Nonparametric estimation of infinite order regression and its application to the risk-return tradeoff

This paper studies nonparametric estimation of the infinite order regression [see code in paper] with stationary and weakly dependent data. We propose a Nadaraya-Watson type estimator that operates with an infinite number of conditioning variables. The established theories are applied to examine the intertemporal risk-return relation for the aggregate stock market, and some new empirical evidence is reported. With a bandwidth sequence that shrinks the effects of long lags, the influence of all conditioning information is modelled in a natural and flexible way, and the issues of omitted information bias and specification error are effectively handled. Asymptotic properties of the estimator are shown under a wide range of static and dynamic regressions frameworks, thereby allowing various kinds of conditioning variables to be used. We establish pointwise/uniform consistency and CLTs. It is shown that the convergence rates are at best logarithmic, and depend on the smoothness of the regression, the distribution of the marginal regressors and their dependence structure in a non-trivial way via the Lambert W function. The empirical studies on S&P 500 daily data from 1950-2016 using our estimator report an overall positive risk-return relation. We also .find evidence of strong time variation and counter- cyclical behaviour in risk aversion. These conclusions are attributable to the inclusion of otherwise neglected information in our method.