In this article we propose a new multivariate generalized autoregressive conditional heteroscedasticity (MGARCH) model with time-varying correlations. We adopt the vech representation based on the conditional variances and the conditional correlations. Whereas each conditional-variance term is assumed to follow a univariate GARCH formulation, the conditional-correlation matrix is postulated to follow an autoregressive moving average type of analog. Our new model retains the intuition and interpretation of the univariate GARCH model and yet satis es the positive-de nite condition as found in the constant-correlation and Baba-Engle-Kraft-Kroner models. We report some Monte Carlo results on the nite-sample distributions of the maximum likelihood estimate of the varying-correlation MGARCH model. The new model is applied to some real data sets.

In this paper we propose a new multivariate GARCH model with timevarying correlations. We adopt the vech representation based on the conditional variances and the conditional correlations. While each conditional-variance term is assumed to follow a univariate GARCH formulation, the conditional-correlation matrix is postulated to follow an autoregressive moving average type of analogue. By imposing some suitable restrictions on the conditional-correlation-matrix equation, we manage to construct a MGARCH model in which the conditional-correlation matrix is guaranteed to be positive de¯nite during the optimisation. Thus, our new model retains the intuition and interpretation of the univariate GARCH model and yet satis¯es the positive-de¯nite condition as found in the constant-correlation and BEKK models. We report some Monte Carlo results on the¯nite-sample distributions of the QMLE of the varying-correlation MGARCH model. The new model is applied to some real data sets. It is found that extending the constant-correlation model to allow for time-varying correlations provides some interesting time histories that are not available in a constant-correlation model.

This paper compares the performances of the hedge ratios estimated from the OLS (ordinary least squares) method and the constant-correlation VGARCH (vector generalized autoregressive conditional heteroscedasticity) model. These methods are evaluated based on the out-of-sample optimal hedge ratio forecasts. A systematic comparison is provided by examining ten spot and futures markets covering currency futures, commodity futures and stock index futures. Using a recently proposed test (Tse, 2000) for the constant-correlation assumption, it is found that the assumption cannot be rejected for eight of the ten series. To gain the maximum benefit of a time-varying hedging strategy the estimation data is kept up-to-date for the re-estimation of the hedge ratios. Both the constant hedge ratio (using OLS) and the timevarying hedge ratio (using constant-correlation VGARCH) are re-estimated on a day-by-day rollover, and the post-sample variances of the hedged portfolios are examined. It is found that the OLS hedge ratio performs better than the VGARCH hedge ratio. This result may be another indication that the forecasts generated by the VGARCH models are too variable.

Most studies of exchange rate exposure of stock returns do not address three relevant aspects simultaneously. They are, namely: sensitivity of stock returns to exchange rate changes; sensitivity of volatility of stock returns to volatility of changes in foreign exchange market; and the correlation between volatilities of stock returns and exchange rate changes. In this paper, we employ a bivariate GJR-GARCH model to examine all such aspects of exchange rate exposure of sectoral indexes in Japanese industries. Based on a sample data of fourteen sectors, we find significant evidence of exposed returns and its asymmetric conditional volatility of exchange rate exposure. In addition, returns in many sectors are correlated with those of exchange rate changes. We also find support for the "averaged-out exposure and asymmetries" argument. Our findings have direct implications for practitioners in formulating investment decisions and currency hedging strategies.

In this paper we consider several tests for model misspeci®cation after a multivariate conditional heteroscedasticity model has been ®tted. We examine the performance of the recent test due to Ling and Li (J. Time Ser. Anal. 18 (1997), 447±64), the Box±Pierce test and the residual-based F test using Monte Carlo methods. We ®nd that there are situations in which the Ling±Li test has very weak power. The residual-based diagnostics demonstrate signi®cant under-rejection under the null. In contrast, the Box±Pierce test based on the cross-products of the standardized residuals often provides a useful diagnostic that has reliable empirical size as well as good power against the alternatives considered.