Multivariate DCC-GARCH Model: -With Various Error Distributions

Abstract

In this thesis we have studied the DCC-GARCH model with Gaussian, Student's $t$ and skew Student's t-distributed errors. For a basic understanding of the GARCH model, the univariate GARCH and multivariate GARCH models in general were discussed before the DCC-GARCH model was considered. The Maximum likelihood method is used to estimate the parameters. The estimation of the correctly specified likelihood is difficult, and hence the DCC-model was designed to allow for two stage estimation. Usually Gaussian distributed errors are assumed in the first stage independent of the choice of the error distribution in the correctly specified likelihood. In the second stage, the parameters $a$ and $b$ of the dynamic correlation matrix, and the parameters of the error distribution, are estimated using the correctly specified likelihood. After the parameters of the DCC-model have been estimated, the forecast of the conditional covariance matrix is obtained by forecasting the conditional variances and the conditional correlation matrix separately. The forecasts of the conditional variances is done by assuming Gaussian distributed errors. The forecast of the conditional correlation matrix can not be directly calculated. In this thesis, two different methods of approximating this matrix have been discussed. An important issue is how to evaluate goodness of fit for the DCC-GARCH model. This might be done by checking both the marginal and multivariate goodness of fit. One specific approach considered is the backtesting of Value-at-Risk, this is used to measure risk of loss of a portfolio of financial asset series. After precenting the theory, DCC-GARCH models were fit to a portfolio consisting of European, American and Japanese stocks assuming three different error distributions; Gaussian, Student's t and skew Student's t. The European, American and Japanese series seemed to have a bit different marginal distributions. The DCC-GARCH model with skew Student's t-distributed errors performed best. But even the DCC-GARCH with skew Student's t-distributed errors did explain all of the asymmetry in the asset series. Hence even better models may be considered. Comparing the DCC-GARCH model with the CCC-GARCH model using the Kupiec test showed that the first model gave a better fit to the data.There are several possible directions for future work. It might be better to use other marginal models such as the EGARCH, QGARCH and GJR GARCH, that capture the asymmetry in the conditional variances. If the univariate GARCH models are more correct, the DCC-GARCH model might yield better results. Other error distributions, such as a Normal Inverse Gaussian (NIG) might also give a better fit. When we fitted the Gaussian, Student's t- and skew Student's t-distibutions to the data, we assumed all the distributions to be the same for the three series. This might be a too restrictive criteria. A model where the marginal distributions is allowed to be different for each of the asset series might give a better fit. One then might use a Copula to link the marginals together