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December, 1990 Testing for Threshold Autoregression
K. S. Chan
Ann. Statist. 18(4): 1886-1894 (December, 1990). DOI: 10.1214/aos/1176347886

Abstract

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.

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K. S. Chan. "Testing for Threshold Autoregression." Ann. Statist. 18 (4) 1886 - 1894, December, 1990. https://doi.org/10.1214/aos/1176347886

Information

Published: December, 1990
First available in Project Euclid: 12 April 2007

zbMATH: 0711.62074
MathSciNet: MR1074443
Digital Object Identifier: 10.1214/aos/1176347886

Subjects:
Primary: 62M10
Secondary: 62J05

Keywords: $\rho$-mixing , asymptotics , autoregressive model , Brownian bridge , contiguity , ergodicity , Gaussian process , least squares , nuisance parameter present only under the alternative hypothesis , stationarity , threshold autoregressive model

Rights: Copyright © 1990 Institute of Mathematical Statistics

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Vol.18 • No. 4 • December, 1990
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