(1) Since the simpler model features less regressor than the larger model, it follows that the VIF of the simpler model will be less than that of the larger model. The reason is that the more variables we include in the model, the greater multicollinearity, and, hence, the greater Rj 2 , unless the omitted variables happen to be orthogonal to the regressors included in the simpler model. The simpler model, which omits relevant variables, produces bias estimates but with smaller variances. Consequently, there appears to be a tradeoff between bias and precision. . | Applied Econometrics 1 Model Selection and the Use of F - Statistic Applied Econometrics Lecture 6 Model Selection and the Use of F - Statistic Say not I have found the truth but rather I have found a truth KAHLIL GIBRAN THE PROPHET 1 Introduction The variance of the least squares estimator of the slope coefficient of a multiple regression is given by V ß Ç2_1 È Sij- S 2 1 - R2 where Rj2 is the coefficient of determination of the auxiliary regression of Xj with all the regressors included in the model. The variance of the estimator of the slope coefficient V 0j depends on the three factors 1 the error variance ct2 2 the total sum of squares of Xj TSS E Slj - y 2 and its variance inflation factor VIF 1 1-Rj2 . 1 Since the simpler model features less regressor than the larger model it follows that the VIF of the simpler model will be less than that of the larger model. The reason is that the more variables we include in the model the greater multicollinearity and hence the greater Rj2 unless the omitted variables happen to be orthogonal to the regressors included in the simpler model. The simpler model which omits relevant variables produces bias estimates but with smaller variances. Consequently there appears to be a tradeoff between bias and precision. 2 We calculate the squared standard error of the regression s2 as the proxy measurement of ct2 . s2 RSS n - k As to the numerator the RSS of the larger model will generally be less than that of the model which omits relevant variables. As to the denominator given the sample size smaller models have more degrees of freedom. In sum whether the inclusion of omitted variables in the model leads to a reduction in s2 depends on whether the RSS falls sufficiently to offset the loss in degrees of freedom. There are two opposite tendencies when we include the relevant variables in the model increase in VIF due to multicollinearity and reduction s2. Written by Nguyen Hoang Bao May 22 2004 Applied Econometrics 2 Model
