Can I use a z-test on heteroscedastic data?

Question

Suppose I have two methods that I test over time on the same data. I suspect that the methods degrade over time, and that this degradation is more or less linear. I want to know if one method degrades faster than the other.

So, what I do is, I fit a line through both sets of data points with ordinary least squares (OLS) and I can do a z-test on the two slope coefficients β₁ and β₂ (see e.g. this post):

$$ z = \frac{\beta_1 - \beta_2}{\sqrt{SE(\beta_1)^2 + SE(\beta_2)^2}} $$

Now suppose I have heteroscedastic data, e.g. because I have far more data points in some time periods than in others. I shouldn't use OLS in that case, but weighted least squares (WLS).

Now my question is: can I still use the same z-test? So using on the coefficients and the SE's I get from the WLS fit?

Answer 1

Note that the $\beta$'s in your $z$ formula should be $\hat \beta$'s (both in the numerator and denominator).

The short answer is 'yes'; as long as (a) the sample sizes are sufficiently large that (i) the $\hat \beta$ terms are close to normal (i.e. the CLT 'kicks in'), and (ii) the two $\hat\sigma$ terms (on which the $SE$ terms are based) are very accurately estimated (i.e. Slutsky's theorem 'kicks in'); and (b) the parameter estimates are independent (i.e. the above formula for the denominator is correct).

With all the appropriate caveats in place for it to work (even unweighted), it doesn't matter that it's weighted.