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Are there any difference in beta coefficients when doing several multiple regression analysis as compared to doing multivariate regression?

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The short answer is "no"

x = matrix(rnorm(200),100)
y = matrix(rnorm(200),100)

bhat = solve(t(x)%*%x)%*%t(x)%*%y
bhat1 = solve(t(x)%*%x)%*%t(x)%*%y[,1]
bhat2 = solve(t(x)%*%x)%*%t(x)%*%y[,2]

all.equal(cbind(bhat1,bhat2),bhat)
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    $\begingroup$ I'd be interested in an explanation that uses some narrative in addition to the code. $\endgroup$ – rolando2 Jan 23 '15 at 0:43

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