Semiparametric Estimation of Risk–Return Relationships

Abstract

This article proposes semiparametric generalized least-squares estimation of parametric restrictions between the conditional mean and the conditional variance of excess returns given a set of parametric factors. A distinctive feature of our estimator is that it does not require a fully parametric model for the conditional mean and variance. We establish consistency and asymptotic normality of the estimates. The theory is nonstandard due to the presence of estimated factors. We provide sufficient conditions for the estimated factors not to have an impact in the asymptotic standard error of estimators. A simulation study investigates the finite sample performance of the estimates. Finally, an application to the CRSP value-weighted excess returns highlights the merits of our approach. In contrast to most previous studies using nonparametric estimates, we find a positive and significant price of risk in our semiparametric setting.