Multivariate Linear Regression

Multivariate Linear Regression Model

The multivariate linear regression model expresses a d-dimensional continuous response vector as a linear combination of predictor terms plus a vector of error terms with a multivariate normal distribution. Let yi=(yi1,,yid) denote the response vector for observation i, i = 1,...,n. In the most general case, given the d-by-K design matrix Xi and the K-by-1 vector of coefficientsβ, the multivariate linear regression model is


where the d-dimensional vector of error terms follows a multivariate normal distribution,


The model assumes independence between observations, meaning the error variance-covariance matrix for the n stacked d-dimensional response vectors is


If y denotes the nd-by-1 vector of stacked d-dimensional responses, and X denotes the nd-by-K matrix of stacked design matrices, then the distribution of the response vector is


Solving Multivariate Regression Problems

To fit multivariate linear regression models of the form


in Statistics and Machine Learning Toolbox™, use mvregress. This function fits multivariate regression models with a diagonal (heteroscedastic) or unstructured (heteroscedastic and correlated) error variance-covariance matrix, Σ, using least squares or maximum likelihood estimation.

Many variations of multivariate regression might not initially appear to be of the form supported by mvregress, such as:

  • Multivariate general linear model

  • Multivariate analysis of variance (MANOVA)

  • Longitudinal analysis

  • Panel data analysis

  • Seemingly unrelated regression (SUR)

  • Vector autoregressive (VAR) model

In many cases, you can frame these problems in the form used by mvregress (but mvregress does not support parameterized error variance-covariance matrices). For the special case of one-way MANOVA, you can alternatively use manova1. Econometrics Toolbox™ has functions for VAR estimation.

    Note:   The multivariate linear regression model is distinct from the multiple linear regression model, which models a univariate continuous response as a linear combination of exogenous terms plus an independent and identically distributed error term. To fit a multiple linear regression model, use

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