We consider time series models in which the conditional mean of the response variable given the past depends on latent covariates. We assume that the covariates can be estimated consistently and use an iterative nonparametric kernel smoothing procedure for estimating the conditional mean function. The covariates are assumed to depend (non)parametrically on past values of the covariates and of the observations. Our procedure is based on iterative fits of the covariates and nonparametric kernel smoothing of the conditional mean function. An asymptotic theory for the resulting kernel estimator is developed and the estimator is used for testing parametric specifications of the mean function. Our leading example is a semiparametric class of GARCH-in-Mean models. In this set-up our procedure provides a formal framework for testing economic theories that postulate functional relations between macroeconomic or financial variables and their conditional second moments. We illustrate the usefulness of the methodology by testing the linear risk-return relation predicted by the ICAPM.
Specification test GARCH-M semiparametric regression risk premium ICAPM