Generalized Spectral Testing with Sample Splitting

Abstract: Residual-based goodness-of-fit tests for parametric time-series models are often complicated by parameter-estimation effects, which can alter the limiting behavior of diagnostic statistics. We propose a sample-splitting generalized spectral test (in the spirit of Escanciano(2006)) for assessing conditional mean specification in linear and nonlinear time-series models. The procedure estimates the model parameter on a fitting subsample and constructs a generalized spectral Cramer-von Mises statistic from residuals computed on a checking/testing subsample. The statistic aggregates pairwise conditional mean restrictions over all lags and is therefore bandwidth-free and free of truncation-lag selection. Under mild regularity conditions and a score-alignment condition, the residual-based process has the same limiting null distribution as the infeasible oracle process based on the true errors. Although the resulting limiting law is still non-pivotal, it can be consistently approximated by a simple multiplier bootstrap that does not require generating bootstrap time series or re-estimating parameters. Such an oracle-equivalence property is in sharp contrast to the original full-sample test, for which parameter estimation contributes an additional first-order term to the limiting process, and requires re-estimating parameters in each bootstrapped sample. We further establish consistency of the proposed test against fixed alternatives and nontrivial power against local alternatives. Extensive simulations and real data analyses show that the proposed test controls size well, has comparable power, and delivers substantial computational savings in models where repeated estimation is costly.