Industry Return Predictability: A Machine Learning Approach


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1 Industry Return Predictability: A Machine Learning Approach David E. Rapach Saint Louis University Jack K. Strauss University of Denver Jun Tu Singapore Management University Guofu Zhou Washington University in St. Louis and CAFR October 8, 2018 Corresponding author. Send correspondence to Guofu Zhou, Olin School of Business, Washington University in St. Louis, St. Louis, MO 63130; phone: This paper supersedes an earlier version that circulated in July 2014 under the title of Industry Interdependencies and CrossIndustry Return Predictability. We are grateful to conference and seminar participants at the 2015 Australasian Finance and Banking Conference, 2015 International Symposium on Forecasting, 2015 Midwest Econometrics Group Meeting, 2016 Midwest Finance Association Meeting, 2016 SMU Summer Camp, 2017 Conference on Financial Predictability and Big Data, 2018 Alliance Bernstein Boston Quantitative Finance Conference, Saint Louis University, West Virginia University, and especially Robert Connolly (MFA discussant), Rossen Valkanov (SMU Summer Camp discussant), and Dacheng Xiu for very helpful comments. Jun Tu acknowledges financial support from the Sim Kee Boon Institute for Financial Economics.
2 Industry Return Predictability: A Machine Learning Approach Abstract We use machine learning tools to analyze industry return predictability based on the information in lagged industry returns from across the entire economy. Controlling for postselection inference and multiple testing, we find significant insample evidence of industry return predictability. Lagged returns for the financial sector and commodityand materialproducing industries exhibit widespread predictive ability, consistent with the gradual diffusion of information across economically linked industries. Outofsample industry return forecasts that incorporate the information in lagged industry returns are economically valuable: controlling for systematic risk using leading multifactor models from the literature, an industryrotation portfolio that goes long (short) industries with the highest (lowest) forecasted returns delivers an annualized alpha of over 8%. The industryrotation portfolio also generates substantial gains during economic downturns, including the Great Recession. JEL classifications: C22, C58, G11, G12, G14 Key words: Predictive regression; OLS postlasso; Postselection inference; Multiple testing; Industryrotation portfolio; Multifactor model
3 1. Introduction A voluminous literature investigates aggregate stock market return predictability. 1 In contrast, a relatively limited number of studies examine stock return predictability along industry lines, even though analyst reports and asset allocations are often industry based. Studies that analyze industry return predictability typically rely on popular predictor variables from the literature on aggregate market return predictability, such as the aggregate dividend yield, nominal yields, and yield spreads (e.g., Ferson and Harvey 1991, 1999; Ferson and Korajczyk 1995; Avramov 2004). In this paper, we examine industry return predictability using a different information set namely, lagged industry returns from across the entire economy. Our study is the first to directly analyze industry return predictability based on lagged industry returns from across the economy. The lack of previous studies investigating this topic is perhaps due to the statistical challenges inherent in estimating regression models with a large number of predictors. 2 Our use of lagged industry returns to forecast individual industry returns is motivated by the theoretical model in Hong et al. (2007), who introduce information frictions into an economy with multiple linked industries. Due to the industry links, cash flow shocks originating in one industry can affect expected cash flows in related industries. In a frictionless rationalexpectations equilibrium, investors readily recognize all of the interindustry implications of a cash flow shock in a particular industry. Consequently, equity prices across all relevant industries immediately adjust to fully impound the interindustry effects of the cash flow shock, and lagged industry returns do not have predictive power. However, incorporating insights from Merton (1987) and Hong and Stein (1999), Hong et al. (2007) posit that investors with limited informationprocessing capabilities spe 1 See Rapach and Zhou (2013) for a survey. 2 Moskowitz and Grinblatt (1999) analyze industry return dynamics in the context of momentum (Jegadeesh and Titman 1993), while we directly test whether lagged industry returns have predictive power for individual industry returns. Cohen and Frazzini (2008) and Menzly and Ozbas (2010) investigate customersupplier links between individual firms and industries; in contrast, our study focuses on how returns across all industries are related to one another. 1
4 cialize in specific market segments. In this environment, when a cash flow shock arises in a particular industry, informationprocessing limitations prevent investors specializing in related industries from quickly working out the full implications of the shock. Information thus diffuses gradually across industries, and the resulting delayed adjustment in equity prices gives rise to industry return predictability on the basis of lagged industry returns. We analyze the predictive ability of lagged industry returns in a general predictive regression framework that allows each industry s return to respond to the lagged returns for all industries, thereby accommodating a wide array of industry links, both direct and indirect. Given the plethora of potential predictors in the predictive regression models, conventional ordinary least squares (OLS) estimation has potential drawbacks. First, if all lagged returns are used, the abundance of predictors makes OLS estimation susceptible to overfitting. Second, if only a few lagged returns are chosen, it is difficult to know a priori which are the most important. Therefore, we employ a machine learning approach to guard against overfitting the data in our highdimensional setting and to select the most relevant predictors. Along this line, Feng et al. (2017), Freyberger et al. (2018), Han et al. (2018), and Kozak et al. (forthcoming) recently utilize machine learning to identify the relevant determinants of crosssectional stock returns. 3 Instead of identifying the relevant determinants of crosssectional returns, we harness machine learning to discover the pertinent lagged industry returns for predicting individual industry returns. Although our research topics differ, we share with Feng et al. (2017), Freyberger et al. (2018), Han et al. (2018), and Kozak et al. (forthcoming) the use of machine learning to deal with the challenges posed by highdimensional data in finance. We use the least absolute shrinkage and selection operator (LASSO, Tibshirani 1996), a powerful and popular tool in machine learning. Similarly to ridge regression (Hoerl and Kennard 1970), the LASSO induces shrinkage in the estimated coefficients by including a 3 In addition, Gu et al. (2018) use a wide variety of machine learning tools to analyze individual firm return predictability based on a large number of firm characteristics and macroeconomic variables, while Chinco et al. (forthcoming) use machine learning to analyze crossfirm return predictability at the oneminute horizon. 2
5 convex penalty term in the objective function for fitting a model. In contrast to the l 2 penalty term in ridge regression, the LASSO relies on an l 1 penalty term, so that it permits shrinkage to exactly zero for some coefficients. It thus performs variable selection, which typically yields a sparse model. Sparsity has two key advantages. First, it helps to avoid overfitting the data by setting unimportant coefficients to zero. Second, it facilitates interpretation of the estimated model by identifying the most relevant predictor variables. Although the LASSO s l 1 penalty term mitigates overfitting the data via sparsity, it also tends to overshrink the coefficients for the selected variables. This tendency can lead to substantial downward biases (in magnitude) in the estimated coefficients (Fan and Li 2001). Recent studies propose OLS postlasso estimation to reduce the biases (e.g., Efron et al. 2004; Meinshausen 2007; Belloni and Chernozhukov 2011, 2013). The idea is to first use the LASSO to reduce the dimension of the model; then, to lessen the biases in the LASSO coefficient estimates, the coefficients for the selected predictor variables are reestimated using OLS. We use OLS postlasso to estimate predictive regression models for each industry, where the set of candidate predictors includes the lagged returns for all 30 industries that we consider. OLS postlasso estimation allows us to identify the most relevant set of lagged industry returns for predicting a given industry s return, while generating more accurate estimates of the coefficients for the relevant lagged industry returns. We use both insample and outofsample tests to analyze the ability of lagged industry returns to predictive individual industry returns. With respect to the insample tests, we address the important issues of postselection inference and multiple testing. In the context of OLS postlasso estimation, postselection inference recognizes that the predictor variables appearing in the regression model are first selected from a large group of candidate predictors, so that conventional inferences for the selected variables coefficients are not necessarily reliable we peeked at the data twice. We incorporate insights from Berk et al. (2013), Leeb et al. (2015), Lee et al. (2016), and Zhao et al. (2017) to make accurate postselection inferences concerning industry return predictability. 3
6 Multiple testing relates to the fact that we test a large number of individual null hypotheses, so that conventional pvalues which implicitly assume that a researcher tests a given null hypothesis in isolation can present a misleading picture of statistical significance. To account for multiple testing, we use the Benjamini and Hochberg (2000) adaptive version of the Benjamini and Hochberg (1995) linear stepup procedure to control the false discovery rate when testing hypotheses concerning individual coefficients for the entire set of LASSOselected lagged industry returns. For the insample analysis, we estimate predictive regression models via OLS postlasso using monthly return data spanning 1960 to 2016 for 30 industry portfolios from Kenneth French s Data Library. The LASSO selects at least one lagged industry return as a predictor for 29 of the individual industries, while multiple lagged industry returns are selected for 22 of the individual industries. Furthermore, the OLS postlasso estimation results indicate that the lagged industry returns selected by the LASSO are often statistically significant predictors of industry returns, even after controlling for postselection inference and multiple testing. Although monthly equity returns inherently contain a relatively small predictable component, the Campbell and Thompson (2008) metric indicates that the degree of return predictability is economically meaningful for nearly all industries. Lagged industry returns also retain their predictive ability when we control for the popular predictor variables used by Ferson and Harvey (1991, 1999), Ferson and Korajczyk (1995), and Avramov (2004). Overall, we find extensive evidence of statistically and economically meaningful industry return predictability based on the information in lagged industry returns. Lagged returns for the financial sector and commodity and materialproducing industries are selected as return predictors for numerous individual industries. The prominent role for these sectors is economically intuitive from the perspective of the theoretical model in Hong et al. (2007). Firms in many industries rely on financial intermediaries for financing: when the financial sector experiences a positive return shock, financial firms have larger capital buffers and thus become more willing to provide credit on favorable terms to industries 4
7 across the economy; borrowers benefit directly from the better terms, while their customers benefit indirectly. In the presence of information frictions, we thus expect lagged financial sector returns to positively affect future returns in many industries, which is what we find. Furthermore, commodity price shocks raise product prices and returns for sectors located in earlier production stages, while they squeeze profit margins and lower returns for sectors located in later production stages. With information frictions, we thus expect returns for lagged commodity and materialproducing industries to negatively affect future returns for industries located in later stages of the production chain, which is again what we find. However, we cannot readily attribute all industry return predictability that we detect to gradual information diffusion across economically related industries. Indeed, it is well known that machine learning is a powerful tool for discovering new relationships and patterns in the data. Some of the predictability patterns that we uncover require further study in future research to better understand them, as is the case as for a number of the interesting discoveries by Feng et al. (2017), Freyberger et al. (2018), and Kozak et al. (forthcoming). The outofsample analysis mimics the situation of an investor in real time and measures the economic value of industry return predictability. We first compute outofsample forecasts of monthly industry returns based on predictive regressions estimated via OLS postlasso. We then sort the 30 industries according to their forecasted returns over the next month and construct a zeroinvestment portfolio that goes long (short) the top (bottom) quintile of sorted industries. For the 1970:01 to 2016:12 outofsample period, the industryrotation portfolio generates an average return of 7.33% per annum, annualized Sharpe (Sortino) ratio of 0.65 (1.16), and Goetzmann et al. (2007) manipulationproof performance measure (MPPM) of 4.84% for a relative risk aversion coefficient of four. The industryrotation portfolio performs especially well during businesscycle recessions, thereby providing a hedge against bad times for the macroeconomy. The portfolio also exhibits negative exposures to the broad equity market and Fama and French (1993) value factors, and it generates a substantial annualized alpha of more than 8% in the context of both the Carhart 5
8 (1997) fourfactor and Hou et al. (2015) qfactor models. Furthermore, the industryrotation portfolio based on the OLS postlasso forecasts produces a considerably higher average return, Sharpe and Sortino ratios, MPPM, and alpha than a benchmark industryrotation portfolio based on prevailing mean forecasts of industry returns, which ignore the information in lagged industry returns. The portfolio based on the OLS postlasso forecasts also outperforms a second benchmark portfolio based on industry return forecasts computed via OLS estimation of the predictive regressions, demonstrating the usefulness of our machine learning approach in an outofsample context. The rest of the paper is organized as follows. Section 2 presents the predictive regression framework and discusses OLS postlasso estimation. Section 3 reports insample results, while Section 4 reports outofsample results for the longshort industryrotation portfolios. Section 5 contains concluding remarks. An Online Appendix accompanying the paper reports results for a variety of robustness checks. 2. Predictive Regression Framework Our basic framework is the following general predictive regression model specification: y i = a i ι T + Xb i + ε i for i = 1,..., N, (2.1) where [ y i = [ X = [ x j = [ b i = ] r i,1 r i,t, (2.2) x 1 x N ], (2.3) ] r j,0 r j,t 1 for j = 1,..., N, (2.4) ] b i,1 b, (2.5) i,n 6
9 [ ε i = ε i,1 ε i,t ], (2.6) r i,t is the montht return on industry portfolio i in excess of the onemonth Treasury bill return, ι T is a T vector of ones, T is the usable number of monthly observations, N is the number of industry portfolios, and ε i,t is a zeromean disturbance term. Equation (2.1) allows for lagged returns for all industries across the economy to affect a given industry s excess return, thereby accommodating general industry links. 4 Due to the highdimensional nature (N = 30) of Equation (2.1), conventional OLS estimation runs the risk of overfitting. To deal with the challenges posed by the high dimensionality of the predictive regression models, we employ the LASSO from machine learning. Tibshirani (1996) introduces the LASSO as a regularization device for performing shrinkage in regressions with a large number of candidate predictor variables. For Equation (2.1), the LASSO objective function can be expressed as ( ) 1 arg min a i R, b i R 2T y i a i ι T Xb i λ i b i 1, (2.7) N where ( T ) 0.5 [ z 2 = zt 2 for z = t=1 z 1 z T ], (2.8) b i 1 = N b i,j, (2.9) j=1 and λ i 0 is a regularization parameter. The first component in parentheses in Equation (2.7) is the familiar sum of squared residuals, so that the objective function reduces to 4 Equation (2.1) can be viewed as a representative equation from a firstorder vector autoregression for the entire set of N industry portfolio excess returns. By including an industry s own return as a regressor in Equation (2.1), we guard against finding spurious evidence of industry return predictability based on lagged industry returns due to autocorrelation in an industry s own return in conjunction with contemporaneously correlated industry returns. Boudoukh et al. (1994), Hameed (1997), and Chordia and Swaminathan (2000) warn of spurious return predictability when analyzing leadlag relationships among portfolios sorted according to firm size and trading volume. 7
10 that for OLS when λ i = 0. The second component in parentheses in Equation (2.7) is an l 1 penalty term that shrinks the slope coefficient estimates to prevent overfitting. Unlike ridge regression, which relies on an l 2 penalty, the l 1 penalty in Equation (2.7) allows for shrinkage to zero (for sufficiently large λ i ), so that it performs variable selection. By reducing the dimension of the model, variable selection helps to guard against overfitting and facilitates interpretation of the estimated model. Powerful algorithms are available for computing the solution to Equation (2.7), such as cyclical coordinate descent (Friedman et al. 2010), making LASSO estimation feasible even when N is large. The LASSO generally performs well in selecting the most relevant predictor variables (e.g., Zhang and Huang 2008; Bickel et al. 2009; Meinshausen and Yu 2009). However, LASSO estimates of the coefficients for the selected predictors suffer from downward biases in magnitude (Fan and Li 2001). Intuitively, the LASSO penalty term overshrinks the coefficients for the selected predictors, and the effect is likely to be more severe for the most relevant predictors. To alleviate the biases in the LASSO estimates, Efron et al. (2004), Meinshausen (2007), and Belloni and Chernozhukov (2011, 2013) propose reestimating the coefficients for the LASSOselected predictors via OLS (OLS postlasso estimation). Because the predictor variables are first selected from a set of candidate predictors, statistical inference based on OLS postlasso estimation constitutes postselection inference, an active area of recent research (e.g., Wasserman and Roeder 2009; Meinshausen et al. 2009; Berk et al. 2013; Leeb et al. 2015; Lee et al. 2016; Tibshirani et al. 2016; Zhao et al. 2017). 5 When making postselection inferences, we first need to identify the inferential target. The traditional target is the set of true regression coefficients for the full model that includes all of the potential predictors (Leeb et al. 2015), which, in our context, corresponds to b i in Equation (2.1). Alternatively, Berk et al. (2013) advocate targeting the true regression 5 See Dezeure et al. (2015) and Taylor and Tibshirani (2015) for recent surveys of the postselection inference literature. 8
11 coefficients for a submodel: y i = a (M) i ι T + X M b (M) i + ε (M) i, (2.10) where M {1,..., N} is the index set of predictors for the submodel and X M is comprised of the columns of X indexed by M. Recognizing that the actual datagenerating process is essentially unknowable, a useful submodel provides a parsimonious representation of the data that focuses on the most relevant predictors. 6 Because they measure different things, the true regression coefficients for the full model and submodel are generally different: b i,j measures the response to a given predictor, conditional on the remaining N 1 predictors in the full model; b (M) i,j measures the response to the predictor, conditional on the other M 1 predictors in the submodel, where M is the cardinality of M. 7 Let CI ( ˆM) i,j for any j ˆM denote the conventional OLS confidence interval for b ( ˆM) i,j the model selected by the LASSO, where ˆM is the index set of predictors for the LASSOselected model. Leeb et al. (2015) use simulations to investigate the coverage properties of CI ( ˆM) i,j. For the case where the true regression coefficients for a submodel are targeted, they find that CI ( ˆM) i,j has approximately correct coverage. This finding is surprising, as the naïve OLS postlasso confidence interval does not explicitly account for the fact that the data are used twice first to select the predictor variables from the complete set of potential predictors via the LASSO and then to estimate the regression coefficients for the LASSOselected predictors via OLS. Zhao et al. (2017) provide a theoretical explanation for this finding. Under reasonably weak assumptions, they show that the set of LASSOselected predictors is deterministic with high probability, so that the conventional OLS postlasso 6 Targeting the submodel is in the spirit of George Box s famous dictum, All models are wrong, but some are useful (Box 1979, p. 202). 7 Measuring the predictor variables in deviations from means, the vectors of true slope coefficients for the full model and submodel are related by b (M) i = (X M X M) 1 X M Xb i, so that b i,m = b(m) i only if X M X Mb i, M = 0, where X M is comprised of the columns of X not in X M and b i,m (b i, M ) is comprised of the elements of b i in (not in) b(m) i (Berk et al. 2013; Zhao et al. 2017). in 9
12 confidence intervals and tstatistics are asymptotically valid. 8 In Section 3, we target the true regression coefficients for the LASSOselected submodel. In line with the results in Leeb et al. (2015) and Zhao et al. (2017), we assess statistical significance using conventional OLS postlasso tstatistics. The results in Leeb et al. (2015) and Zhao et al. (2017) demonstrate that the pvalue for the conventional OLS postlasso tstatistic is valid for making postselection inferences for an individual coefficient b ( ˆM) i,j. However, when analyzing the significance of a large number of individual coefficient estimates, as we do in Section 3, the issue of multiple testing remains. When testing many individual null hypotheses, conventional pvalues can present a misleading picture of statistical significance. 9 Suppose that we test m individual null hypotheses (m = 167 in our application in Section 3), each at the significance level q. One approach for addressing multiple testing is to control the familywise error rate: FWER = Pr(V 1) q, where V is the number of wrongly rejected null hypotheses. The familiar Bonferroni procedure controls the FWER by multiplying each of the unadjusted pvalues by m. However, FWER control tends to be extremely conservative, with little power to detect false null hypotheses. Instead of the FWER, Benjamini and Hochberg (1995) propose to control the false discovery rate: FDR = E(V/R) q, where R is the number of rejections. They develop a popular linear stepup procedure based on adjusted pvalues to control the FDR. The Benjamini and Hochberg (1995) procedure orders the unadjusted pvalues and their corresponding null hypotheses (p (1),..., p (m) and H (1),..., H (m), respectively) and rejects the first k null hypotheses (H (1),..., H (k) ), where { k = max 1 j m : p BH (j) q } (2.11) j 8 Asymptotically, it is as if the data are only used once. 9 Harvey et al. (2016), Green et al. (2017), Hou et al. (2018), and Linnainmaa and Roberts (2018) highlight the relevance of multiple testing in the context of identifying significant firm characteristics in the cross section of stock returns. 10
13 and p BH (j) = m j p (j) (2.12) is the adjusted pvalue for H (j). Benjamini and Hochberg (1995) show that the linear stepup procedure controls the FDR when p (1),..., p (m) are independent, while Benjamini and Yekutieli (2001) demonstrate that the procedure controls the FDR for a particular type of dependence (positive regression dependence) in the pvalues. 10 When the number of true null hypotheses m 0 is less than m, the Benjamini and Hochberg (1995) procedure controls the FDR at too low of a level: FDR π 0 q, where π 0 = m 0 /m. To improve its performance, a number of studies (e.g., Benjamini and Hochberg 2000; Storey et al. 2004; Benjamini et al. 2006; Blanchard and Roquain 2009) adapt the Benjamini and Hochberg (1995) procedure to the data by replacing p BH (j) in Equation (2.12) with p adapt (j) = ˆπ 0 m j p (j), (2.13) where ˆπ 0 is a conservative estimate of π 0. The adaptive procedures can be shown to control the FDR when p (1),..., p (m) are independent. Theoretical results concerning FDR control for adaptive procedures under dependent pvalues are more limited (e.g., Farcomeni 2007; Blanchard and Roquain 2009), and simulations are typically used to analyze FDR control and power. In Section 3, we control for multiple testing via the FDR using the Benjamini and Hochberg (2000) adaptive procedure, which employs the Hochberg and Benjamini (1990) estimator for π 0. For values of m relevant to our application, simulations in Benjamini et al. (2006) indicate that the Benjamini and Hochberg (2000) adaptive procedure delivers good overall performance in terms of FDR control and power, including for dependent pvalues. Finally, to implement OLS postlasso estimation, we need to choose the regularization parameter λ i in Equation (2.7). The most popular approach for choosing λ i is Kfold cross 10 Benjamini and Yekutieli (2001) modify the Benjamini and Hochberg (1995) procedure so that it controls the FDR under arbitrarily dependent pvalues. However, the modified procedure is typically very conservative (at times more so than the Bonferroni procedure), so that it has limited ability to identify false null hypotheses. 11
14 validation. However, this approach can be sensitive to the number of folds K, as well as to how the folds are defined. To avoid these drawbacks, we choose λ i using the Hurvich and Tsai (1989) corrected version of the Akaike information criterion (AIC, Akaike 1973). In the realistic case where the true model is not included among the candidate models, Flynn et al. (2013) show that the corrected AIC (AICc) is asymptotically efficient for choosing λ i for OLS postlasso estimation, meaning that the AICc asymptotically selects the bestperforming model from among the candidate models in terms of the l 2 loss criterion for predictive performance. In addition, Flynn et al. (2013) find that the AICc performs well for selecting the bestperforming model in finitesample simulations InSample Results We estimate predictive regressions via OLS postlasso using monthly excess return data for 30 valueweighted industry portfolios from Kenneth French s Data Library, 12 where the industries are defined based on the Standard Industrial Classification (SIC) system. Table 1 reports summary statistics for the industry portfolio excess returns for 1959:12 to 2016:12. We refer to the industries by their Data Library abbreviations, which are given in the notes to Table 1. Along with the fact that the industry portfolios are value weighted, starting the sample in 1959:12 mitigates illiquidity and thintrading concerns. 13 Smoke (Tobacco Products) displays the highest annualized average excess return and Sharpe ratio, 11.79% and 0.56, respectively, in Table 1, while Steel (Steel Works, Etc.) has the lowest annualized average excess return and Sharpe ratio, 3.49% and 0.14, respectively. Table 2 reports the OLS postlasso coefficient estimates for each industry. 14 After accounting for the lagged predictors, the available estimation sample covers 1960:01 to 2016:12 11 In extensive simulations, Taddy (2017) also finds that the AICc performs well for choosing λ i for various LASSObased procedures. 12 Available at library.html. 13 We start the sample in 1959:12 to account for the lagged predictors when estimating the predictive regressions. 14 All computations are performed using R (R Core Team 2017). We use the glmnet package (Friedman et al. 2017) to compute the LASSO solution. Following convention, the predictor variables are standardized 12
15 (684 observations). As discussed Section 2, we target the true regression coefficients for the LASSOselected submodel. To conserve space, we use a bold (italicized bold) entry to indicate that a coefficient estimate is significant at the 10% (5%) level based on the conventional OLS postlasso tstatistic. Overall, the OLS postlasso estimates in Table 2 highlight the importance of lagged industry returns for predicting individual industry returns. The LASSO selects 167 lagged industry returns (out of a possible 900) as individual industry return predictors. At least one lagged industry return is selected as a return predictor by the LASSO for 29 of the 30 individual industries, and multiple lagged industry returns are selected for 22 individual industries. According to the conventional OLS postlasso tstatistics, 82 (53) of the 167 LASSOselected lagged industry returns are significant predictors at the 10% (5%) level. 15 In general, autocorrelation plays a limited role in Table 2, as an industry s own lagged return is only selected by the LASSO for seven industries. Figure 1 controls for multiple testing across all 167 of the OLS postlasso coefficient estimates in Table 2. The figure depicts the first 85 elements of the sequence of sorted unadjusted pvalues for the conventional OLS postlasso tstatistics, along with corresponding adjusted pvalues based on the Benjamini and Hochberg (2000) adaptive procedure. In line with the bold (italicized bold) entries in Table 2, there are 82 (53) rejections of H 0 : b ˆM i,j = 0 at the 10% (5%) significance level based on the unadjusted pvalues. According to the Benjamini and Hochberg (2000) adjusted pvalues, 72 (21) of these rejections remain when we control for multiple testing via the FDR. 16 In conjunction with the outofsample results in Section 4, the numerous significant coefficient estimates in Table 2 that survive multiplebefore solving Equation (2.7). The estimated coefficients in Table 2 correspond to the scales of the original predictor variables. 15 As shown in Table A1 of the Online Appendix, inferences are similar for tstatistics based on White (1980) heteroskedasticityrobust standard errors. 16 There are 72 (17) rejections at the 10% (5%) significance level when we control for multiple testing using the Storey et al. (2004) adaptive procedure with the tuning parameter set to the significance level. Simulations in Blanchard and Roquain (2009) indicate that the Storey et al. (2004) adaptive procedure with a tuning parameter of q delivers good overall performance in terms of FDR control and power for both independent and dependent pvalues. 13
16 testing control suggest that our evidence of industry return predictability is not simply an artifact of data mining. Many of the coefficient estimates in Table 2 appear economically plausible from the perspective of gradual information diffusion across economically related industries (Hong et al. 2007). For example, lagged Fin (Banking, Insurance, Real Estate, and Trading) returns are selected by the LASSO for 19 of the 30 individual industries, and eleven (seven) of the coefficient estimates are significant at the 10% (5%) level according to the conventional OLS postlasso tstatistics. 17 Furthermore, all of the coefficient estimates for lagged Fin returns are positive. This is economically reasonable, as firms in many industries rely extensively on financial intermediaries for financing. A positive return shock in the financial sector increases financial firms capital buffers, so that financial firms become more willing to make credit available to firms throughout the economy; in contrast, adverse shocks to the financial sector curtail intermediaries capacity to lend, thereby driving up borrowing costs and driving down returns for many sectors. Financial sector shocks have direct effects for firms that borrow from financial intermediaries, as well as indirect effects for the customers of the borrowing firms. The coefficient estimates for lagged Fin returns are typically sizable in Table 2, reaching as high as 0.18 (Txtls, Textiles). Another noteworthy pattern in Table 2 involves industries located in different stages of the production process. Lagged returns for commodity and materialproducing industries located in earlier stages of the production chain such as Coal (Coal) and Oil (Petroleum and Natural Gas) are often negatively related to returns for industries located in later stages of the production chain such as Smoke, Books (Printing and Publishing), Txtls, Paper (Business Supplies and Shipping Containers), Whlsl (Wholesale), and Meals (Restaurants, Hotels, and Motels). Lagged Coal and Oil returns are selected by the LASSO for 16 and 13, respectively, of the individual industries in Table 2. Eleven and 13 (seven and nine) of the coefficient estimates for lagged Coal and Oil returns, respectively, are significant at the 17 Nine (three) of the coefficient estimates are significant at the 10% (5%) level after controlling for multiple testing using the Benjamini and Hochberg (2000) adaptive procedure. 14
17 10% (5%) level based on the conventional OLS postlasso tstatistics. 18 The estimated coefficients for lagged Coal and Oil returns are all negative in Table 2, with the exception of the autoregression coefficient for Coal. These negative relationships presumably stem from supply shocks that raise product prices and returns for sectors located in earlier production stages but squeeze profit margins and lower returns for sectors located in later production stages. The significant coefficient estimates are again sizable in magnitude. While additional predictive relationships in Table 2 readily accord with gradual information diffusion across related industries, there are other relationships that are more challenging to explain; for example, it is not obvious what economic channel links lagged Beer (Beer and Liquor) to future Coal returns. It is well known that machine learning is an effective means for discovering new relationships in the data, and we uncover numerous unusual relationships in Table 2. Future research is needed to better understand their economic underpinnings, a common issue for machine learning studies (e.g., Feng et al. 2017; Freyberger et al. 2018; Kozak et al. forthcoming). For the industries with at least one lagged return selected by the LASSO, the R 2 statistics at the bottom of Table 2 range from 0.78% (Chems, Chemicals) to 7.93% (Clths, Apparel). Because monthly stock returns inherently contain a sizable unpredictable component, the degree of monthly stock return predictability will necessarily be limited. To assess the economic importance of the R 2 statistics, we use the convenient metric suggested by Campbell and Thompson (2008): ( )( R 2 CT i = i 1 + S 2 i 1 Ri 2 Si 2 ), (3.1) where R 2 i is the R 2 statistic for the predictive regression when r i,t is the regressand and S i is the unconditional Sharpe ratio for r i,t. Equation (3.1) measures the proportional increase in average excess return for a meanvariance investor who allocates between the industry i 18 After controlling for multiple testing using the Benjamini and Hochberg (2000) adaptive procedure, eight and twelve (three and two) of the coefficient estimates for lagged Coal and Oil returns, respectively, are significant at the 10% (5%) level. 15
18 equity portfolio and riskfree bills when the investor utilizes return predictability relative to when the investor ignores return predictability. The parentheses below the R 2 statistics in Table 2 report the CT i metric for each industry based on its monthly Sharpe ratio in Table 1 and R 2 statistic in Table According to the CT i metrics at the bottom of Table 2, industry return predictability based on lagged industry returns increases the average excess return for a meanvariance investor by proportional factors ranging from 0.64 (ElcEq, Electrical Equipment) to (Autos, Automobiles and Trucks). The vast majority of the CT i measures are above one, so that the average excess return for a meanvariance investor more than doubles. The substantial proportional increases in average excess return indicate that the R 2 statistics represent an economically meaningful degree of return predictability for nearly all of the industries. 20 Compared to OLS estimation of the full model, the LASSO sets numerous unimportant coefficients to zero, thereby reducing the noise in the data to better identify the predictive signal in lagged industry returns. As previously discussed, however, the LASSO tends to overshrink the fullmodel OLS coefficient estimates for the predictors selected by the LASSO. Indeed, on average, the LASSO coefficient estimates are less than half as large (in magnitude) as the corresponding fullmodel OLS estimates for the lagged industry returns selected by the LASSO. 21 The degree of shrinkage induced by OLS postlasso estimation is more limited: on average, the OLS postlasso estimates are only 15% smaller (in magnitude) than the corresponding fullmodel OLS estimates for the LASSOselected predictors. 22 In 19 We divide the annualized Sharpe ratio in Table 1 by 12 to arrive at the monthly Sharpe ratio. 20 We use the Campbell and Thompson (2008) metric in Table 2 to analyze the economic significance of return predictability for each industry individually. In Section 4, we further analyze the economic value of industry return predictability by constructing an industryrotation portfolio that simultaneously invests across multiple industries. 21 The LASSO and fullmodel OLS coefficient estimates are reported in Tables A2 and A3, respectively, of the Online Appendix. 22 Of course, the R 2 statistics for the full models estimated via OLS are larger than the corresponding statistics at the bottom of Table 2, as OLS estimation of the full model maximizes the R 2 statistic by construction. However, the R 2 statistics for the full models likely overstate the actual degree of return predictability, as OLS estimation fails to adequately filter the noise in the data. We provide outofsample evidence of the benefits of the noise reduction afforded by OLS postlasso visàvis OLS estimation in Section 4. 16
19 sum, OLS postlasso estimation more accurately measures the predictive signal in lagged industry returns by avoiding both the overfitting associated with fullmodel OLS estimation and overshrinking of coefficient estimates engendered by the LASSO for the LASSOselected predictors. Informed by Leeb et al. (2015) and Zhao et al. (2017), we base postselection inferences on conventional OLS postlasso tstatistics in Table 2. To check the robustness of the postselection inferences, we also compute confidence intervals using the approach of Lee et al. (2016). Demonstrating that the LASSO selection event can be characterized as a union of polyhedra, Lee et al. (2016) construct valid confidence intervals conditional on the event M = ˆM. Many of the coefficient estimates that are significant in Table 2 are also significant based on Lee et al. (2016) confidence intervals. 23 Instead of primarily capturing information frictions across industries, the return predictability in Table 2 could reflect time variation in risk premia. To shed light on this issue, we augment Equation (2.1) with four lagged predictor variables similar to those used by Ferson and Harvey (1991, 1999), Ferson and Korajczyk (1995), and Avramov (2004): the S&P 500 dividend yield, threemonth Treasury bill yield, difference between yields on a tenyear Treasury bond and threemonth Treasury bill (term spread), and difference between yields on BAA and AAArated corporate bonds (credit spread). 24 These variables represent popular return predictors from the literature, and they are often viewed as capturing timevarying risk premia. We again use OLS postlasso to estimate the augmented predictive regression models. When the lagged economic variables are included in Equation (2.1), the lagged industry returns selected by the LASSO and corresponding OLS postlasso coefficient estimates are typically quite similar to those reported in Table Popular measures 23 The complete results are reported in Table A4 of the Online Appendix. We compute Lee et al. (2016) confidence intervals using the selectiveinference package (Tibshirani et al. 2017). 24 Data for these variables are from Global Financial Data at 25 Complete results for the augmented predictive regressions are reported in Table A5 of the Online Appendix. 17
20 of timevarying risk premia thus do not readily account for the predictive power of lagged industry returns. Another potential concern is that the LASSO tends to select one predictor variable from among a group of correlated predictors while dropping the other predictors. To address this concern, we select the lagged industry returns in Equation (2.1) using the elastic net (ENet, Zou and Hastie 2005). The penalty term for the ENet is a convex combination of l 1 (ridge) and l 2 (LASSO) components. For a value of 0.5 for the parameter that blends the l 1 and l 2 penalty components, the ENet has a stronger tendency than the LASSO to select correlated predictor variables as a group. In our application, the ENet (based on a blending parameter value of 0.5) and LASSO typically select similar sets of predictor variables for the individual industries. Indeed, the two procedures select the same set of lagged industry returns as predictors for many individual industries. 26 Practically speaking, the LASSO s simpler penalty term thus appears sufficient for identifying the relevant lagged industry returns for predicting individual industry returns. 4. OutofSample Results This section reports outofsample measures of the economic value of industry return predictability in the context of a monthly longshort industryrotation portfolio. We construct a longshort industryrotation portfolio for 1970:01 to 2016:12 using outofsample forecasts of monthly industry excess returns based on OLS postlasso estimation of predictive regressions for each industry. We construct the longshort industryrotation portfolio as follows. We first use data from the beginning of the sample through 1969:12 to estimate Equation (2.1) for each industry via OLS postlasso and generate a set of 30 outofsample industry excess return forecasts for 1970:01. We sort the industries in ascending order according to the excess return forecasts and form equalweighted quintile portfolios; we then create a zeroinvestment portfolio that 26 The complete ENet results are reported in Table A6 of the Online Appendix. 18
21 goes long (short) the top (bottom) quintile portfolio. Next, we use data through 1970:01 to compute an updated set of outofsample industry excess return forecasts for 1970:02 based on OLS postlasso estimation of Equation (2.1), sort the industries according to the forecasts, form equalweighted quintile portfolios, and the zeroinvestment portfolio again goes long (short) the top (bottom) quintile portfolio. Continuing in this fashion, we construct a monthly longshort industryrotation portfolio based on the OLS postlasso industry excess return forecasts for the 1970:01 to 2016:12 outofsample period (564 months). 27 In generating the outofsample excess return forecasts, observe that we only use data available at the time of forecast formation when selecting the predictor variables via the LASSO and estimating the predictive regressions. Our forecasts thus do not entail lookahead bias, so that the outofsample exercise mimics the situation of an investor in real time. For the purpose of comparison, we construct a pair of benchmark longshort industryrotation portfolios. The first benchmark is constructed in the same manner as the portfolio described above, except that we use outofsample industry excess return forecasts based on the prevailing mean. The prevailing mean forecast corresponds to the constant expected excess return model Equation (2.1) with b i,j = 0 for j = 1,..., N which assumes that individual industry excess returns do not depend on lagged industry returns. The forecast is simply the mean industry excess return based on data from the beginning of the sample through the month of forecast formation. 28 We use this benchmark to assess portfolio performance under the assumption of no industry excess return predictability (apart from the mean excess return). The second benchmark portfolio is again constructed in the same manner, but it relies on outofsample industry excess return forecasts based on OLS estimation of Equation (2.1). Comparing the performance of the benchmark portfolio based on the OLS forecasts to that of the portfolio based on the OLS postlasso forecasts provides 27 Starting the outofsample period in 1970:01 provides a reasonably long initial insample period for estimating the parameters of the predictive regressions used to generate the initial outofsample industry excess return forecasts. 28 The prevailing mean forecast is a popular benchmark in outofsample tests of return predictability (e.g., Goyal and Welch 2008). 19
22 a sense of the ability of machine learning to increase the economic value of outofsample industry return forecasts. Table 3 reports performance measures for the industryrotation portfolio based on the OLS postlasso forecasts and two benchmark portfolios. In addition to the annualized mean, volatility, and Sharpe ratio, the table reports the annualized downside risk and Sortino ratio, 29 maximum drawdown, and annualized Goetzmann et al. (2007) MPPM for a relative risk aversion coefficient of four. MPPM measures the continuously compounded certainty equivalent return in excess of the riskfree return for a powerutility investor. The benchmark portfolio based on the prevailing mean forecasts generates an annualized average return of 2.22%, so that simply going long (short) industries with the highest (lowest) historical returns is not a profitable strategy over the outofsample period. Due to the negative average return, the Sharpe and Sortino ratios are both negative. The prevailing mean benchmark portfolio also has a sizable annualized downside risk (8.34%), large maximum drawdown (73.97%), and negative annualized MPPM ( 4.73%). In contrast to the prevailing mean benchmark, the benchmark portfolio based on the OLS forecasts produces a positive annualized average return of 5.52%. Although it has somewhat higher volatility than the prevailing mean benchmark, the OLS benchmark evinces a smaller downside risk (7.17%), much lower maximum drawdown (29.39%), positive Sharpe and Sortino ratios (0.47 and 0.77, respectively), and positive MPPM (2.84%). Overall, incorporating the information in lagged industry returns, as captured by the OLS forecasts, substantially improves portfolio performance relative to ignoring such information. Although the OLS benchmark portfolio generally outperforms the prevailing mean benchmark in Table 3, OLS estimation of the predictive regressions is vulnerable to overfitting, which potentially detracts from the usefulness of the OLS forecasts as inputs for constructing the industryrotation portfolio. The performance measures for the industryrotation port [ 29 Downside risk is measured by the semistandard deviation: DR = (1/T ) ] T 0.5, t=1 min(r p,t, 0) 2 where r p,t is the portfolio return and T is the number of outofsample observations. The Sortino ratio is the average portfolio return dividend by the downside risk. 20
23 folio based on the OLS postlasso forecasts indicate that our machine learning approach increases the economic value of industry returns forecasts, as the OLS postlasso forecasts improve portfolio performance visàvis the OLS forecasts across all measures in Table 3. The annualized average return and volatility of 7.33% and 11.29%, respectively, for the OLS postlasso portfolio translate into a substantive Sharpe ratio of The downside risk and maximum drawdown (6.31% and 25.65%, respectively) are smaller for the OLS post LASSO portfolio visàvis the OLS portfolio, while the Sortino ratio (1.16) is considerably larger. Furthermore, the annualized MPPM increases by 200 basis points (to 4.84%) when we allocate across industries using the OLS postlasso forecasts in lieu of the OLS forecasts. Figure 2 depicts log cumulative returns for the longshort industryrotation portfolios. In line with its negative average return in Table 3, the prevailing mean benchmark portfolio (dashed line) experiences prolonged periods of poor performance in Figure 2. The OLS benchmark (dotted line) produces gains on a relatively consistent basis, and it typically performs well during businesscycle recessions as dated by the National Bureau of Economic Research (NBER). The industryrotation portfolio based on the OLS postlasso forecasts (solid line) also delivers sizable gains on a consistent basis and appears to perform even better than the OLS benchmark during recessions, especially the recent Great Recession. Table 4 examines portfolio performance over the business cycle in greater detail. The table reports annualized average portfolio returns during good and bad macroeconomic states, measured as months when the economy is in an NBERdated expansion or recession, respectively. As a robustness check, we also measure the state of the economy using the Chicago Fed national activity index (CFNAI), where months in the bottom quintile (top 80%) of CFNAI observations are deemed a bad (good) state. 30 The average returns for the benchmark portfolio based on the prevailing mean forecasts are negative during both expansions and recessions (although neither is significant), and the difference in average returns during recessions visàvis expansions is insignificant. Confirm 30 Monthly indicator variables for NBER recessions and CFNAI observations are from Federal Reserve Economic Data at 21
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