Does multivariate regression produce the same results as multiple, single regressions? [duplicate]

I've read a variety of answers on this topic and as far as I can see, the consensus is that multivariate regression is different from multiple, individual linear regressions. I also understand that in multivariate regression, the results are estimated simultaneously, which in principle could result in different outcomes.

However, I believe I am missing something in my understanding because I can't reproduce an example where they produce different results.

For example, using R syntax, 3 regressions with correlated errors

error <- rnorm(100,0,1)

x <- rnorm(100,0,1)

y <- rnorm(100,0,1)

z1 <- 1*x + 2*y + error

z2 <- 1.5*x + 0.5*y + error

z3 <- 0.8*x + 1*y + error

Result:

lm(cbind(z1, z2, z3) ~ x + y)

gives the exact same results as:

lm(z1 ~ x + y) lm(z2 ~ x + y) lm(z3 ~ x + y)

The results here are the same for both methods and I cant think of an example where the results wouldn't be the same.

Simply, could you give an example where multivariate regression produces different results to multiple, individual regressions?

• Feb 27 '20 at 14:03