I derive a point estimate given value $x_0$ using an estimated linear regression as follows:
$$\hat{y_0} = x_0^T\hat{\beta}.$$
I know that a prediction interval for a given value $x_0$: $$\hat{y}_0\pm t_{n-p}^{(\alpha/2)}\hat{\sigma}\sqrt{x_0^T(X^TX)^{-1}x_0 + 1},$$
where $\hat{\sigma}$ is the sample standard deviation of the residuals and $t_{n-p}^{\alpha/2}$ is a t-statistic with $n-p$ degrees of freedom at the $\alpha/2$ quantile.
I have the following question:
Thus far, I have assumed $\hat{\sigma}$ to be constant - the "standard" case. How do I estimate prediction intervals if my estimate of $\sigma$ changes when $x$ changes, e.g. instead of predicting conditional on $x_0$ $(\hat{\sigma_0})$, I predict conditional on $x_1$ $(\hat{\sigma_1})$. Here, the index may indicate changes across the cross-section or changes in time periods.
