Tarun Chordia Paper

Cross-Sectional Asset Pricing with Individual Stocks: Betas versus Characteristics*

Tarun Chordia, Amit Goyal, and Jay Shanken

January 2015

Abstract

We develop a methodology for bias-corrected return-premium estimation from cross-sectional regressions of individual stock returns on betas and characteristics. Over the period from July 1963 to December 2013, there is some evidence of positive beta premiums on the profitability and investment factors of Fama and French (2014), a negative premium on the size factor and a less robust positive premium on the market, but no reliable pricing evidence for the book-to-market and momentum factors. Firm characteristics consistently explain a much larger proportion of variation in estimated expected returns than factor loadings, however, even with all six factors included in the model.

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*We thank Joe Chen, Wayne Ferson, Ken Singleton, and seminar participants at the Financial Research Association Meetings, Frontiers of Finance Conference at Warwick Business School, Deakin University, Erasmus University, Goethe University Frankfurt, Laval University, McGill University, Singapore Management University, State University of New York at Buffalo, Tilburg University, University of Missouri, and the University of Washington summer conference for helpful suggestions. Special thanks to Jon Lewellen for his insightful comments. Amit Goyal would like to thank Rajna Gibson for her support through her NCCR-FINRISK project.

A fundamental paradigm in finance is that of risk and return: riskier assets should earn higher expected returns. It is the systematic or nondiversifiable risk that shouldbe priced, and under the Capital Asset Pricing Model (CAPM) of Sharpe(1964), Lintner (1965), and Mossin (1966) this systematic risk is measured by an asset’s market beta. While Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973) do find a significant positive cross-sectional relation between security betas and expected returns, more recently Fama and French (1992) and others find that the relation between betas and returns is negative, though not reliably different from zero. This calls into question the link betweenrisk and expected returns.

There is also considerable evidence of cross-sectional patterns (so-called anomalies) in stock returns that raises doubts about the risk-return paradigm. Specifically, price momentum, documented by Jegadeesh and Titman (1993), represents the strong abnormal performance of past winners relative to past losers. The size and book-to-market effects have been empirically established by, among others, Fama and French (1992). In particular, small market capitalization stock returns have historically exceeded big market capitalization stock returns, and high book-to-market (value) stocks have outperformed their low book-to-market (growth) counterparts. Brennan, Chordia, and Subrahmanyam (1998) find that investments based on anomalies result in reward-to-risk (Sharpe) ratios that are about three times as high as thatobtained by investing in the market, too large it would seem, to be consistent with a risk-return model (also see MacKinlay (1995)).

The behavioral finance literature points to psychological biases on the part of investors to explain the breakdown of the risk-return relationship. In contrast, Fama and French (1993) propose a three-factor model that includes risk factors proxying for the size- and value-effects, in addition to the market excess-return factor, Mkt.The size factor, SMB, is a return spread between small firms and big firms, while the value factor, HML, is a return spread between high and low book-to-market stocks. There is controversy in the literature as to whether these two additional factors are really riskfactors, however, i.e., whether the factors can be viewed as hedge portfolios in an intertemporal CAPM along the lines of Merton (1973). Greater still, we suspect, is skepticism about a risk-based interpretation of the momentum factor MOM. This (winner-loser) spread factor is often included in a four-factor model along with the three Fama and French (1993) factors, e.g., Carhart (1997) and Fama and French (2012). More recently, Fama and French (2014) have proposed a five-factor model that adds CMA (conservative minus aggressive investment) and RMW (robust minus weakprofitability) factors to the original three.

While some researchers are inclined to viewexpected return variation associated with factor loadings (betas) as dueto risk, and variation captured by characteristics like book-to-market as due to mispricing, we believe that a more agnostic perspective on this issue is appropriate. One reason is that the betas on an ex-ante efficient portfolio(a potential “factor”) will alwaysfully “explain” expected returns as a mathematical proposition (see Roll (1977)), whatever the nature of the underlying economic process. This makes it difficult to infer that a beta effect is truly driven by economic risk unless there is evidence that the factor correlates with some plausiblenotion of aggregate marginal utility in an intertemporal CAPM or other economic setting.

For the usual spread factors, it is also important to recognize that there is a mechanical relation between, say, the book-to-market ratio and loadings on HML: a weighted average of the loadings for stocks in the high book-to-market portfolio must exceed that for stocks in the low book-to-market portfolio.Therefore, the relation between loadings and expected returns can be mechanical as well. In fact, Ferson, Sarkissian, and Simin (1998) construct an example in which expected returns are determined entirely by a characteristic, but one that is nearly perfectly correlated with loadings on the associated spread factor. In general, though, there need not be a simple relation between loadings and characteristicsat the individual stock level. For example, at the end of 2013, Comcast’s book-to-marketratio of 3.4 placed it at the 99thpercentile, extreme value territory, while its negative loading on the HML factor was at the 30thpercentile, suggestive of a growth tilt.Empirically, we find relatively low correlations (less than 0.5) between characteristics and the corresponding loadings, even adjusting for estimation-error noise. Therefore, it is legitimate to ask whether the underlying firm characteristics or the factor loadings do a better job of tracking expected returns in the cross-section. Answering this question is the main objective of our paper.

1 See related work by Haugen and Baker (1996), Titman, Wei, and Xie (2004) and Cooper, Gulen, Schill (2008) and Hou, Xue, and Zhang (2014) among others. 2

The regression of HML on the Fama-French factors must produce a perfect fit, with a loading of one on itself and zero on the other factors. Since the HML loading equals the difference between the value (H) and growth (L) portfolio loadings, that difference must equal one. But, ofcourse, each of these portfolio loadings is a weighted average of the loadings for the stocks in the portfolio.

While the economic interpretation of beta pricing can be unclear, determining the underlying causation for the cross-sectional explanatory power of a characteristic can likewise be challenging. For one thing, it is hard to rule out the possibility that the significance of a stock characteristic reflects the fact that it happens to line up well with the betas on some omitted risk factor. But we need not think solely in terms ofrisk. For example, Fama and French (2014) use observations about the standard discounted cash flow valuation equation to derive predictions about the relation between expected returns and stock characteristics: market equity, the book-to-market ratio, and the expected valuesof profitability and investment.This approach is more in the spirit of an implied cost of capital and, as they note, the predictions are the same whether the price is rational or irrational.

Understanding what determines observed pricing patterns is undoubtedly important, but it is not the focus of this paper. Whatever the appropriate economic interpretation, important gaps remain in our knowledge about the relevant empiricalrelations. We fill some of those gaps. Whereas Fama and French (1993) and Davis, Fama, and French (2000) argue that it is factor loadings that explain expected returns, Daniel and Titman (1997) contend that it is characteristics. On the other hand, Brennan, Chordia, and Subrahmanyam (1998) present evidence that firm characteristics explain deviationsfrom the three-factor model, whereas Avramov and Chordia (2006) find that size and book-to-market have no incremental effect (momentum and liquidity do) when the model’s loadings are time varying. However, despite the considerable literature on this subject, we know of no study that directly evaluates how much of the cross-sectional variation in expected returns is accounted for by betas and how much by characteristics in a head-to-head competition. The main goal of this paper is to provide evidence on this issue using appropriate econometric methods.

A number of methodological issues arise in this setting. Indeed, the lack of a consensus on the betas versus characteristics question stems, in part, from issues of experimental design. For example, Brennan Chordia, and Subrahmanyam and Avramov and Chordia work with individual stocks and employ risk-adjustedreturns as the dependent variable in their cross-sectional regressions (CSRs). In computing the risk-adjustment, the prices of riskfor the given

3Similarly, Liu, Whited and Zhang (2009) relate expectedreturns to stock characteristics in a framework based on q-theory. 4

While linear functions of the lagged values of profitability and investment may serve as rough proxies for the required expectations, a justification for substituting the corresponding factor loadings for the characteristics in this discounted cash flow (or the related q-theoretic) context has, to our knowledge, yet to be articulated.

factors are constrained to equal the factor means and the zero-beta rate is taken to be the risk-free rate. A virtue of this approachis that the well-known errors-in-variables (EIV) problem is avoided since the betas do not serve as explanatory variables. However, while this can be useful for the purpose of model testing, the relative contributions of loadings and characteristics cannot be inferred from such an experiment.

Unlike these papers, we do not impose restrictions on the prices of risk or document patterns of model misspecification. Rather, we evaluate the role of loadings and of characteristics in the cross-sectional return relation that best fits the data when both are included as explanatory variables. Since (excess) returns,not risk-adjusted returns, serve as the dependent variable, in this context, it is important to address the EIV problem. Typically, in asset pricing empirical work, stocks are grouped into portfolios to improve the estimates of beta and thereby mitigate the EIV problem. However, the particular method ofportfolio grouping can dramatically influence the results (see Lo and MacKinlay (1990) and Lewellen, Nagel, and Shanken (2010)). Using individual stocks as testassets avoids this somewhat arbitrary element.

Ang, Liu, and Schwarz (2010) also advocate the use of individual stocks, but from a statistical efficiency perspective, arguing that greater dispersion in the cross-section of factor loadings reduces the variability of the risk-premium estimator. Simulation evidence in Kim (1995) indicates, though, that mean-squared error is higher with individual stocksthan it is with portfolios, due to the greater small-sample bias, unless the risk premium estimator is corrected for EIV bias.In this paper, we employ EIV corrections that build on the early work of Litzenberger and Ramaswamy (1979), perhaps the first paper to argue for the use of individual stocks, and extensions by Shanken (1992). We alsocorrect for a potential bias that can arise when characteristics are time-varying and influencedby past returns, as is the case for size and several other characteristics. This influence can induce cross-sectionacorrelation between characteristics and the measurement errors in betas, a complication that, to our knowledge, has not previously been considered.

We conduct our tests for a comprehensive sample of NYSE, AMEX, and NASDAQ stocks over the period 1963-2013. The independent variables in our CSRs consist of loadings as

Ang, Liu, and Schwarz (2010) use an MLE framework with constant betas to develop analytical formulas for EIV correction to standard errors, but they do not address the bias in the estimated coefficients. Also, they seem to implicitly assume that the factor mean is known, which might explain the huge t-statistics that they report (see Jagannathan and Wang (2002) for a similar critique in the context of SDF models).

well as firm characteristics. The asset pricing model betas examined in the paper are those of the CAPM, the Fama-French three- and five-factor models, and models that include a momentum factor along with the Fama-French factors. The firm characteristics that we examine are the “classic” characteristics firm size, book-to-market ratio, and pastsix-month returns, and the additional characteristics investment and the ratio of operating profitability to book equity.

The results point to some evidence of a positive beta premium on the profitability (RMW) and investment factors (CMA), a negative premium on the size factor (SMB), and a less robust positive premium on the market (multifactor, not CAPM beta), but no evidence for the book-to-market (HML) or the momentum (MOM) factors. Also, the estimated zero-beta rates exceed the risk-free rate by at least 6 percentage points (annualized), even with the additional factors and characteristics in the models. Our main finding is that firm characteristics consistently explain a much larger fraction of the variation in estimated expected returns than factor loadings, even in the case of the six-factor model that includes the Fama-French five-factor model augmented by the momentum factor. Moreover, all of the characteristics are reliably different from zero, with the familiar signs.

The rest of the paper is organized as follows. The next section presentsthe methodology. Section II provides simulation evidence on the finite-sample behavior of the EIV correction that we employ. Section III presents the data and Section IV discusses the results. Section V explores the impact of time-varying premia. Section VI concludes.

I. Methodology

We run CSRs of individual stock returns on their factor loadings and characteristics, correcting for the biases discussed above.

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Panel B of Table 2 presents the time-series averages of the cross-sectional correlations between the different factor loadings and the characteristics. These are also corrected for EIV bias. As expected, the beta for SMB is negatively correlated with firm size, the HML beta is positively correlated with the book-to-market ratio, the profitability beta is positively correlated with operating profits and the investment beta is negatively correlated with investments. The respective correlations are −0.43, 0.33, 0.26, and−0.12 (−0.33, 0.22, 0.16, and −0.08 without EIV correction). Thus, there is considerable independent variation of the characteristics and corresponding factor loadings, permitting identification of their separateeffects on expected return.

In addition, size and book-to-market have a correlation of −0.31 and the past six-month return has a correlation of −0.25 with book-to-market. Firm size is positively correlated and book-to-market is negatively correlated with operating profits and investments. The correlation between profitability and investment is 0.27, suggesting that the profitable firms have better opportunities as well as access to internal or external financing.

IV. Cross-sectional Results

We present results for the one-factor CAPM, the Fama and French (1993) three-factor model FF3, the four-factor model which augmentsthe Fama and French (1993) model with the momentum factor (MOM), the five-factor Fama and French (2014)model FF5, and the six-factor model which augments the Fama and French (2014)model with the momentum factor. Separate analysis of these factor models helps in analyzing the additional importance of the various factors. We present the standard Fama and MacBeth (1973) coefficients as well as bias-corrected coefficients side by side in all our results. This facilitates an evaluation ofthe importance of bias correction to the estimated premia. Finally, we report the Fama and MacBeth (1973) t-statistics.

In the next subsection, we present the results for the sample of all stocks and later we will present the results for the sample of non-microcap stocks.

IV.A. All stocks

Since our goal in this paper is to examine the relative contributions offactor loadings and characteristics to expected returns, we will present results for the Fama-MacBeth (1973) regressions that include characteristics along with the betas. However, we have examined the 17 factor models in the absence of the characteristics and first discuss these results.15Across all models, from the single-factor CAPM to the six-factor model, the risk premium on the market is negative and statistically insignificant. The risk premiums on SMB, MOM, and RMW are also statistically indistinguishable from zero. This contrasts with the significance of the corresponding means for these factors in Fama and French (2014) and suggests that the associated expected return relation is violated for these models. The risk premium on HML is positive and significant in FF3 and FF5, but is no longer significant when MOM is included in the four- and six-factor models. For instance, the risk premium estimate for HML is 0.35% per month in FF3. The risk premium on CMA is positive and significant. In FF5 this risk premium is 0.27% and it is 0.23% per month in the six-factor model.

Panel A of Table 3 reports results for the factor models when the characteristics Sz, B/M, and Ret6 are included in the Fama-MacBeth regressions. Panel B of Table 3 adds the firm-level characteristics Profit and Invest. With uncorrelated factors, estimation error in the betas would bias all of the estimated risk premiums towardzero. While there is some correlation between the factors, we find nonetheless that correcting the EIV bias generally increases the risk premium estimates, sometimes by over 100%.

Consider first the results in Panel A. In the one-factor and FF3 models, the market beta is not priced when the characteristics are included in the CSRs.16In the case of FF5, the market risk premium is 0.42% per month with a t-statistic of 2.16. For comparison, the sample average market excess return is0.50% per month. The beta premium on SMB is negative across all factor models despite its positive sample mean. For instance, in FF3, the premium is −0.29% per month with a t-statistic of −2.21. The negative premium may seem odd, but it is importantto note that this premium captures the partialeffect on return of the SMB beta, controlling for the size characteristic and the other variables (similarly for the other factors). With nonzero characteristic premiums, the usual restriction that the beta premiums equal the factor means need not hold under the cross-sectional model.

Unlike the case where the firm-level characteristics were not included in the regressions, the beta premium for HML is nowno longer significant, possibly due to competition between the HML beta and the book-to-market ratio. The beta premiums on RMW and CMA are both

15
These results are available upon request. 16

For conciseness, we refer to FF3 or FF5 to identify the factors, but the models always include characteristics as well from this point on.

significant, with respective estimates of 0.31 (t-statistic=2.46) and 0.22 (t-statistic=2.33)in FF5 and estimates of 0.26 (t-statistic=2.34) and 0.18 (t-statistic=2.00) in the case of the six-factor model.

The intercepts in second-pass regressions are around 6% to 8% per year, with t-statistics of about four or more. Since characteristics are measured as deviations from NYSE means, the intercepts can be interpreted as the expected return on a zero-beta portfolio with weighted characteristics equal to the NYSE average. Such large differences between the zero-beta rate and the risk-free rate, common in the literature goingback to Black, Jensen and Scholes (1972) and Fama and MacBeth (1973), are hard to fully reconcile with more general versions of the CAPM that incorporate restrictions on borrowing.

The premia on firm characteristics are also noteworthy-as usual, large firms earn lower returns, value firms earn higher returns, and firms with higher past returns continue to earn higher returns and the estimates are statisticallysignificant. In economic terms, for the bias- corrected six-factor model, a one standard deviation increase in firm size decreases monthly returns by 28 basis points, a one standard deviation increase in the book-to-market ratio leads to an increase in returns of 24 basis points per month, and a standard deviation increase in the past six month returns raises returns by 43 basis points per month.

The CAPM and FF3 results for the firm characteristics are similar to those in Brennan, Chordia, and Subrahmanyam (1998) and imply rejection of those beta-pricing models. However, Brennan, Chordia, and Subrahmanyam relate beta-adjustedreturns to characteristics, with risk premiums restricted to equal the factor means and the zero-beta rate equal to the riskless rate. In contrast, we let the loadings and characteristics compete without constraints on the risk premia or the zero-beta rate. What we learn from the new results is that the premia on firm characteristics (specifically Sz, B/M, and Ret6) remain significant even without those constraints and the addition of the factors RMW, CMA and MOM.

There is a controversy in the literature about the interpretation of the size- and value-effects. Fama and French (1993) and Davis, Fama, and French (2000) argue that these empirical phenomena point to the existence of other risk factors, proxied for by SMB and HML. In other words, these studies claim that factor loadingsexplain cross-sectional variation in expected returns. Daniel and Titman (1997), on the other hand, show that portfolios of firms with similar

17See also Frazzini and Pedersen (2013) who show that high zero-beta returns are obtained for most countries.

characteristics but different loadings on the Fama and French factors have similar average returns. They conclude from this finding that itis characteristics that drive cross-sectional variation in expected returns. None of the studies, however, runs a direct horse race between these two competing hypotheses. Our approach using individual stocks is designed to directly address this controversy. We allow both factor loadings and characteristics to jointly explain the cross-section of returns.

The average cross-sectional adj-R2 values (not reported) are higher when the characteristics are included as independent variables in the cross-sectional regressions than when they are not. This might seem to provide prima-facie evidence about the additional explanatory power of characteristics (beyond market beta) in the cross-section of returns. However, one cannot draw conclusions about the relative explanatory power ofcharacteristics and betas by comparing these adj-R2s. To see this, consider a scenario in which the ex-postcoefficient on an explanatory variable is positive (+x, for instance) and significant in half the sample and negative (−x, for instance) and significant in the other half. The computed average of the cross-sectional adj-R2s could be high even though the coefficient is zero on average and carries no ex-ante premium.

To address these problems withadj-R2s, it is common in the literature to report the adj-R2from a single regression of averagereturns on unconditional betas for a set of test asset portfolios (see Kan, Robotti, and Shanken (2013)). This is problematic in our context, as our regressions are for individual stocks with an unbalanced panel dataset. One approach would be to report theadj-R2for a regression of average returns on average betas and average characteristics.However, a momentum characteristic averaged over time would display minimal cross-sectional variation and, therefore, its highly significant explanatory powerfor expected returns would essentially be neglected by such an adj-R2measure. For these reasons, we do not report adj-R2sfor our regressions. Instead, we report measures of the relative contributions of loadings or characteristics make toward explaining the variation in expected returns, as discussed in Section I.C.

The last four rows of Table 3 present the contributions made by factor loadings and characteristics, followed by the contribution differences and a 95% bootstrap confidence interval

for the latter computedfollowing the procedure in Section I.C.Focusing on the bias-corrected coefficients, we find that the CAPM beta explains only 0.8% and the characteristics explain 104.2% of the cross-sectional variation; in the case of FF3, the betas explain 12% and the characteristics explain 110%; with the four-factor model, the betas explain about 11% and the characteristics 109%; with FF5, it is betas 31% and characteristics 97%; and for the six-factor model, betas explain 24% and characteristics 102%.Clearly, the characteristics explain an overwhelming majority of the variation in expected returns. This is confirmed by the 95% confidence intervals, which, in each case, indicatethat the difference is significantly positive at the 5% level. The best showing for beta is inFF5, but even there the point estimate of the difference is 67% and the confidence interval indicates a difference of at least 36%.

The findings when we include the additional firm-level characteristics Profit and Invest in Panel B of Table 3 are very similar. The risk premium on the market betais not significantly different from zero in the CAPM and the four-factor model, but it is significant in the other cases. For instance, the market risk premium is 0.47% in FF5. The premiums for SMB are still negative, but significant at the 5% level only in the case of FF5. The premium on RMW remains significant in the five- and six-factor models, but the premiums on HML, MOM and CMA are never reliably different from zero. As compared toPanel A, the CMA beta loses its significance, probably due to competition with the corresponding characteristic Invest.

Even with the additional factorloadings included, the characteristic premiums for size, book-to-market, past six-month return, profitability and investment growth are all consistent with the prior literature and highly significant. In economic terms, for the six-factor model, a one standard deviation increase in Sz, B/M, Ret6, Profit, and Invest increases returns by −31, 23, 40, 21, and −22 basis points per month, respectively. Once again, the characteristics explain most of the variation in expected returns for this specification. Similarly, the bootstrap confidence intervals are consistent with a significantly larger fraction of the variation in returns being explained by characteristics as compared to the factor loadings. IV.B. Non-microcap stocks

18A comparison between our results and those in Daniel and Titman (1997) is complicated by the fact that we use past returns as an additional characteristic in our cross-sectional regressions.
19Recall that the total percent explained can differ from 100% because of correlation between the components of expected returns due to beta
s and due to characteristics.

Next, we turn our attention to non-microcap stocks which, following Fama and French (2008), includes all stocks whose market capitalization is larger than that of the 20thpercentile of NYSE stocks. Table 4 shows the second-stage CSR beta-premium estimates for the different models as well as the characteristic premiums. Panel A presents the results with the characteristics Sz, B/M, and Ret6 included in the regressions and Panel B includes Profit and Invest along with the characteristics in Panel A. The bias-corrected beta premiums for the market, HML and MOM are not statistically significant in either of the two Panels. However, the premium on SMB is significantly negative in the four-, five-, and six-factor models in Panel A and in the four- and six-factor models in Panel B of Table 4. The premiums on the RMW and CMA betas are generally significant in Panel A of Table 4, but in Panel B only the premium of RMW is significant and that only in the six-factor model. This suggests that the factors RMW and CMA are robustly priced only in the absence of the firm-level characteristics Profit and Invest. All of the characteristic premiums, i.e., those for size, book-to-market, past return, profitability and investment, are statistically and economically significant. The bias-corrected estimates all have t-statistics greater than two (and often much larger) in both panels of Table 4.

The economic magnitudes and statistical significance reported thus far indicate that both factor loadings and characteristics matter for non-microcap stocks.But how much variation does each explain? Note, first, that the contribution of factor loadingsto the variation in expected returns, as shown in Table 4, increases with the number of factors in the asset pricing models. This contrasts with the all-stock results, where the contribution of betas declined with the addition of MOM to FF5. However, as in Table 3, the contribution of characteristics far exceeds that of the factor loadings in all cases presented in Panels A and B ofTable 4. The corresponding differences are statistically significant except for FF5 in Panel A, where the difference of 39.7% is not quite distinguishable fromzero at the 5% level, given the wide confidence interval.

IV.D. Additional robustness checks

Recall that, in implementing EIV correction, we to switch to OLS estimation in a given month if the “correction” leads to an X’Xestimate that is not positive definite or if the premium estimator is an “outlier,” i.e., differs from the factor realization by more than 20%. These issues are encountered only with four or more factors and occur in at most nine months with less than six factors. For the six-factor model, there are 23not-positive-definite months and eight outliers.

We have also explored 10% and 50% outlier criteria. Not surprisingly, there are many more outliers with 10%, but our main conclusion, that characteristics explain much more variation in expected returns than betas is not sensitive to the treatment of outliers. Individual beta-premium coefficients are occasionally materially affected, however. For example, the premium for RMW in the five-factor model with all characteristics goes from 0.24 (t-statistic=2.01) to 0.16 (t-statistic=1.45) with a 10% outlier cutoff. There isa larger change for the MOM beta premium in the six-factor model, but none of the estimates is statistically significant.

We have also conducted the analysis withoutincluding the correction, described in the appendix, for time-varying characteristics. Whilethe tenor of the results is unchanged, the impact on the magnitude of return premia is occasionally non-trivial (around 30% up or down).

Finally, a conditional time-series regression framework for estimating betas with monthly returns has also been explored. Here, each individual stock beta is allowed to vary as a linear function (for simplicity) of the corresponding characteristic and each stock alpha is a linear function of all the characteristics, similar to the approach in Shanken (1990). Thus, the beta on SMB depends on size, the beta on HML depends on book-to-market, etc. Details are provided in Appendix C. This approach is appealing (in principle), since it directly addresses the possibility that, with betas assumedto be constant, the appearance of significant pricing ofa characteristic such as size may actually be a reflection of the premium for a time-varying SMB beta.In practice, however, we encountered the not-positive definite problem withgreater frequency and found no evidence of beta pricing other than a t-statistic of 2.0 on the RMW beta in the six-factor specification.Again, characteristics dominate.

V. Time-Varying Premia

In this section, we consider the possibility thatthe expected return premia for loadings or cross-sectional characteristics are time varying and we examine the impact that this has on our measures of the relative contributions to cross-sectional expected-return variation.Following Ferson and Harvey (1991), we estimate changing premia via time-series regressions of the monthly CSR estimates on a set of predictive variables. The idea is that the premium estimate for

20See related work by Ferson and Harvey (1998), Lewellen (1999), and Avramov and Chordia (2006)
21Concerned about the possibility of noise related to the large number of parameters that must be estimated in these time-series regressions for individual stocks, we also tried zeroing-out estimates of the interaction terms with t-statistics less than one. This made little difference in the results.
22Gagliardinia, Ossola, and Scaillet (2011) also consider time-varying premia in large cross sections.

a given month is equal to the true conditional premium plus noise. Therefore, regressing that series on relevant variables known at the beginning of each month identifies the expected component.

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