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.

翻译：我们研究时变系数回归模型中长程依赖性的检验问题，其中协变量和误差项具有局部平稳性，允许存在复杂的时变动态性和异方差性。基于非参数残差，我们构建了KPSS、R/S、V/S和K/S型统计量。在零假设、局部备择假设以及固定备择假设下，我们推导了检验统计量的极限分布。由于当时变均值、方差、误差的长程方差、协变量及截距项位于特定超平面时，这四类检验统计量可能出现退化，因此我们证明在退化与非退化情形下，基于 bootstrap 的检验具有一致性。特别地，在存在协变量的情况下，基于 bootstrap 的检验的精确局部渐近功效可达到与经典严格平稳序列长记忆KPSS检验相同的阶数。渐近理论建立在针对含短记忆协变量的局部平稳长记忆过程的新型高斯逼近技术之上，该技术本身具有独立研究价值。通过大量模拟研究和真实数据分析，我们验证了所提出检验方法的有效性。