Collinearity and the least squares predictor

Подпись: var( D0) = a2x0( X'X) 1x0 = a2x0CЛ 1C'x0 = a: Подпись: (X0C1)2 + (x0c2)2 image279

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

(12.8)

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.

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