How can I prove that the f-statistic does not follow an F distribution in the context of step-wise regression? There is a good number of threads about the deficiencies of step-wise regression, and particularly on the shortcomings of the partial F test as a tool for step selection.
However I find it difficult to see why the test is not F-distributed. I would be grateful if someone could sketch a quick proof.
 A: The issue isn't just whether the test is F-distributed. The interpretation of the p-values in the context of multiple testing and the optimism bias introduced in the variable selection procedure are major sources of difficulty.
For p-values, there's a reasonable analogy to post-hoc testing in classic ANOVA. When the overall F-test for a classic one-way ANOVA with multiple treatment groups is significant, multiple-testing adjustments need to be made for post-hoc comparisons among the groups. You do not use unadjusted p-values. Similarly, when you have a significant full regression model, you need to correct for multiple testing when you effectively compare regression coefficients during the subsequent selection process.
For optimism bias, note that stepwise selection will favor the coefficients that happened to be largest in the particular sample that was examined. (Similarly, important coefficients that happened to be small in the particular sample will be discounted.) Thus the coefficients are likely to be biased toward values that are too great in magnitude.
For the F-test itself, there is an issue in what the degrees of freedom should be. The total number of candidate variables is a better choice than the number in the selected subset. See this page and references cited there.
That said, step-down selection can be useful in some circumstances. See Chapter 4 of Harrell's rms course notes or of his Regression Modeling Strategies book for more detail and for references.
