Economics Introduction to Econometrics - Fall 2007 Final Exam - Answers

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1 Student Name: Economics Introduction to Econometrics - Fall 2007 Final Exam - Answers SHOW ALL WORK! Evaluation: Problems: 3, 4C, 5C and 5F are worth 4 points. All other questions are worth 3 points. 1. Answer the following questions: A) What is the consequence of specifying a model with a varable in log form, if in the population model, the variable is in level form? This is a case of Functional Form Misspeci cation, which causes the OLS estimators to be biased. B) Suppose the true income-consumption model is cons = inc + 2 inc 2 + u. What is the consequence, if we estimate the model without the quadratic term inc 2? This is another case of Functional Form Misspeci cation, which causes the OLS estimators to be biased. It is also a case of omitted variable. C) Why does the simple regression model y = x + u typically fail to uncover the ceteris paribus e ect of x on y? There are typically unobserved factors of y, which are correlated with x, and enter into the error term, causing the OLS estimators to be biased. D) Is a regression with a low R 2 useless? Explain. What does a low R 2 imply about the speci ed regression model? Not necessarily. However, the low R 2 implies that the included regressors do not explain much of the variation in y. That is, there are important omitted factors. E) What would be the likely sign of the bias of the coe cient on IQ if we omitt edu from themodel: log (wage) = IQ + 2 edu + 3 tenure + u? Bias f1 = e1 2 2 is expected to be > 0 e 1 = Corr (IQ; edu) is expected to be > 0 Bias f1 is expected to be > 0 F) What is the e ect of increasing the sample size on se cj? Explain. 1

2 Increasing the sample size decreases the error variance V ar (u) = 2 and so r se cj b = 2 SST j(1 Rj) will decrease, where 2 b2 is the estimator of 2. G) What is the e ect of increasing the sample size on bias cj? Explain. The bias in c j remains as the sample size increases. 2. Suppose you have estimated the following linear probability model explaining 401(k) eligibility in terms of of income, age, and gender: \e401k = :506 + :0124inc :000062inc 2 + :0265age :00031age 2 :0035male (:081) (:0006) (:000005) (:0039) (:00005) (:0121) n = 9; 275; R 2 = :094 where e401k is a binary variable for eligibility in a 401(k) plan (e401k = 1 if eligble for 401(k), and = 0 otherwise), inc denotes family annual income (in $1; 000), age denotes the individual s age (in years), and male is a binary variable for gender (male = 1 if male individual, and = 0 otherwise). A) Based on the reported regression results, would you say that 401(k) eligibility depends on income in a statistically signi cant way? Explain. (show test statistic, critical value, rejection rule, etc.) 401(k) eligibility clearly depends on income. Each of the two terms involving inc have very signi cant t statistics. B) What is the estimated e ect of income on 401(k) eligibility? (Be speci c. Use the estimated coe cients). Interpret your inc = : (:000062) inc C) Holding other factors xed, for an individual with family income of $200; 000, if annual income increases by $1; 000, what happens to the probability of 401(k) inc = : (:000062) (200) = 0:012 4 So, for an individual with family income of $200; 000, if annual income increases by $1; 000 (inc increases by 1), the estimated 401(k) eligibility decreases by 0: D) Interpret the coe cient on male. The di erence in the predicted 401(k) eligibility between male and female individuals. 2

3 E) Is there statistically signi cant evidence of gender discrimination? (show test statistic, critical value, rejection rule, etc.) t male = :0035 :0121 = 0: t 5% = 1:96 Reject H 0 : male = 0 if j 0:289 26j > 1:96. So, cannot reject H 0 or male is not statistically di erent from 0. Therefore, there isn t su cient evidence of gender discrimination. F) We add the variable pira, a binary variable representing whether a family member has an individual retirement account (IRA) (pira = 1 if a family member has an IRA, and = 0 otherwise), as an explanatory variable to the model. Its estimated coe cient is [ pira = :0198 with a two-sided p value = :105. Is that estimated coe cient statistically signi cant at the 1%; 5% or 10% level? No, pira is not statistically signi cant even at the 10% level. G) Now, we specify the following version of the above model: e401k = inc + 2 inc age + 4 age black_male + 7 black_female + 8 nonblack_female + u What is the interpretation of 6? It is the predicted di erence in e401k for black male and nonblack male (the base group). H) Refering to the model in part G, what is the interpretation of ( 6 7 )? It is the predicted di erence in e401k for black male and black female. I) Write the model that you would estimate to determine whether ( 6 7 ) is statistically signi cant. Choose black_f emale as base group. J) Using the estimation results from the original model \e401k = :506 + :0124inc :000062inc 2 + :0265age :00031age 2 :0035male (:081) (:0006) (:000005) (:0039) (:00005) (:0121) predict the 401(k) eligibility of a person with the following characteristics: inc = 100; age = 30, and female. Does the predicted probability make sense? What is the reason for obtaining this result? Explain what exactly is going on. 3

4 0:63 \e401k = :506 + :0124 (100) : (100) 2 + :0265 (30) :00031 (30) 2 :0035 (0) = Yes, it is in the range [0; 1]. 3. Suppose you want to estimate the following model y = x x x 3 + u Describe how you would test for heteroskedasticity using the special case of the White test? Describe each step, state the null and the alternative hypotheses, the test statitsic and the rejection rule. (i) Estimate y = x x x 3 + u and obtain the resduals bu (ii) Estimate bu 2 = by + 2 by 2 + u (iii) Test H 0 : 1 = 0; 2 = 0; F = R 2 bu 2 =k 1 R 2bu 2 =(n k 1) 4. Suppose you want to estimate the following capital asset pricing model for some company: R f t = R m t + u t where R f t is the company s return at time t and Rm t is the market return at time t. A) How would you augment this model to test whether the announcement of the retirement of the company s CEO has had a signi cant efect on the company s stock return? What will be the null and the alternative hypothesis, what test would you use? R f t = R m t + 2 d t + u t Include a dummy variable, say d t, for the day of and a few days after the announcement. Test H 0 : 2 = 0 against H 1 : 2 6= 0. using a t-test. B) How would you further augment the model to test whether there has been insider trading on the information of the CEO retirement? R f t = R m t + 2 d t + 3 b t + u t Include another dummy variable, say b t, for a few days before the announcement. Test H 0 : 3 = 0 against H 1 : 3 6= 0 using a t-test. 4

5 C) Assuming that Rt m is strictly exogenous, how would you test whether the errors in the model R f t = Rt m + u t are serially correlated? Describe each step, the null and the alternative hypotheses, the test statistic and the rejection rule. (i) Estimate R f t = Rt m + u t and obtain the residuals bu t (ii) Estimate bu t = du t 1 + e t and obtain b (iii) Test H 0 : = 0 against H 1 : 6= 0 using a t-test. 5. Suppose you have esimated the following model of fertility and personal exemption: dgfr t = 92:05 + :089pe t :0040pe t 1 + :0074pe t 2 21:34ww2 t 31:08pill t (3:33) (:126) (:1531) (:1651) (11:54) (3:90) n = 68; R 2 = :537 where gfr births per 1; 000 women in child bearing age, and pe is average real dollar value of the personal tax exemption. A) Interpret the coe cient on pe t. Be speci c. It is the impact multiplier. It implies that a one-dollar increase in pe t at time t, is predicted to increase fertility at time t by :089 births per 1; 000 women. B) What is the long-run e ect of pe on gfr? (:089 : :0074) = 0:092 4 C) How would you test whether the long-run e ect is statistically signi cant? Let s denote the coe cients of pe t, pe t 1 and pe t 2 by 1, 2 and 3, respectively. De ne a new parameter = , which equals the long-run multiplier. Solve for 1 : 1 = ( ) : Plug the result in the original regression and rearrange: gfr t = pe t + 2 pe t pe t ww2 t + 5 pill t + u t gfr t = 0 + [ ( )] pe t + 2 pe t pe t ww2 t + 5 pill t + u t gfr t = 0 + pe t + 2 (pe t 1 pe t ) + 3 (pe t 2 pe t ) + 4 ww2 t + 5 pill t + u t Use a t test for the statistical signi cance of (H 0 : = 0 against H 1 : 6= 0): 5

6 D) How would you augment the model if you suspect that some of the variables exhibit a linear time trend? Add a linear trend regressor, t: gfr t = pe t + 2 pe t pe t ww2 t + 5 pill t + 6 t + u t E) Suppose that gfr and pe both have upward time trends, what would be a potential consequence of not including a time trend in the regression model? What is this phenomenon called? Spurious correlation between gf r and pe. F) Suppose that gfr and pe both have upward time trends. How would you augment the model to obtain a goodness-of- t measure, which is not a ected by the presence of time trend. Show all work. (i) Regress gfr = t + e t and obtain the residuals detrended gf r. (ii) Replace gfr with :: gfr in the original regression. The R 2 regression is not a ected by the trending of gfr. :: gfr, which are the from the last town 6. Suppose you have tested a model of rent rates and student population in a college \ log (rent) = 1:39 + :066 log (pop) + :507 log (avginc) + :0056pctstu + u (:844) (:039) (:081) (:0017) n = 64; R 2 = :458 where rent is the average monthly rent paid on rental units in a college town in the United States, pop denotes the total city population, avginc denotes the average city income, and pctstu denotes the student population as a percentage of the total population. A) Suppose you want to test H 0 : log(avginc) = 0:5 against H 1 : log(avginc) 6= 0:5. Construct the 95% con dence interval for the parameter log(avginc). The 5% critical value for a two-tailed test with df = 64 4 = 60 is 2:00. The 95% con dence interval for 2 is: :507 2 (:081) = [0:345 ; 0:669] : B) Test the hypothesis in part A using the calculated con dence interval for log(avginc). Since :5 is within the 95% con dence interval for 2, then we fail to reject H 0 : 2 = 0:5 against H 1 : 2 6= 0:5 at the 5% level. 6

7 C) Write down the t statistic for testing H 0 : log(pop) = 2 pctstu against H 1 : log(pop) > 2 pctstu at the 5% level. (Do not try to calculate or simplify.) t = c 1 c 3 se c1 c 3 D) Write the rejection rule for the test in part C. Make sure to report the critical value for the test. (Do not try to perform the test.) t < 1:671 E) Test the joint signi cance of log(pop), 2 pctstu and log(avginc) at the 5% level, that is, test H 0 : log(pop) = 0; 2 pctstu = 0 and log(avginc) = 0: This is a test for overall regression signi cance, that is, we test whether all slope coe cients are equal to 0 and so Rr 2 = 0. R F = u=k 2 (1 Ru 2 )=(n k 1) = :458=3 (1 :458)=(64 4) = 16: 9 > t 5% Reject H 0. The coe cients log(pop), 2 pctstu and log(avginc) are jointly signi - cant at the 5% level. 7

Exercise sheet 3 The Multiple Regression Model

Exercise sheet 3 The Multiple Regression Model Note: In those problems that include estimations and have a reference to a data set the students should check the outputs obtained with Gretl. 1. Let the