  Long-run covariance matrix estimation is the building block of time series
inference. The corresponding difference-based estimator, which avoids
detrending, has attracted considerable interest due to its robustness to both
smooth and abrupt structural breaks and its competitive finite sample
performance. However, existing methods mainly focus on estimators for the
univariate process while their direct and multivariate extensions for most
linear models are asymptotically biased. We propose a novel difference-based
and debiased long-run covariance matrix estimator for functional linear models
with time-varying regression coefficients, allowing time series
non-stationarity, long-range dependence, state-heteroscedasticity and their
mixtures. We apply the new estimator to (i) the structural stability test,
overcoming the notorious non-monotonic power phenomena caused by piecewise
smooth alternatives for regression coefficients, and (ii) the nonparametric
residual-based tests for long memory, improving the performance via the
residual-free formula of the proposed estimator. The effectiveness of the
proposed method is justified theoretically and demonstrated by superior
performance in simulation studies, while its usefulness is elaborated via real
data analysis. Our method is implemented in the R package mlrv.
