30.3 31.5 30.9 39.2 34.3 36.7 T-Costs 0.24 0.30 0.54 0.27 0.35 0.62 0.23 0.28 0.52 0.32 0.29 0.61 Consistency ALL 53 58 59 55 59 63 57 51 60 51 45 45 1980s 42 60 52 48 60 62 55 45 55 50 45 40 1990s 62 70 70 55 65 65 55 65 70 48 50 55 2000s 54 49 56 59 54 62 60 46 57 54 41 43 ABMA DMSFE MEAN BMi Long Short L−S Long Short L−S Long Short L−S Long Short L−S E(rgrosse) 11.72 2.43 9.29 13.65 2.06 11.59 14.05 3.20 10.85 9.35 9.99 −0.64 αgross 2.22 −4.38** 6.60** 2.67 −5.47*** 8.15*** 4.02** −2.56 6.59** 1.78 −1.01 2.79 (1.38) (−2.05) (2.39) (1.59) (−2.75) (3.11) (2.38) (−1.18) (2.35) (0.84) (−0.65) (1.06) β 1.16*** 1.06*** 0.10 1.23*** 1.08*** 0.15* 1.22*** 1.03*** 0.19** 1.01*** 1.18*** −0.17* (21.52) (14.79) (1.14) (21.92) (16.28) (1.71) (21.59) (14.12) (2.06) (14.20) (22.51) (−1.95) s 0.14* 0.43*** −0.29** 0.10 0.46*** −0.36*** 0.00 0.44*** −0.43*** 0.24** 0.34*** −0.10 (1.79) (4.21) (−2.21) (1.18) (4.78) (−2.87) (0.05) (4.17) (−3.21) (2.37) (4.50) (−0.76) h −0.20*** 0.34*** −0.54*** −0.14* 0.43*** −0.57*** −0.27*** 0.26*** −0.54*** 0.20** 0.09 0.11 (−2.76) (3.46) (−4.29) (−1.76) (4.78) (−4.75) (−3.55) (2.64) (−4.20) (2.04) (1.23) (0.91) m 0.19*** −0.39*** 0.58*** 0.24*** −0.37*** 0.62*** 0.23*** −0.47*** 0.70*** −0.15** 0.16*** −0.31*** (3.69) (−5.84) (6.68) (4.59) (−5.94) (7.44) (4.32) (−6.86) (7.93) (−2.21) (3.29) (−3.72) Adjusted R2 0.82 0.73 0.35 0.82 0.76 0.41 0.82 0.72 0.42 0.66 0.83 0.12 E(rnete) 10.75 −3.64 7.12 12.56 −3.45 9.11 13.13 −4.34 8.79 8.08 −11.16 −3.08 αnet 2.03 4.97** 7.08*** 3.13** 5.68*** 8.87*** 4.34*** 3.67* 7.99*** 0.00 0.00 0.00 (1.33) (2.44) (2.70) (1.99) (3.00) (3.54) (2.71) (1.74) (2.97) TO 31.5 34.6 33.0 36.6 39.0 37.8 30.3 31.5 30.9 39.2 34.3 36.7 T-Costs 0.24 0.30 0.54 0.27 0.35 0.62 0.23 0.28 0.52 0.32 0.29 0.61 Consistency ALL 53 58 59 55 59 63 57 51 60 51 45 45 1980s 42 60 52 48 60 62 55 45 55 50 45 40 1990s 62 70 70 55 65 65 55 65 70 48 50 55 2000s 54 49 56 59 54 62 60 46 57 54 41 43 Table IV. Performance of strategies using forty-four industry portfolios Each column presents statistics for the long, short, and long-minus-short (L−S) industry-rotation strategies based on the three different combination forecast methods described in Section 5 (ABMA, DMSFE, and MEAN) or the benchmark real-time univariate predictive regression that uses industry-own BM (BMi). E(rgrosse) denotes annualized gross average excess return and αgross, β, s, and h and m denote estimates from the Fama–French–Carhart four-factor model: rgross,te=αgross+β*MKTt+s*SMBt+h*HMLt+m*MOMt+ϵt. (16) E(rnet) denotes the annualized average returns net of transaction costs and αnet denotes the generalized net-of-costs α of Novy-Marx and Velikov (2016) (which is non-negative by construction). The bottom four rows present the percentage of quarters in which the strategy beat the market benchmark after transaction costs over the whole sample (ALL) and subsamples defined in by the row heading. TO and T-Costs denote average turnover and transaction costs (%/quarter). The sample is 1980:1–2015:4 (N = 144). *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. ABMA DMSFE MEAN BMi Long Short L−S Long Short L−S Long Short L−S Long Short L−S E(rgrosse) 11.72 2.43 9.29 13.65 2.06 11.59 14.05 3.20 10.85 9.35 9.99 −0.64 αgross 2.22 −4.38** 6.60** 2.67 −5.47*** 8.15*** 4.02** −2.56 6.59** 1.78 −1.01 2.79 (1.38) (−2.05) (2.39) (1.59) (−2.75) (3.11) (2.38) (−1.18) (2.35) (0.84) (−0.65) (1.06) β 1.16*** 1.06*** 0.10 1.23*** 1.08*** 0.15* 1.22*** 1.03*** 0.19** 1.01*** 1.18*** −0.17* (21.52) (14.79) (1.14) (21.92) (16.28) (1.71) (21.59) (14.12) (2.06) (14.20) (22.51) (−1.95) s 0.14* 0.43*** −0.29** 0.10 0.46*** −0.36*** 0.00 0.44*** −0.43*** 0.24** 0.34*** −0.10 (1.79) (4.21) (−2.21) (1.18) (4.78) (−2.87) (0.05) (4.17) (−3.21) (2.37) (4.50) (−0.76) h −0.20*** 0.34*** −0.54*** −0.14* 0.43*** −0.57*** −0.27*** 0.26*** −0.54*** 0.20** 0.09 0.11 (−2.76) (3.46) (−4.29) (−1.76) (4.78) (−4.75) (−3.55) (2.64) (−4.20) (2.04) (1.23) (0.91) m 0.19*** −0.39*** 0.58*** 0.24*** −0.37*** 0.62*** 0.23*** −0.47*** 0.70*** −0.15** 0.16*** −0.31*** (3.69) (−5.84) (6.68) (4.59) (−5.94) (7.44) (4.32) (−6.86) (7.93) (−2.21) (3.29) (−3.72) Adjusted R2 0.82 0.73 0.35 0.82 0.76 0.41 0.82 0.72 0.42 0.66 0.83 0.12 E(rnete) 10.75 −3.64 7.12 12.56 −3.45 9.11 13.13 −4.34 8.79 8.08 −11.16 −3.08 αnet 2.03 4.97** 7.08*** 3.13** 5.68*** 8.87*** 4.34*** 3.67* 7.99*** 0.00 0.00 0.00 (1.33) (2.44) (2.70) (1.99) (3.00) (3.54) (2.71) (1.74) (2.97) TO 31.5 34.6 33.0 36.6 39.0 37.8 30.3 31.5 30.9 39.2 34.3 36.7 T-Costs 0.24 0.30 0.54 0.27 0.35 0.62 0.23 0.28 0.52 0.32 0.29 0.61 Consistency ALL 53 58 59 55 59 63 57 51 60 51 45 45 1980s 42 60 52 48 60 62 55 45 55 50 45 40 1990s 62 70 70 55 65 65 55 65 70 48 50 55 2000s 54 49 56 59 54 62 60 46 57 54 41 43 ABMA DMSFE MEAN BMi Long Short L−S Long Short L−S Long Short L−S Long Short L−S E(rgrosse) 11.72 2.43 9.29 13.65 2.06 11.59 14.05 3.20 10.85 9.35 9.99 −0.64 αgross 2.22 −4.38** 6.60** 2.67 −5.47*** 8.15*** 4.02** −2.56 6.59** 1.78 −1.01 2.79 (1.38) (−2.05) (2.39) (1.59) (−2.75) (3.11) (2.38) (−1.18) (2.35) (0.84) (−0.65) (1.06) β 1.16*** 1.06*** 0.10 1.23*** 1.08*** 0.15* 1.22*** 1.03*** 0.19** 1.01*** 1.18*** −0.17* (21.52) (14.79) (1.14) (21.92) (16.28) (1.71) (21.59) (14.12) (2.06) (14.20) (22.51) (−1.95) s 0.14* 0.43*** −0.29** 0.10 0.46*** −0.36*** 0.00 0.44*** −0.43*** 0.24** 0.34*** −0.10 (1.79) (4.21) (−2.21) (1.18) (4.78) (−2.87) (0.05) (4.17) (−3.21) (2.37) (4.50) (−0.76) h −0.20*** 0.34*** −0.54*** −0.14* 0.43*** −0.57*** −0.27*** 0.26*** −0.54*** 0.20** 0.09 0.11 (−2.76) (3.46) (−4.29) (−1.76) (4.78) (−4.75) (−3.55) (2.64) (−4.20) (2.04) (1.23) (0.91) m 0.19*** −0.39*** 0.58*** 0.24*** −0.37*** 0.62*** 0.23*** −0.47*** 0.70*** −0.15** 0.16*** −0.31*** (3.69) (−5.84) (6.68) (4.59) (−5.94) (7.44) (4.32) (−6.86) (7.93) (−2.21) (3.29) (−3.72) Adjusted R2 0.82 0.73 0.35 0.82 0.76 0.41 0.82 0.72 0.42 0.66 0.83 0.12 E(rnete) 10.75 −3.64 7.12 12.56 −3.45 9.11 13.13 −4.34 8.79 8.08 −11.16 −3.08 αnet 2.03 4.97** 7.08*** 3.13** 5.68*** 8.87*** 4.34*** 3.67* 7.99*** 0.00 0.00 0.00 (1.33) (2.44) (2.70) (1.99) (3.00) (3.54) (2.71) (1.74) (2.97) TO 31.5 34.6 33.0 36.6 39.0 37.8 30.3 31.5 30.9 39.2 34.3 36.7 T-Costs 0.24 0.30 0.54 0.27 0.35 0.62 0.23 0.28 0.52 0.32 0.29 0.61 Consistency ALL 53 58 59 55 59 63 57 51 60 51 45 45 1980s 42 60 52 48 60 62 55 45 55 50 45 40 1990s 62 70 70 55 65 65 55 65 70 48 50 55 2000s 54 49 56 59 54 62 60 46 57 54 41 43 Table IV shows that each of the combination methods generates substantial spreads in average returns of about 9.3–11.6% per annum between the long and short legs of the forty-four industry combo strategies. Further, the long–short strategies earn significant four-factor αgross of about 6.6–8.2% as exposure to the risk factors explains only a few percent of the spread in average returns.8 The BMi strategy does not generate a significant spread in returns or αgross. The combo strategy returns do not arise from a value effect. The HML loadings on the long leg of the combo strategies are lower than those of the short leg. In contrast to the HML loadings, the MOM loadings are significantly higher for the long legs of the combo strategies than for the short legs. Further, the MOM exposure of the long–short combo strategies actually lowers αgross relative to αnet by about 0.5–1.2% because the transaction costs on the combo strategies are significantly lower than those of MOM. While the momentum factor is correlated with our strategies returns, it does not explain the returns on our strategies as the α is not 0. In the Online Appendix, we also show that a factor model similar to the Fama–French–Carhart model that includes an industry-momentum factor explains even less of our strategies’ returns than the Fama–French–Carhart model. Further, the spread in MOM loadings is driven by other industries; the opposite pattern occurs with the BMi strategy. The more likely explanation of the MOM loadings is that our rotation strategies take long and short positions, respectively, in industries with high and low predicted returns. Thus, we should expect our long–short strategy returns to be positively correlated with characteristics that are themselves positively correlated with expected returns, such as the market beta and momentum. Hou, Xue, and Zhang (2015) also find that the momentum premium is explained by profitability and investment, so the momentum loadings in Table IV likely proxy for more fundamental characteristics that are correlated with expected returns. The bottom four rows of Table IV panel demonstrate that for both sets of industries, the long–short combo strategies earn positive net-of-costs returns over 59–63% of quarters over the whole sample. For most decades, the long, short, and long–short combo strategies outperform the market benchmarks as well. Thus, the combo strategies generally perform reliably over the sample, exhibiting no sign of deterioration over time. The high αnet relative to αgross in Table IV demonstrate how the value-weighting of the industries in our strategies economize on transaction costs. The value-weighting of the industry portfolios ensures that the short legs of our strategies are feasible to execute. D’Avolio (2002), for example, finds that stocks that cannot be shorted combine to less than 1% of the aggregate stock market. A related concern is that short-sale fees would eliminate the profitability of our strategies. However, D’Avolio (2002) also finds the short-sale costs on 91% of stocks are less than 1% per annum and the average shorting cost on the remaining 9% averages only 4.9%. Drechsler and Drechsler (2016) similarly show that stocks that are expensive to short have the smallest market capitalizations. While accounting for short-sale fees could create a modest drag on the absolute profitability of our industry-rotation strategies, it should improve the net-of-costs α for the same reason that accounting for effective spreads does. Long and short positions in HML and MOM both involve short-selling portfolios of expensive-to-short small-cap stocks.9 Similarly, our forty-four industry strategies are generally net short SMB, which would further involve short-selling small-cap stocks. Overall, investors facing transaction costs can earn positive abnormal returns relative to the Fama–French–Carhart four-factor model by trading industries based on all three combination forecasts. 5.1.a. Reliability and economic significance to investors Figure 2 depicts the time series of the natural log of the net-of-costs accumulated value of $1 invested in the different each leg of the forty-four industry combo strategies and the market (MKT) benchmark over the 36 year out-of-sample period. For ease of visualization, the short positions in Figure 2 are expressed as long positions. The long legs of the three combo strategies steadily gain relative to the market bench, and exhibit no obvious break in performance. Similarly, the short legs of the combo strategies generally under-perform the market benchmark throughout the entire 36-year period. Although transaction costs reduce final payoffs, the combo strategies steadily outperform the market over several decades. Figure 2. View largeDownload slide Cumulative log returns, 1980:1–2015:4. This figure depicts the natural log of the accumulated net-of-costs value of $1 invested in each of the long and short positions associated with the combination forecast-trading strategies described in Section 4 applied to the Fama–French forty-eight industries. BMi denotes the analogous trading strategy based on a real-time univariate predictive regression of returns on industry i on the lagged BM of industry i. Figure 2. View largeDownload slide Cumulative log returns, 1980:1–2015:4. This figure depicts the natural log of the accumulated net-of-costs value of $1 invested in each of the long and short positions associated with the combination forecast-trading strategies described in Section 4 applied to the Fama–French forty-eight industries. BMi denotes the analogous trading strategy based on a real-time univariate predictive regression of returns on industry i on the lagged BM of industry i. Figure 3 sorts industries in ascending order by average BMi, and then plots the percentage of quarters that each industry is chosen in the ABMA-based strategy (other strategies are not shown for brevity, but look qualitatively similar). The figure shows no clear relationship between BMi and industry selection in the ABMA strategy. In fact, the ABMA chooses the industry with the lowest average BMi (DRUGS) most often. Moreover, no industry is picked most of the time and most industries are chosen at least once. Hence, the industry selection of the ABMA strategy reflects meaningful time-variation in relative predicted returns across industries. Figure 3. View largeDownload slide Frequency each industry is picked in ABMA strategy BM ratios with industries sorted by B/M. Panel (A) depicts the percentage of times each industry is selected in the long leg of the rotation portfolio based on the ABMA forecasts. Panel (B) depicts the short leg. Industries are sorted by their time-series average BM along the horizontal axis from smallest on the left and highest on the right. Figure 3. View largeDownload slide Frequency each industry is picked in ABMA strategy BM ratios with industries sorted by B/M. Panel (A) depicts the percentage of times each industry is selected in the long leg of the rotation portfolio based on the ABMA forecasts. Panel (B) depicts the short leg. Industries are sorted by their time-series average BM along the horizontal axis from smallest on the left and highest on the right. Investors choose from many assets and their utility depends on the Sharpe ratio of their whole portfolio. Table V presents the ex post mean–variance frontier and maximum Sharpe ratio attainable by investing in the benchmark Fama–French–Carhart factors and each of our long–short industry-combo strategies in the presence of transaction costs. The forty-four industry combo strategies improve the maximum Sharpe ratio by 0.15–0.24 per year relative to the four-factor model, which is comparable to the gains from adding the Fama–French and Carhart factors to the market return. Moreover, given access to the forty-four industry combo strategies, the momentum factor has effectively zero weight in the ex post tangency portfolio. Hence, access to our industry-rotation strategies is an economically significant improvement in investment opportunities transaction costs. Table V. Ex post mean–variance efficient portfolios net of transaction costs This table reports ex post tangency portfolio weights and Sharpe ratios (SRs) on the net-of-costs returns on the Fama–French–Carhart factors and one of each of the different long–short strategies based on the four combination forecast methods described in Section 3 and constructed from the forty-four Fama–French industries. Panel A presents results for the Fama–French–Carhart factors. Panel B adds the long–short strategies. Panel A: Fama–French factors MKT SMB HML MOM L−S SR CAPM 1.00 0.46 FF3 0.51 0.00 0.49 0.61 FF4 0.38 0.00 0.41 0.21 0.71 Panel B: Forty-four industries ABMA 0.29 0.03 0.44 0.01 0.23 0.86 DMSFE 0.25 0.06 0.44 0.00 0.25 0.95 MEAN 0.27 0.06 0.45 0.00 0.23 0.89 Panel A: Fama–French factors MKT SMB HML MOM L−S SR CAPM 1.00 0.46 FF3 0.51 0.00 0.49 0.61 FF4 0.38 0.00 0.41 0.21 0.71 Panel B: Forty-four industries ABMA 0.29 0.03 0.44 0.01 0.23 0.86 DMSFE 0.25 0.06 0.44 0.00 0.25 0.95 MEAN 0.27 0.06 0.45 0.00 0.23 0.89 Table V. Ex post mean–variance efficient portfolios net of transaction costs This table reports ex post tangency portfolio weights and Sharpe ratios (SRs) on the net-of-costs returns on the Fama–French–Carhart factors and one of each of the different long–short strategies based on the four combination forecast methods described in Section 3 and constructed from the forty-four Fama–French industries. Panel A presents results for the Fama–French–Carhart factors. Panel B adds the long–short strategies. Panel A: Fama–French factors MKT SMB HML MOM L−S SR CAPM 1.00 0.46 FF3 0.51 0.00 0.49 0.61 FF4 0.38 0.00 0.41 0.21 0.71 Panel B: Forty-four industries ABMA 0.29 0.03 0.44 0.01 0.23 0.86 DMSFE 0.25 0.06 0.44 0.00 0.25 0.95 MEAN 0.27 0.06 0.45 0.00 0.23 0.89 Panel A: Fama–French factors MKT SMB HML MOM L−S SR CAPM 1.00 0.46 FF3 0.51 0.00 0.49 0.61 FF4 0.38 0.00 0.41 0.21 0.71 Panel B: Forty-four industries ABMA 0.29 0.03 0.44 0.01 0.23 0.86 DMSFE 0.25 0.06 0.44 0.00 0.25 0.95 MEAN 0.27 0.06 0.45 0.00 0.23 0.89 5.2 Alternative Test Assets We apply the portfolio-rotation strategies from the previous section to each of the twenty-five size/BM, size/investment, and size/profitability portfolios. However, we relegate the size/investment and size/profitability results and discussion to the Online Appendix for brevity. Characteristic portfolios are interesting for a variety of reasons. Many asset pricing studies use them (e.g., Fama and French, 1993, 2015, 2016). Further, the characteristic portfolios allow us to check the robustness of combination forecast methods to returns on test assets formed on economically different criterion than industry. The evidence from Section 4 also shows that return predictability varies with characteristics. Moreover, strategies based on characteristic sorts are potentially useful to investors who allocate funds across characteristic classes (e.g., Dimensional Fund Advisors, or Morningstar Equity Style Box10). The characteristic portfolios have several drawbacks, however, which lead us to advocate using industries in rotation strategies. First, multiple portfolios in the characteristic sorts consist entirely of small-cap stocks that are expensive to trade. Second, expected returns on characteristic portfolios have a well-known high correlation between characteristic ranks and expected returns. Our combo strategies should invest in the portfolios with the highest expected returns, but these portfolios tend to be easy to predict with their characteristic rank (e.g., small-cap and high BM). Verifying this pattern would confirm that the combination forecasts work properly, but is less informative than forming combo strategies with the industry returns whose expected return ranks are time-varying. Characteristic portfolios also have a tight factor structure (e.g., Lewellen, Nagel, and Shanken, 2010). The Fama–French factors in particular explain most of the variation in returns across the characteristic-sorted portfolios, which make it difficult to construct strategies with them that earn significant alpha. To illustrate this point, Figure 4 shows the distribution of the standard deviation ( σ(ϵt)) of the residuals from the Fama and French (2015) five-factor model: rte=α+β*MKTt+s*SMBt+h*HMLt+c*CMAt+r*RMWt+ϵt, (17) where rte denotes the gross excess returns one of the forty-four industry portfolios or the characteristic portfolios. Figure 4 shows that relative to the industry portfolios, the characteristic portfolios have a fraction of the variation in returns that is not explained by exposure to the Fama–French factors. Hence, the characteristic portfolios provide less opportunity than industry portfolios to demonstrate the effectiveness of the combination forecasts to improve the investment opportunity set spanned by these factors. Figure 4. View largeDownload slide Standard deviations of residuals from regressions of portfolio returns on Fama–French factors. This figure depicts histograms of the standard deviation of residuals from regressions of portfolio excess returns on the relevant Fama–French factors; MKT, SMB, HML, and MOM for the industry and size/BM portfolios, and the four factors plus CMA and RMW for the size/investment and size/profitability portfolios. The sample is 1980:1–2015:4. Figure 4. View largeDownload slide Standard deviations of residuals from regressions of portfolio returns on Fama–French factors. This figure depicts histograms of the standard deviation of residuals from regressions of portfolio excess returns on the relevant Fama–French factors; MKT, SMB, HML, and MOM for the industry and size/BM portfolios, and the four factors plus CMA and RMW for the size/investment and size/profitability portfolios. The sample is 1980:1–2015:4. We apply our long–short strategies to the characteristic-sorted portfolios following the industry strategies in Section 5. Because of the tight factor structure of these portfolios, we use fewer portfolios to long and short than with the industries to get the most extreme predicted returns. For the size and BM portfolios, we take a long position in the two portfolios with the highest predicted returns, and vice versa for the short position. Table VI presents average returns, Fama–French–Carhart four-factor αgross and factor loadings ignoring transaction costs, average net returns, four-factor αnet, turnover (%/quarter), and transaction costs (%/quarter) of the size and BM combo strategies as well as the benchmark BMi strategy. The combo strategies all generate a sizable spread in returns of about 8.7–10.7% per annum. Conversely, the BMi strategy yields less than half the spread in average returns, verifying the effectiveness of the combination return forecasts. The long–short size/BM combo strategies earn significant αgross in spite of the expected significant exposure to HML. The size and significance of the long–short α also parallel the patterns in ROS2 from Table III. ABMA generated the highest ROS2, followed by DMSFE and then MEAN. Similarly, the rotation strategies’ α is highest for ABMA, followed by DMSFE and then MEAN. Table VI. Performance of strategies using twenty-five size and BM portfolios Each column presents statistics for the long, short, and long-minus-short (L−S) rotation strategies described in Section 5 based on one of the three different combination forecast methods (ABMA, DMSFE, and MEAN) or the benchmark univariate real-time predictive regression that uses portfolio-own BM (BMi). The strategies use the Fama–French twenty-five size and BM portfolios. E(rgrosse) denotes annualized gross average excess return and αgross, β, s, and h and m denote estimates from the Fama–French–Carhart four-factor model: rgross,te=αgross+β*MKTt+s*SMBt+h*HMLt+m*MOMt+ϵt. (18) E(rnet) denotes the annualized average returns net of transaction costs and αnet denotes the generalized net-of-costs α of Novy-Marx and Velikov (2016). TO and T-Costs denote average turnover and transaction costs (%/quarter). The sample is 1980:1–2015:4 (N = 144). *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. ABMA DMSFE MEAN BMi Long Short L−S Long Short L−S Long Short L−S Long Short L−S E(rgrosse) 13.45 2.73 10.71 13.26 3.38 9.88 12.86 4.16 8.69 10.07 5.96 4.11 αgross 2.62** −4.54*** 7.16*** 2.67** −4.35*** 7.01*** 2.12 −3.05** 5.18** −0.13 −3.46** 3.33 (2.11) (−3.13) (3.29) (2.06) (−2.98) (2.87) (1.58) (−2.11) (2.19) (−0.13) (−2.24) (1.60) β 0.97*** 1.01*** −0.04 0.98*** 1.01*** −0.03 1.02*** 0.98*** 0.04 0.95*** 1.05*** −0.10 (23.22) (20.77) (−0.61) (22.52) (20.56) (−0.35) (22.55) (20.17) (0.51) (27.72) (20.32) (−1.51) s 0.74*** 0.54*** 0.20* 0.67*** 0.64*** 0.03 0.56*** 0.57*** −0.01 0.59*** 0.74*** −0.15 (12.48) (7.79) (1.93) (10.75) (9.08) (0.27) (8.67) (8.26) (−0.11) (12.00) (9.94) (−1.51) h 0.53*** −0.31*** 0.84*** 0.62*** −0.18*** 0.80*** 0.60*** −0.26*** 0.86*** 0.62*** −0.02 0.64*** (9.34) (−4.62) (8.43) (10.47) (−2.63) (7.13) (9.66) (−3.98) (7.93) (13.16) (−0.27) (6.68) m 0.07* 0.00 0.06 0.00 −0.04 0.04 0.01 0.00 0.01 −0.01 0.05 −0.05 (1.70) (0.06) (0.93) (0.07) (−0.78) (0.50) (0.34) (0.01) (0.19) (−0.25) (0.95) (−0.83) Adjusted R2 0.88 0.86 0.34 0.86 0.86 0.28 0.85 0.85 0.32 0.90 0.85 0.33 E(rnete) 11.68 −3.87 7.80 10.74 −5.22 5.52 11.46 −4.91 6.55 11.06 −5.90 5.16 αnet 1.49 3.44** 5.43** 1.44 2.67* 4.44* 1.00 1.42 2.56 0.00 0.00 0.00 (1.08) (2.44) (2.58) (0.91) (1.88) (1.88) (0.63) (0.95) (1.12) TO 38.7 24.9 31.8 41.4 31.7 36.6 35.5 37.8 36.7 49.6 45.3 47.5 T-Costs 0.44 0.28 0.73 0.45 0.38 0.83 0.45 0.43 0.88 0.52 0.54 1.06 ABMA DMSFE MEAN BMi Long Short L−S Long Short L−S Long Short L−S Long Short L−S E(rgrosse) 13.45 2.73 10.71 13.26 3.38 9.88 12.86 4.16 8.69 10.07 5.96 4.11 αgross 2.62** −4.54*** 7.16*** 2.67** −4.35*** 7.01*** 2.12 −3.05** 5.18** −0.13 −3.46** 3.33 (2.11) (−3.13) (3.29) (2.06) (−2.98) (2.87) (1.58) (−2.11) (2.19) (−0.13) (−2.24) (1.60) β 0.97*** 1.01*** −0.04 0.98*** 1.01*** −0.03 1.02*** 0.98*** 0.04 0.95*** 1.05*** −0.10 (23.22) (20.77) (−0.61) (22.52) (20.56) (−0.35) (22.55) (20.17) (0.51) (27.72) (20.32) (−1.51) s 0.74*** 0.54*** 0.20* 0.67*** 0.64*** 0.03 0.56*** 0.57*** −0.01 0.59*** 0.74*** −0.15 (12.48) (7.79) (1.93) (10.75) (9.08) (0.27) (8.67) (8.26) (−0.11) (12.00) (9.94) (−1.51) h 0.53*** −0.31*** 0.84*** 0.62*** −0.18*** 0.80*** 0.60*** −0.26*** 0.86*** 0.62*** −0.02 0.64*** (9.34) (−4.62) (8.43) (10.47) (−2.63) (7.13) (9.66) (−3.98) (7.93) (13.16) (−0.27) (6.68) m 0.07* 0.00 0.06 0.00 −0.04 0.04 0.01 0.00 0.01 −0.01 0.05 −0.05 (1.70) (0.06) (0.93) (0.07) (−0.78) (0.50) (0.34) (0.01) (0.19) (−0.25) (0.95) (−0.83) Adjusted R2 0.88 0.86 0.34 0.86 0.86 0.28 0.85 0.85 0.32 0.90 0.85 0.33 E(rnete) 11.68 −3.87 7.80 10.74 −5.22 5.52 11.46 −4.91 6.55 11.06 −5.90 5.16 αnet 1.49 3.44** 5.43** 1.44 2.67* 4.44* 1.00 1.42 2.56 0.00 0.00 0.00 (1.08) (2.44) (2.58) (0.91) (1.88) (1.88) (0.63) (0.95) (1.12) TO 38.7 24.9 31.8 41.4 31.7 36.6 35.5 37.8 36.7 49.6 45.3 47.5 T-Costs 0.44 0.28 0.73 0.45 0.38 0.83 0.45 0.43 0.88 0.52 0.54 1.06 Table VI. Performance of strategies using twenty-five size and BM portfolios Each column presents statistics for the long, short, and long-minus-short (L−S) rotation strategies described in Section 5 based on one of the three different combination forecast methods (ABMA, DMSFE, and MEAN) or the benchmark univariate real-time predictive regression that uses portfolio-own BM (BMi). The strategies use the Fama–French twenty-five size and BM portfolios. E(rgrosse) denotes annualized gross average excess return and αgross, β, s, and h and m denote estimates from the Fama–French–Carhart four-factor model: rgross,te=αgross+β*MKTt+s*SMBt+h*HMLt+m*MOMt+ϵt. (18) E(rnet) denotes the annualized average returns net of transaction costs and αnet denotes the generalized net-of-costs α of Novy-Marx and Velikov (2016). TO and T-Costs denote average turnover and transaction costs (%/quarter). The sample is 1980:1–2015:4 (N = 144). *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. ABMA DMSFE MEAN BMi Long Short L−S Long Short L−S Long Short L−S Long Short L−S E(rgrosse) 13.45 2.73 10.71 13.26 3.38 9.88 12.86 4.16 8.69 10.07 5.96 4.11 αgross 2.62** −4.54*** 7.16*** 2.67** −4.35*** 7.01*** 2.12 −3.05** 5.18** −0.13 −3.46** 3.33 (2.11) (−3.13) (3.29) (2.06) (−2.98) (2.87) (1.58) (−2.11) (2.19) (−0.13) (−2.24) (1.60) β 0.97*** 1.01*** −0.04 0.98*** 1.01*** −0.03 1.02*** 0.98*** 0.04 0.95*** 1.05*** −0.10 (23.22) (20.77) (−0.61) (22.52) (20.56) (−0.35) (22.55) (20.17) (0.51) (27.72) (20.32) (−1.51) s 0.74*** 0.54*** 0.20* 0.67*** 0.64*** 0.03 0.56*** 0.57*** −0.01 0.59*** 0.74*** −0.15 (12.48) (7.79) (1.93) (10.75) (9.08) (0.27) (8.67) (8.26) (−0.11) (12.00) (9.94) (−1.51) h 0.53*** −0.31*** 0.84*** 0.62*** −0.18*** 0.80*** 0.60*** −0.26*** 0.86*** 0.62*** −0.02 0.64*** (9.34) (−4.62) (8.43) (10.47) (−2.63) (7.13) (9.66) (−3.98) (7.93) (13.16) (−0.27) (6.68) m 0.07* 0.00 0.06 0.00 −0.04 0.04 0.01 0.00 0.01 −0.01 0.05 −0.05 (1.70) (0.06) (0.93) (0.07) (−0.78) (0.50) (0.34) (0.01) (0.19) (−0.25) (0.95) (−0.83) Adjusted R2 0.88 0.86 0.34 0.86 0.86 0.28 0.85 0.85 0.32 0.90 0.85 0.33 E(rnete) 11.68 −3.87 7.80 10.74 −5.22 5.52 11.46 −4.91 6.55 11.06 −5.90 5.16 αnet 1.49 3.44** 5.43** 1.44 2.67* 4.44* 1.00 1.42 2.56 0.00 0.00 0.00 (1.08) (2.44) (2.58) (0.91) (1.88) (1.88) (0.63) (0.95) (1.12) TO 38.7 24.9 31.8 41.4 31.7 36.6 35.5 37.8 36.7 49.6 45.3 47.5 T-Costs 0.44 0.28 0.73 0.45 0.38 0.83 0.45 0.43 0.88 0.52 0.54 1.06 ABMA DMSFE MEAN BMi Long Short L−S Long Short L−S Long Short L−S Long Short L−S E(rgrosse) 13.45 2.73 10.71 13.26 3.38 9.88 12.86 4.16 8.69 10.07 5.96 4.11 αgross 2.62** −4.54*** 7.16*** 2.67** −4.35*** 7.01*** 2.12 −3.05** 5.18** −0.13 −3.46** 3.33 (2.11) (−3.13) (3.29) (2.06) (−2.98) (2.87) (1.58) (−2.11) (2.19) (−0.13) (−2.24) (1.60) β 0.97*** 1.01*** −0.04 0.98*** 1.01*** −0.03 1.02*** 0.98*** 0.04 0.95*** 1.05*** −0.10 (23.22) (20.77) (−0.61) (22.52) (20.56) (−0.35) (22.55) (20.17) (0.51) (27.72) (20.32) (−1.51) s 0.74*** 0.54*** 0.20* 0.67*** 0.64*** 0.03 0.56*** 0.57*** −0.01 0.59*** 0.74*** −0.15 (12.48) (7.79) (1.93) (10.75) (9.08) (0.27) (8.67) (8.26) (−0.11) (12.00) (9.94) (−1.51) h 0.53*** −0.31*** 0.84*** 0.62*** −0.18*** 0.80*** 0.60*** −0.26*** 0.86*** 0.62*** −0.02 0.64*** (9.34) (−4.62) (8.43) (10.47) (−2.63) (7.13) (9.66) (−3.98) (7.93) (13.16) (−0.27) (6.68) m 0.07* 0.00 0.06 0.00 −0.04 0.04 0.01 0.00 0.01 −0.01 0.05 −0.05 (1.70) (0.06) (0.93) (0.07) (−0.78) (0.50) (0.34) (0.01) (0.19) (−0.25) (0.95) (−0.83) Adjusted R2 0.88 0.86 0.34 0.86 0.86 0.28 0.85 0.85 0.32 0.90 0.85 0.33 E(rnete) 11.68 −3.87 7.80 10.74 −5.22 5.52 11.46 −4.91 6.55 11.06 −5.90 5.16 αnet 1.49 3.44** 5.43** 1.44 2.67* 4.44* 1.00 1.42 2.56 0.00 0.00 0.00 (1.08) (2.44) (2.58) (0.91) (1.88) (1.88) (0.63) (0.95) (1.12) TO 38.7 24.9 31.8 41.4 31.7 36.6 35.5 37.8 36.7 49.6 45.3 47.5 T-Costs 0.44 0.28 0.73 0.45 0.38 0.83 0.45 0.43 0.88 0.52 0.54 1.06 The αnet on the size/BM combo strategies are lower than the αgross for each of the combo strategies, and also decrease going from the ABMA to DMSFE to MEAN. The αnet are significant for the ABMA strategy, marginally significant for the DMSFE strategy, and are insignificant for the MEAN strategy. The lower magnitude and significance of the αnet than αgross reflects the fact that the transaction costs are higher for smaller stocks that are selected relatively often because they have the most extreme returns on average. Results in the Online Appendix show the long–short size/investment combo strategies have similar average returns and α as the size/BM strategies. In contrast, the long–short size/profitability combo strategies have similar average returns, but only marginally significant α. Regardless of choice of test assets, the long–short spreads in average returns on the portfolio-rotation strategies are large, ranging from 7.7% to 14.2% depending on method and assets. These spreads in returns need not accompany Fama–French–Carhart α, but the long–short combo strategies do mostly earn significant α, even net of transaction costs. Similarly, adding the industry long–short combo strategies to the Fama–French–Carhart factors substantially increases the maximum attainable Sharpe ratio. Thus, combining information in the cross-section of BM ratios improves investment opportunities for investors net of transaction costs. 6. Conclusion Predictive regressions show that the cross-section of industry BM ratios has greater explanatory power for forecasting both individual industry returns and CFs than industry-own or aggregate BM ratios. We use combination forecast methods to capture this cross-industry predictive information in real time. Since 1980, the combination forecasts predict quarterly industry returns with out-of-sample R2 that average 1.6–2.1%. Trading strategies that buy and sell industries with the highest and lowest out-of-sample predicted returns, respectively, generate significant a Fama–French–Carhart alpha of 7.1–8.9% per year net of transaction costs. Combination forecasts based on the cross-section of characteristic portfolios’ BM ratios also generate significant out-of-sample R2 and their trading strategies frequently result in significant Fama–French–Carhart alpha. Overall, our results reveal that a stock’s BM ratio can provide useful predictive information for the expected returns on other stocks. This finding can be explained by reexamining present value models. An asset’s BM ratio is a noisy proxy for that asset’s expected return because it is a joint function of the asset’s expected future return and CFs. Given commonality in expected returns and imperfect correlation of CFs across stocks, BM ratios of any one stock portfolio can provide relevant but non-redundant signals about the expected returns on other stocks or portfolios. Future studies that predict returns on industry or characteristic portfolios should therefore consider the cross-section of valuation ratios. Supplementary Material Supplementary data are available at Review of Finance online. Footnotes 1 From the thirty-eight industries, we exclude Agric, Stone, Garbg, Steam, Water, and Govt. From the forty-eight industries, we exclude Coal, Guns, Smoke, and Soda. 2 The industry MINES has a very low first-autocorrelation coefficient of 0.06. This is caused by MINES having a relatively small number of firms, several of which have volatile market values and only report COMPUSTAT accounting data every other quarter. In the Online Appendix, we demonstrate that all results are robust to an alternative construction of BM that corrects for this problem by using the prior quarter’s book equity if the current quarter’s is missing. 3 Hasbrouck provides SAS code to estimate effective spreads at http://people.stern.nyu.edu/jhasbrou/. This code runs directly on the WRDS server after only changing relevant file paths. 4 Novy-Marx and Velikov (2016) present an analogous plot for all but CMA and RMW. 5 In the Online Appendix, we discuss how relaxing the Campbell and Thompson (2008) restriction impacts the combination forecasts. 6 The first principal component of industry BM has a 93% correlation with BMM, so principal components 2–6 are effectively common factors in the cross-section of BM beyond the market BM. 7 We present results for analogous strategies using the thirty-two industry portfolios in the Online Appendix. Inferences are unchanged. 8 We also construct combo strategies using BM ratios with annual accounting data. 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Published by Oxford University Press on behalf of the European Finance Association. All rights reserved. For Permissions, please email: journals.permissions@oup.com

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Review of FinanceOxford University Press

Published: Jul 10, 2017

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