Sup-tests against time-varying GARCH models

Testing GARCH models against time-varying GARCH models involves nuisance parameters which are not identified under the null hypothesis. Asymptotic distribution theory is used for additive nonlinear regression models to derive misspecification tests against a new GARCH model with a deterministic time-varying intercept. First, we linearise the GARCH model by an ARMA representation. Second, we use testing theory for regression models with additive nonlinearity to derive test statistics. The asymptotic distributions of test statistics can be expressed as functionals of chi-squared processes. The supremum (sup) and average (ave) functionals are used to derive test statistics. The asymptotic distributions of the test statistics are approximated by simulation. In a Monte Carlo study, we find that the proposed sup and ave tests have good size and power properties. The results show that the tests tend to be slightly conservative but have higher power than tests based on auxiliary regressions. The power loss implied by the Taylor expansion in auxiliary regression-based tests is substantial against a time-varying GARCH model with an intercept
that is a smooth function of time.