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

Question

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?

Answer 1

I've read the related link noted above, particularly the comment regarding the UCLA example. I wanted to highlight a section as an answer in case someone else was struggling with the differentiation.

Quoted from the UCLA webpage: "You could analyze these data using separate OLS regression analyses for each outcome variable. The individual coefficients, as well as their standard errors will be the same as those produced by the multivariate regression. However, the OLS regressions will not produce multivariate results, nor will they allow for testing of coefficients across equations."

Therefore, it seems from an estimation perspective that the results are exactly the same. The difference appears only on whether you want to test the coefficients across the equations, against a hypothesis, simultaneously.