Multiple Regression & Inferenceintermediate

Prediction and Prediction Intervals

Q: Why is the prediction interval for a single new outcome wider than the confidence interval for the mean at the same x?

Because it adds the individual error variance σ^2 to the uncertainty in the estimated mean. A new unit scatters around the regression line, so its interval includes the irreducible error σ^2 on top of the coefficient uncertainty captured by the mean interval.

Q: As the sample size grows very large, the width of a prediction interval for a single new outcome:

Approaches a positive width determined by the error variance. Estimation uncertainty vanishes with more data, but the individual error variance σ^2 remains, so the interval converges to a positive width, not zero.

Regression supports two distinct forecasts. One is the mean outcome $E(y\mid x)$ at given regressor values, the other an individual outcome for a new unit. The interval for a single new $y$ is wider than the confidence interval for the mean, because it adds the irreducible error variance $\sigma^2$ on top of the uncertainty in the estimated coefficients. Predictions far outside the sample range of the regressors are unreliable.

Why it matters

Estimating where the regression line sits at a point is easier than guessing one person’s actual value, since individuals scatter around the line. So a prediction interval for a new observation has to be padded for that personal randomness. And the model has no warrant once you extrapolate beyond the data it learned from.

Before you read on — recall

Why is the prediction interval for a single new outcome wider than the confidence interval for the mean at the same $x$ ?

Formulas

Confidence interval for the mean

\hat{y}^0 \pm c\cdot\mathrm{se}(\hat{y}^0)

Captures uncertainty in the estimated regression function at

x=x^0

Prediction interval for a new outcome

\hat{y}^0 \pm c\cdot\sqrt{\mathrm{se}(\hat{y}^0)^2+\hat{\sigma}^2}

Adds

\hat{\sigma}^2

for the individual error, so it is wider.

Worked examples

Scenario

Using `regress lwage educ exper`, you predict log wage for a worker with 16 years of school and 5 of experience.

Solution

In Stata, `margins, at(educ=16 exper=5)` gives the mean prediction with its confidence interval, while a prediction interval for one such worker is wider because it also carries $\hat{\sigma}^2$ . The point estimate is the same; only the band differs.

NotePredicting log wage and then exponentiating needs a retransformation adjustment for the mean of the level.

Common mistakes

✗Using the confidence interval for the mean as the interval for a single new outcome. The individual interval is wider by the error variance $\sigma^2$ .
✗Believing prediction intervals shrink toward zero width as $n\to\infty$ . They converge to a positive width set by the irreducible error, not to zero.
✗Trusting predictions far outside the sample range. Extrapolation assumes the fitted relationship holds where there are no data, which is rarely safe.
✗Forgetting the retransformation issue when the model is in logs. Exponentiating the log prediction underestimates the mean level without a correction.

Revision bullets

•Distinguish predicting the mean $E(y\mid x)$ from predicting a single new $y$ .
•The prediction interval for a new outcome is wider, adding $\sigma^2$ .
•Individual prediction intervals do not shrink to zero as $n$ grows.
•Extrapolation beyond the sample range of the regressors is unreliable.
•Log-model predictions need a retransformation adjustment for the level mean.

Quick check

Why is the prediction interval for a single new outcome wider than the confidence interval for the mean at the same $x$ ?

As the sample size grows very large, the width of a prediction interval for a single new outcome:

Connected topics

Fitted / Resid R-squared CLM normality t test Conf. intervals Logs & Elastic.

More in Multiple Regression & Inference

MLR model Partialling out Omitted var bias MLR assumptions Gauss-Markov Asymptotics F test

Sources

Wooldridge, Introductory Econometrics, Ch. 6
Wooldridge (2019), Introductory Econometrics: A Modern Approach, 7th ed., Sec. 6.4 (prediction and prediction intervals)
Hill, Griffiths & Lim, Ch. 4
Hill, Griffiths & Lim (2018), Principles of Econometrics, 5th ed., Ch. 4

How to cite this page

Dr. Phil's Quant Lab. (2026). Prediction and Prediction Intervals. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-prediction-intervals