Question

Problem 1

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

$\begin{bmatrix} y_1 \\ y_2\\.\\.\\.\\y_n \end{bmatrix}=\begin{bmatrix} 1 & x_{11} & x_{12}&...&x_{1k} \\ 1 & x_{21} & x_{22} & ... & x_{2k}\\. & . & . & &.\\. & . & . & &.\\.& . & . & & .\\1 & x_{n1} & x_{n2} & ...&x_{nk} \end{bmatrix}\begin{bmatrix} \beta_0\\ \beta_1\\.\\.\\.\\\beta_k \end{bmatrix}+\begin{bmatrix} \in_1\\ \in_2\\.\\.\\.\\\in_n \end{bmatrix}.$