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We introduce a robust method for multivariate regression based on robust estimation of the joint location and scatter matrix of the explanatory and response variables. As a robust estimator of location and scatter, we use the minimum covariance determinant (MCD) estimator of Rousseeuw. Based on simulations, we investigate the finite-sample performance and robustness of the estimator. To increase the efficiency, we propose a reweighted estimator selected from several possible reweighting schemes. The resulting multivariate regression does not need much computation time and is applied to real datasets. We show that the multivariate regression estimator has the appropriate equivariance properties, has a bounded influence function, and inherits the breakdown value of the MCD estimator. These theoretical robustness properties confirm the good finite-sample results obtained from the simulations.