John Cai Jesus College Self-test questions for Econometrics Lecture Slides 1: Multiple Linear Regressions 1. What are the F.O.C.s to obtain the coefficients for the regression wagei = Bo + B1Edi + B2IQi + Mi a. We want to minimize >(u2) - Sum of Squared Residuals so we choose Bo, B1, B2 to minimize E(wage; - wage;)2 = E(wage; - Bo - B1 Ed; - B2IQi) b. FOC: i. E(wage ;- Bo -B1Ed; - B2IQi) = 0 ii. EEd;(wage; - Bo - BIEd; - BzIQi)=0 iii. ?IQi (wage; - ?0 - BjEdi -PzIQi)=0 2. What are the 4 conditions to ensure that the estimators in Multiple Linear Regression are unbiased? a. Linear in Parameters b. Random Sampling - iid c. No Perfect Collinearity d. Zero Conditional Mean 3. What is the Frisch-Wold Theorem and how do you perform a partialling out of the MLR (2 ways)? Use the Multiple Linear Regression of wagei = Bo + B1Ed + B2IQ + u to demonstrate your answer. (find a way to obtain ß1 after regressing Ed on IQ a. Regress Ed on IQ i. Edi = do + a1IQi + Vi ii. Édi = â0 + â1IQi iii. Obtain îi = Edi - Edi b. Regress wage; on vi i. wage; = Yo + Y1Vi + e; ii. 1 is theoretically the same as 31 - it is the part of education that explains wage after the effects of IQ are "partialled-out" 4. What is the Omitted Variable Bias and prove that it exists by taking expectations of a regression that omits the variable IQ when the true regression is wage = Bo + B1Ed + B2IQ + v a. wage = Bo + ß1Ed + u b. E[1] = Cov(wage,Ed) = Cov(Bo+B1Ed+B2IQ+v, Ed) = B1Cov(Ed,Ed) + B2 Cov(Ed,IQ) = B1 + Var(Ed) Var(Ed) B2 Cov(Ed,IQ) Var(Ed) (bias) Var(Ed) Var(Ed) 5. What are the 2 conditions for the existence of an omitted variable bias? And what are the intuitive interpretation for each a. B2 + 0 i. IQ has an effect on wage b. Cov(Ed, IQ) + 0 i. IQ and Education are correlated 6. What are the conditions for upward/ downward bias of the estimate B1 a. Upward i. B2 > 0, Cov(Ed, IQ) > 0 ii. B2 < 0, Cov(Ed, IQ) < 0 b. Downward i. B2 > 0, Cov(Ed, IQ) < 0 ii. B2 < 0, Cov(Ed, IQ) > 0 7. What is the additional assumption required to obtain an estimate of the variance of the OLS estimator? a. Homoskedasticity 1
John Cai Jesus College 8. Write out the equation for the variation of the OLS estimator (for MLR) and explain the different components in it. a. Var(B1) = 04/ (1 - R2)(ST1) b. ô2 = Zu? /(n- k - 1) c. Where R2 is the R2 obtained from the regression of the independent variable on all other independent variables i. E.g. Regression of Ed on IQ d. SST1 = >(Edi - Ed) 9. What are the conditions for