3.6 ( Section 3.1.6) B Suppose we wish to explain a variable y and that the number of possible explanatory variables is so large that it is tempting to take a subset. In such a situation some researchers apply the so-called Theil criterion and maximize the adjusted R defined by =1-(1-R) where n is the number of observations and k the number of explanatory variables. R = a. Prove that R never decreases by including an additional regressor in the model. b. Prove that the Theil criterion is equivalent with minimizing s, the standard error of regression. c. Prove that the Theil criterion implies that an explanatory variable will be Xj, maintained if and only if the F-test statistic for the null hypothesis ; = 0 is larger than one. d. Show that the size (significance level) of such a test is larger than 0.05. 3.6 ( Section 3.1.6) B Suppose we wish to explain a variable y and that the number of possible explanatory variables is so large that it is tempting to take a subset. In such a situation some researchers apply the so-called Theil criterion and maximize the adjusted R defined by =1-(1-R) where n is the number of observations and k the number of explanatory variables. R = a. Prove that R never decreases by including an additional regressor in the model. b. Prove that the Theil criterion is equivalent with minimizing s, the standard error of regression. c. Prove that the Theil criterion implies that an explanatory variable will be Xj, maintained if and only if the F-test statistic for the null hypothesis ; = 0 is larger than one. d. Show that the size (significance level) of such a test is larger than 0.05