In this work, multivariate heterogeneous autoregressive-realized volatility (HAR-RV) models are discussed with their least squares estimations. We consider multivariate HAR models of order p with q multiple assets to explore the relationships between two or more assets’ volatility. The strictly stationary solution of the HAR(p,q) model is investigated as well as the asymptotic normality theories of the least squares estimates are established in the cases of i.i.d. and correlated errors. In addition, an exponentially weighted multivariate HAR model with a common decay rate on the coefficients is discussed together with the common rate estimation. A Monte Carlo simulation is conducted to validate the estimations: sample mean and standard error of the estimates as well as empirical coverage and average length of confidence intervals are calculated. Lastly, real data of volatility of Gold spot price and S&P index are applied to the model and it is shown that the bivariate HAR model fitted by selected optimal lags and estimated coefficients is well matched with the volatility of the financial data.
This paper is devoted to modelling and predicting COVID-19 confirmed cases through a multiple linear regression. Especially, prediction intervals of the COVID-19 cases are extensively studied. Due to long-memory feature of the COVID-19 data, a heterogeneous autoregression (HAR) is adopted with Growth rates and Vaccination rates; it is called HAR-G-V model. Top eight affected countries are taken with their daily confirmed cases and vaccination rates. Model criteria results such as root mean square error (RMSE), mean absolute error (MAE),
, AIC and BIC are reported in the HAR models with/without the two rates. The HAR-G-V model performs better than other HAR models. Out-of-sample forecasting by the HAR-G-V model is conducted. Forecast accuracy measures such as RMSE, MAE, mean absolute percentage error and root relative square error are computed. Furthermore, three types of prediction intervals are constructed by approximating residuals to normal and Laplace distributions, as well as by employing bootstrap procedure. Empirical coverage probability, average length and mean interval score are evaluated for the three prediction intervals. This work contributes three folds: a novel trial to combine both growth rates and vaccination rates in modeling COVID-19; construction and comparison of three types of prediction intervals; and an attempt to improve coverage probability and mean interval score of prediction intervals via bootstrap technique.
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