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I have 2 response variables (Y1, Y2) and some independent variables. I need to predict both Y1 and Y2 using the same set of predictors.In other words I need to fit the model:
$Y=XB+E$
where:
$Y$ has $n$ rows and $q$ columns,
$X$ has $n$ rows and $p$ columns,
$B$ has $p$ rows and $q$ columns,
$E$ has $n$ rows and $q$ columns.
In this case, $q=2$.

One way to do this in R is by calling the lm() function:

m1=lm(cbind(Y1,Y2)~x1+x2+x3)

This is equivalent to fitting two univariate multiple regression models, one for Y1, and the other for Y2:

m1.1=lm(Y1~x1+x2+x3)
m1.2=lm(Y2~x1+x2+x3)

How can I fit a multivariate multiple linear regression model for both Y1 and Y2 which permits correlation between Y1 and Y2?
Thank you in advance.

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In this type of problem you have two covariance matrices involved, one that is $n \times n$ over the rows of $Y$, and another that is $q \times q$ over the columns of $Y$. If the $q \times q$ covariance matrix is the identity matrix, then you're right, your solution for $B$ is equivalent to running separate regressions.

Check out R's systemfit package.

Also consider using a SUR (Seemingly Unrelated Regressions) model, which would allow you to use different X matrices for each column of Y. Here is a good resource for using systemfit to do a SUR model.

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