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In this paper we introduce and develop two different novel multivariate regression models with Power Exponential (PE) random errors for the ?rst time. Our ?rst model assumes that the observations are independent and the second model assumes that the observations are dependent. These two models coincide
only when the shape parameter of the multivariate Power exponential (MPE) distribution is equal to one which corresponds to the multivariate normal distribution. We develop method of moments (MOM) and the maximum likelihood (ML) methods to estimate the model parameters. The model selection criteria such
as AIC and ICOMP(IFIM) for both models are derived. Two simulation examples and a real example on a benchmark data set are given to show the applications of these two models in subset selection of the best predictors. A genetic algorithm (GA) approach is used to obtain the estimates of the model parameters and to carry out the subset selection of the best predictors under these two different model types.