Klüppelberg, Claudia and Maller, R. A. and Van De Vyver, M. and Wee, D.
Testing for Reduction to Random Walk in Autoregressive Conditional Heteroskedasticity Models.
Collaborative Research Center 386, Discussion Paper 266
The AR-ARCH and AR-GARCH models, which allow for conditional heteroskedasticity and autoregression, reduce to random walk or white noise for some values of the parameters. We consider generalised versions of the AR-ARCH(1) and AR-GARCH(1,1) models, and, under mild assumptions, calculate the asymptotic distributions of pseudo-likelihood ratio statistics for testing hypotheses that reect these reductions. These hypotheses are of two kinds: the conditional volatility parameters may take their boundary values of zero, or the autoregressive component may take the form of a unit root process or not in fact be present. The limiting distributions of the resulting test statistics can be expressed in terms of functionals of Brownian motion related to the Dickey-Fuller statistic, together with independent chi-square components. The finite sample performances of the test statistics are assessed by simulations, and percentiles are tabulated. The results have applications in the analysis of financial time series and random coefficient models.