Regression with correlation structure

I have a theoretical question about regression models. Let's say I measured multiple responses from $n$ subjects and these responses are correlated with each other. For example, let's say I measured heart rate and body temperature from $n$ individuals with the following categorical factors: sex (male/female) and age (young/adult).

It's quite possible that heart rate and body temperature are correlated to some extent in any individual.

Ignoring the correlation between heart rate and body temperature I'll model each response independently as:

$y_{heart\_rate} = X\beta_{heart\_rate} + \epsilon_{heart\_rate}$

$y_{body\_temperature} = X\beta_{body\_temperature} + \epsilon_{body\_temperature}$

where $X$ is identical for the two models and is of dimensions $n \times 2$, and:

$\epsilon_{heart\_rate} \sim N(0,\sigma^2_{heart\_rate})$

$\epsilon_{body\_temperature} \sim N(0,\sigma^2_{body\_temperature})$

If I do want to take the correlation into account I would define $X^*$ as a block diagonal matrix of $X$'s. I.e., $X^* = \left( \begin{array}{cc} X & 0 \\ 0 & X \\ \end{array} \right)$

and $\epsilon^* \sim N_2(0,\left( \begin{array}{cc} \sigma^2_{heart\_rate} & \rho \\ \rho & \sigma^2_{body\_temperature} \\ \end{array} \right))$

For simplicity let's assume that $\rho$ (the correlation between heart rate and body temperature) is given.

And my model will be:

$\left( \begin{array}{c} y_{heart\_rate}\\ y_{body\_temperature}\\ \end{array} \right) = X^* \left( \begin{array}{c} \beta_{heart\_rate}\\ \beta_{body\_temperature}\\ \end{array} \right) + \epsilon^*$

My question is whether $\hat{\beta}_{heart\_rate}$ and $\hat{\beta}_{body\_temperature}$ and their standard errors will be different under the two different models.

The answer is no. This is a standard result in the theory of what economists call Seemingly Unrelated Regressions. If you have an SUR which has the exact same $X$ in each equation, then the results are identical to OLS. Same estimates, same standard errors.