Multivariate volatility models

Correlations between asset returns are important in many financial applications. In recent years, multivariate volatility models have been used to describe the time-varying feature of the correlations. However, the curse of dimensionality quickly becomes an issue as the number of correlations is $k(k-1)/2$ for $k$ assets. In this paper, we review some of the commonly used models for multivariate volatility and propose a simple approach that is parsimonious and satisfies the positive definite constraints of the time-varying correlation matrix. Real examples are used to demonstrate the proposed model.Comment: Published at http://dx.doi.org/10.1214/074921706000001058 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Semiparametric multivariate volatility models

Estimation of multivariate volatility models is usually carried out by quasi maximum likelihood (QMLE), for which consistency and asymptotic normality have been proven under quite general conditions. However, there may be a substantial efficiency loss of QMLE if the true innovation distribution is not multinormal. We suggest a nonparametric estimation of the multivariate innovation distribution, based on consistent parameter estimates obtained by QMLE. We show that under standard regularity conditions the semiparametric efficiency bound can be attained. Without reparametrizing the conditional covariance matrix (which depends on the particular model used), adaptive estimation is not possible. However, in some cases the e?ciency loss of semiparametric estimation with respect to full information maximum likelihood decreases as the dimension increases. In practice, one would like to restrict the class of possible density functions to avoid the curse of dimensionality. One way of doing so is to impose the constraint that the density belongs to the class of spherical distributions, for which we also derive the semiparametric efficiency bound and an estimator that attains this bound. A simulation experiment demonstrates the e?ciency gain of the proposed estimator compared with QMLE. --Multivariate volatility,GARCH,semiparametric efficiency,adaptivity

Temporal aggregation of multivariate GARCH processes

This paper derives results for the temporal aggregation of multivariate GARCH processes in the general vector specification. It is shown that the class of weak multivariate GARCH processes is closed under temporal aggregation. Fourth moment characteristics turn out to be crucial for the low frequency dynamics for both stock and flow variables. The framework used in this paper can easily be extended to investigate joint temporal and contemporaneous aggregation. Discussing causality in volatility, I find that there is not much room for spurious instantaneous causality in multivariate GARCH processes, but that spurious Granger causality will be more common however numerically insignificant. Forecasting volatility, it is generally advisable to aggregate forecasts of the disaggregate series rather than forecasting the aggregated series directly, and unlike for VARMA processes the advantage does not diminish for large forecast horizons. Finally, results are derived for the distribution of multivariate realized volatility if the high frequency process follows multivariate GARCH. A numerical example illustrates some of the resultsmultivariate GARCH, temporal aggregation, causality in volatility, forecasting volatility, realized volatility

Semiparametric multivariate volatility models

Estimation of multivariate volatility models is usually carried out by quasi maximum likelihood (QMLE), for which consistency and asymptotic normality have been proven under quite general conditions. However, there may be a substantial efficiency loss of QMLE if the true innovation distribution is not multinormal. We suggest a nonparametric estimation of the multivariate innovation distribution, based on consistent parameter estimates obtained by QMLE. We show that under standard regularity conditions the semiparametric efficiency bound can be attained. Without reparametrizing the conditional covariance matrix (which depends on the particular model used), adaptive estimation is not possible. However, in some cases the efficiency loss of semiparametric estimationwith respect to full information maximum likelihood decreases as the dimension increases.In practice, one would like to restrict the class of possible density functions to avoid the curse of dimensionality. One way of doing so is to impose the constraint that the density belongs to the class of spherical distributions, for which we also derive the semiparametric efficiency bound and an estimator that attains this bound. A simulation experiment demonstrates the efficiency gain of the proposed estimator compared with QMLE.

Do Jumps Matter? Forecasting Multivariate Realized Volatility allowing for Common Jumps

Realized volatility of stock returns is often decomposed into two distinct components that are attributed to continuous price variation and jumps. This paper proposes a tobit multivariate factor model for the jumps coupled with a standard multivariate factor model for the continuous sample path to jointly forecast volatility in three Chinese Mainland stocks. Out of sample forecast analysis shows that separate multivariate factor models for the two volatility processes outperform a single multivariate factor model of realized volatility, and that a single multivariate factor model of realized volatility outperforms univariate models.Realized Volatility, Bipower Variation, Jumps, Common Factors, Forecasting

Multivariate Realized Stock Market Volatility

We present a new matrix-logarithm model of the realized covariance matrix of stock returns. The model uses latent factors which are functions of both lagged volatility and returns. The model has several advantages: it is parsimonious; it does not require imposing parameter restrictions; and, it results in a positive-definite covariance matrix. We apply the model to the covariance matrix of size-sorted stock returns and find that two factors are sufficient to capture most of the dynamics. We also introduce a new method to track an index using our model of the realized volatility covariance matrix.Econometric and statistical methods; Financial markets