In the empirical analysis of financial time series, multivariate GARCH models have been used in various forms. In most cases it is not well understood how the use of a restricted model has to be paid with loss of valuable information. We investigate the structural implications of two alternative models for the response of the conditional (co-)variances to independent shocks. The impulse response analysis, adopted to volatility models, appears to be a convenient methodology to obtain information on the interaction of financial series. We define volatility impulse response functions and provide an empirical analysis for a bivariate exchange rate series. For the analyzed series, the impulse response function of the correlation reveals strong discrepancies between the estimated diagonal and BEKK models. This indicates that the diagonality restriction may hide important structural properties of the series.