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WISE International Masters ECONOMETRICS Instructor: Brett Graham INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This examination paper contains 32 questions.

1 WISE International Masters ECONOMETRICS Instructor: Brett Graham INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This examination paper contains 32 questions. You are REQUIRED to answer ALL questions. No marks will be deducted for wrong answers. 3 For each multiple choice question there is one and ONLY ONE suitable answer. 4 All numerical answers should be rounded to 3 decimal places. I will accept every answer to within of the correct answer. All probabilities should be expressed in decimal form. 5 This examination paper contains 17 pages including this instruction sheet, an answer sheet for the first 30 questions, an answer sheet for question 31, an answer sheet for question 32 and a blank page at the end of the exam. 6 This is a closed-book examination. You are allowed to bring one handwritten 105mm by 75mm piece of paper to the exam. You are also allowed to use a financial calculator. 7 You are required to return all examination materials at the end of the examination. 8 Where required, please use the following critical values, Tail-end Probabilities of the Normal Distribution z Pr(Z z) (%) % Significance Level Critical Values of the χ 2 m Distribution Degrees of Freedom (m) Critical Value

2 Regressor (a) (b) (c) (d) (e) (f) Age (0.002) (0.042) (0.042) (0.056) (0.056) (0.064) Age (0.001) (0.001) (0.001) (0.001) (0.001) F em (0.010) (0.010) (0.014) (1.230) (0.014) (1.240) F em Age (0.084) (0.084) F em Age (0.001) (0.001) Bach (0.010) (0.010) (0.014) (0.014) (1.228) (1.240) Bach Age (0.083) (0.084) Bach Age (0.001) (0.001) F em Bach (0.021) (0.021) (0.021) (0.021) Intercept (0.054) (0.613) (0.612) (0.819) (0.819) (0.945) SER R n Table 1: Regression results for Questions 1 to 6. The dependent variable is ln(ahe). Page 2

3 Use the information found in Table 1 to answer the next 6 questions (1-6): The data set used to estimate the regressions in Table 1 consists of information on 4000 full-time full-year workers. The variables are defined as follows: AHE = average hourly earnings (in 1998 dollars), College = binary variable (1 if college, 0 if high school), F emale = binary variable (1 if female, 0 if male), Age = age (in years). 1. Using Column (c) in Table 1, test if the coefficient on F em Bach is significant at the 8% level. Show the value of your test statistic and your conclusion (significant or not significant). Solution: t = = The coefficient is significant. 2. Using Column (a) in Table 1, construct a 96% confidence interval for the predicted change in ln(ahe) if Age increases from 25 to 35. Solution: UCL = 10 ( ) = LCL = 10 ( ) = Alexis is a 30-year-old female with a bachelor s degree. What does the regression in Column (d) in Table 1 predict for her value of ln(ahe)? Solution: ln(ahe) = = Page 3

4 4. Using Column (d) in Table 1, and assuming homoskedastic errors, test if the effect of Age on earnings is different for males than for females at the 5% significance level. Show the value of your test statistic and your conclusion (same or different). Solution: From the regression in Column (d), R 2 = k + R 2 (n k 1) n 1 = (7986 8) = From the regression in Column (c), R 2 = k + R 2 (n k 1) n 1 = (7986 6) = F = (R2 unrestricted R2 restricted )/q ( )/2 = (1 Runrestricted 2 )/(n k 1) /(7986 8) The effect of Age on earnings is the same for males and females. = The regression shown in which Column in Table 1 could be used for testing whether the effect of Age on earnings is different for high school graduates than college graduates? Solution: Either Column (e) or Column (f) is an acceptable answer. 6. Using Column (b) in Table 1, test if the effect of Age on earnings is nonlinear at the 4% significance level. Show the value of your test statistic and your conclusion (linear or nonlinear). Solution: t = = 2. The effect of Age on earnings is nonlinear. Page 4

5 Use the following information to answer the next 3 questions (7-8): The demand for a commodity is given by Q = P +u, where Q denotes quantity, P denotes price, and u denotes factors other than price that determine demand. Supply for the commodity is given by Q = P + v, where v denotes factors other than price that determine supply. Suppose that u and v both have a mean of zero, have variance of 4 and 3 and are uncorrelated. 7. What is the covariance between Q and v. Solution: cov(q, v) = (0.5/0.5) = What is the covariance between Q and P. Solution: cov(q, P ) = ( ) 2 = The entity fixed effects regression model a. has n different intercepts. b. allows the slope coefficient to differ across entities, but has the intercept fixed. c. in a log-log model may include logs of the binary variables, which control for the fixed effects. d. has fixed (repaired) the effect of heteroskedasticity. 10. Time fixed effects regression is useful in dealing with omitted variables a. even if you only have a cross-section of data available. b. if these omitted variables are constant across entities but vary over time. c. if these omitted variables are constant over time but vary across entities. d. when there are more than 100 observations. Page 5

6 11. If cov(u it, u is X it, X is ) = 0 for t s, then a. the statistical analysis is externally valid. b. conditional on the regressors, the errors are uncorrelated over time. c. the division of errors by regressors in different time periods is always zero. d. there is no correlation over time in the residuals. e. there is no perfect multicollinearity in the errors. 12. The binary dependent variable model is an example of a a. limited dependent variable model. b. model that cannot be estimated by OLS. c. model where the left-hand variable is measured in base 2. d. model with homoskedastic errors. e. regression model, which has as a regressor, among others, a binary variable. 13. The major flaw of the linear probability model is that a. the observed values of the dependedent variable are only 0 and 1, but the predicted values are almost never equal to 0 or 1. b. the predicted values can lie above 1 and below 0. c. people do not always make clear-cut decisions. d. the regression R 2 cannot be used as a measure of fit. 14. The following problems could be analyzed using probit and logit estimation with the exception of whether or not a. applicants will default on a loan. b. a college student decides to study abroad for one semester. c. a college student will attend a certain college after being accepted. d. being a female has an effect on earnings. Page 6

7 15. Consider the following linear probability model, where Pr(Y i = 1 X i ) = X i. What is var(u i X i = 5)? Solution: var(u i X i ) = var(y i β 0 β 1 X i X i ) = var(y i X i ) = E(Y 2 i X i ) (E(Y i X i )) 2 = 1 2 P r(y i = 1 X i ) P r(y i = 0 X i ) (1 P r(y i = 1 X i ) + 0 P r(y i = 0 X i )) 2 = β 0 + β 1 X i (β 0 + β 1 X i ) 2 = (β 0 + β 1 X i )[1 (β 0 + β 1 X i )]. Hence, var(u i X i = 5) = 0.6 (1 0.6 = Maximum likelihood estimation yields the values of the coefficients that a. are always different to those from OLS estimation. b. are typically larger than those from OLS estimation. c. come from a probability distribution and hence have to be positive. d. maximize the likelihood function. e. minimize the sum of squared prediction errors. 17. The logit model can be estimated and yields consistent estimates if you are using a. differences in means between those individuals with a dependent variable equal to one and those with a dependent variable equal to zero. b. maximum likelihood estimation. c. the linear probability model. d. OLS estimation. Page 7

8 18. The two conditions for a valid instrument are a. corr(z i, X i ) = 0 and corr(z i, u i ) = 0. b. corr(z i, X i ) = 0 and corr(z i, u i ) 0. c. corr(z i, X i ) 0 and corr(x i, u i ) 0. d. corr(z i, X i ) 0 and corr(z i, u i ) = 0. e. corr(z i, X i ) 0 and corr(z i, u i ) When there is a single instrument and a single regressor, the TSLS estimator for the slope SLS can be calculated as follows: ˆβT 1 = s a. ZX s s ZY. b.. c. ZY s s ZX. d. XY s ZY. e.. s ZY s 2 Z s XY s 2 X 20. Estimation of the IV regression model a. requires exact identification. b. allows only one endogenous regressor, which is typically correlated with the error term. c. is only possible if the number of instruments is the same as the number of regressors. d. requires exact identification or overidentification. 21. The rule-of-thumb for checking for weak instruments is as follows: for the case of a single endogenous regressor, a. the t-statistic on each of the instruments must exceed at least b. a first stage F > 1.96 indicates that the instruments are weak. c. a first stage F < 10 indicates that the instruments are weak. d. a first stage F must be statistically significant to indicate a strong instrument. 22. In practice, the most difficult aspect of IV estimation is a. finding instruments that are both relevant and exogenous. b. calculating the J-statistic. c. that you have to use two stages in the estimation process. d. finding instruments that are exogenous. Relevant instruments are easy to find. Page 8

9 23. Program evaluation a. establishes rating systems for television programs in a controlled experiment framework. b. is the field of study that concerns estimating the effect of a program, policy, or some other intervention or treatment. c. is conducted for most departments in your university/college about every seven years. d. tries to establish which statistical software program works best for your econometrics course. 24. The major distinction between the experiments and quasi-experiments chapter and earlier chapters is the a. superiority of TSLS over OLS. b. use of heteroskedasticity-robust standard errors. c. type of data analyzed and the special opportunities and challenges posed when analyzing experiments and quasi-experiments. d. frequent use of binary variables. 25. Assume that data are available on other characteristics of the subjects that are relevant to determining the experimental outcome. Then including these determinants explicitly results in a. the differences in means test. b. large scale equilibrium effects. c. the multiple regression model. d. the limited dependent variable model. 26. In a quasi-experiment a. the t-statistic is no longer normally distributed in large samples. b. quasi differences are used, i.e., instead of Y you need to use Ȳ after λȳ before, where 0 < λ < 1. c. randomness is introduced by variations in individual circumstances that make it appear as if the treatment is randomly assigned. d. the causal effect has to be estimated through quasi maximum likelihood estimation. Page 9

10 27. The slope coefficient in the model ln(y i ) = β 0 + β 1 X i + u i is interpreted as follows: a. a change in X by one unit is associated with a 100 β 1 % change in Y. b. a 1% change in X is associated with a change in Y of 0.01β 1. c. a 1% change in X is associated with a β 1 % change in Y. d. a change in X by one unit is associated with a β 1 change in Y. 28. To decide whether Y i = β 0 + β 1 X + u i or ln(y i ) = β 0 + β 1 X + u i fits the data better, you cannot consult the regression R 2 because a. ln(y ) may be negative for 0 < Y < 1. b. R 2 does not adjust for the number of explanatory variables. c. the regression R 2 can be greater than one in the second model. d. the second model is a non-linear function of X. e. the slope no longer indicates the effect of a unit change of X on Y in the log-linear model. f. the TSS are not measured in the same units between the two models. 29. The components of internal validity are a. a large sample, and an estimator with the BLUE property. b. a regression R 2 above 0.75 and serially uncorrelated errors. c. equivalence between the population and setting studied and the population and setting of interest. d. nonstochastic explanatory variables, and prediction intervals close to the sample mean. e. unbiasedness and consistency of the estimator, and the desired significance level of hypothesis testing. Page 10

11 30. You try to explain the number of IBM shares traded in the stock market per day in As an independent variable you choose the closing price of the share. This is an example of a. simultaneous causality. b. a situation where homoskedasticity-only standard errors should be used since you only analyze one company. c. sample selection bias since you should analyze more than one stock. d. invalid inference due to a small sample size. Page 11

12 Long Answers 31. You are interested in estimating the effect of adopting a value-added tax on economic growth. You collect a panel data set of economic growth for 100 randomly selected countries over 20 years from 1990 to 2010 and another dataset that identifies the year that the country adopted the VAT, if at all, and the rate of the adopted tax. You estimate the following economic model: Growth it = β 0 + β 1 V AT it + u it. where Growth it is economic growth in country i in year t and V AT it is a dummy variable that equals 1 if t is after the country adopted a VAT and 0 before the country adopted a VAT. (a) A friend reads your study and claims that your estimate is biased. He claims that individual country characteristics are related to whether or not a country adopts a VAT and are also related to that country s economic performance. How do you respond to your friend s claim, in particular how might you change your economic model to address your friend s concern? Solution: You could add entity fixed effects: Growth it = β 0 + β 1 V AT it + α i + u it. (b) Your friend also claims that your model ignores the fact that, over time, more countries have adopted a VAT. How do you respond to your friend s claim, in particular how might you change your economic model to address your friend s concern? Solution: You could add time fixed effects: Growth it = β 0 +β 1 V AT it +α i +δ t +u it. (c) Another friend claims that the effect of adopting a VAT on economic growth is positive if the rate of the adopted tax rate is small but negative if the adopted tax rate is large. How might you change your economic model to test your friend s hypothesis? Solution: You could construct the variable T R it which is equal to the value-added tax rate in country i in year t and estimate the following model: Growth it = β 0 + β 1 V AT it + β 2 T R it V AT it + β 3 T R 2 it V AT it + α i + δ t + u it. (d) Another friend claims that your model is misspecified since the effect of adopting a VAT on economic growth is different for developed and developing countries. How might you change your economic model to test your friend s hypothesis?. Solution: You could construct the variable H i which is equal to one if the country is a developed country and zero otherwise (you might like to use the country s initial level of GDP to define what a developed country is) and estimate the following model: Page 12 Growth it = β 0 +β 1 V AT it +β 2 T R it V AT it +β 3 T Rit V 2 AT it +β 3 H it +β 4 H it V AT it +α i +δ t +u it.

13 32. Consider observations (Y it, X it ) from the linear panel data model (5) Y it = β 1 X it + α i + λ i t + u it, t = 1..., T, i = 1..., N, where α i +λ i t is an unobserved individual-specific time trend. Construct an OLS model that can generate an unbiased estimate of β 1 or show how it is not possible to estimate β 1 using OLS regression. Solution: Y i,t Y i,t 1 = β 1 (X i,t X i,t 1 ) + λ i + (u i,t u i,t 1 ), that is Y i,t = β 1 X i,t + λ i + u i,t. Thus Y i,t Y i,t 1 = β 1 ( X i,t X i,t 1 ) + ( u it u it 1 ). By regressing Y i,t Y i,t 1 on X i,t X i,t 1, β 1 can be estimated. Page 13

14 Answer Sheet Econometrics Mid-term Exam Name: Question Points Answer Total: 15 Question Points Answer Total: 15

15 Answer Sheet Econometrics Mid-term Exam Name: Question 31 (5 points) Page 15

16 Answer Sheet Econometrics Mid-term Exam Name: Question 32 (5 points) Page 16

WISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, Academic Year Exam Version: A

WISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, 2015-16 Academic Year Exam Version: A INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This