Abstract
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.
Highlights
Volatility in financial asset returns is one of the most important components in the financial market for optimization decisions such as portfolio selection, risk management and asset pricing.Over recent decades, extensive research works by statisticians and econometricians have analyzed volatility alongside time series modelings
The following theorem states the asymptotic normality of the ordinary least squares estimator (OLSE)
We list some critera of MSE, R2, AIC, BIC in Table 4 for the OLSEs of the bivariate HAR model of order p = 3, 4 by examining the volatility of Gold and
Summary
Volatility in financial asset returns is one of the most important components in the financial market for optimization decisions such as portfolio selection, risk management and asset pricing. Because multiple assets are correlated with each other in the financial markets, the cross-correlation of two or more asset returns and the spillover effect of the volatility have been represented in multivariate time series models rather than the univariate models. Various adaptive versions of the HAR model are used to analyze the volatility along with empirical data analysis. We consider an exponentially weighted multivariate HAR model and discuss the estimation of a common decay rate on coefficients in the exponentially weighted. HAR model and estimate the coefficient parameters. The remainder of the paper is organized as follows: In Section 2, the multivariate HAR model is presented with discussion on strictly stationary solution and in Section 3 the LSEs and their multivariate normalities are derived.
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