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Model checks using residual marked empirical processes. (English) Zbl 1145.62071
Summary: This article proposes omnibus goodness-of-fit tests of a parametric regression time series model. We use a general class of residual marked empirical processes as the buildingblocks for our testing problem. First, we establish a new weak convergence theorem under mild assumptions, one that extends previous existing asymptotic results and which may be of independent interest. This result allows us to study the asymptotic null distribution of the test statistics and their asymptotic behavior against Pitman’s local alternatives in a unified way.
To approximate the asymptotic null distribution of the test statistics we give a theoretical justification of a bootstrap procedure. Our bootstrap tests are robust to conditional higher moments of unknown form, in particular to conditional heteroskedasticity. Finally, a Monte Carlo study shows that the bootstrap and the asymptotic results provide good approximations for small sample sizes and an empirical application to the Canadian lynx data set is considered.

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F05 Asymptotic properties of parametric tests
62E20 Asymptotic distribution theory in statistics
60F05 Central limit and other weak theorems
62F40 Bootstrap, jackknife and other resampling methods
62G30 Order statistics; empirical distribution functions
62F03 Parametric hypothesis testing