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Long-run covariance matrix estimation is the building block of time seriesinference problems. The corresponding difference-based estimator, which avoidsdetrending, has attracted considerable interest due to its robustness to bothsmooth and abrupt structural breaks and its competitive finite sampleperformance. However, existing methods mainly focus on estimators for theunivariate process while their direct and multivariate extensions for mostlinear models are asymptotically biased. We propose a novel difference-basedand debiased long-run covariance matrix estimator for functional linear modelswith time-varying regression coefficients, allowing time seriesnon-stationarity, long-range dependence, state-heteroscedasticity and theirmixtures. We apply the new estimator to i) the structural stability test,overcoming the notorious non-monotonic power phenomena caused by piecewisesmooth alternatives for regression coefficients, and (ii) the nonparametricresidual-based tests for long memory, improving the performance via theresidual-free formula of the proposed estimator. The effectiveness of theproposed method is justified theoretically and demonstrated by superiorperformance in simulation studies, while its usefulness is elaborated by meansof real data analysis.

Difference-based covariance matrix estimate in time series nonparametric regression with applications to specification tests