For three variables, e.g. $X$, $Y$, and $Z$, it is easy for me to deduce the partial and semi-partial correlation coefficients (in general), however I cannot do the same for more than three variables. If, $$ X = \alpha + \beta Z + \delta, \quad \text{where} \; \alpha = \bar{X} - \beta \bar{Z} \; \text{and} \; \beta = \frac{\sigma_{ZX}}{\sigma_Z^2}, $$ and $$ Y = \alpha^* + \beta^* Z + \varepsilon, \quad \text{where} \; \alpha^* = \bar{Y} - \beta^* \bar{Z} \; \text{and} \; \beta^* = \frac{\sigma_{ZY}}{\sigma_Z^2}. $$ Given that $\rho_{AB} = \sigma_{AB}/\sigma_A \sigma_B$ for any random variables $A$ and $B$, then $$ \sigma_\delta^2 = \operatorname{Cov}\left( X - \alpha - \beta Z, X - \alpha - \beta Z \right) = \sigma_X^2 - 2 \beta \sigma_{ZX} + \beta^2 \sigma_Z^2 = \sigma_X^2 \left( 1-\rho^2_{ZX} \right), $$ $$ \sigma_\varepsilon^2 = \operatorname{Cov}\left( Y - \alpha^* - \beta^* Z, Y - \alpha^* - \beta^* Z \right) = \sigma_X^2 - 2 \beta^* \sigma_{ZY} + {\beta^*}^2 \sigma_Z^2 = \sigma_Y^2 \left( 1-\rho^2_{ZY} \right), $$ $$ \sigma_{\delta Y} = \operatorname{Cov}\left( X - \alpha - \beta Z, Y \right) = \sigma_{XY} - \beta \sigma_{ZY} = \sigma_X \sigma_Y \left( \rho_{XY} - \rho_{ZX} \rho_{ZY} \right), $$ and $$ \sigma_{\delta \varepsilon} = \operatorname{Cov}\left( X - \alpha - \beta Z, Y - \alpha^* - \beta^* Z \right) = \sigma_{XY} - \beta \sigma_{ZY} - \beta^* \sigma_{ZX} + \beta \beta^* \sigma_Z^2 = \sigma_X \sigma_Y \left( \rho_{XY} - \rho_{ZX} \rho_{ZY} \right). $$ Therefore, the partial correlation coefficient is $$ \rho_{(XY).Z} = \frac{\sigma_{\delta \varepsilon}}{\sigma_\delta \sigma_\varepsilon} = \frac{\sigma_X \sigma_Y \left( \rho_{XY} - \rho_{ZX} \rho_{ZY} \right)}{\sqrt{\sigma_X^2 \left( 1-\rho^2_{ZX} \right)} \sqrt{\sigma_Y^2 \left( 1-\rho^2_{ZY} \right)}} = \frac{\rho_{XY} - \rho_{ZX} \rho_{ZY}}{\sqrt{1-\rho^2_{ZX}} \sqrt{1-\rho^2_{ZY}}}, $$ and the semi-partial correlation coefficient is $$ \rho_{Y (X.Z)} = \frac{\sigma_{\delta Y}}{\sigma_\delta \sigma_Y} = \frac{\sigma_X \sigma_Y \left( \rho_{XY} - \rho_{ZX} \rho_{ZY} \right)}{\sqrt{\sigma_X^2 \left( 1-\rho^2_{ZX} \right)} \sigma_Y} = \frac{\rho_{XY} - \rho_{ZX} \rho_{ZY}}{\sqrt{1-\rho^2_{ZX}}}. $$ How can I do the same for the general form: $$ X = \alpha + \beta_1 Z_1 + \dots + \beta_p Z_p + \delta \quad \text{and} \quad Y = \alpha^* + \beta_1^* Z_1 + \dots + \beta_p^* Z_p + \varepsilon? $$ How can I do the same in matrix/vector form, i.e. when $\delta = X - Z\hat{\beta} = X - Z\left( Z^TZ\right)^{-1} Z^TX$ and $\varepsilon = Y - Z\hat{\beta^*} = Y - Z\left( Z^TZ\right)^{-1} Z^TY$?
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