We suggest a sequential monitoring scheme to detect changes in the parameters of a GARCH~p, q! sequence+ The procedure is based on quasi-likelihood scores and does not use model residuals+ Unlike for linear regression models, the squared residuals of nonlinear time series models such as generalized autoregressive conditional heteroskedasticity~GARCH! do not satisfy a functional central limit theorem with a Wiener process as a limit, so its boundary crossing probabilities cannot be used+ Our procedure nevertheless has an asymptotically controlled size, and, moreover, the conditions on the boundary function are very simple; it can be chosen as a constant+ We establish the asymptotic properties of our monitoring scheme under both the null of no change in parameters and the alternative of a change in parameters and investigate its finite-sample behavior by means of a small simulation study+
Autoregressive time series models of order p have p + 2 parameters, the mean, the variance of the white noise and the p autoregressive parameters. Change in any of these over time is a sign of disturbance that is important to detect. The methods of this paper can test for change in any one of these p + 2 parameters separately, or in any collection of them. They are available in forms that make one-sided tests possible, furthermore, they can be used to test for a temporary change. The test statistics are based on the efficient score vector. The large sample properties of the change-point estimator are also explored.
We study the asymptotics of maximum-likelihood ratio-type statistics for testing a sequence of observations for no change in parameters against a possible change while some nuisance parameters remain constant over time. We obtain extreme value as well as Gaussian-type approximations for the likelihood ratio. We get necessary and sufficient conditions for the weak convergence of supremum and L p -functionals of the likelihood ration process. We also approximate the maximum likelihood ratio with Ornstein Uhlenbeck processes and obtain bounds for the rate of approximation. We show that the Ornstein Uhlenbeck approach is superior to the extreme value limit in case of moderate sample sizes.1996 Academic Press, Inc.
We propose some testa for detecting possible change in the variance of independent observations. We obtain their asymptotic properties under the nochange null hypothesis. We also investigate the limit distributions of estimators for the time of change.
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