Dissecting Characteristics Nonparametrically

Abstract

In his presidential address, Cochrane (2011) argues the cross-section of the expected return “is once again descending into chaos.” Harvey et al. (2016) identify “hundreds of papers and factors” that have predictive power for the cross-section of expected returns. Many economic models, such as the consumption capital asset pricing model (CAPM) of Lucas (1978), Breeden (1979), and Rubinstein (1976), instead predict that only a small number of state variables suffice to summarize cross-sectional variation in expected returns.

Researchers typically employ two methods to identify return predictors: (1) (conditional) portfolio sorts based on one or multiple characteristics, such as size or book-to-market, and (2) linear regression in the spirit of Fama and MacBeth (1973). Both methods have many important applications, but they fall short in what Cochrane (2011) calls the multidimensional challenge: “Which characteristics really provide independent information about average returns? Which are subsumed by others?” Portfolio sorts are subject to the curse of dimensionality when the number of characteristics is large, and linear regressions make strong functional form assumptions and are sensitive to outliers.¹ In addition, in many empirical settings, most of the variation in characteristic values and returns are in the extremes of the characteristic distribution and the association between characteristics and returns appears nonlinear (see Fama and French 2008). Cochrane (2011) speculates, “to address these questions in the zoo of new variables, I suspect we will have to use different methods.”

We propose a nonparametric method to determine which firm characteristics provide incremental information for the cross-section of expected returns without making strong functional form assumptions. Specifically, we use a group LASSO (least absolute shrinkage and selection operator) procedure developed by Huang, Horowitz, and Wei (2010) for model selection and nonparametric estimation. Model selection deals with the question of which characteristics have incremental predictive power for expected returns, given the other characteristics. Nonparametric estimation deals with estimating the effect of important characteristics on expected returns without imposing a strong functional form.

We show three applications of our proposed framework. First, we study which characteristics provide incremental information for the cross-section of expected returns. We estimate our model on 62 characteristics including size, book-to-market, beta, and other prominent variables and anomalies on a sample period from July 1965 to June 2014. Only 13 variables, including size, total volatility, and past return-based predictors, have incremental explanatory power for expected returns for the full sample period and all stocks. A hedge portfolio going long stocks with high predicted returns and shorting stocks with low predicted returns has an in-sample Sharpe ratio of more than 3. Only 11 characteristics have predictive power for returns in the first half of our sample. In the second half, instead, we find 14 characteristics are associated with cross-sectional return premiums. For stocks whose market capitalization is above the 20% NYSE size percentile, only nine characteristics, including changes in shares outstanding, past returns, and standardized unexplained volume, remain incremental return predictors. The in-sample Sharpe ratio is still 2.25 for large stocks.

Second, we compare the out-of-sample performance of the nonparametric model with a linear model. Estimating flexible functional forms raises the concern of in-sample overfitting. We estimate both a linear and the nonparametric model over a period until 1990 and select return predictors. We then use 10 years of data to estimate the models on the selected characteristics. In the first month after the end of our estimation period, we take the selected characteristics, predict 1-month-ahead returns, and construct a hedge portfolio similar to our in-sample exercise. We roll the estimation and prediction period forward by 1 month and repeat the procedure until the end of the sample.

Specifically, we perform model selection once until December 1990 for both the linear model and the nonparametric model. Our first estimation period is from December of 1981 until November of 1990, and the first out-of-sample prediction is for January 1991 using characteristics from December 1990.² We then move the estimation and prediction period forward by 1 month. The nonparametric model generates an out-of-sample Sharpe ratio of 2.75 compared to 1.06 for the linear model.³ When we adjust Sharpe ratios for transaction costs, we find the nonlinear model still compares favorably to the linear model with Sharpe ratios of 1.56 and 0.29, respectively. The characteristics we study are not a random sample, but have been associated with cross-sectional return premiums in the past. Therefore, we focus mainly on the comparison across models rather than emphasizing the overall magnitude of the Sharpe ratios.

The linear model selects 30 characteristics in-sample compared to only eleven for the nonparametric model, but performs worse out-of-sample and nonlinearities are important. We find an increase in out-of-sample Sharpe ratios relative to the Sharpe ratio of the linear model when we employ the nonparametric model for prediction but use the 30 characteristics the linear model selects. The linear model appears to overfit the data in-sample. We find an identical Sharpe ratio for the linear model when we use the 11 characteristics selected by the nonparametric model, as we do with the 30 characteristics selected by the linear model. These results underscore once more the importance of nonlinearities. With the same set of 11 characteristics the nonlinear model selects, we find the nonparametric model has a Sharpe ratio that is larger by a factor of 2.5 relative to the Sharpe ratio of the linear model using the same set of characteristics.

Third, we study whether the predictive power of characteristics for expected returns varies over time. We estimate the model using 120 months of data on all characteristics we select in our baseline analysis, and then estimate rolling 1-month-ahead return forecasts. We find substantial time variation in the predictive power of characteristics for expected returns. As an example, momentum returns conditional on other return predictors vary substantially over time, and we find a momentum crash similar to Daniel and Moskowitz (2016) as past losers appreciated during the recent financial crisis. Size conditional on the other selected return predictors, instead, has a significant predictive power for expected returns throughout our sample period similar to the findings in Asness, Frazzini, Israel, Moskowitz, and Pedersen (2018).

The method we propose has several “tuning” parameters and one might be concerned that our conclusions depend on some of the choices we have to make. We document in an extensive simulation study both aspects of our proposed method: model selection and return prediction. Across a wide array of choices regarding the tuning parameters, we find the adaptive group LASSO performs well along both dimensions, that is, it has a high probability to select the “right” set of characteristics and performs well in predicting returns out of sample. We also compare the performance of the nonlinear adaptive group LASSO for model selection and return prediction to a linear LASSO and popular recent proposals like increased thresholds for t-statistics or p-value adjustments for false-discovery rates (FDRs). We find along both dimensions that allowing for nonlinearities improves performance substantially.

The paper provides a new method in empirical asset pricing to understand which of the previously published firm characteristics provide information for expected returns conditional on other characteristics. We see this exercise as a natural first step in the “multidimensional challenge.” Once we understand which characteristics indeed provide incremental information, we can aim to relate characteristics to factor exposures, estimate factors and stochastic discount factors directly, or relate characteristics and factors to economic models.

The capital asset pricing model (CAPM) of Sharpe (1964), Lintner (1965), and Mossin (1966) predicts that an asset’s beta with respect to the market portfolio is a sufficient statistic for the cross-section of expected returns. Subsequently, researchers identified many variables that contain additional independent information for expected returns. Fama and French (1992) synthesize these findings, and Fama and French (1993) show that a 3-factor model with the market return, a size factor, and a value factor can explain cross-sections of stocks sorted on characteristics that appeared anomalous relative to the CAPM. In this sense, Fama and French (1993) achieve a significant dimension reduction: researchers who want to explain the cross-section of stock returns only have to explain the size and value factors.

In the 20 years following the publication of Fama and French (1992), many researchers joined a “fishing expedition” to identify characteristics and factor exposures that the 3-factor model cannot explain. Harvey, Liu, and Zhu (2016) provide an overview of this literature and list over 300 published papers that study the cross-section of expected returns. They propose a |$t$|-statistic of 3 for new factors to account for multiple testing on a common data set. However, even employing the higher threshold for the |$t$|-statistic still leaves approximately 150 characteristics as useful predictors for the cross-section of expected returns.

The large number of significant predictors is not a shortcoming of Harvey et al. (2016), who address the issue of multiple testing. Instead, authors in this literature usually consider their proposed return predictor in isolation without conditioning on previously discovered return predictors. Haugen and Baker (1996) and Lewellen (2015) are notable exceptions. They employ Fama and MacBeth (1973) regressions to combine the information in multiple characteristics. Lewellen (2015) jointly studies the predictive power of 15 characteristics and finds that only few are significant predictors for the cross-section of expected returns. Green, Hand, and Zhang (2017) adjust Fama-MacBeth regressions to avoid overweighting microcaps, adjust |$p$|-values for a data snooping bias and find for a sample starting in 1980 that many return predictors do not provide independent information. Although Fama-MacBeth regressions carry a lot of intuition, they do not offer a formal model selection method. We build on Lewellen (2015) and provide a framework that allows for nonlinear associations between characteristics and returns, provide a formal framework to disentangle important from unimportant return predictors, and study many more characteristics.

We also build on a large literature in economics and statistics using penalized regressions. Horowitz (2016) gives a general overview of model selection in high-dimensional models, and Huang, Horowitz, and Wei (2010) discuss variable selection in a nonparametric additive model similar to the one we implement empirically. Recent applications of LASSO methods in finance are Rapach, Strauss, and Zhou (2013), who use a LASSO to investigate the lead-lag relationship between US and international stock returns, Huang and Shi (2016), who use an adaptive group LASSO in a linear framework and construct macro factors to test for determinants of bond risk premiums, and Chinco, Clark-Joseph, and Ye (2019), who use a linear model for high-frequency return predictability using past returns of related stocks, and find their method increases predictability relative to OLS. Goto and Xu (2015) use a LASSO to obtain a sparse estimator of the inverse covariance matrix for mean variance portfolio optimization. Chinco, Neuhierl, and Weber (2019) invert the best-fit tuning parameter in penalized regressions to estimate the anomaly base rate.

Gagliardini, Ossola, and Scaillet (2016) develop a weighted two-pass cross-sectional regression method to estimate risk premiums from an unbalanced panel of individual stocks. Giglio and Xiu (2016) instead propose a three-pass regression method that combines principal component analysis and a two-stage regression framework to estimate consistent factor risk premiums in the presence of omitted factors when the cross-section of test assets is large. DeMiguel et al. (2016) extend the parametric portfolio approach of Brandt et al. (2009) to study which characteristics provide valuable information for portfolio optimization. Kelly, Pruitt, and Su (2017) generalize standard PCA to allow for time-varying loadings and extract common factors from the universe of individual stocks. Kim, Korajczyk, and Neuhierl (2018) develop a new method to estimate arbitrage portfolios by utilizing information contained in firm characteristics for both abnormal returns and factor loadings. Kozak, Nagel, and Santosh (2017) exploit economic restrictions relating expected returns to covariances to construct stochastic discount factors. Lettau and Pelger (2018) generalizes PCA by including a penalty on the pricing error in expected returns that allows them to identify “weak” factors with high Sharpe ratios.

We, instead, are mainly concerned with formal model selection, that is, which characteristics provide incremental information in the presence of other characteristics.

1. Current Methods and Nonparametric Models

1.1 Expected returns and current methods

We often use portfolio sorts to approximate |$m_t$| for a single characteristic. We typically sort stocks into 10 portfolios and compare mean returns across portfolios. Portfolio sorts are simple, straightforward, and intuitive, but they also suffer from several shortcomings. They suffer from the curse of dimensionality, they do not offer formal guidance to discriminate between characteristics, and they assume returns do not vary within portfolio.

Linear regressions allow us to study the predictive power for expected returns of many characteristics jointly, but they also have potential pitfalls. Most importantly, no a priori reason exists why the conditional mean function should be linear.

We discuss many of these shortcomings in more detail in Section A.2 of the Online Appendix and how researchers typically address some of the shortcomings. Cochrane (2011) synthesizes many of the challenges that portfolio sorts and linear regressions face in the context of many return predictors, and suspects “we will have to use different methods.”

1.2 Nonparametric estimation

Cochrane (2011) conjectures in his presidential address, “[P]ortfolio sorts are really the same thing as nonparametric cross-sectional regressions, using nonoverlapping histogram weights.” We establish a formal equivalence result between portfolio sorts and regressions in the Online Appendix. Specifically, suppose we have a single characteristic |$C_{1,it-1}$| and we sort stocks into |$L$| portfolios depending on the value of the characteristic. We show in the Online Appendix a one-to-one relationship exists between the portfolio returns and regression coefficients in a regression of returns on |$L$| indicator functions, where indicator function |$l$| is equal to |$1$| if stock |$i$| is in portfolio |$l$| for |$l = 1, \ldots, L$|⁠. Hence, portfolio sorts are equivalent to approximating the conditional mean function with a step function. The nonparametric econometrics literature also refers to these functions as constant splines. Our proposed framework is a natural generalization of portfolio sorts. Specifically, we use a smooth extension of this estimation strategy with many possible regressors.

Although the assumption of an additive model is somewhat restrictive, it provides desirable econometric advantages. In addition, this assumption is far less restrictive than assuming additivity and linearity, as we do in Fama-MacBeth regressions. Another major advantage of an additive model is that we can jointly estimate the model for a large number of characteristics, select important characteristics, and estimate the summands of the conditional mean function, |$m_{t}$|⁠, simultaneously, as we explain in subsection 1.3 below. In addition, the additive structure allows us to extrapolate to regions with very few data points on the joint support of characteristics from regions with more data, because we can average over the marginal distributions of characteristics. Without any additional structure, we would get very imprecise estimates when the joint distribution of characteristics is sparse.

Hence, knowledge of the conditional mean function |$m_t$| is equivalent to knowing the transformed conditional mean function |$\tilde{m}_t$|⁠, which is the function we estimate.⁶ Similar to portfolio sorting, we are typically not interested in the actual value of a characteristic in isolation, but rather in the rank of the characteristic in the cross-section. Consider firm size. Size grows over time, and a firm with a market capitalization of USD 1 billion in the 1960s was considered a large firm, but today it is not. Our normalization considers the relative size in the cross-section rather than the absolute size, similar to portfolio sorting.

1.3 Adaptive group LASSO

The idea of the group LASSO is to estimate the functions |$\tilde{m}_{ts}$| nonparametrically, while setting functions for a given characteristic to |$0$| if the characteristic does not help predict returns. Therefore, the procedure achieves model selection; that is, it discriminates between the functions |$\tilde{m}_{ts}$|⁠, which are constant, and the functions that are not constant.⁷

The first part of Equation (5) is just the sum of the squared residuals as in ordinary least squares regressions; the second part is the LASSO group penalty function. Rather than penalizing individual coefficients, |$b_{sk}$|⁠, the group LASSO penalizes all coefficients associated with a given characteristic. Thus, we can set the point estimates of an entire expansion of |$\tilde{m}_t$| to 0 when a given characteristic does not provide incremental information for expected returns. Because of the penalty, the LASSO is applicable even when the number of characteristics is larger than the sample size. Yuan and Lin (2006) propose to choose |$\lambda_1$| in a data-dependent way to minimize Bayesian information criterion (BIC), which we follow in our application.

However, the first step of the LASSO may select too many characteristics. Informally speaking, the LASSO selects all characteristics that predict returns, but also selects some characteristics that have no predictive power. A second step introduces characteristic-specific weights in the LASSO group penalty function as a function of first-step estimates to address this problem. The Online Appendix discusses in Section A.3 the second step, the consistency conditions, and the efficiency properties of the resultant estimates in detail.

If the cross-section is sufficiently large, we could perform model selection and estimation period by period. Hence, the method allows for the importance of characteristics and the shape of the conditional mean function to vary over time. For example, some characteristics might lose their predictive power for expected returns over time. McLean and Pontiff (2016) show that for 97 return predictors, predictability decreases by 58% post publication. However, if the conditional mean function was time-invariant, pooling the data across time would lead to more precise estimates of the function and therefore more reliable predictions. In our empirical application in Section 2, we estimate our model not only over the whole sample in the baseline but also over subsamples and estimate rolling specifications to investigate the variation in the conditional mean function over time.

1.4 Interpretation of the conditional mean function

Therefore, the summands of the transformed conditional mean function, |$\tilde{m}_{s}$|⁠, are only identified up to a constant. The model-selection procedure, expected returns, and the portfolios we construct do not depend on these constants. However, the constants matter when we plot an estimate of the conditional mean function for one characteristic.

We report estimates of the functions using the common normalization that the functions integrate to |$0$|⁠, which is identified.

Section A.6 of the Online Appendix discusses how we construct confidence bands for the reported figures and how we select the number of interpolation points in the empirical application of Section 2.

2. Empirical Application

We now discuss the universe of characteristics we use in our empirical application and study which of the 62 characteristics provide incremental information for expected returns, using the adaptive group LASSO for selection and estimation.

2.1 Data

Stock return data come from the Center for Research in Security Prices (CRSP) monthly stock file. We follow standard conventions and restrict the analysis to common stocks of firms incorporated in the United States trading on NYSE, Amex, or Nasdaq.

Balance sheet data are from the Standard and Poor’s Compustat database. We use balance sheet data from the fiscal year ending in calendar year |$t-1$| for estimation starting in June of year |$t$| until May of year |$t+1$| predicting returns from July of year |$t$| until June of year |$t+1$|⁠.

Table 1 provides an overview of the 62 characteristics we use. We group them into six categories: past return based predictors, such as momentum (⁠|$\mathbf{r_{12-2}}$|⁠) and short-term reversal (⁠|$\mathbf{r_{2-1}}$|⁠); investment-related characteristics, such as the annual percentage change in total assets (Investment) or the change in inventory over total assets (IVC); profitability-related characteristics, such as gross profitability over the book-value of equity (Prof) or return on operating assets (ROA); intangibles, such as operating accruals (OA) and tangibility (Tan); value-related characteristics, such as the book-to-market ratio (BEME) and earnings-to-price (E2P); and trading frictions, such as the average daily bid-ask spread (Spread) and standard unexplained volume (SUV). We follow Hou, Xue, and Zhang (2015) in the classification of characteristics.

To alleviate a potential survivorship bias due to backfilling, we require that a firm has at least 2 years of Compustat data. Our sample period is July 1965 until June 2014. Table 2 reports summary statistics for various firm characteristics and return predictors. We calculate all statistics annually and then average over time. We have 1.6 million observations in our baseline analysis.

Section A.1 in the Online Appendix contains a detailed description of the characteristics, the construction, and the relevant references.

2.2 Selected characteristics and their influence

The purpose of this section is to show different applications of the adaptive group LASSO. We do not aim to exhaust all possible combinations of characteristics, sample periods, and firm sizes, or all possible applications but rather aim to provide some insights into the flexibility of the method in actual data. Section 3 contains an extensive simulation to study the choices researchers have to make when implementing the method, such as the number of interpolation points, the order of the spline functions, or the information criterion for model selection. Another goal of the simulation is to compare in detail the performance to alternative (linear) models and model selection techniques, such as the |$t$|-statistic adjustment of Harvey et al. (2016) or the FDR |$p$|-value adjustment of Green et al. (2017).

Table 3 reports average annualized returns with standard errors in parentheses of 10 equally weighted portfolios sorted on the characteristics we study. Most of the 62 characteristics individually have predictive power for expected returns in our sample period and result in large and statistically significant hedge portfolio returns and alphas relative to the Fama and French 3-factor model (Table 4). Thirty-one sorts have annualized hedge returns of more than 5%, and 13 characteristics are even associated with excess returns of more than 10%. Thirty-six characteristics have a |$t$|-statistic above 2. Correcting for exposure to the Fama-French 3-factor model affects these findings minimally. The vast majority of economic models, that is, the ICAPM (Merton (1973)) or consumption-based models, as surveyed in Cochrane (2007), suggest a low number of state variables can explain the cross-section of returns. Therefore, all characteristics are unlikely to provide incremental information for expected returns.