Regression and Random Confounding

An ordinary least squares regression estimate for the slope, regardless of its strength, can have its sign reversed through adjustment for a random confounding vector of data. The assumption of a rotionally invariant distribution, on the space of centered, random, confounding vectors of data, makes calculation of probabilities for these reversals possible. Here these probabilities are shown to decrease exponentially, as the sample size increases. This analytic result leads to some asymptotic comparison between regular sampling error and the error due to a mis-specified model.