Visualizing and Testing the Multivariate Linear Regression Model

Abstract

Recent results make the multivariate linear regression model much easier to use. This model has $m \geq 2$ response variables. Results by Kakizawa (2009) and Su and Cook (2012) can be used to explain the large sample theory of the least squares estimator and of the widely used Wilks' $\Lambda$, Pillai's trace, and Hotelling Lawley trace test statistics. Kakizawa (2009) shows that these statistics have the same limiting distribution. This paper reviews these results and gives two theorems to show that the Hotelling Lawley test generalizes the usual partial $F$ test for $m = 1$ response variable to $m \geq 1$ response variables. Plots for visualizing the model are also given, and can be used to check goodness and lack of fit, to check for outliers and influential cases, and to check whether the error distribution is multivariate normal or from some other elliptically contoured distribution.