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 widely used Brock, Dechert, andScheinkman (BDS) test when applied to the logarithm of the squaredstandardized residuals of an estimated GARCH(1,1) model as atest for the adequacy of this specification. We review the conditionsderived by De Lima (1996; Econometric Reviews 15, 237–259)for the nuisance-parameter-free property to hold and addressthe issue of their necessity, using the flexible framework offeredby the GARCH(1,1) model in terms of moment, memory, and timeheterogeneity properties. By means of Monte Carlo simulations,we show that the BDS test statistic still approximates the standardnull distribution even for mildly explosive processes that violatethe majority of the conditions. Thus the test performs reasonablywell, its empirical size being rather close to the nominal one.As a by-product of this study, we also shed light on the relatedissue of the consistency of the QML estimators of the conditionalvariance parameters under various parameter configurations andalternative distributional assumptions on the innovation process.