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Testing for threshold autoregression. (English) Zbl 0711.62074

Summary: 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.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62E20 Asymptotic distribution theory in statistics
62F05 Asymptotic properties of parametric tests
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