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We introduce a new equivariant estimation method of the parameters of the multivariate regression model with q responses and p regressors. The estimate matrix is derived from the lift rank covariance matrix (LRCM) where the lift rank vectors are based on the Oja criterion function. The k = p + q variate ranks and k + 1 variate lift ranks are constructed using hyperplanes (or fits) going through k observations. The new LRCM regression estimate and the least squares (LS) estimate are shown to be weighted sums of the elemental estimates based on these hyperplanes. The LRCM regression estimate is equivariant and convergent, has a limiting multinormal distribution, and is highly efficient in the multivariate normal case. For heavy-tailed distributions, it performs better than the standard LS estimate. Estimation of the variance-covariance matrix of the LRCM estimate is briefly discussed. The theory is illustrated by simulations and a real data example. [ABSTRACT FROM AUTHOR]