Infinite-order Long Memory Heterogeneous Autoregressive Models

Abstract

An infinite order extension of the HAR-RV model, denoted by HAR(∞), is developed. The autocorrelation function of the model is shown to be algebraically decreasing and thus the model is a long-memory model if and only if the HAR coefficients decrease exponentially. For a finite sample, prediction is made using coefficients estimated by ordinary least squares (OLS) fitting for a finite order model, HAR(p), say. The ordinary least squares estimator (OLSE) is shown to be consistent and asymptotically normal. The approximated one-step ahead prediction mean square error is derived. The analysis shows that prediction error is mainly due to estimating the HAR(p) coefficients rather than to the errors made in approximating HAR(∞) by HAR(p). This result provides a theoretical justification for the wide use of the HAR(3) model for predicting long-memoriedy realized volatility. The theoretical result is also confirmed by a finite sample Monte-Carlo experiment. A real data set is analyzed using the developed theory. Keywords: HAR-RV model; least squares estimator; asymptotic property; prediction mean squared error; realized volatility.