Influence of HAC estimates to p-value of t-test

Question

I have a linear regression model and because of heteroskedasticity or autocorrrelation I use HAC (Newey-West) estimates. This influences also p-values of significance t-tests of estimated coefficients as they are then different than the normal p-values would be. My question is: should this influence also the p-value of intercept or should this p-value remain the same?

Answer 1

Keep in mind what the $t$-ratio is: it's $\hat{\alpha}/S(\hat{\alpha})$, where $\hat{\alpha}$ is the estimated intercept and $S(\hat{\alpha})$ its standard error. What you do by choosing to apply HAC is using a different function $S(\cdot)$.

Assume that $\hat{\alpha}$ is the first regressor in the regressor matrix $X$. Then, while for OLS

$S_{OLS}(\hat{\alpha}) = \sqrt{\text{Var}_{OLS}(\hat{\alpha})_{(1,1)} }= \sqrt{s^2(X'X)^{-1}_{(1,1)}} = \sqrt{(s^2)(1/n)}$,

this is not the case if you use HAC standard errors. They are constructed diffferently (see e.g. here: https://en.wikipedia.org/wiki/Heteroscedasticity-consistent_standard_errors). In particular, for the $i$th estimated regression residual $\hat{u_i} = Y_i - X_i\hat{b}$,

$S_{HAC}(\hat{\alpha}) = \sqrt{\left( (X'X)^{-1}(X'\text{diag}(\hat{u_1}, ... \hat{u_n})X)(X'X)^{-1}\right)_{(1,1)} }$.

Notice that if we replaced $\text{diag}(\hat{u_1}, ... \hat{u_n})$ by $I_n \cdot s^2$, the HAC would become identical to the OLS standard error estimate. But unless $\hat{u_1} = ... = \hat{u_n}$, this will not happen.

This applies to the intercept as it does to all other coefficients. Hence, the denominator of your $t$-ratio changes depending on the choice of OLS vs HAC unless $\hat{u_1} = ... = \hat{u_n}$, which happens with probability $0$.

Answer 2

All p-values will change, including the intercept when you use HAC. They may change to various degrees, of course. Also, sometime, the p-values are calculated to certain precision, so you might not notice the change if it was very small.