Multivariate GARCH with dynamic beta

Abstract

We investigate a solution for the problems related to the application of multivariate GARCH models to markets with a large number of stocks by restricting the form of the conditional covariance matrix and by introducing a system of recursion formals. The model is based on a decomposition of the conditional covariance matrix into two components and requires only six parameters to be estimated. The first component can be interpreted as the market factor, all remaining components are assumed to be equal. This allow the analytical calculation of the inverse covariance matrix. The factors are dynamic and therefore enable to describe dynamic beta coefficients. We compare the estimated covariances for the S&P500 market with those of other GARCH models and find that they are competitive, despite the low number of parameters. As applications we use the daily values of beta coefficients to confirm a transition of the market in 2006. Furthermore we discuss the relationship of our model with the leverage effect.

Keywords:

• Model construction and estimation < econometric modeling portfolio choice
• investment decisions < general financial markets
• asset pricing
• trading volume
• bond interest rates < general financial markets

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Note that in order to ease notation for the rest of the paper we directly use r in these recursion. Hence, we pretend that these r are identical to the increments of some vector stochastic process $\{y_t\}$. Note also that (3(3) $H_t=H_{t-1}+\sum _{k=0}^{K-1}[A_k(r_{t-1}{r}_{t-1}^{\prime }-H_{t-1})A_k+G_k(\overline{H}-H_{t-1})G_k]$(3) ) only becomes a properly defined process for the covariance matrix once we define the process for the market volatility, which indirectly imposes restrictions on A, G and $H(0)$ and leads to the recursion in (14(14) $\begin{array}{rl}{H}_t& =H_{t-1}+\sum _{k=0,1}{P}_{kt-1}[{\alpha }_k(r_{t-1}{r}_{t-1}^{\prime }-H_{t-1})+{\gamma }_k(\overline{H}-H_{t-1})]P_{kt-1}\\ & +P_{0t-1}[{\alpha }_{01}{r}_{t-1}{r}_{t-1}^{\prime }+{\gamma }_{01}\overline{H}]P_{1t-1}\\ & +P_{1t-1}[{\alpha }_{01}{r}_{t-1}{r}_{t-1}^{\prime }+{\gamma }_{01}\overline{H}]P_{0t-1}\end{array}$(14) ), see also Appendices 1 and 3.

2 Our qualitative findings hold for variations of the length of the sample size. Similar conclusion on the effect of the sample size can also be drawn from comparing the relative size of the leading eigenvalue.

3 We excluded stocks which price did not change for more than 8% of the trading days, or which were exempt from trading or for which no trading was recorded for more than 10 days in a row. We manually deleted 15 stocks which price movements at some point showed similarities to penny stocks and/or which market capitalization was very low.