Estimation of a threshold autoregressive model under misspecification

Abstract

This paper obtains an asymptotic distribution for the least squares estimator of the self-exciting threshold autoregressive model, which was introduced by Tong (1983), under the assumption that the model is an approximation to a more complicated nonparametric system. Under some moderate assumptions on the true data generating process, it is shown that the least squares estimator is mere n^{1/3}-consistent to a pseudo true value, where n is the sample size, and the limit distribution is characterized by the minimizer of a non-zero-mean Gaussian process as in Kim and Pollard (1990). This is in sharp contrast to Chan (1993), which obtained n-consistency for the threshold parameter estimate under the correct specification. The slower rate of convergence implies that the confidence region is much wider and can be useful to represent the case where it is difficult to locate the true threshold value. Some examples illustrates the case