Partial regression (${\hat{\beta }}_j=\frac{X_j^T(I-P_{-j})Y}{X_j^T(I-P_{-j})X_j}$)

Partial regression first projects a response and a selected predictor orthogonally off the remaining design columns. Regressing the residualized response on the residualized predictor yields its coefficient in the full ordinary least squares fit.