Nonstationary Density Estimation and Kernel Autoregression

An asymptotic theory is developed for the kernel density estimate of a random walk and the kernel regression estimator of a nonstationary first order autoregression. The kernel density estimator provides a consistent estimate of the local time spent by the randon walk in the spatial vicinity of a point that is determined in part by the argument of the density and in part by initial conditions. The kernel regression estimator is shown to be consistent and to have a mixed normal limit theory. The limit distribution has a mixing variate that is given by the reciprocal of the local time of a standard Brownian motion. The permissible range for the bandwidth parameter hn includes rates which may increase as well as decrease with the sample size n, in contrast to the case o a stationary autoregression. However, the convergence rate of the kernel regression estimator is at most n1/4, and this is slower than that of a stationary kernel autoregression, in contrast to the parametric case. In spite of these differences in the limit theory and the rates of convergence between the stationary and nonstationary cases, it is shown that the usual formulae for confidence intervals for the regression function still apply when hn → 0.

Key words and phrases: Brownian sheet, kernel regression, local time, martingale embedding, mixture normal, nonstationary density, occupation time, quadratic variation, unit root autoregression.