Time-varying parameter regressions with stationary persistent data

We consider local level and local linear estimators for estimation and inference in time-varying parameter (TVP) regressions with general stationary covariates. The latter estimator also yields estimates for parameter derivatives that are utilized for the development of time invariance tests for the regression coefficients. Our theoretical framework is general enough to allow for a wide range of stationary regressors, including stationary long memory. We demonstrate that neglecting time variation in the regression parameters has a range of adverse effects in inference, in particular, when regressors exhibit long-range dependence. For instance, parametric tests diverge under the null hypothesis when the memory order is strictly positive. The finite sample performance of the methods developed is investigated with the aid of a simulation experiment. The proposed methods are employed for exploring the predictability of SP500 returns by realized variance. We find evidence of time variability in the intercept as well as episodic predictability when realized variance is utilized as a predictor in TVP specifications.