The BDS test as a test for the adequacy of a GARCH (1,1) specification: A Monte Carlo study

In this study, we examine the Brock, Dechert and Scheinkman (BDS) test when applied to the standardised residuals of an estimated GARCH(1,1) model as a test for the adequacy of this specification. We review the conditions derived by De Lima (1996, Econometric Reviews, 15, 237-259) for the nuisance-parameter free property to hold, and address the issue of their necessity, using the GARCH(1,1) model. By means of Monte Carlo simulations, we show that, provided that the unconditional mean exists, the BDS test statistic still approximates the standard null distribution even when the majority of the conditions are violated. Further, the test performs reasonably well, as its empirical size is rather close to the nominal one. As a by-product of this study, we also examine the related issue of consistency of the QML estimators of the conditional variance parameters under various parameter configurations and alternative distributional assumptions on the innovation process.