Answer to Question #203372 in Economics of Enterprise for Leta Lemessa

Question

Consider a k-variables linear regression model, i.e., Y = X 1β1 + X 2 β2 + ε,  Where, X1 is (N  k1 ) , X 2 is (N  k2 ) and k = k1 + k2 . As you may recall, adding columns to the X matrix (including additional regressors in the model) gives positive definite increase in R2. The adjusted R2 ( R 2 ) attempts to avoid this phenomenon of ever increase in R2. Show that the additional k2 number of variables (regressors) in this model increases R 2 if the calculated F-statistic in testing the joint statistical significance of coefficients of these additional regressors (β2 ) is larger than one. 

Answer 1

$\begin{bmatrix} y_1 \\ y_2\\.\\.\\. \\y_T \end{bmatrix}=\begin{bmatrix} 1 & x_{11}&...&Ax_{1k}\\ 1 & x_{21}&...&x_{2k}\\.&.&&.\\.&.&&.\\.&.&&.\\1&x_{T1}&...&x_{Tk} \end{bmatrix}\begin{bmatrix} \beta _0 \\ \beta _1 \\.\\.\\.\\\beta _k \end{bmatrix}+\begin{bmatrix} \in _1 \\ \in _2\\.\\.\\.\\\in_T, \end{bmatrix}.$