We consider the problem of testing for long-range dependence in time-varying coefficient regression models, where the covariates and errors are locally stationary, allowing complex temporal dynamics and heteroscedasticity. We develop KPSS, R/S, V/S, and K/S-type statistics based on the nonparametric residuals, and propose bootstrap approaches equipped with a bias-corrected difference-based long-run covariance matrix estimator for practical implementation. Under the null hypothesis, the local alternatives as well as the fixed alternatives, we derive the limiting distributions of the test statistics and establish the uniform consistency of the difference-based long-run covariance estimator. As the four types of test statistics could degenerate when the time-varying mean, variance, long-run variance of errors, covariates, and intercept lie in certain hyperplanes, we propose bootstrap-assisted tests and establish their consistency under both degenerate and non-degenerate scenarios. In particular, in the presence of covariates the exact local asymptotic power of the bootstrap-assisted tests can enjoy the same order as that of the classical KPSS test of long memory for strictly stationary series. The asymptotic theory is built on a new Gaussian approximation technique for locally stationary long-memory processes with short-memory covariates, which is of independent interest. The effectiveness of our tests is demonstrated by extensive simulation studies and real data analysis.
翻译:我们考虑在时间变化系数回归模型中测试长期依赖性的问题,在这种模型中,共差和差错是当地固定的,允许复杂的时间动态和异变性。我们根据非参数剩余量开发了KPSS、R/S、V/S和K/S型统计。我们根据非参数剩余量开发了KPSS、R/S、V/S和K/S型统计,并提议了配有基于偏差的差异长期共差矩阵估测器的靴套式方法,以实际实施。在无效假设、当地替代物以及固定的替代物中,我们得出测试统计数据的有限分布,并确立基于差异的长期变异性长期变异性估算器的统一一致性。由于四种类型的测试统计数据在时间变化平均值、差异、长期误差、变异性以及拦截等某些超常平中可能会退化,我们提议了靴套带辅助测试,并在退化和非衰减的假设情景下确立其一致性。特别是,在存在变异的情况下,我们测算短期恒定测测测测测测测测测测测测测测测测测测测测测测测测测测测测测的短期测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测测