Collinearity and the least squares predictor
Another bit of conventional wisdom is that while collinearity may affect the precision of the least squares estimator, it need not affect the reliability of predictions based on it, if the collinearity in the sample extends to the forecast period. Suppose we wish to predict the value of y0, given by y0 = x0P + e0, where x0 is a 1 x K vector of regressor values, and e0 is a random disturbance with zero-mean, constant variance a2, and which is uncorrelated with the regression disturbances. Using equation (12.6), the best linear unbiased predictor of E(y0), y0 = x’0b, has variance
If, for example, there is a single linear dependence among the columns of X, then XK ~ 0, and XcK ~ 0. A small eigenvalue could make the last term of the sum in equation (12.8) large, producing a large prediction variance. However, if (and this is a big if) the new observation x0 obeys the same collinearity pattern as the sample data, then it may also be true that x’0cK ~ 0, effectively negating the small eigenvalue.