Finite Sample and Optimal Inference in Possibly Nonstationary ARCH Models with Gaussian and Heavy-Tailed Errors

Most of the literature on testing ARCH models focuses on the null hypothesis of no-ARCH effects. In this paper, we consider the general problem of testing any possible set of coefficient values in ARCH models, which may be non-stationary, with Gaussian and non-Gaussian errors, as well as with any number exogenous regressors in the mean equation. Both Engle-type and point-optimal tests are studied. Special problems considered include the hypothesis of no-ARCH effects and IARCH structure. We propose exact inference based on pivotal Monte Carlo tests [as in Dufour and Kiviet (1996, 1998) and Dufour, Khalaf, Bernard and Genest (2004)] and maximised Monte Carlo tests [Dufour (2004))], depending on whether nuisance parameters are present. This will allow the introduction of dynamics in the mean equation as well. We show that the method suggested provides provably valid tests in both finite and large samples, in cases where standard asymptotic and bootstrap methods may fail in the presence of heavy-tailed errors [as shown by Hall and Yao (2003)]. The performance of the proposed procedures with both Gaussian and non-Gaussian errors is analyzed in a simulation experiment. Our results show that the proposed procedures work well from the viewpoints of size and power. The powers gains provided by the point optimal procedures are in many cases spectacular. The tests also exhibit good behaviour outside the stationarity region [following the work of Jensen and Rahbek (2004)]. Finally, the technique is applied to the US inflation