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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 $${y}_{i}={\left({y}_{i1},\dots ,{y}_{id}\right)}^{\prime}$$ 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

$${y}_{i}={X}_{i}\beta +{\epsilon}_{i},$$

$${\epsilon}_{i}\sim MV{N}_{d}\left(0,\Sigma \right).$$

$${I}_{n}\otimes \Sigma =\left(\begin{array}{ccc}\Sigma & & 0\\ & \ddots & \\ 0& & \Sigma \end{array}\right).$$

$$y\sim MV{N}_{nd}(X\beta ,{I}_{n}\otimes \Sigma ).$$

To fit multivariate linear regression models of the form

$${y}_{i}={X}_{i}\beta +{\epsilon}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\epsilon}_{i}\sim MV{N}_{d}(0,\Sigma )$$

`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.

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`

.

`LinearModel.fit`

| `manova1`

| `mvregress`

| `mvregresslike`

- Set Up Multivariate Regression Problems
- Multivariate General Linear Model
- Fixed Effects Panel Model with Concurrent Correlation
- Longitudinal Analysis

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