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 ≥ 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' Λ, 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 ≥ 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.

A Relationship Between the One-Way MANOVA Test Statistic and the Hotelling Lawley Trace Test Statistic

“…Note that the common covariance matrix assumption implies that each of the p treatment groups or populations has the same covariance matrix Σ i = Σ ϵ for i = 1, ..., p, an extremely strong assumption. Kakizawa (2009) and Olive et al (2015) show that similar results hold for the multivariate linear model. The common covariance matrix assumption, Cov(ϵ k ) = Σ ϵ for k = 1, ..., n, is often reasonable for the multivariate linear regression model.…”

“…are widely used Olive et al (2015). explains the large sample theory of the Wilks' Λ, Pillai's trace, and Hotelling Lawley trace test statistics and gives two theorems to show that the Hotelling Lawley test generalizes the usual partial F test for m = 1 response variable to m ≥ 1 response variables.…”

Money attitudes, financial capabilities, and impulsiveness as predictors of wealth accumulation

“…This statistic can be shown to generalise the partial F test used for ordinary least squares regression, with one dependent variable, to a test for regression with multiple dependent variables [ 46 ].…”