Hidden extrapolations in linear regression On page 261 of Kutner, the author cautions readers about hidden extrapolations in linear regression. As shown in the figure, the allowed region for prediction is not a rectangle delimited by the ranges of $x_1$ and $x_2$, but a ellipse. Why is this the case and how is this ellipse generated?

 A: I believe the ellipse drawn is just done by hand, but it is supposed to illustrate something like the convex hull (or here) of the data support.  There are multiple ways of deciding where exactly the data support lies.  For instance, you could just find the "outermost" points in your data cloud and connect them with lines, or you could have an algorithm that attempts to "smoothly" join the outer points, ignoring any points that the more naive algorithm would see as "outer" but which are too far interior to produce a nice smoothed shape.
Example in R
# Example from ?mvrnorm
library(MASS)
Sigma <- matrix(c(10,3,3,2),2,2)
Sigma
set.seed(1)
dat <- mvrnorm(n=1000, rep(0, 2), Sigma)
plot( dat )


# Find convex hull and plot: http://stats.stackexchange.com/questions/11919/convex-hull-in-r#11921
library(grDevices) # load grDevices package
con.hull.pos <- chull(dat)
con.hull <- rbind(dat[con.hull.pos,],dat[con.hull.pos[1],])
plot(dat) # plot data
lines(con.hull,col="red")


The idea here is that the underlying function outside of where you have data could continue as you expect--or it could verge off elsewhere.  So at $(-5,4)$, for example, the relationship between $(X1,X2)$ and $f(X1,X2)$ might not be what you're expecting, and you're extrapolating despite being within the range of both X1 and X2.
A: The standard error of prediction for a new observation is 
$$
s_{pred}=s_e\sqrt{\left(1+\frac{1+D^2(x)}{N}\right)},
$$
where $D^2(x)$ is the squared Mahalanobis distance of the predictor vector $x$ for the new case from the centroid of the training set predictor matrix $X$. This implies that the contours of constant standard error are elliptical and centered at the centroid of the training set, regardless of the shape of the joint distribution of the training set. Although such an ellipse will generally not be tangent to all four sides of the box defined by the extrema of the training set, the general impression conveyed by the picture is correct.
A: My guess would be that $X_1$ and $X_2$ are supposed to have a Gaussian (Normal) distribution, defined by a mean ($\mu_i$) and a variance ($\sigma^2_i$). The joint distribution of $X_1$ and $X_2$ would thus be an ellipse whose size is determined at some level of confidence.
In this case, the "Range of $X_1$" may well be something like the 95% CI, which corresponds to $\mu_1 \pm 1.96\sigma_1$, and similarly for $X_2$ ($\mu_2 \pm 1.96\sigma_2$). The ellipse corresponds to the 95% CI for the joint distribution of $X_1$ and $X_2$, and depends on their correlation.
Look at the graph at the top of the Wikipedia entry for Joint Distribution.
