Estimating critical values for testing the i.i.d. in standardized residuals from GARCH models in finite samples

Abstract

Taking into account that the BDS test--which is used as a misspecification test applied to standardized residuals from the GARCH(1,1) model--is characterized by size distortion and departure from normality in finite samples, this paper obtains the critical values for the finite sample distribution of the BDS test. We focus on bootstrap simulation to avoid the sampling uncertainty of parameter estimation and make use of estimated response surface regressions (RSR) derived from the experimental results. We consider an extensive grid of models to obtain critical values with the results of the bootstrap experiments. The RSR used to estimate them is an artificial neural network (ANN) model, instead of the traditional linear regression models. Specifically, we estimate critical values by using a bootstrap aggregated neural network (BANN) and by employing functions of the sample size and parameters used in the experiment as the embedding dimension and proximity parameters in the BDS statistic, GARCH parameters and even the q-quantiles of the BDS distributions. The main results confirm that the sample size and BDS parameters play a role in size distortion. Finally, an empirical application to three price indexes is performed, to highlight the differences between decisions made using the asymptotic or our predicted critical values for the BDS test in finite samples.