17. This problem shows that multicollinearity also affects precision of estimation of partial correlations.

a) Suppose the true correlations are px,x = .85, prx, = .65. and pix; = .65. Show that pyx, x- Prx; x = .244.

b) In a sample. x, x = .9. ryx = .7 and Frx, = .6. Unless the sample is very large, these are well within the limits of sampling enor for the true values Show that ryx. X> .46 and ryx, x=-10 Note how small differences in ryx, and ryx, yield large differ- ences in partial correlations when multicollinearity exists. (This illustrates that partial correlations have large standard errors when multicollinearity exists. For these values, an unwary observer might conclude that the partial effects of X, and X2 have opposite signs and that the partial effect of X, is much stronger. when in fact they are identical in the population of interest.)

c) For comparison. compute the partial correlations in

(b) when ryx = .7 and yx = .6, but (i) rx,x=0. (ii) rx,x; = .6.