## 1. INTRODUCTION

Understanding and predicting the temporal dependence in the second-order moments of asset returns is important for many issues in financial econometrics. It is now widely accepted that financial volatilities move together over time across assets and markets. Recognizing this feature through a multivariate modelling framework leads to more relevant empirical models than working with separate univariate models. From a financial point of view, it opens the door to better decision tools in various areas, such as asset pricing, portfolio selection, option pricing, hedging and risk management. Indeed, unlike at the beginning of the 1990s, several institutions have now developed the necessary skills to use the econometric theory in a financial perspective.

Since the seminal paper of Engle (1982), traditional time series tools such as autoregressive moving average (ARMA) models (Box and Jenkins, 1970) for the mean have been extended to essentially analogous models for the variance. Autoregressive conditional heteroscedasticity (ARCH) models are now commonly used to describe and forecast changes in the volatility of financial time series. For a survey of ARCH-type models, see Bollerslev *et al.* (1992, 1994), Bera and Higgins (1993), Pagan (1996), Palm (1996) and Shephard (1996), among others.

The most obvious application of MGARCH (multivariate GARCH) models is the study of the relations between the volatilities and co-volatilities of several markets.1 Is the volatility of a market leading the volatility of other markets? Is the volatility of an asset transmitted to another asset directly (through its conditional variance) or indirectly (through its conditional covariances)? Does a shock on a market increase the volatility on another market, and by how much? Is the impact the same for negative and positive shocks of the same amplitude? A related issue is whether the correlations between asset returns change over time.2 Are they higher during periods of higher volatility (sometimes associated with financial crises)? Are they increasing in the long run, perhaps because of the globalization of financial markets? Such issues can be studied directly by using a multivariate model, and raise the question of the specification of the dynamics of covariances or correlations. In a slightly different perspective, a few papers have used MGARCH models to assess the impact of volatility in financial markets on real variables like exports and output growth rates, and the volatility of these growth rates.3

Another application of MGARCH models is the computation of time-varying hedge ratios. Traditionally, constant hedge ratios are estimated by OLS as the slope of a regression of the spot return on the futures return, because this is equivalent to estimating the ratio of the covariance between spot and futures over the variance of the futures. Since a bivariate MGARCH model for the spot and futures returns directly specifies their conditional variance–covariance matrix, the hedge ratio can be computed as a byproduct of estimation and updated by using new observations as they become available. See Lien and Tse (2002) for a survey on hedging and additional references.

Asset pricing models relate returns to ‘factors’, such as the market return in the capital asset pricing model. A specific asset excess return (in excess of the risk-free return) may be expressed as a linear function of the market return. Assuming its constancy, the slope, or β coefficient, may be estimated by OLS. Like in the hedging case, since the β is the ratio of a covariance to a variance, an MGARCH model can be used to estimate time-varying β coefficients. See Bollerslev *et al.* (1988), De Santis and Gérard (1998), Hafner and Herwartz (1998) for examples.

Given an estimated univariate GARCH model on a return series, one knows the return conditional distribution, and one can forecast the value-at-risk (VaR) of a long or short position. When considering a portfolio of assets, the portfolio return can be computed directly from the asset shares and returns. A GARCH model can be fit to the portfolio returns for given weights. If the weight vector changes, the model has to be estimated again. On the contrary, if a multivariate GARCH model is fitted, the multivariate distribution of the returns can be used directly to compute the implied distribution of any portfolio. There is no need to re-estimate the model for different weight vectors. In the present state of the art, it is probably simpler to use the univariate framework if there are many assets, but we conjecture that using a multivariate specification may become a feasible alternative. Whether the univariate ‘repeated’ approach is more adequate than the multivariate one is an open question. The multivariate approach is illustrated by Giot and Laurent (2003) using a trivariate example with a time-varying correlation model.

MGARCH models were initially developed in the late 1980s and the first half of the 1990s, and after a period of tranquillity in the second half of the 1990s, this area seems to be experiencing again a quick expansion phase. MGARCH models are partly covered in Franses and van Dijk (2000), Gourieroux (1997) and most of the surveys on ARCH models cited above, but none of them presents, as this one, a comprehensive and up-to-date survey of the field, including the most recent findings.

The paper is organized in the following way. In Section 2, we review existing MGARCH specifications. Section 3 is devoted to estimation problems and Section 4 to diagnostic tests. Finally, we offer our conclusions and ideas for further developments in Section 5.