Nonparametric estimation of the distribution of the autoregressive coefficient from panel random-coefficient AR(1) data

Abstract

We discuss nonparametric estimation of the distribution function $G\left(x\right)$ of the autoregressive coefficient $a\in (-1,1)$ from a panel of $N$ random-coefficient AR(1) data, each of length $n$, by the empirical distribution function of lag 1 sample autocorrelations of individual AR(1) processes. Consistency and asymptotic normality of the empirical distribution function and a class of kernel density estimators is established under some regularity conditions on $G\left(x\right)$ as $N$ and $n$ increase to infinity. The Kolmogorov–Smirnov goodness-of-fit test for simple and composite hypotheses of Beta distributed $a$ is discussed. A simulation study for goodness-of-fit testing compares the finite-sample performance of our nonparametric estimator to the performance of its parametric analogue discussed in Beran et al. (2010).