Discussion
If \(r_{12.3}\) is the correlation coefficient between the variables \(X_{1}\) and \(X_{2}\) after eliminating the linear effect of \(X_{3}\), then which of the following are correct ?
(1) \(-1 \le r_{12.3} \le 1\)
(2) \(r_{12}^{2} + r_{13}^{2} + r_{23}^{2} - 2r_{12}r_{13}r_{23} \le 1\)
(3) \(r_{12.3}^{2} = b_{12.3}\, b_{21.3}\), where b's are partial regression coefficients
Answer: D
All three statements are correct: (1), (2) and (3).
(1) \(r_{12.3}\) is an ordinary correlation, between the residuals of \(X_{1}\) and \(X_{2}\) after regressing each on \(X_{3}\), so it lies in [-1, 1].
(2) The correlation matrix of \(X_{1}, X_{2}, X_{3}\) is positive semi-definite, so its determinant is non-negative:
\[ \begin{aligned} &1 - r_{12}^{2} - r_{13}^{2} - r_{23}^{2} \\ &\quad + 2r_{12}r_{13}r_{23} \ge 0 \end{aligned} \]
which rearranges to statement (2).
(3) As with simple regression, where \(r^{2} = b_{yx}b_{xy}\), the partial correlation is the geometric mean of the two partial regression coefficients: \(b_{12.3} = r_{12.3}\sigma_{1.3}/\sigma_{2.3}\) and \(b_{21.3} = r_{12.3}\sigma_{2.3}/\sigma_{1.3}\), whose product is \(r_{12.3}^{2}\).
Since all three hold, every option naming only two of them is incomplete.
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