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 $${y}_{i}={\left({y}_{i1},\dots ,{y}_{id}\right)}^{\prime}$$ denote the response vector for observation i, i = 1,...,n. In the most general case, given the d-by-K design matrix $${X}_{i}$$ and the K-by-1 vector of coefficients$$\beta $$, the multivariate linear regression model is
$${y}_{i}={X}_{i}\beta +{\epsilon}_{i},$$
where the d-dimensional vector of error terms follows a multivariate normal distribution,
$${\epsilon}_{i}\sim MV{N}_{d}\left(0,\Sigma \right).$$
The model assumes independence between observations, meaning the error variance-covariance matrix for the n stacked d-dimensional response vectors is
$${I}_{n}\otimes \Sigma =\left(\begin{array}{ccc}\Sigma & & 0\\ & \ddots & \\ 0& & \Sigma \end{array}\right).$$
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
$$y\sim MV{N}_{nd}(X\beta ,{I}_{n}\otimes \Sigma ).$$
To fit multivariate linear regression models of the form
$${y}_{i}={X}_{i}\beta +{\epsilon}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\epsilon}_{i}\sim MV{N}_{d}(0,\Sigma )$$
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, $$\Sigma ,$$ 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
| manova1
| mvregress
| mvregresslike