If the multiple correlation coefficient of X 1on X 2and X 3is zero, then:
r12 = 0, r13 = 0
\(R_{1.23}\) is the correlation between \(X_{1}\) and its best linear predictor based on \(X_{2}, X_{3}\). If \(R_{1.23}=0\), no linear combination of \(X_{2}, X_{3}\) correlates with \(X_{1}\).
Using \[R^{2}_{1.23}=\frac{r_{12}^{2}+r_{13}^{2}-2r_{12}r_{13}r_{23}}{1-r_{23}^{2}}=0,\] the numerator must vanish, giving \((r_{12}-r_{13}r_{23})^{2}+r_{13}^{2}(1-r_{23}^{2})=0\). Both squared terms are non-negative, so each is zero, forcing \(r_{13}=0\) and then \(r_{12}=0\).
Hence \(R_{1.23}=0\) requires \(r_{12}=0\) and \(r_{13}=0\).
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