# Documentation

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

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

### Solving Multivariate Regression Problems

To fit multivariate linear regression models of the form

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

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`.