Why both intervals widen as you leave the data

Question

You fit a simple linear regression and draw the interval band around the fitted line. It hugs the line tightly near the middle of your data and fans out toward the edges.

Explain the term that makes both the confidence and prediction intervals widen away from the center, and why it penalizes extrapolation.

Accepted Answer

For simple linear regression, the variance of the fitted value at a new input x0 is \operatorname{Var}(\hat y0) = \sigma^2\left[\frac{1}{n} + \frac{(x0 - \bar x)^2}{S{xx}}\right], \qquad S{xx} = \sumi (xi - \bar x)^2. The confidence interval uses exactly this; the prediction interval adds the fresh-noise "+1" inside the bracket. Either way, both carry the leverage term (x0 - \bar x)^2 / S{xx}. Why it flares. That term is zero at x0 = \bar x and grows with the square of the distance from the center. Intuitively, the line pivots about (\bar x, \bar y): a small wobble in the estimated slope bare…

Why both intervals widen as you leave the data

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