Asset Pricing Tests Using Random Portfolios

Transcription

1 Asset Pricing Tests Using Random Portfolios Frank Ecker Abstract Results from two-stage asset pricing tests vary with the type and number of test assets. First, implied factor premia from systematic portfolios as test assets depend on the level of aggregation. Randomly generated portfolios yield results that are consistent with a reduction in measurement error and robust to the arbitrary level of aggregation. Qualitative results generally converge when more assets are used, i.e., when aggregation decreases. Second, the market and HML factor premia are strongly dependent on the distinction between explaining and predicting returns. January 2013 Keywords: Asset pricing, two-stage cross-sectional test, random portfolios, Fama-French portfolios. Duke University, The Fuqua School of Business, 100 Fuqua Drive, Durham, NC 27708, frank.ecker@duke.edu. I am grateful for the valuable comments from Michael Brandt, Per Olsson, Tjomme Rusticus (discussant), Katherine Schipper, participants at the Duke/UNC Fall Camp, Duke University, and the 2012 AAA meeting.

2 1. Introduction Recent research in finance and accounting frequently fails to provide evidence that supports risk factors being priced in two-stage cross-sectional asset pricing tests (see Fama and Mac- Beth 1973) on realized returns. 1 For example, Core, Guay, and Verdi (2008) do not find a significant premium for the Fama-French (1993) size and book-to-market factors, nor for a factor based on accruals quality (AQ), in tests on the firm level. Portfolio-level tests in Petkova (2006) and Core et al. (2008) also raise doubts about the market factor being priced. In some specifications, the market factor appears to be negatively associated with returns. In this paper, I examine how the power of two-stage cross-sectional asset pricing tests varies with the construct validity of factor betas in the first-stage and with the choice of test assets. I find that results are highly sensitive to research design choices, and that there are economically plausible research designs that result in significant factor premia. Firmlevel results highlight the importance of differentiating between tests that aim to explain contemporaneous returns and tests that aim to predict future returns. Complementing the firm-level tests, results from randomly generated portfolios are consistent with measurement error reduction. Results from standard portfolios in the literature, such as the Fama-French size/book-to-market portfolios, are not. First, I probe the effects of varying the (time-series) estimation of the factor betas, i.e. the independent variables in the (cross-sectional) second stage tests. Prior research appears indifferent between constant betas (i.e., betas estimated over the entire sample period) and rolling betas, with the vast majority of studies using constant betas. I hypothesize that rolling betas of individual firms have higher construct validity for two reasons. First, they do not incorporate future returns information and are therefore not conditional on the firm s 1 The two-stage (or two-pass) cross-sectional test estimates asset-specific betas in a time-series regression in the first stage. The second stage consists of cross-sectional regressions of excess returns on the firststage betas as explanatory variables. Implied factor premium refers to the coefficient estimates from the second stage tests, which are interpretable as the month-specific factor premia implied in the crosssectional covariation of the first-stage betas with (excess) returns. 1

3 survival in the sample. Second, rolling betas will be a better representation of the state of the firm and therefore its expected return, which likely varies over the firm s life cycle. For example, a young firm s return may be only weakly associated with a beta estimated over its entire sample life, which includes future changes in maturity, size, business model or leverage. Therefore, risk sensitivities that are held constant over a firm s life are likely to be less precise proxies for risk in the cross-sectional second stage test. In addition, rolling betas support a distinction between explaining (contemporaneous) returns and predicting (future) returns. The former implicitly holds the information in betas and returns constant, while return prediction tests allow for the return to impute more and newer information. Second, I complement the returns explanation tests on the firm level by tests on the portfolio level. In the extant literature, using portfolios as test assets (in lieu of firms) is mainly motivated by a reduction in the measurement error in firm-specific first-stage betas. However, there is no consensus in the literature about how such portfolios should be formed (see, e.g. Fama and French 1993, Lo and MacKinlay 1990, Cochrane 2005, Ahn et al. 2009, Lewellen et al. 2010). As a consequence, prior research has exclusively focused on portfolio-level tests (e.g., Petkova 2006, Ahn, Conrad, and Dittmar 2009, Lewellen, Nagel, and Shanken 2010), and often only on one type of portfolios. This is problematic for at least two reasons: First, conclusions about the pricing of a given factor are sensitive to the choice of the characteristic(s) on which the test portfolios are based (see, e.g., Ahn et al. 2009, Core et al. 2008, Lewellen et al. 2010, Ogneva 2012). Second, there are sharp differences between results on the firm level and results from the systematic portfolios suggested by prior literature, e.g., portfolios based on size and the book-to-market ratio (size/bm ). In principle, a reduction in measurement error in the first-stage betas should lead to a lower attenuation bias in the implied factor premia, i.e., higher absolute point estimates of the same sign as on the firm level. The stark differences between firmlevel and portfolio-level results, therefore, seem inconsistent with a reduction in measurement 2

4 error. Therefore, I introduce and validate randomly generated portfolios as test assets for crosssectional asset pricing tests. I hypothesize that naïve diversification in random portfolios is sufficient to reduce the measurement error; i.e., aggregation does not need to be systematic. By construction, random portfolios cannot be affected by prominent critiques against systematic portfolios in the literature. Specifically, random portfolios cannot suffer from any data snooping biases (Lo and MacKinlay 1990), from issues with inflated explanatory power (Lewellen et al. 2010), or from issues with implicitly sorting on unexpected returns (Elton 1999, Ogneva 2012). I explore these two questions on a sample of firm-specific monthly returns from April 1971 to December 2009, using the three factors in the Fama-French (1993) model, the market excess return (market factor), a size factor (SMB), and a book-to-market factor (HML), as well as an accruals-quality mimicking factor, AQfactor, as developed by Francis et al. (2005). The choice of those example factors is motivated by the prevalence of the threefactor model in the literature, and by the dispute on the pricing of the AQfactor in the accounting literature. In addition to these four test factors, I use a randomly generated factor (RandomFactor) and a factor based on ex-post realized returns themselves (ReturnFactor). These two factors (spreads) serve as lower and upper benchmark factors, respectively, in univariate tests, to gauge the power of the test design. In the (more common) tests using constant betas, estimated over the firm s entire sample life, only the implied market factor (the only long-only factor considered) is significant at the 10% level, all univariate and the multivariate test show significant intercepts. All factors that are constructed as returns spreads, including ReturnFactor as the upper benchmark factor, are insignificant in the constant-betas design. ReturnFactor turns highly significant in all designs with rolling betas, as do the implied SMB and AQfactor premia. The market factor is significant at the 10% level in only one of the multivariate specifications. In contrast, 3

5 the implied HML premium is consistently insignificant in all specifications of this test on contemporaneous returns. I proceed by highlighting that portfolio-level results depend on the portfolio creation rule, and the results are hard to reconcile with firm-level results under a measurement error reduction hypothesis, which provided the initial motivation for using portfolios. In tests on rolling betas that use the Fama-French 25 size/bm portfolios, qualitative results on each of the four example factors are opposite to the firm-level results. While HML is the only insignificant factor on the firm level, it is the only significant factor when the 25 Fama- French size/bm portfolios are used. I vary the aggregation level (portfolio size) of size/bm portfolios and show that the results converge towards the firm-level results when aggregation is reduced. The magnitude and significance of implied factor estimates from cluster portfolios, introduced by Ahn et al. (2009), are also inconsistent with a reduction in measurement error, and qualitative results change also with the aggregation level. The SMB factor and AQfactor are significant only at low levels of aggregation. Interestingly, at the aggregation levels used in Ahn et al. s original study (10 and 25 portfolios), none of the four factors is significant. Similarly, qualitative results on industry portfolios, as suggested by Lewellen et al. (2010), are also highly dependent on the industry definition. I proceed by testing the asset pricing models on random portfolios, which are formed by randomly grouping all firms in the cross section. In sharp contrast to above, I find that using random portfolios yield implied factor premia that are consistent with a reduction in measurement error. Specifically, estimates from random portfolios are always of the same sign, and larger in absolute values compared to the firm-level results. I document further advantages of random portfolios compared to these three types of systematic portfolios in the literature: First, results are robust to changes in the (arbitrarily chosen) number of portfolios. In contrast, size/bm portfolios, cluster portfolios and industry portfolios yield results that 4

6 are highly sensitive to the number of portfolios. The fact that the aggregation level impacts qualitative results also seems inconsistent with a simple measurement error reduction story. Second, results from random portfolios are more robust to the design choice between constant and rolling betas, particular at low levels of aggregation. Specifically, and while constantbeta results are consistently weaker point estimates are numerically similar, and qualitative conclusions do not change if up to 12 firms are aggregated to a random portfolio. In contrast, when systematic portfolios are used, the differences are more pronounced and sometimes lead to opposite qualitative conclusions. Viewed broadly, the results suggest that aggregation per se, as in the random portfolios, does not induce a deviation from qualitative firm-level results. In fact, increases in estimated factor premia and significance levels are consistent with a reduction in measurement error. Systematic aggregation to portfolios, however, yields results that show considerable deviations from firm-level results and become sensitive to the level of aggregation. When rolling betas are used to predict firm-specific returns, the results differ sharply from the explanation tests above. Coefficients on the HML betas are consistently positive and significant across specifications that vary both the length of the first-stage estimation windows and the horizon for the future return. Evidence on the other betas is mixed, with SMB betas and AQfactor betas significant when estimated over a longer history or when the future returns are aggregated over longer horizons. The market factor betas are insignificant in all specifications that use future returns. To summarize, I report three main findings that show how asset pricing test results on four risk factors depend on research design choices. First, rolling betas appear more powerful than constant betas, which incorporate future information. Second, results using randomly generated portfolios as test assets for cross-sectional asset pricing tests, in contrast to systematic portfolios, do not vary with the portfolio aggregation level, and are consistent with a reduction of measurement error in firm-specific betas estimates. Results on random 5

7 portfolios are also more robust to the choice between a constant-beta design and a rollingbeta design. Third, risk premia estimates, particularly for the market factor and the HML factor, depend on whether the test aims to explain contemporaneous returns or to predict future returns. My results show that there is a role for each of the four example factors in this study, with market factor betas, SMB betas and AQfactor betas being able to explain contemporaneous returns, and HML betas being the strongest and most consistent predictors of future returns. The remainder of the paper is organized as follows. Section 2 discusses prior literature and develops the hypotheses. Section 3 discusses the methodology, the data and some initial results. Section 4 presents the results of the main tests on both the firm level and the portfolio level. I discuss sensitivity tests in Section 5. Section 6 concludes. 2. Prior Literature and Hypotheses Development The cross-sectional asset pricing test based on Fama and MacBeth (1973) uses two stages. The first stage is a time-series regression of excess returns on risk factors. The secondstage tests are month-specific cross-sectional regressions of excess returns on these first-stage betas as independent variables. The slope coefficients from these second-stage regressions are interpretable as monthly implied factor premia. In this paper, I examine if the power of a cross-sectional asset pricing test depends on the estimation of the first-stage betas and on the choice of test assets. As example factors, I use the three Fama-French factors from the finance literature and AQfactor from the accounting literature. 2.1 Constant Betas versus rolling betas The standard implementation of the cross-sectional test uses data from the entire firm s sample life to estimate first-stage betas, yielding a firm-specific constant throughout the 6

8 firm s life for the cross-sectional tests. In this section, I contrast designs that impose constant betas with designs that allow for rolling betas. Rolling betas can have superior construct validity compared to constant betas because they do not incorporate information from future returns observations. On a related note, and equally important, they do not depend on how long the firm survives in the sample. In addition, rolling betas are more descriptive of a firm s risk profile that is likely to vary over time. Both total risk and betas vary predictably with firm age; for example, younger firms tend to have higher returns variances and higher CAPM betas than older firms. The empirical evidence is consistent with these arguments. Blume (1971, 1975) documents that CAPM betas converge over a firm s life; Ecker et al. (2006) show that firm-specific betas on AQfactor decrease with firm age. As a first step, I test for a linear relation between firm age and overall returns variability. The slope coefficient from a regression of the timeseries standard deviation, averaged across all firms in the same firm-age month, on firm age is about per month (t = ). 2 Figure 1 follows the design of that test and depicts the average univariate betas over time for the full sample. Average 1-year betas (constant betas) are displayed as a solid red line (dashed blue line). 3 Consistent with Blume s (1971, 1975) evidence, CAPM betas converge towards 1, and betas on factors constructed as return spreads converge towards 0. Coefficients on firm age are negative (positive for h-betas, which start from a negative value) and highly significant (p-values below ). Taken together, prior literature and the evidence in Figure 1 suggests that betas vary predictably over time, inconsistent with constant betas. 2 Specifically, the regression is σ it = τ 0 + τ 1 F irmage t + ε t, whereby F irmage t indicates the age of the firms, in months. σ it is the average firm-specific standard deviation of returns at firm age t, estimated over the prior 24 months. By construction, the number of portfolio firms for this test declines from a maximum number of firms at the required minimum age of twelve months. The sample consists of all CRPS observations from April 1971 to December 2009, restricted to firm-age months (t) with a minimum of 1,000 firms. This yields firm-age cross sections from 12 to 391 months. 3 Constant betas will change only as the sample composition changes. 7

9 Some prior studies, such as Shanken and Weinstein (2006) and Kim and Qi (2010) have used both constant and rolling betas. Shanken and Weinstein (2006), in tests on the portfolio level and for five macro-economic factors, find generally consistent results, and only one of the five factors significant in the rolling-betas design, but none in the constant betas design. Kim and Qi limit their discussion to the impact of a control for low-priced stocks on the question whether AQfactor is a priced risk factor. There are, however, differences in the magnitude and significance of other factors in their Table 4. The implied HML factor (market factor) is positive and significant when rolling (constant) betas are used, but not with constant betas (rolling betas). The constant-betas design aims to explain contemporaneous returns, i.e., the estimation window for the betas includes all returns observations for a firm; these same returns are also used in the second-stage cross-sectional tests. The rolling-betas design in the literature typically uses only past returns to estimate the first-stage betas, 4 As such, the design implicitly switches from explaining contemporaneous returns to predicting future returns, testing the joint hypotheses of an association between betas and returns and the (over-time) stability of that association. As such, the return (the variable to be explained) incorporates new information, and the betas, estimated on past returns only, do not. By definition, returns prediction is not feasible with constant betas. To compare the two designs, contemporaneous return observations should be included in the first stage if the cross-sectional variation of returns is to be explained rather than predicted by the betas. 5 Tests on rolling betas might also prove more powerful as rolling betas have higher crosssectional dispersion, a direct consequence of using fewer returns observations in the first-stage estimation. While the precision of the betas estimate that is assumed constant over time must decrease when fewer observations are used, Ang et al. (2010) hypothesize that the 4 An exception is the ( ex-post ) portfolio test in Ang et al. (2010), where betas estimation and returns measurement period coincide. 5 Note that the sorting characteristics are already lagged in the construction of SMB, HML and AQfactor. 8

10 power in the second stage may depend on the cross-sectional dispersion of the betas as well. 2.2 Firm-Level Tests versus Portfolio-Level Tests As discussed by Ahn, Conrad, and Dittmar (2009), and others, the main argument for testing asset-pricing models on portfolios rather than on individual firms is the hypothesized reduction of measurement error in firm-level first-stage betas estimates. In turn, measurement error in the first-stage betas will result in an attenuation bias in the second-stage implied risk premia estimates. 6 In the context of this study, an additional argument for the use of portfolios is that the concern about constant betas on the firm level is potentially mitigated, albeit in an indirect and incomplete way, because the frequent resorting of the cross section into portfolios ensures that the portfolio compositions reflect the hypothesized relative riskiness of the constituent firms. Assuming the betas of frequently recreated portfolios are constant over the entire sample period may therefore be less restrictive than assuming a specific firm s beta is constant. The literature has proposed several distinct portfolio creation rules. First, following Fama and French (1993), portfolios can be created based on firm fundamentals (characteristics), such as size and book-to-market ratios. Fama and French (1993) construct their 5 5 portfolios to span the same dimensions as their SMB and HML factors (which are based on 2 3 portfolios) and perform time-series tests of their asset pricing model. These time-series tests, therefore, provided a sense about how well their factors explain the previously documented differences in returns of size-based and book-to-market-based portfolios. To the extent the intercepts from their time-series regressions are statistically zero, the results implicitly validate the authors choices in the construction of the SMB and HML factors. It is unclear, however, if these size/bm portfolios are appropriate for purposes beyond validating 6 Note that the focus here is on correcting for a bias in the premia estimate. In contrast, some research on the errors-in-variables problem proposed adjustments to the cross-sectional standard errors of the estimate (see, e.g., Shanken (1992), and Cochrane (2005)). 9

11 the factor construction in time-series tests. The question of how test portfolios should be formed has been extensively debated in the literature. Lo and MacKinlay (1990) and Conrad et al. (2003) argue that the use of characteristics-based portfolios in asset pricing tests results in a data-snooping bias, which is increasing in the number of sorting dimensions. In essence, these papers question the robustness of the characteristics-return relation over time. In addition, Ogneva (2012) documents that portfolios based on AQ are implicitly sorted on the magnitude of unexpected returns from future fundamental news about cash flows. Lewellen et al. (2010) document that it is too simple to arrive at high cross-sectional explanatory power using size- and book-to-market-based portfolios, even with theoretically very different asset pricing factors, and therefore advocate expanding the set of test assets, e.g. with industry portfolios. Ahn et al. (2009) dismiss characteristics-based portfolios altogether and propose portfolios constructed directly from historical returns data. Specifically, they form portfolios based on a cluster analyses of historical (pairwise) correlations in stock returns ( cluster portfolios ). 7 Characteristics-based portfolios, on the other hand, aim to (implicitly) sort on the magnitude of returns. The authors argue that cluster portfolios are more robust against small changes in the data compared to characteristics-based portfolios. This discussion highlights that, ex ante, the role of portfolio-level tests is to complement firm-level tests, whose results may suffer from attenuation biases due to the measurement error in firm-specific betas. In the context of the two-stage test, a reduction of attenuation bias implies that firm-level and portfolio-level implied risk premia have the same sign, but the latter are either comparable or higher in absolute terms. 8 To the extent this relation between 7 The correlations are transformed into a matrix of distance measures in order to satisfy the necessary conditions for cluster analysis input variables. 8 While the reduction in attenuation bias in the implied factor premia would predict, all else equal, an increase in significance levels as well, Ang et al. (2010) argue that standard errors from portfolio tests are actually higher than firm-level standard errors due to the smaller number of test assets. I discuss this proposition when I present the main results in Section 4. 10

12 firm-level and portfolio-level results does not hold, the portfolio creation rules and/or other related design choices (e.g., the number of portfolios created or the weighting scheme) must have induced results that seem inconsistent with measurement error reduction, i.e. the original motivation for using portfolios in the first place. In short, a comparison of portfoliolevel implied factor premia to firm-level implied factor premia can be used to evaluate the portfolio creation rule itself. Results from two-stage tests in prior research differ strongly between firm-level and portfolio-level tests. Kim and Qi (2010), for example, report a positive implied market factor on the firm level, and a negative implied market factor when the 25 Fama-French size/bm portfolios are used. The size factor and book-to-market factor are insignificant on the firm level, but on the portfolio level, the size factor is significantly negative and the book-to-market factor significantly positive. Similar evidence on the 25 size/bm portfolios in Petkova (2006) suggests that only the book-to-market factor (HML) is priced. While this result is broadly consistent with earlier research, Petkova concludes that the negative implied market factor in particular is a puzzle that warrants further investigation. Ahn et al. (2009) restrict their tests to the portfolio level. Specifically, they compare results based on the 25 (characteristics-based) size/bm portfolios and their proposed cluster portfolios and find striking differences in results on the eight factors and eight asset pricing models considered. In fact, only the consumption beta, and only in the univariate model, yields a consistently significant result. Notably, the two-stage test using cluster portfolios yields insignificant premia for all three factors in the Fama-French model. In sum, Ahn et al. document that conclusions about the pricing of most of risk factors depend on how the systematic portfolios have been constructed. Therefore, truth remains unknown due to the lack of a benchmark. My approach benchmarks the portfolio-level results against the results on the firm level to assess their (ex-post construct) validity as test assets. I examine if random portfolios, 11

13 in contrast to systematic portfolios, yield implied factor results that are consistent with a reduction of the measurement error bias in firm-level results. If the goal is to reduce measurement error in the first-stage betas, it seems, ex ante, unnecessary and potentially counterproductive to create portfolios systematically, sorted either on fundamental characteristics or on return correlations. Instead, I propose and test the use of random portfolios, whereby measurement error is reduced by naïve diversification across firms only; that is, by pooling random stocks, not stocks with similar fundamentals or return properties. Consequently, random portfolios cannot, by definition, suffer from datasnooping biases, nor can they be implicitly sorted on the direction and magnitude of future news, the concern raised by Elton (1999) and probed by Ogneva (2012). Both random and systematic portfolios can be frequently recreated, so as to yield a full panel of data. 9 They both potentially mitigate some of the concern about holding the firststage betas constant over time. 10 There is no theoretical guidance on the optimal number of portfolios to be used. The upper limit is the number of firms in the cross section (5,788 firms on average in this study); the practical lower limit is the degrees of freedom necessary for the second-stage cross-sectional regression. I form between 2,000 and 8 portfolios by randomly assigning an equal number of firms each month. To reconcile the aforementioned differences documented in prior literature between results on the firm level and results from systematic portfolios, I explore whether results based on, for example, the 25 Fama-French size/bm portfolios are driven by an underlying covariance structure in returns that is induced by these characteristics, or whether the results are driven by the arbitrarily chosen number of portfolios, i.e., the level of aggregation. I hypothesize 9 The probability of arriving at an incomplete panel is increasing in the length of the holding period of the portfolio, and decreasing in the minimum number of firms within portfolios. As my simulations allow for portfolios with as few as two firms, I eliminate the possibility of an incomplete panel by re-balancing each month. When the number of systematic portfolios increases, some monthly cross sections might consist of fewer than the requested number of assets. 10 Note that the betas of random portfolios are, in expectation, reset to the (cross-sectional) average beta. 12

14 that characteristics-based portfolio results will converge towards the firm-level results as more portfolios are formed. 3. Methodology and Sample Statistics 3.1 Methodology My tests use the approach of two-stage cross-sectional regressions. 11 The first stage of this approach estimates slope coefficients in a firm-specific time-series regression of excess returns on four risk factors (in the univariate approaches, on one risk factor). R i,t R f,t = α i + β Market i (R M,t R f,t ) + β SMB i SMB t + βi HML HML t +β AQfactor i AQfactor t + ε i,t (1) where R i,t R f,t is the asset-specific excess return for asset i, R M,t R f,t is the market excess return (market factor), SMB t and HML t are the Fama-French (1993) size and book-tomarket factors, respectively. AQfactor t is the Francis et al. (2005) factor based on accruals quality. The subscript t indicates the (sample) month. For the constant betas, the estimation window is the entire firm s (sample) life (i.e., all available returns for that firm). The average firm in my sample has 106 monthly returns observations. Various forms of rolling betas (β Market i,t, β SMB i,t, β HML i,t, β AQfactor ) are obtained from Regression (1) when estimated either for rolling windows with a fixed length (1 year, 5 years, or 10 years) or using all historical returns up to Month t. The former (latter) are subsequently denoted rolling betas ( firm-age betas ). Note that firm-age betas will converge towards the constant betas as the length of the estimation window increases towards the 11 As pointed out by many researchers, a significant implied factor (i.e., the second-stage coefficient estimate) in these tests seems a necessary, but not sufficient criterion to answer the pricing question for a specific risk factor, as even a significant implied factor is subject to interpretation, either as a risk factor or as evidence of mispricing. i,t 13

15 entire sample life of the firm. In the second stage, I estimate month-specific cross-sectional regressions of the excess returns on the four first-stage factor betas (β Market i, β SMB i, β HML i, β AQfactor ). I include all four factor betas in the multivariate approach, and one factor beta at a time in the univariate approaches. i R i,t R f,t = γ 0,t + γ Market t βi Market + γt SMB βi SMB + γ HML t βi HML + γ AQfactor t β AQfactor i + ϑ i,t (2) The second-stage coefficient estimates (γ Market t, γ SMB t, γ HML t, γ AQfactor ) are interpretable as implied risk premia (implied by the first-stage betas) in Month t. Following Fama and MacBeth (1973), the test statistic is the average monthly coefficient estimate, relative to the time-series standard error of the monthly estimates. I report t-statistics based on Newey- West (1987) standard errors with two lags. 12 In the univariate tests, I introduce two additional pseudo asset pricing factors as benchmarks for assessing the power of the tests. Both benchmark factors are constructed using the entire returns sample and follow the design choices of AQfactor. The first factor is sorted on a randomly generated number (RandomFactor). This benchmark is informative about the power of the test for a factor that is pure noise. The second factor,returnfactor, uses the month-specific returns themselves as the sorting criterion. The use of this factor is motivated by the following two-part argument: If realized returns are valid proxies for the expected return, and if there was a perfect risk characteristic on which we could sort ex ante, the result would be ReturnFactor. In this sense, ReturnFactor is formed with perfect foresight. Comparing the magnitude of ReturnFactor with the magnitudes of the three ex-ante factors (SMB, HML, AQfactor) provides a sense of, first, how much of the cross-sectional dispersion 12 Considering only two lags yields conservative t-statistics. Serial correlations in implied factor premia of higher order (than two) are generally negative; their inclusion would therefore increase the significance level. In attempt to quantify the impact of this adjustment and for comparison with prior research, Table 2 also shows Fama-MacBeth (1973) standard errors without correction for serial dependence. t 14

16 in returns ex-ante risk characteristics actually explain, and, second, about the power of the two-stage tests for return spreads. 13 The market factor is essentially also a perfect-foresight factor based on contemporaneous returns. In contrast to ReturnFactor, it is a value-weighted long-only portfolio. Therefore, differences in results are informative about the effect of these factor design choices. Viewed differently, it should be noted that the return spreads SMB, HML, and AQfactor are constructed as implementable trading strategies on publicly available (lagged) characteristics. Therefore, their construction already reflects a hypothesized inter-temporal link between the characteristics on one hand and (future) returns as well as the betas based on those returns on the other hand. I perform the initial tests on the firm level, and complement these tests by the portfoliolevel tests in Section 4.2, where I use both random portfolios and three types of systematic portfolios (size/bm based on Fama and French 1993, cluster portfolios from Ahn et al. 2009, and industry portfolios as suggested by Lewellen et al In each run, random portfolios are generated by randomly assigning the entire cross section of firms into a pre-specified number of portfolios each month. The portfolio return is the equal-weighted return of the constituent firms in that month. 14 these portfolios returns are used in the first-stage beta estimation and the monthly second-stage cross-sectional regressions. For the random portfolio tests, I report average results and ranges of t-statistics from 100 independent runs for each aggregation level. The portfolio identifiers of random portfolios, as well as the cluster portfolios (see below), are not consistently assigned over time, i.e. Portfolio 1 of one month is not related or compa- 13 Kan and Zhang (1999) provide evidence that the inclusion of useless factors in the asset pricing tests might bias the results on the sensible factors. For that reason, I use RandomF actor and ReturnF actor in univariate asset pricing tests only. 14 All else equal, naïve diversification through equal-weighting returns will decrease measurement error more compared to value-weighting; in fact, value-weighting based on the highly skewed empirical distribution of (lagged) market capitalizations might mitigate the diversification effects and lead to decreased comparability with firm-level results. 15

17 rable to Portfolio 1 in the next month. Ahn et al. devise an auxiliary algorithm that assigns the same portfolio identifier for adjacent portfolios when the number of overlapping member firms is maximized. Note that this approach assumes a constant relative risk sensitivity over time for each firm, and migration across portfolios is therefore minimized. Ahn et al. probe the sensitivity of their results to using unsorted and such time-consistent portfolios and, interestingly, find qualitatively similar results (their Footnote 6). In other words, randomizing how portfolio identifiers are assigned at each formation time (once a year) has no impact on the conclusions. 15 The random portfolios are also unsorted in this sense. I discuss results from random portfolios that are not reset during the (first-stage) estimation period in the sensitivity section. To enable direct comparisons across portfolios creation rules, I also standardize the construction of the systematic portfolios. For the standardized portfolios based on size and book-to-market, I independently sort the whole cross section of stocks on the lagged values of both variables in each sample month, and the portfolio return is the equal-weighted return across all member firms. The original size/bm portfolios by Fama and French (1993) use breakpoints from the subsample of NYSE-listed firms for both sorting dimensions, are value-weighted portfolios and resorted only once a year. For the standardized cluster portfolios, I follow an approach similar to Ahn et al. (2009) by estimating the pairwise correlations of stock returns each year over a rolling window of 60 months (requiring at least 36 months 16 ). The estimation period ends in the month prior to the portfolio formation. After transforming the correlations into distance measures, the clustering algorithm defines portfolio memberships in each month. 15 I replicated this analysis for my sample period and test designs with the standardized cluster portfolios. While the explanatory power in the first-stage regression generally increases slightly, second-stage test results are qualitatively similar, and there is no systematic relation between the implied factor premia from unsorted portfolios and those from time-consistent portfolios. Overall, I conclude that introducing a risk constancy assumption to arrive at time-consistent portfolio identifiers does not seem to be of importance for two-stage test results. 16 Setting the number of required returns higher than half the estimation period ensures that a correlation of all firm-pairs is estimable. 16

18 The standardized industry portfolios are also reformed each month, using month-specific industry membership data, and are equal-weighted. Similar to the random portfolios, I form systematic portfolios of various sizes. Industry portfolios increase monotonically in size when the industry definition is modified from 4-digit, 3-digit, and 2-digit historical SIC codes on CRSP, followed by the three Fama-French classifications into 49, 30, and 17 industries. 17 Note that both cluster portfolios and industry portfolios cannot be standardized to consist of an equal number of firms in any given month. The independent sorts on size and BM also yield unequal portfolio sizes over time, but it is possible via sequential sorts to force constant portfolio sizes each month. Results from sensitivity tests on sequentially sorted portfolios are discussed in Section Data and Descriptive Statistics My tests use CRSP returns data from April 1971 to December 2009 (465 monthly cross sections). I require a minimum of 12 returns observations to estimate time-series factor betas, yielding a total of 2,691,776 observations, for 25,373 distinct firms. Data on the riskfree rate, the market factor (market excess return) as well as the size factor (SMB) and the book-to-market factor (HML) are from Kenneth French s website. The estimation of accruals quality (AQ, see Dechow and Dichev 2002, McNichols 2002) and the construction of the monthly factor-mimicking portfolio (AQfactor) follow the procedure detailed in Francis et al. (2005). Table 1 provides descriptive statistics on the monthly asset pricing factors (Panel A), and average statistics from the 465 cross-sectional (monthly) distributions. On average, each cross section consists of 5,788 firms, ranging from 2,230 to 8,233 over the sample period. All variables are defined in Appendix A. The mean excess return is 0.72% per month, with an 17 The Fama-French industry classifications are extensions of the original classification in Fama and French (1997) and available on Ken French s website in updated versions. 17

19 average dispersion (i.e., cross-sectional standard deviation) of 15.66%. Excess returns are skewed, with the median being -0.35%. The mean and median firm-specific autocorrelation in returns, estimated over the entire sample life, is zero, with an average dispersion of Expectedly, constant betas, estimated over all return observations, have a considerably lower cross-sectional dispersion (ranging from 0.57 to 1.02) compared to the rolling 1-year betas (ranging from 1.85 to 3.59). 3.3 Factors versus Implied Factors From Cross-Sectional Tests There has been some discussion in recent literature about the implications of positive factor realizations over time for the question of that factor being priced. By factor realizations, I refer to the factor based on risk characteristics as constructed by the researcher (such as SMB and HML in Fama and French 1993, and AQfactor in Francis et al. 2005). The implied factor premia from cross-sectional pricing tests, in contrast, are based on the (first-stage) betas. For example, Mashruwala and Mashruwala (2011, p. 1350) argue that cross-sectional pricing tests with factor realizations that are insignificant over time is moot, and consequently omit cross-sectional tests on the factor betas in their study. Therefore, as a preliminary step, I focus on the question whether a significant mean factor realization over time is a sufficient criterion for that factor being priced. Table 2 contrasts the test results on the realized factor means (Panel A) and implied factor means from twostage tests (Panels B and C). To facilitate a direct comparison of the results across panels, these tests are performed separately as univariate two-stage tests (i.e., both Regression (1) and (2) are estimated as one-factor models). Also, I employ the firm s entire sample life in the first stage to obtain constant betas. Panel A shows that only the realized market and HML factors are reliably positive over time. A comparison with Panel B indicates that the implied market factor is of similar magnitude (0.43% in Panel A versus 0.47% in Panel B), and also significantly positive at the 18

20 10% level. The implied HML factor, however, turns insignificant. 18 SMB and AQfactor are consistently insignificant in both tests. ReturnFactor, constructed to be the upper benchmark factor realization, drops 98% in magnitude and turns insignificant at conventional levels in the two-stage test. I speculate that this result is mainly driven by the low autocorrelation in realized returns, which, in turn, leads to first-stage betas estimates that are close to zero. The hurdle for detecting any significant implied factor premia, therefore, seems high when realized returns and constant first-stage betas are used. There is some discussion in the literature, briefly outlined in Cochrane (2005, p. 236), that the cross-sectional (second-stage) regression should, in theory, not include an intercept term, and that its omission would result in greater efficiency of the test. Intuitively, the conditional mean of the excess return when all risk sensitivities are zero should be zero as well. Panel B, however, indicates that all intercepts are reliably non-zero (at the 10% level or higher). Nevertheless, as a sensitivity test, I repeat the second-stage tests suppressing the intercept term, and report the results in Panel C. All factors premia are now significant, with the exception of the random factor. The implied HML factor premium is significantly negative. Taken together, the significance or insignificance of the average factor realizations over time is not conclusive about the significance of the corresponding implied factor premium from two-stage tests. The market factor is the only factor that yields results of comparable magnitude and significance in both tests. In contrast, HML and even the benchmark ReturnFactor, which is constructed with perfect foresight about returns, are highly significant over time, but do not yield significant premia in the cross-sectional tests. 18 Without trying to enter the characteristics versus betas discussion (see Daniel and Titman 1997, Lewellen 1999 and Davis et al. 2000), I highlight that my results on the HML betas (denoted h) are not, in expectation, comparable to tests that use the book-to-market characteristic as explanatory variable. In fact, the latter might yield opposite results for two related reasons. First, in my sample, h-betas are actually negatively correlated with the book-to-market ratio. Second, they are strongly negatively correlated with overall risk as measured by the firm-specific returns variance (results untabulated). 19

21 4. Main Empirical Analyses and Results 4.1 Explaining Firm-Level Returns As laid out in Section 2, I distinguish between designs that, in the first stage, include or exclude the month of the second-stage cross-sectional regression. I first include this month in the return explanation tests of this section. Conceptually, this design choice more closely resembles the constant-betas approach (where only return explanation tests are feasible by construction). It is also possible to use rolling betas to predict future returns, and the results of these prediction tests are in Section 4.3. Briefly, I probe if asset pricing tests on constant first-stage betas are less powerful compared to tests on betas that are allowed to vary over time. As discussed in Section 2.1, I hypothesize this to be the case for at least two reasons: First, they contain future information as they are estimates across the firm s entire (sample) life. Second, constant risk sensitivities are inconsistent with changes in the firm s risk profile over time. The trade-off is that rolling betas are estimated with fewer returns observations, which can lead to lower precision in the first-stage estimates. I first turn to the asset pricing tests on the firm level. Table 3 reports the average coefficient estimates from Regression (2) for constant betas and six variants of rolling betas. Tests are performed separately for the four-factor asset pricing model and two univariate models with RandomFactor and ReturnFactor, respectively. These results are reported in the rightmost columns of the Table 3. The results on constant betas, in the first row of Table 3, indicate that only the implied market factor is significant at the 10% level (p = 0.066), which is consistent with results reported by Core et al. (2008) and Kim and Qi (2010). 19 The intercept is significant (p = 0.014) and the explanatory power is 8.17%. Note that all three factors that are constructed as 19 I verify that the (small) differences in numerical results are almost entirely due to the extended sample period employed here. 20

22 return spreads, and even the implied ReturnFactor reprinted from Table 2), are insignificant. I find that second-stage regressions on rolling four-factor betas are better specified in that the intercept estimates are consistently insignificant (no t-statistic exceeds 0.82). At the same time, the explanatory power is considerably higher than 8%. The implied SMB and AQfactor are consistently positive and significant at the 10% level. The estimate for the implied market factor, while positive throughout, is only significant in one out of the four specifications (p = in the rolling 1-year specification). The explanatory power of the regression is highest in the rolling 1-year design, where all factors are significant with the exception of the implied HML factor, which remains insignificant across all specifications. 20 In line with the implied SMB and implied AQfactor in the four-factor model, the implied ReturnFactor (which was also insignificant in the constant-betas design) turns significantly positive when rolling betas are used. As the lower benchmark, the implied RandomFactor is consistently insignificant across all specifications, while the regression intercept is consistently significant. A comparison between the constant betas and the firm-age betas in particular highlight the importance of betas not being conditioned on future returns observations. A possible concern about this test design is that the inclusion of the contemporaneous return in both stages might cause a spurious correlation between returns and betas and that this will translate into the magnitude of the implied factors. Compared to the case of constant betas, this correlation might be higher due to that fact that fewer returns observations are used in the first stage and, therefore, the contemporaneous return observation is given a higher weight. Ang et al. (2010) argue that, while a trading strategy on ex-post information is not implementable, overlapping estimation and return periods do not invalidate an asset 20 The test statistics in this paper use Fama-MacBeth (time-series) standard errors, corrected for serial correlation. As such, the approach is designed to control for cross-sectional dependence in residuals. Nevertheless, I also probe the extent of the errors-in-variables problem for cross-sectional standard errors, following Shanken (1992). As can be expected (see,e.g., the discussion in Cochrane 2005), the adjustments are small when using monthly returns. In the constant-beta design, for example, the correction factor for the standard error is

23 pricing test. Furthermore, Shanken (1992) discusses the potential spurious correlation, and concludes that, under standard assumptions, contemporaneous betas and mean returns are statistically independent. 21. I examine the potential impact of spurious correlation empirically on the results of the second-stage test. I first vary the length of the estimation windows (10 years, 5 years and 1 year). I find that the explanatory power is indeed (and mechanically) increasing with shorter estimation windows (i.e., fewer observations). In an attempt to quantify such spurious correlation between time-series betas and current returns, I note that the explanatory power on the RandomFactor increases from 1.05% in the constant-betas design to 9.35% when oneyear rolling betas are used. An increase in explanatory power, however, does not translate into implied factor premia. Specifically, there is no monotonic relation between the length of the estimation window and the magnitude (or significance) of the implied factors as the main focus of the cross-sectional test. Second, I probe whether the magnitude and significance of the implied factor premia in the second stage are sensitive to a recency adjustment in the first-stage betas estimation. Spurious correlations would increase when more weight is given to returns that are closer in time to the month of the second-stage test. For betas estimated over the firm s entire returns history ( firm age betas ), I put increasing weights on more recent observations, and the highest weight on the contemporaneous month. These betas variants are estimated using weighted least squares in the first stage, with weights either linearly or exponentially increasing from the first returns observation to Month t. For the linear weights (exponential weights), I assign the first month a weight of 1, increasing by increments of 1 (by 20% of the prior value) for each subsequent month up until the month of the cross-sectional regression. 21 His discussion focuses on portfolio tests where time-series mean returns can be regressed on constant betas in a single cross-sectional regression. In that specific case, the implied factor premia from a single cross-sectional regressions on average returns are equal to the average implied factor premia from monthly cross-sectional regressions, the approach taken here. This equivalence no longer holds when either rolling betas are used, or the sample of test assets changes over time, as is the case with firms (see, e.g., Cochrane (2005)) 22