Finite Sample Analysis of Two-Pass Cross-Sectional Regressions

In this paper, we investigate finite sample properties of the two-pass cross-sectional regression (CSR) methodology, which is popular for estimation of risk premia and testing of beta pricing models. We find that the finite sample distribution of the estimated risk premium differs significantly from the asymptotic distribution. In particular, the risk premium estimates obtained from the second pass CSR of average returns on estimated betas can have serious bias even when the number of time series observations is reasonably large. In addition, the standard error of the estimated risk premium based on the asymptotic distribution overstates the actual standard error. A simple biasadjustment on the estimated zero-beta rate and risk premium is proposed and the adjusted version is shown to have better finite sample properties than the unadjusted one. In empirical asset pricing literature, the popular two-pass cross sectional regression (CSR) methodology developed by Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973) is often used for estimation of risk premia and testing of beta asset pricing models. Although there are many variations of this two-pass methodology, its basis setup always involves two steps. In the first pass, the betas of the test assets are estimated using the usual ordinary least squares (OLS) time series regression of returns on some common factors. In the second pass, the returns on test assets are regressed on a constant term and the estimated betas obtained from the first pass. By running this second pass CSR on a period-by-period basis, we obtain a time series of the intercept and the slope coefficients. The average values of the intercept and the slope coefficients are then used as estimates of the zero-beta rate and the risk premia. Since betas are estimated with errors in the first pass time series regression, errors-in-variables (EIV) problem is introduced in the second pass CSR. Measurement errors in the estimated betas cause two problems. The first problem is that the estimated zero-beta rate and risk premium are biased. However, as the length of the times series used to estimate betas increases to infinity, Shanken (1992) shows that the estimation errors on the betas go to zero and hence the estimated zero-beta rate and risk premium from the second pass CSR are still consistent. The second problem is that the standard errors of the estimated zero-beta rate and risk premium that are due to Fama and MacBeth (1973) are inconsistent. Shanken (1992) proposes an asymptotically valid EIV adjustment to the standard errors. Unlike the asymptotic results which are nicely presented by Shanken (1992), finite sample analysis of the two-pass CSR is largely unavailable in literature. Aside from some simulation studies like Affleck-Graves and Bradfield (1993) and Shanken and Zhou (2000), we have very little understanding of the finite sample properties of the estimated zero-beta rate and risk premium from the second pass CSR. For example, while we know that there is a bias in the estimated risk premium, we do not know the magnitude of this bias, whether the bias is positive or negative, or what the parameters determining it are. While asymptotically the estimated zero-beta rate and risk premium from the second pass CSR are consistent, it is not clear if their finite sample biases are negligible. In many applications, betas of test assets are estimated using as few as 60 monthly observations, so it is reasonable to suspect that the asymptotic results may not be entirely relevant