Testing for threshold autoregression with conditional heteroscedasticity

This paper addresses the null distribution of the Lagrange-multiplier statistic for the threshold autoregression with conditional heteroscedasticity. The problem is nonstandard because the threshold parameter is a nuisance parameter which is absent under the null hypothesis. We generalise the results of Chan (1990) and Chan & Tong (1990) to show that the asymptotic null distribution of the Lagrange-multiplier statistic is a functional of a zero-mean Gaussian process. The generalisation is not direct as the conditional variance is changing and the unconditional distribution of the process variable is no longer normal. In some special cases, we can reduce the problem to the asymptotic distribution of certain functions of Brownian bridges and the upper percentage points can be tabulated as in Chan (1991). Monte Carlo experiments show that the approximation and the power of the test are quite good. We also demonstrate the importance of using our test if the true model has conditional heteroscedasticity.