  This work is concerned with the estimation of multidimensional regression and
the asymptotic behaviour of the test involved in selecting models. The main
problem with such models is that we need to know the covariance matrix of the
noise to get an optimal estimator. We show in this paper that if we choose to
minimise the logarithm of the determinant of the empirical error covariance
matrix, then we get an asymptotically optimal estimator. Moreover, under
suitable assumptions, we show that this cost function leads to a very simple
asymptotic law for testing the number of parameters of an identifiable and
regular regression model. Numerical experiments confirm the theoretical
results.
