# Obtaining a formula for prediction limits in a linear model (i.e.: prediction intervals)

Let's take the following example:

set.seed(342)
x1 <- runif(100)
x2 <- runif(100)
y <- x1+x2 + 2*x1*x2 + rnorm(100)
fit <- lm(y~x1*x2)


This creates a model of y based on x1 and x2, using a OLS regression. If we wish to predict y for a given x_vec we could simply use the formula we get from the summary(fit).

However, what if we want to predict the lower and upper predictions of y? (for a given confidence level).

How then would we build the formula?

• The Confidence Interval on New Observations section of this page may help. – GaBorgulya Apr 3 '11 at 18:32
• @Tal Sorry, but it's not really clear to me what you actually mean by "predict the lower and upper predictions of y". Does it have something to do with prediction or tolerance bands? – chl Apr 3 '11 at 19:43
• @Tal - a couple of queries. When you say " .. y based on x1 and x2, using a OLS regression." , you mean your create a linear model and estimate parameters using OLS. Am I right? and @chl's question -- do you want to predict the lower and upper bounds for the prediction interval? – suncoolsu Apr 3 '11 at 19:55
• @chl, sorry for not being more clear. I am looking for two formulas that will give an interval for that will "catch" the "real" value of y 95% of the time. I feel how I'm using definitions for the CI for the mean, when there is probably some other term I should be using, sorry about that... – Tal Galili Apr 3 '11 at 20:02
• @suncoolsu - yes and yes. – Tal Galili Apr 3 '11 at 20:03

## 3 Answers

You will need matrix arithmetic. I'm not sure how Excel will go with that. Anyway, here are the details.

Suppose your regression is written as $\mathbf{y} = \mathbf{X}\mathbf{\beta} + \mathbf{e}$.

Let $\mathbf{X}^*$ be a row vector containing the values of the predictors for the forecasts (in the same format as $\mathbf{X}$). Then the forecast is given by $$\hat{y} = \mathbf{X}^*\hat{\mathbf{\beta}} = \mathbf{X}^*(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{Y}$$ with an associated variance $$\sigma^2 \left[1 + \mathbf{X}^* (\mathbf{X}'\mathbf{X})^{-1} (\mathbf{X}^*)'\right].$$ Then a 95% prediction interval can be calculated (assuming normally distributed errors) as $$\hat{y} \pm 1.96 \hat{\sigma} \sqrt{1 + \mathbf{X}^* (\mathbf{X}'\mathbf{X})^{-1} (\mathbf{X}^*)'}.$$ This takes account of the uncertainty due to the error term $e$ and the uncertainty in the coefficient estimates. However, it ignores any errors in $\mathbf{X}^*$. So if the future values of the predictors are uncertain, then the prediction interval calculated using this expression will be too narrow.

• +1, excellent answer. I should note though, that regression model always estimates conditional expectation, so it is as good as its regressors are. So the last comment although is very good, it is not strictly necessary, since if you build regression model you must trust the regressors. – mpiktas Apr 4 '11 at 3:51
• why the 1 comes up in the formula? We have $\hat{y}=X^*\beta+X^*(X'X)^{-1}X'e$. Then $var \hat{y}=var X^*(X'X)^{-1}X'e=\sigma^2X^*(X'X)^{-1}(X^*)'$? – mpiktas Apr 7 '11 at 14:25
• The 1 is for prediction intervals. Leave it off for confidence intervals. Var($\hat{y}$) relates to confidence intervals. – Rob Hyndman Apr 12 '11 at 14:44
• @RobHyndman thank you for your excelent answer (one year ago ;)) however, am I missing something or is the term in the square root $N \times N$? – Seb Sep 28 '12 at 6:06
• @Seb. $X^*$ is a row vector, so the term is scalar. – Rob Hyndman Sep 28 '12 at 23:26

Are you by chance after the different types of prediction intervals? The predict.lm manual page has

 ## S3 method for class 'lm'
predict(object, newdata, se.fit = FALSE, scale = NULL, df = Inf,
interval = c("none", "confidence", "prediction"),
level = 0.95, type = c("response", "terms"),
terms = NULL, na.action = na.pass,
pred.var = res.var/weights, weights = 1, ...)


and

Setting ‘intervals’ specifies computation of confidence or prediction (tolerance) intervals at the specified ‘level’, sometimes referred to as narrow vs. wide intervals.

Is that what you had in mind?

• Hi Dirk, that is indeed what I wish to find, but I want the upper and lower bonds to be in the form of a formula (so to later implement in some low form of statistical software, for example, excel...) – Tal Galili Apr 3 '11 at 22:16
• p.s: I now see that there was an edit to the title of my question that might had led you to think I was asking about predict.lm interval parameter (which I am not) :) – Tal Galili Apr 3 '11 at 22:18
• You are abusing terminology here. Excel is not statistical software. – Dirk Eddelbuettel Apr 3 '11 at 22:23
• You're right, my bid, how about "a spreadsheet application" ? – Tal Galili Apr 3 '11 at 22:42
• I can live with that; it calls the devil by its name ;-) – Dirk Eddelbuettel Apr 3 '11 at 22:52

@Tal: Might I suggest Kutner et al as a fabulous source for linear models.

There is the distinction between 1) a prediction of Y from an individual new observation X_vec, 2) the expected value of a Y conditioned on X_vec, $E(Y|X_{vec})$ and 3) Y from several instances of x_vec - all covered in detail in the text.

I think you are looking for the formula for the confidence interval around $E(Y|X_{vec})$ and that is $\hat{Y}$ $\pm$ t(1-$\alpha$ /2)s{$\hat{Y}$} where t has n-2 d.f. and s{$\hat{Y}$} is the standard error of $\hat{Y}$, $\frac{\sigma^{2}}{n}$+($X_{vec}-\bar{X})^{2}\frac{\sigma^{2}}{\sum(X_{i}-\bar{X})^{2}}$

• (+1) for making the distinction. However, I believe the OP is asking for (1), not (2) (and I have edited the question's title accordingly). Also note that your formula appears to assume the regression depends only on one variable. – whuber Apr 4 '11 at 14:48