Testing for Threshold Autoregression

We consider the problem of determining whether a threshold autoregressive model fits a stationary time series significantly better than an autoregressive model does. A test statistic $\lambda$ which is equivalent to the (conditional) likelihood ratio test statistic when the noise is normally distributed is proposed. Essentially, $\lambda$ is the normalized reduction in sum of squares due to the piecewise linearity of the autoregressive function. It is shown that, under certain regularity conditions, the asymptotic null distribution of $\lambda$ is given by a functional of a central Gaussian process, i.e., with zero mean function. Contiguous alternative hypotheses are then considered. The asymptotic distribution of $\lambda$ under the contiguous alternative is shown to be given by the same functional of a noncentral Gaussian process. These results are then illustrated with a special case of the test, in which case the asymptotic distribution of $\lambda$ is related to a Brownian bridge.