  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. Under the null hypothesis, the local alternatives as well as the
fixed alternatives, we derive the limiting distributions of the test
statistics. As the four types of test statistics could degenerate when the
time-varying mean, variance, long-run variance of errors, covariates, and the
intercept lie in certain hyperplanes, we show the bootstrap-assisted tests are
consistent 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.
