In this dissertation we introduce two variable selection procedures for multivariate responses. Our procedures are based on sufficient dimension reduction concepts and are model-free. In the first procedure we consider the dual marginal coordinate hypotheses, where the role of the predictor and the response is not important. Motivated by canonical correlation analysis (CCA), we propose a CCA-based test for the dual marginal coordinate hypotheses, and devise a joint backward selection algorithm for dual model-free variable selection. The second procedure is based on ordinary least squares (OLS). We derive and study the asymptotic properties of the OLS-based test under the normality assumption of the predictors as well as an asymmetry assumption. When these assumptions are violated, the asymptotic test with elliptical trimming and clustering is still valid with desirable numerical performances. A backward selection algorithm for the predictor is also provided for the OLS-based test. The performances of the proposed tests and the variable selection procedures are evaluated through synthetic examples and a real data analysis. / Statistics