Nonparametric Estimation of Time-Varying Parameters

Summary

A sequence of observations y_t, t = 1, 2,..., N, is generated by the time-varying multiple regression model

$${y_t} = {\beta '_t}{x_t} + {\sigma _t}{u_t}, t = 1,2, \ldots ,N,$$

where, for t = 1, 2,..., N, u _t is an unobservable random variable with zero mean and unit variance, x _t is an observable p-vector-valued variable, and σ _t and β _t are, respectively, unobservable scalar and p-vector-valued parameters. No model (stochastic or nonstochastic) is assumed for the σ _t or β _t ; instead they are assumed to be smoothly varying over t, in a certain sense. A class of estimators of the β _t , σ _t is proposed, for each value of t; the estimators optimize a criterion prompted by Gaussian maximum likelihood considerations, and may be viewed as analogous to certain nonparametric function fitting estimators, employing a kernel function and band-width parameter, both selected by the practitioner. Consistency and asymptotic normality are established in case of independent u _t , and a consistent estimator of the asymptotic covariance matrix of the β _t estimators is given. Such results are also possible for serially correlated u _t . We discuss questions of implementation, in particular the choice of kernel function and band-width. Generalization of the class of estimators to include certain robust estimators is possible, as is generalization of the methods to more general models involving time-varying parameters.