Detecting long-range dependence for linear models with time-varying coefficients and a difference-based covariance estimator

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.