Partial correlation

Remarks

Example: let $Z, U, V$ be uncorrelated with variance $1$ and mean $0$, and let $X = Z + U$, $Y = Z + V$. Then $\Sigma_{XZ} = \Sigma_{YZ} = 1$ and $\Sigma_{ZZ} = 1$, so $e_X = U$ and $e_Y = V$, and $\rho_{XY \cdot Z} = 0$. Yet $\operatorname{Corr}(X, Y) = 1/2$. Non-example: if $X = Z$, then $e_X = 0$, so $\operatorname{Var}(e_X) = 0$ and the partial correlation is not defined.

For a single control variable $Z$, with $\rho_{XY}$, $\rho_{XZ}$, $\rho_{YZ}$ the pairwise correlations, $$\rho_{XY \cdot Z} = \frac{\rho_{XY} - \rho_{XZ} \rho_{YZ}}{\sqrt{(1 - \rho_{XZ}^2)(1 - \rho_{YZ}^2)}}.$$ If $P = (p_{ij})$ is the inverse of the covariance matrix of $(X, Y, Z)$, then $\rho_{XY \cdot Z} = -p_{12} / \sqrt{p_{11} p_{22}}$. The sample partial correlation uses the sample covariances, or equivalently the residuals of least squares fits on the data.