08 Aug Econometrics 3220 Spring 2020Problem Set 2This ass
Econometrics 3220 Spring 2020Problem Set 2This assignment is due on Thursday, the 5th of March. A hard copy has to behanded in at the beginning of lecture.Question 1This exercise asks you to prove some theoretical results. It is important to show all thesteps in your proofs.Consider the following simple linear regression model:Yi = 0 + 1Xi + ui i = 1; : : : ; nassuming E(ujX) = 0 and (Yi;Xi), i = 1 : : : n are iid.(a) What is the criterion for obtaining ordinary least squares estimators for 0 and 1?Write it using mathematical notation.(b) Using the objective function specified in (a), derive the first order conditions andsolve them simultaneously to recover the following formulae for the OLS estimators:b 0 = Y b 1 Xand b 1 =Pni=1Xi X Yi YPni=1Xi X2with: X= 1nPni=1 Xi and Y = 1nPni=1 Yi:(c) What does it mean, in words, for an estimator to be unbiased? Prove that b 1 is anunbiased estimator under the given assumptions. The following steps will guideyou through the proof:(i) Using b 1 =Pni=1Xi XYi Pni=1Xi X2 , plug in Yi and simplify to show thatb 1 = 0(0) + 1(1) +P(Xi XP )ui(Xi X)2.You might be wondering why we are not using the expression for b 1 derived in(b). In fact, both the expressions are identical as you can easily show the twonumerators are equal to each otherPni=1Xi XYi =Pni=1Xi X Yi Y.For unbiasedness, it is more convenient to use the expression in (i) which isjust one of the several equivalent forms(ii) Condition on X (which allows you to treat X as non-random) and take expectationon both sides. Use the assumption E(uijXi) = 0 to show thatE( b 1) = 1.(d) We further assume now that V ar(ujX) = 2, assumed to be known. This is theassumption of homoskedasticity. Under this assumption, it can be shown that,V ar( b 1jX) =2Pni=1Xi X2Discuss intuitively how the variance of the estimator b 1 depends on the error varianceand the variation in X.Question 2Consider the following Cobb-Douglas production function Yi = AK 1i L 2i eui (where Yis output, A is the level of technology, K is the capital stock, L is the labor force and uis the error term)Using data on (Yi;Ki; Li) for i = 1; :::; n firms (where n is a large number) explainhow you would test for constant returns to scale. Write down the regression you wouldrun (how would you interpret the coefficients in this regression?) and the null hypothesisfor constant returns to scale. Form the t-statistic for your test and show how you wouldget the numerator and the denominator for this t statistic. When would you reject thenull of constant returns to scale at the 5% significance level?Question 3Suppose a researcher, using wage data on 200 randomly selected male workers and 240female workers, estimates the OLS regression (standard errors in parentheses belowcoefficients)Wage = 10:73(0:16)+ 1:78(0:60)Male;R2 = 0:09; SER = 3:8where Wage is measured in dollars per hour and Male is a binary variable that isequal to 1 if the person is a male and 0 if the person is a female. Define the wage-gendergap as the difference in mean earnings between men and women.(a) What is the estimated gender gap?(b) Do men earn significantly more than women? Compute the p-value for testing thenull hypothesis that there is no gender gap against the one-sided alternative that menearn more (refer to the normal table on pg. 804 of your textbook).(c) Construct a 95% confidence interval for the gender gap.(d) In the sample, what is the mean wage of women? What is the mean wage of men?(e) Another researcher uses these same data but regresses Wages on Female, a variablethat is equal to 1 if the person is female and 0 if the person is male. What are thecoefficients (intercept and slope) calculated from this regression? What is the R2value? How do you interpret it?(f) Does it make sense to assume that E(ujMale) = 0 where u is the error term in themodel,Wage = 0 + 1Male + uDiscuss the implications of this assumption being violated. Based on your answerin (b), can you claim gender discrimination against females if E(ujMale) 6= 0?Explain.Question 4Consider the following multiple regression modelYi = 0 + 1X1i + 2X2i + uiYou want to consider certain hypotheses involving more than one parameter, and youknow that the regression error is homoskedastic. You decide to test the joint hypothesesusing the homoskedasticity-only F-statistics. For each of the cases below specify arestricted model and indicate how you would compute the F-statistic to test for thevalidity of the restrictions at the 1% level.1. 1 = 2; 3 = 02. 1 + 2 + 3 = 13. 1 = 2; 3 = 0
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