Estimating the Integrated Parameter of the Locally Parametric Model in High-Frequency Data

In this paper, we give a general time-varying parameter model, where the multidimensional parameter follows a continuous local martingale. As such, we call it the locally parametric model (LPM). The quantity of interest is defined as the integrated value over time of the parameter process $\Theta := T^{-1} \int_0^T \theta_t^* dt$. We provide a local parametric estimator (LPE) of $\Theta$ based on the original (non time-varying) parametric model estimator and conditions under which we can show the central limit theorem. As an example of how to apply the limit theory provided in this paper, we build a time-varying friction parameter extension of the (semiparametric) model with uncertainty zones (Robert and Rosenbaum (2012)) and we show that we can verify the conditions for the estimation of integrated volatility. Moreover, practical applications in time series, such as the optimal block length and local bias-correction, are discussed and numerical simulations are carried on the local MLE of a time-varying parameter MA(1) model to illustrate them.