Specification Tests for Time Series Models with GARCH-Type Conditional Variance

Abstract

Specification tests are developed for the conditional distribution of a dependent process {Xi} in a family of nonlinear time-series models. The family includes several Generalized AutoRegressive Conditional Heteroscedastic [GARCH] models that are widely used in finance and economics. Such tests are essential for ensuring the validity of density forecasts that are based on the assumed model. The tests are implemented using a bootstrap procedure, because the test statistics are not asymptotically pivotal. A novel method is developed for verifying the validity of the bootstrap tests for a given model; this involves simultaneously embedding the process under the null hypothesis and the bootstrap process in a system of stochastic recurrence equations. The method is illustrated for a well-known GARCH model. Within the specification testing literature, a feature that distinguishes this paper is that the conditional distribution of Xi under the null model, depends on the unobservable past values {. . . , X−2, X−1} extending back to the infinite past. The theory for testing the specification of the conditional distribution of Xi in such parametric models has not been developed in the general setting considered in this paper. The tests performed well in a simulation study, and a data example illustrates the tests.