Consider the multiple regression model y = Xβ + ϵ. Let X = [X1,X2]. Assume there is no intercept in the model. Regress y on X1 and obtain the residuals y∗ from this regression. Then regress each column of X2 on X1 to get a matrix of residuals X∗ 2. Finally regress the residuals from the first regression on the residuals from the second regression to obtain the partial coefficient ˆβ2.1. Show that y∗ − X∗ 2 ˆβ2.1 is orthogonal to X.
Added by Elizabeth M.
Step 1
The residuals are calculated as: \[ y^* = y - \hat{y} = y - X_1 \hat{\beta}_1 \] where \( \hat{\beta}_1 \) is the estimated coefficient from the regression of \( y \) on \( X_1 \). Show more…
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