Abstract: The testing problem for the hypothesis of linearity against the double threshold autoregressive conditional heteroscedastic model is addressed. The problem is nonstandard as the threshold parameter is a nuisance parameter which is absent under the null hypothesis. We will show that the asymptotic null distribution of the Lagrange-multiplier test statistic is a functional of a zero-mean Gaussian process. In some cases, we give the upper percentage points of the test statistic. The performance of the test statistic is illustrated by extensive simulation experiments and an example.
Key words and phrases: Conditional heteroscedasticity, Gaussian process, Lag- range-multiplier test, threshold time series model.