Given, \begin{equation} \rho = Corr[X1, (X2 + X3)] \end{equation} and \begin{equation} \hat{\rho} = Corr[X2, X3] \end{equation}
where Corr[] stands for Pearson correlation, how can Corr[X1, X2] and Corr[X1, X3] be attributed such that the resulting 3x3 correlation matrix is positive semidefinite?
Let´s assume E[X1]=E[X2]=E[X3]=0 for simplicity please (E[] stands for expectation).
Thanks in advance.
Sorry for the question since I see now the answer is quite easy/straitforward.
Denoting by \begin{equation} \rho_{12}, \rho_{13} \end{equation}
the quantities of interest, and \begin{equation} \sigma_1,\sigma_2, \sigma_3 \end{equation}
the standard deviations of the three variables X1, X2, X3, we have that
\begin{equation} \rho = \frac{E[X_1 (X_2 + X_3)] - E[X_1]E[X_2 + X_3]}{\sigma_1 \sqrt{\sigma_2^2 + \sigma_3^2 + 2\hat{\rho}\sigma_2 \sigma_3}} \end{equation}
Therefore the answer is the set of pairs
\begin{equation} \rho_{12}, \rho_{13} \end{equation}
that fullfils
\begin{equation} \rho = \frac{\rho_{12}\sigma_2 + \rho_{13}\sigma_3}{\sqrt{\sigma_2^2 + \sigma_3^2 + 2 \hat{\rho} \sigma_2 \sigma_3}} \end{equation}
with \begin{equation} 0 \leq \rho_{1,2}, \rho_{13} \leq 1 \end{equation}
I will work more on what solutions from this set form a positive semidefinite matrix.