# Prediction intervals in the case of changing variance

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.

• Nov 25, 2022 at 19:45
• You need a model for the variance of the conditional response. Otherwise, you can't even construct prediction intervals for unobserved values of $x.$