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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 denote the response vector for observation i, i = 1,...,n. In the most general case, given the d-by-K design matrix and the K-by-1 vector of coefficients, 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 denotes the nd-by-1 vector of stacked d-dimensional responses, and denotes the nd-by-K matrix of stacked design matrices, then the distribution of the response vector is
To fit multivariate linear regression models of the form
in Statistics Toolbox™, use mvregress. This function fits multivariate regression models with a diagonal (heteroscedastic) or unstructured (heteroscedastic and correlated) error variance-covariance matrix, 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)
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.