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.
Bibliographical noteFunding Information:
We are very grateful for the valuable comments of Professor Wayne A. Fuller and two anonymous referees that improved the paper considerably. We thank Ms Soyoung Park for providing data analysis. This work was supported by the National Research Foundation of Korea ( NRF-2012-2046157 ) through the National Research Foundation of Korea (NRF) funded by the Ministry of Education Science and Technology.
- Asymptotic property
- HAR-RV model
- Least squares estimator
- Prediction mean-squared error
- Realized volatility