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
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?