This problem shows that multicollinearity also affects precision of estimation of partial correlations.
a) Suppose the true correlations are $\rho_{X_1 X_2}=.85, \rho_{Y X_1}=.65$. and $\rho_1 X_z=.65$. Show that $\rho_{Y x_1, x-}=\rho_\gamma x_2 \delta=.244$.
b) In a sample. $r_{X_1 \lambda_1}=.9$. $r_{Y X_1}=.7$ and $r_1 x_2=.6$. Unless the sample is very large, these are well within the linits of sampling enor for the true valucs Slow that $r_r x_1 x$, $=$ .46 and $r_{Y X_2 \lambda_1}=-.10$ Note how small differences in $r_{Y X_1}$ and $r_{Y X_2}$ yield large differences in partial correlations when multicollineanity exists. (This illustrates that partial correlations have large standard errors when multicollinearity exists. For these values, an unwary observer mught conclude that the partial effects of $X_t$ and $X_2$ have opposite signs and that the partial effect of $X_t$ is much stronger. when in fact they are identical in the population of interest.)
c) For companson. compute the purtial correlations in (b) when $r_{Y X_1}=.7$ and $/ y X=.6$, but (i) $r_{X_1 x_2}=0$. (ii) $r_{X_1 X_2}=.6$.