Semiparametric Inference in a GARCH-in-Mean Model

A new semiparametric estimator for an empirical asset pricing model with general nonparametric risk-return tradeoff and a GARCH process for the underlying volatility is introduced. The estimator does not rely on any initial parametric estimator of the conditional mean function, and this feature facilitates the derivation of asymptotic theory under possible nonlinearity of unspecified form of the risk-return tradeoff. Besides the nonlinear GARCH-in-mean effect, our specification accommodates exogenous regressors that are typically used as conditioning variables entering linearly in the mean equation, such as the dividend yield. Using the profile likelihood approach, we show that our estimator under stated conditions is consistent, asymptotically normal, and efficient, i.e. it achieves the semiparametric lower bound. A sampling experiment provides evidence on finite sample properties as well as comparisons with the fully parametric approach and the iterative semiparametric approach using a parametric initial estimate proposed by Conrad and Mammen (2008). An empirical application to the daily S&P 500 stock market returns suggests that the linear relation between conditional expected return and conditional variance of returns from the literature is misspecified, and this could be the reason for the disagreement on the sign of the relation.