Academic journal article Federal Reserve Bank of Atlanta, Working Paper Series

A Note on the Estimation of Asset Pricing Models Using Simple Regression Betas

Academic journal article Federal Reserve Bank of Atlanta, Working Paper Series

A Note on the Estimation of Asset Pricing Models Using Simple Regression Betas

Article excerpt

Working Paper 2009-12

March 2009

Abstract: Since Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973), the two-pass cross-sectional regression (CSR) methodology has become the most popular tool for estimating and testing beta asset pricing models. In this paper, we focus on the case in which simple regression betas are used as regressors in the second-pass CSR. Under general distributional assumptions, we derive asymptotic standard errors of the risk premia estimates that are robust to model misspecification. When testing whether the beta risk of a given factor is priced, our misspecification robust standard error and the Jagannathan and Wang (1998) standard error (which is derived under the correctly specified model) can lead to different conclusions.

JEL classification: G12

Key words: two-pass cross-sectional regressions, risk premia, model misspecification, simple regression betas, multivariate betas


In the 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 estimating risk premia and testing pricing models that relate expected security returns to security betas on economic factors (beta pricing models). Although there are many variations of this two-pass methodology, its basic approach 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 the estimated betas obtained from the first pass. By running this second-pass CSR on a period-by-period basis, we obtain 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.

Usually, asset betas are defined as the OLS slope coefficients in the multiple regression of asset returns on factors and are referred to as multiple regression or multivariate betas. However, there is a potential issue with the use of multiple regression betas: unless the factors are uncorrelated, the beta of an asset with respect to a particular factor in general depends on what other factors are included in the first-pass time series OLS regression. As a result, a factor can possess additional explanatory power for the cross-sectional differences in expected returns but yet have a zero risk premium in a model with multiple factors. This makes it problematic to use the risk premium of a factor for the purpose of model selection. To overcome this problem, Chen, Roll, and Ross (1986) and Jagannathan and Wang (1996, 1998) define the beta of an asset with respect to a given factor as the OLS slope coefficient in a simple regression of its return on the factor. These betas are normally referred to as simple regression or univariate betas. In models with simple regression betas, adding or deleting a factor in a model will not change the values of the betas corresponding to the other factors and selecting models based on risk premia becomes more meaningful. (1)

Jagannathan and Wang (1998) present an asymptotic theory for models with simple regression betas. (2) However, their asymptotic results rest on the assumption that expected returns are exactly linear in the betas, i.e., the beta pricing model is correctly specified. It is difficult to justify this assumption when estimating the zero-beta rate and risk premia parameters for many different models because some (if not all) of the models are bound to be misspecified. Since asset pricing models are, at best, approximations of reality, it is inevitable that we will often, knowingly or unknowingly (since asset pricing tests have limited power), estimate an expected return relation that departs from exact linearity in the betas. …

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