8.12 Let yi = + xi + e1i, zi = a + bxi + e2i, i

8.12 Let yi = α + βxi + e1i, zi = a + bxi + e2i, i = 1, . . . , n, denote standard regressions of y on x and z on x, respectively; that is, E(e1i) = E(e2i) = 0, all errors are independent and homoscedastic. Define the new variables “corrected for x” as where carets denote least squares estimators. The (sample) partial correlation coefficient of y and z “holding x fixed” is defined as the correlation between y* and z*; that is, since Eyi* = Ezi* = 0, Now suppose (X,Y,Z) follow a trivariate Normal distribution and xi, yi, zi are their observed values. Find the conditional distribution of Y and Z given X and relate the correlation coefficient in this distribution to that found above.