Realized Beta Garch: A Multivariate Garch Model With Realized Measures of Volatility

Abstract

We introduce a multivariate generalized autoregressive conditional heteroskedasticity (GARCH) model that incorporates realized measures of variances and covariances. Realized measures extract information about the current levels of volatilities and correlations from high-frequency data, which is particularly useful for modeling financial returns during periods of rapid changes in the underlying covariance structure. When applied to market returns in conjunction with returns on an individual asset, the model yields a dynamic model specification of the conditional regression coefficient that is known as the beta. We apply the model to a large set of assets and find the conditional betas to be far more variable than usually found with rolling-window regressions based exclusively on daily returns. In the empirical part of the paper, we examine the cross-sectional as well as the time variation of the conditional beta series during the financial crises.REALIZED BETA GARCH 775 we avoid such issues by incorporating realized measures and the use of measurement equations. The measurement equations tie realized measures to latent volatilities and correlations, and this leads to a useful regularization of the model. This particular structure was chosen for a number of reasons. First, the model provides a good empirical fit for the wide range of assets used in our empirical study; second, the structure of the model is amenable to a deeper analysis of secondary quantities such as betas; third, the model is simple to estimate, which is particularly important when a large set of assets are to be analyzed, as is the case in our empirical analysis.The proposed model has a hierarchical structure in which the market return and a corresponding realized measure form the core of the model. The model can be extended to an arbitrarily large set of individual returns, by adding a conditional model for an individual return and two realized measures, one being a realized measure of return volatility, and the other a realized measure of the correlation between the individual return and the market. This yields a flexible model with a dynamic covariance structure that is constantly revised by using the information contained in the realized measures.The concept of realized betas is not new. Bollerslev and Zhang (2003) carried out a large-scale estimation of the Fama-French three-factor model using high-frequency (5-minute) data on 6400 stocks over a period of 7 years. Their analysis showed that high-frequency data can improve the pricing accuracy of asset pricing models. Their approach differs from ours in important ways. For instance, they model raw realized factor loadings and use simple time series processes to forecast these. Thus there is no explicit link between realized and conditional moments of returns in their framework. Nor do they explicitly account for the measurement error (sampling error) in the realized quantities. Another related paper is Andersen et al. (2006), who study the time variation in realized variances, covariances an...

Macro-Driven VAR Forecasts: From Very High to Very Low-Frequency Data

“…Using the sequence of simulated residuals, a path for the volatility for the next H steps can be constructed by using equation (8) i+k,t ). An equivalent procedure is applied to the Realized LGARCH following insights from Hansen et al (2014) and we use the usual recursions to produce forecasts for the GARCH(1,1). In our procedure, we will use N = 1000 simulations for each horizon.…”

“…Later, Hansen et al (2012) proposed the class of Realized GARCH models that generalizes the GARCH-X by including a measurement equation for the realized measure of volatility. Hansen and Huang (2015), Hansen et al (2014), andVander Elst (2015) completed the class of Realized GARCH models with the Realized EGARCH, the Realized Beta GARCH, and the FloGARCH, respectively. Shephard and Sheppard (2010) proposed the HEAVY model that also focuses on modeling the conditional volatility of returns.…”

The Realized Hierarchical Archimedean Copula in Risk Modelling

“…The formal definition of the realized measures is given in Appendix B. Despite the constantly growing research on incorporating the realized measures into multivariate Gaussian models, discussed in Chiriac and Voev (2011) and Bauer and Vorkink (2011), and into GARCH type models, for example, Hansen et al (2014) and Bollerslev et al (2016), there is still a gap in the literature on how the parameters of non-Gaussian copula can be estimated daily based on high-frequency observations. It is important to note here that such standard copula estimation techniques as the Maximum Likelihood (ML) method or the inversion of Kendall's τ can not be directly applied to tick-by-tick observations.…”

“…Many researchers have implemented the obtained realized measures to model financial time series. Most of those studies, however, employ models where the realized correlation matrix directly characterizes the multivariate distribution, see, for example, Bauer and Vorkink (2011), Chiriac and Voev (2011), Jin and Maheu (2012), or address GARCH type models, for example, Hansen et al (2014), Bauwens et al (2012), Noureldin et al (2012), Bollerslev et al (2016). There are only a limited number of studies which discuss the implementation of high-frequency data in copula models.…”

Dynamic copula models and high frequency data

“…We build on Patton (2006a), Jondeau and Rockinger (2006) and Creal, et al (2012), who consider models of time-varying copulas where a parametric functional form is assumed, and the parameter is allowed to vary through time as a function of lagged information, similar to the ARCH model for volatility, see Engle (1982). 3 We attempt to bridge the gap between the existing timevarying copula models, which have almost exclusively used lower frequency data, and models from the volatility and correlation forecasting literature, which have successfully used high frequency data, see Shephard and Sheppard (2010), Noureldin et al (2012), Hansen et al (2011Hansen et al ( , 2013 for example. In a recent related paper, Fengler and Okhrin (2012) use a method-of-moments approach to match the covariance structure implied by a copula-based multivariate model with that estimated using high frequency data.…”