# What is the Formula for Prediction Interval in Multivariate Case? [duplicate]

I am using linear model to do prediction, and I would like to calculate my prediction's prediction interval, which, when there is only one predictor, is

However, my model has three predictors. What is the formula for multivariate case?

I have been searching for a while, but I cannot find a formula for this.

What is denoted by $$\mathrm{MSE}$$ in your formula is the residual mean square which some people, confusingly, call error mean square or even mean squared error$$-$$hence the abbreviation in your formula. The residual mean square is an unbiased estimator of the (conditional) error or disturbance variance $$\sigma^2$$.
It is given by $$\hat\sigma^2 = \frac{\sum_{i=1}^n (y_i-\hat y_i)^2}{n-k-1},$$ where $$k$$ is the number of regressors. Note that we have $$k=1$$ in the simple linear regression model, and with three regressors (not counting the intercept) we have $$k=3$$.
The formula of the prediction interval for the future observation $$y_h$$ at location $$\mathbf{x}_h={(1,x_{h1},\ldots,x_{hk})}^\top$$ gets only slightly more complicated in the general case with $$k$$ regressors: $$\hat y_h\mp t_{(1-\alpha/2,\,n-k-1)}\times\sqrt{\hat{\sigma}^2\times\left(1+\mathbf{x}_h^{\top}\left(\mathbf{X}^{\top}\mathbf{X}\right)^{-1}\mathbf{x}_h\right)},$$
where $$\hat y_h =\mathbf{x}_h^\top \hat{\boldsymbol{\beta}}$$, $$\mathbf{X}$$ is the design matrix, and $$t_{(1-\alpha/2,\,n-k-1)}$$ the $$(1-\alpha/2)$$ quantile of the $$t$$-distribution with $$n-k-1$$ degrees of freedom.
For $$k=1$$ this reduces to the formula in your question if we identify the second entry of $$\mathbf{x}_h={(1,x_{h1})}^\top$$, i.e. $$x_{h1}$$, with $$x_h$$.
• Thank you for your answer. Should we add $1/n$ in the square root on your second equation? Commented Aug 16, 2022 at 14:38
• @user398843 No, $\sigma^2\left(x_h^{\top}\left(X^{\top}X\right)^{-1}x_h\right)$ is the variance of $\hat y_h$ and reduces to $\sigma^2 \left(1/n + \left(x_h-\bar x \right)^2 / \sum_{i=1}^n \left( x_i-\bar x\right)^2 \right)$ in the case of only one regressor. Commented Aug 16, 2022 at 15:37
• @PascalIv Then $\mathbf X$ doesn't have full column rank and $\hat{\boldsymbol{\beta}}$ is not uniquely defined. Commented Jul 9, 2023 at 1:52