# 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.

## 1 Answer

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.

The only real advantage to running SUR in a case like this is if you are interested in testing hypotheses or making confidence intervals for coefficients in both equations at the same time. For example, suppose you want to test the null hypothesis that sex affects neither heart rate nor body temperature. In order to test this null hypothesis, you are going to need, among other things, an estimate of the covariance between the OLS estimators of these two coefficients. An SUR model will give you that.

See Greene, Econometric Analysis (any edition), and look for Seemingly Unrelated Regressions in the table of contents.