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I have several regression models as below:

$Y_1 = \beta_{11} X_{11} + \beta_{12} X_{12} + \epsilon_1$

$Y_2 = \beta_{21} X_{21} + \beta_{22} X_{22} + \epsilon_2$

$Y_3 = \beta_{31} X_{31} + \beta_{32} X_{32} + \epsilon_3$

$Y_4 = \beta_{41} X_{41} + \beta_{42} X_{42} + \epsilon_4$

$Y_i = \beta_{i1} X_{i1} + \beta_{i2} X_{i2} + \epsilon_i$

When I look at the correlation matrix of $Y$ I noticed that there are strong positive and negative correlations among $Y$ dependent variables.

My question is how do I use this in the regression? I want the predicted variables to retain the same correlation structure?

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    $\begingroup$ Read about seemingly unrelated regression (different from what the current answer suggests). $\endgroup$ – Richard Hardy Dec 18 '16 at 6:30
  • $\begingroup$ @RichardHardy That is exactly what I was looking for. $\endgroup$ – pavybez Apr 4 '17 at 1:12
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Because you can run correlation among the $y$-variables, it indicates that the $y$-variables share the same records. In this case, run multivariate linear regression in which all the $y$-variables are regressed simultaneously on the $x$-variables considered. MVREG will account for between y-variable correlation during the run.

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  • $\begingroup$ Thanks. Will it work even if some Y variables are not correlated? $\endgroup$ – pavybez Dec 18 '16 at 1:28
  • $\begingroup$ Yes. Recall, "orthogonal regression" was introduced for regressing a single $y$ on uncorrelated x-variables that were orthogonalized via PCA. So you could probably replace the $y$-variables with orthogonalized $y$ from the $p$ principal component score vectors which represent $Y_1, Y_2, \ldots, Y_p$. $\endgroup$ – wrktsj Dec 18 '16 at 1:37
  • $\begingroup$ What you suggest will account for correlation of residuals, not of $y$s themselves. $\endgroup$ – Richard Hardy Mar 20 '17 at 13:53

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