Which of the following types of empirical size estimation suffers with individual bias?

A fundamental precept of financial economics is that investors earn higher average returns by bearing systematic risks. While this idea is well accepted, there is little agreement about the identities of systematic risks or the magnitudes of the supposed compensations. This is not due to a lack of efforts along two lines of enquiry. First, numerous candidates have been proposed as underlying risk factors. Second, empirical efforts to estimate risk premiums have a long and varied history.

Starting with the single-factor CAPM (Lintner, 1965, Sharpe, 1964) and the multifactor arbitrage pricing theory (APT) (Ross, 1976), the first line of enquiry has brought forth an abundance of risk factor candidates. Among others, these include the Fama and French size and book-to-market factors, human capital risk (Jagannathan and Wang, 1996), productivity and capital investment risks (Cochrane, 1996, Eisfeldt and Papanikolaou, 2013, Hou et al., 2015), different components of consumption risk (Ait-Sahalia et al., 2004, Lettau and Ludvigson, 2001, Li et al., 2006), cash flow and discount rate risks (Campbell and Vuolteenaho, 2004), and illiquidity risks (Acharya and Pedersen, 2005, Pastor and Stambaugh, 2003).

The papers that propose the new risk factors also typically present evidence suggesting that the factors command risk premiums. Their empircal tests mostly follow the methodology originally introduced by Black et al. (1972), (BJS), and refined by Fama and MacBeth (1973), (FM). A prominent feature of this methodolgy is that it uses portfolios rather than individual stocks as test assets.

The BJS and FM methods involve two-pass regressions: the first pass is a time-series regression of individual asset returns on the proposed factors to estimate factor loadings, or “betas.”1 The second pass regresses the cross-section of asset returns on betas obtained from the first-pass regression. The explanatory variables in the second-pass regressions are beta estimates from the first pass, which estimate true betas with error and therefore have an errors-in-variables (EIV) problem. As a result, the risk premium estimates from the second-pass regressions are biased and inconsistent; and the directions of the biases are unknown when there are multiple factors involved in the two-pass regressions.

With a large number of individual stocks, the EIV bias can be reduced by using portfolios rather than individual stocks as test assets. This process begins by forming diversified portfolios classified by some individual asset characteristics, such as a beta estimated over a preliminary sample period. It then estimates portfolio betas using data for a second period. Finally, it runs the cross-sectional regressions on estimated portfolio betas using data for a third period. BJS, Blume and Friend (1973), and FM note that portfolio betas are estimated more precisely than individual stock betas; so the EIV bias is reduced with portfolios as test assets, which can be entirely eliminated as the number of stocks in the sample grows indefinitely.

But using portfolios as test assets has its own shortcomings. There is an immediate issue of test power since the dimensionality is reduced; i.e., average returns vary with fewer explantory variables across portfolios than across individual stocks. Perhaps more troubling is that diversification into portfolios can mask cross-sectional phenomena in individual stocks that are unrelated to the portfolio grouping procedure. For example, advocates of fundamental indexation (Arnott et al., 2005) argue that assets with high market values are overpriced and vice versa, but any portfolio grouping by an attribute other than market value itself could diversify away such potential mispricing, rendering any mispricing undetectable.

Another disquieting result of portfolio masking involves the cross-sectional relation between average returns and betas. Take the single-factor CAPM as an illustration, though the same effect is also at work for any linear factor models. The linear relation between expected returns and betas holds exactly, if and only if, the market index used for computing betas is on the mean/variance frontier of the individual asset universe. Rejection of this linear relation would imply that the index is not on the frontier. But if the individual assets are grouped into portfolios sorted by beta, any asset pricing errors across individual assets not related to beta are unlikely to be detected. Therefore, this procedure could lead to a mistaken inference if the index is on the efficient frontier constructed with beta-sorted portfolios.

Test portfolios are typically organized by firm characteristics related to average returns, such as size and book-to-market. Sorting on characteristics that are known to predict returns helps generate a reasonable variation in average returns across test assets. But Lewellen et al. (2010) point out sorting on characteristics also imparts a strong factor structure in test portfolio returns. Lewellen et al. (2010) show that as a result even factors weakly correlated with the sorting characteristics could explain the differences in average returns across test portfolios, regardless of the economic merits of the theories that underlie the factors.

Finally, the statistical significance and economic magnitudes of risk premiums are likely to depend critically on the choice of test portfolios. For example, the Fama and French size and book-to-market risk factors are significantly priced when test portfolios are sorted based on the corresponding characteristics. However, they do not command significant risk premiums when test portfolios are sorted only on momentum.

We develop a new procedure to estimate risk premiums and to test their statistical significance using individual stocks that avoids the EIV bias. Our method adopts the instrumental variables technique, a standard econometric solution to the EIV problem. We refer to our approach as the IV method, which first estimates betas for individual stocks from a subset of the observations in the data sample. These betas are the “independent” variables for the second-stage cross-sectional regressions. Then, we estimate betas again using a disjoint data sample, and these betas are the “instrumental” variables in the second-stage regressions. Since we estimate the independent and instrumental variables from disjoint data samples, their measurement errors are uncorrelated.2 For some of our empirical tests, we modify our IV method and employ stock characteristics as additional instruments for betas.

We show that the IV estimator consistently estimates the ex post risk premiums as the number of stocks in the sample (N) increases indefinitely. We refer to this property of the estimator as N-consistency, following Shanken (1992). Since consistency is a large sample property, it is important to examine the estimator's small sample properties as well. We do so using a number of simulation experiments. We match the simulation parameters to those in the actual data. We find that the IV estimates in the simulation are not different from the true parameters even when betas are estimated using a relatively short time-series of data. In contrast, the standard approach that fits the the second-stage regressions using ordinary least squares (OLS) (hereafter we will refer to this standard approach as the OLS method) suffers from severe EIV biases. For example, in simulations with a single-factor model, we find that the OLS estimates with individual stocks are significantly biased towards zero, even when betas are estimated with about ten years of daily data. In contrast, the IV estimates are not different from the true parameters when we use only about one year of daily data to estimate betas.

In terms of test size (i.e., type I error) and power (i.e., type II error), we find that the conventional t-tests based on the IV estimator are well-specified, and they are reasonably powerful even in small samples for the CAPM and for the Fama–French three-factor model. We also show analytically that our IV estimator is consistent, even if betas of individual stocks vary over time, as long as they follow covariance stationary processes. The simulation evidence with time-varying betas is similar to that with constant betas.

We apply the IV method to empirically test whether the risks proposed by the CAPM, the three-factor and five-factor models of Fama and French (1993, 2014), the q-factor asset pricing model of Hou et al. (2015), and the liquidity-adjusted capital asset pricing model (LCAPM) of Acharya and Pedersen (2005) command risk premiums that are different from zero. All these papers find significant premiums for the risks they propose. However, they all use portfolios as test assets and, hence, these tests potentially suffer from low dimensionality problems. In contrast to the original papers, we find that none of these risks are associated with a significant risk premium in the cross-section of individual stock returns after controlling for corresponding stock characteristics.

This failure to find significant risk premiums is not due to the lack of test power of the IV method. Our simulation evidence indicates the t-tests based on the IV method provide reasonably high power under the alternative hypotheses that the true risk premiums equal the sample means of factor realizations observed in the data. For example, when the true high-minus-low (HML) risk premium equals the sample risk premium (4.36% per year), the IV-test rejects the null hypothesis that the risk premium is zero with 91.5% probability.

Several papers in the literature, including Berk et al., 1999, Carlson et al., 2004, Carlson et al., 2006), Zhang (2005), and Novy-Marx (2013) argue that firm characteristics appear to be priced because they may serve as proxies for betas. For example, consider firms A and B that are identical except for their risk. If firm A were riskier than firm B, then firm A would have a bigger book-to-market ratio than firm B because the market would discount its expected cash flows at a bigger discount rate. If error-ridden betas were used along with book-to-market ratios, it may appear that difference in returns is related to book-to-market ratios even when betas are the true measures of risk.

We modify our tests to investigate this alternative explanation. Specifically, we allow for time-varying betas and characteristics, and we let the characteristics anticipate future changes in betas. In the second-stage cross-sectional regression, we use average returns over a long sample period as the dependent variable. We use both betas and characteristics as instruments. We show that this modified IV estimator is consistent, and we find that it is well-specified in small samples. We find that the factor risk premiums are not statistically significant with this modified IV approach as well.

Litzenberger and Ramaswamy (1979) is an early paper that uses individual stocks to test asset pricing models. That paper assumes that stock returns follow a single-factor model. A contemporaneous paper by Chordia et al. (2015) generalizes that approach to multifactor settings. These papers derive the asymptotic bias due to the EIV problem and analytically undo the bias. Kim (1995) corrects the EIV bias using lagged betas to derive a closed-form solution for the maximum likelihood estimator (MLE) of market risk premium. The solution proposed by Kim is based on the adjustment by Theil (1971). In contrast, we use the well-known IV approach to address the EIV problem. We show that our approach is consistent even with time-varying betas. We also modify our approach to address the concerns in Berk et al., 1999, Novy-Marx, 2013, and others that characteristics may be proxies for true betas. The other papers do not address such concerns.

Risk-return models and IV estimation

A number of asset pricing models predict that expected returns on risky assets are linearly related to their covariances with certain risk factors. A general specification of a K-factor asset pricing model can be written as:E(ri )=γ0+∑k=1Kβi,kγk where E(ri) is the expected excess return on stock i, βi, k is the sensitivity of stock i to factor k, and γk is the risk premium on factor k. γ0 is the excess return on the zero-beta asset. If riskless borrowing and lending are allowed, then the

Small sample properties of the IV method − simulation evidence

To evaluate the small sample properties of the IV method, we conduct a battery of simulations using the parameters matched to real data. We first investigate the bias and the root-mean-squared error (RMSE) of the IV estimator and then we examine the size and power of the associated t-test, which we refer to as the IV test.

IV risk premium estimates for selected asset pricing models

This section employs the IV method to estimate the premiums for risk factors proposed by prominent asset pricing models.

Additional tests

This section examines the robustness of our findings to a number of variations in the test specifications and also evaluates the strength of the instruments.

Conclusion

We propose a method for estimating risk premiums using individual stocks as test assets. It overcomes concerns about risk premiums estimated with test portfolios, which have been employed in almost all previous research to mitigate an inherent errors-in-variables problem in testing asset pricing models. Estimated betas from one sample period can serve as effective instruments for estimated betas from a disjoint sample period that serve as the explanatory variables in cross-sectional

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