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.

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