Semi-Parametric Estimation of Risk-Return Relationships

Abstract

This article proposes semi-parametric least squares estimation of parametric risk-return relationships, i.e. parametric restrictions between the conditional mean and the conditional variance of excess returns given a set of unobservable parametric factors. A distinctive feature of our estimator is that it does not require a parametric model for the conditional mean and variance. We establish consistency and asymptotic normality of the estimates. The theory is non-standard due to the presence of estimated factors. We provide simple sufficient conditions for the estimated factors not to have an impact in the asymptotic standard error of estimators. A simulation study investigates the nite 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 non-parametric estimates, we find a positive and significant price of risk in our semi-parametric setting.