Cross validation (CV) seems to be a simple and useful tool for estimating the out-of-sample error of a linear regression model, even though it is rarely used for this purpose. Why that? Is there a better ways for error estimation? Is it possible to estimate the out-of-sample error by only using the training data?
I see it used quite often in fact. Note that the popular predicted residual error sum of squares (PRESS) statistic calculated from the residuals & diagonal elements of the hat matrix is what you'd get by performing leave-one-out cross validation.
Alternatives to cross-validation include splitting the sample into training & test sets, & bootstrapping to estimate the optimism in any measure of fit & then correcting for it. See Steyerberg et al. (2001), "Internal validation of predictive models: efficiency of some procedures for logistic regression analysis", Journal of Clinical Epidemiology, 54, pp 774 – 781 for a comparison. Splitting the sample isn't a good idea unless you have a large sample size (in the tens of thousands, I've heard).