23
$\begingroup$

What is the algebraic notation to calculate the prediction interval for multiple regression?

It sounds silly, but I am having trouble finding a clear algebraic notation of this.

$\endgroup$
29
$\begingroup$

Take a regression model with $N$ observations and $k$ regressors: $$\mathbf{y=X\beta+u} \newcommand{\Var}{\rm Var}$$

Given a vector $\mathbf{x_0}$, the predicted value for that observation would be $$E[y \vert \mathbf{x_0}]=\hat y_0 = \mathbf{x_0} \hat \beta.$$ A consistent estimator of the variance of this prediction is $$\hat V_p=s^2 \cdot \mathbf{x_0} \cdot(\mathbf{X'X})^{-1}\mathbf{x'_0},$$ where $$s^2=\frac{\Sigma_{i=1}^{N} \hat u_i^2}{N-k}.$$ The forecast error for a particular $y_0$ is $$\hat e=y_0-\hat y_0=\mathbf{x_0}\beta+u_0-\hat y_0.$$ The zero covariance between $u_0$ and $\hat \beta$ implies that $$\Var[\hat e]=\Var[\hat y_0]+\Var[u_0],$$ and a consistent estimator of that is $$\hat V_f=s^2 \cdot \mathbf{x_0} \cdot(\mathbf{X'X})^{-1}\mathbf{x'_0} + s^2.$$

The $1-\alpha$ $\rm confidence$ interval will be: $$y_0 \pm t_{1-\alpha/2}\cdot \sqrt{\hat V_{p}}.$$ The $1-\alpha$ $\rm prediction$ interval will be wider: $$y_0 \pm t_{1-\alpha/2}\cdot \sqrt{\hat V_{f}}.$$

$\endgroup$
  • $\begingroup$ The above answer is very well done, but I think this source helps provide some context to the question. $\endgroup$ – June Skeeter Dec 4 '18 at 8:13
  • $\begingroup$ @Dimitriy I believe your second eqn should have a carrot/hat, ‘^’, over the $\beta$. $\endgroup$ – Don Slowik May 27 at 18:04
  • $\begingroup$ Isn’t the forecast error the residual: $\hat{e}=\hat{u}$? $\endgroup$ – Don Slowik May 28 at 11:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.