Bootstrapping the transformed goodness-of-fit test on heavy-tailed GARCH models

We study the bootstrap inference on the goodness-of-fit test for generalized autoregressive conditional heteroskedastic (GARCH) models. Note that the commonly-used portmanteau tests for model adequacy checking necessarily impose moment conditions on innovations, we hence construct the test on the sample autocorrelations of a bounded transformation of absolute residuals, which are obtained by the least absolute deviation estimation from a fitted GARCH model. Specifically, we employ the empirical distribution function of absolute residuals as the transformation. Thus the corresponding portmanteau tests are applicable for very heavy-tailed innovations with only finite fractional moments. We bootstrap both the estimation equation and sample autocorrelations of transformed residuals to approximate the test statistics. The asymptotic validity of the bootstrap procedure is established. Monte Carlo experiments compare the finite-sample performance of the proposed bootstrap-based test with other existing tests. An empirical analysis of modeling exchange rates illustrates its usefulness.