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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

yi=Xiβ+εi,

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

εiMVNd(0,Σ).

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

InΣ=(Σ00Σ).

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

yMVNnd(Xβ,InΣ).

Solving Multivariate Regression Problems

To fit multivariate linear regression models of the form

yi=Xiβ+εi,εiMVNd(0,Σ)

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 LinearModel.fit.

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