Infinite-order, long-memory heterogeneous autoregressive models

We develop an infinite-order extension of the HAR-RV model, denoted by HAR(∞). We show that the autocorrelation function of the model is algebraically decreasing and thus the model is a long-memory model if and only if the HAR coefficients decrease exponentially. For a finite sample, a prediction is made using coefficients estimated by ordinary least squares (OLS) fitting for a finite-order model, HAR(p), say. We show that the OLS estimator (OLSE) is consistent and asymptotically normal. The approximate one-step-ahead prediction mean-square error is derived. Analysis shows that the prediction error is mainly due to estimation of the HAR(p) coefficients rather than to errors made in approximating HAR(∞) by HAR(p). This result provides a theoretical justification for wide use of the HAR(3) model in predicting long-memory realized volatility. The theoretical result is confirmed by a finite-sample Monte Carlo experiment for a real data set.