Abstract :
This article investigates model checks for a class of possibly nonlinear heteroskedastic
time series models, including but not restricted to ARMA-GARCH models. We
propose omnibus tests based on functionals of certain weighted standardized residual
empirical processes. The new tests are asymptotically distribution-free, suitable
when the conditioning set is infinite-dimensional, and consistent against a class of
Pitman’s local alternatives converging at the parametric rate n−1/2, with n the sample
size. A Monte Carlo study shows that the simulated level of the proposed tests is
close to the asymptotic level already for moderate sample sizes and that tests have
a satisfactory power performance. Finally, we illustrate our methodology with an
application to the well-known S&P 500 daily stock index. The paper also contains
an asymptotic uniform expansion for weighted residual empirical processes when
initial conditions are considered, a result of independent interest.
