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Antony Hudson
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1 1. (25 points) In 1992, there was an increase in the (state) minimum wage in one U.S. state (New Jersey) but not in a neighboring location (eastern Pennsylvania). The study provides you with the following information, PA NJ FTE Employment before FTE Employment after Where FTE is full time equivalent and the table reports the average FTEs per fast food restaurant. (a) Calculate the change in the treatment group, the change in the control group, and finally ˆ diff in diff (b) Since minimum wages represent a price floor, did you expect positive or negative? ˆ diff in diff to be (c) The standard error for ˆ diff in diff is Test whether or not the coefficient is statistically significant, given that there are 410 observations. (d) If you believed that the benefit from small minimum wage increases outweighed the cost in terms of employment loss, would finding that this coefficient was not statistically significant discourage you? (e) Do you have any concerns about forming a conclusion about the effect of the minimum wage law using this approach? (Stronger answers will make specific reference to the information presented.)

2 2. ( 20 points) You have estimated the effect of X on Y using a number of different models: Parameter Estimates and Standard Errors Variable Pooled Between First Difference Fixed Xit (0.0842) (0.0542) (0.0321) (0.0242) Random (0.0223) (a) What is the Hausman test statistic for the null hypothesis that u i and X it are uncorrelated? (b) Can you reject the null? (c) What does that mean about which estimator you would prefer to use? (d) Explain the intuition for this test. [Why does the answer to (b) imply the estimator you provided in (c)?]

3 3. (20 points) A classmate is interested in estimating the variance of the error term in the following equation y i =β 0 + β 1 x i + u i where i denotes entities, y is the dependent variable, and x is an explanatory variable for each entity and z is an instrument. 2 Suppose that she uses the following estimator for u from the second-stage regression of TSLS: 2 ˆ 1 n 2 ˆ TSLS ˆ TSLS ( y i xˆ ) 0 1 i 2 where xˆ i is the fitted value from the first-stage regression. Is this a consistent 2 u estimator for? 4. Evans and Schwab (1995) studied the effect of attending Catholic high school on the probability of attending college. Let college be a binary variable equal to 1 if a student attends college and zero otherwise. Let CathHS be a binary variable equal to 1 if a student attends a Catholic high school and zero otherwise. A linear probability model is college i = β 0 + β 1 CathHS i + u i (i) (5 pts) Interpret the magnitudes of the OLS coefficient estimates of β 0 and β 1 (I am looking for more substantive answers than β 0 is the constant. ) (ii) (5 pts) Why might CathHS be correlated with u? (iii) (10 pts) Let CathRel be a binary variable equal to 1 if a student is Catholic. How would you interpret the 2SLS estimate of β 1, using this variable as an IV for CathHS? Use the definition of the Wald estimator in your answer.

4 (iv) (5 pts) Do you think CathRel is a convincing instrument for CathHS? Explain.

5 5. You would like to study the impact of price on demand. By luck you have access to sales and prices for fish sales for 97 days. Since fish prices change often this seems like a good market to look at. Given the data at hand you write down the following model: q i =β 0 +β 1 p i +β 2 Mon i +β 3 Tues i +β 4 Wed i +β 5 Thurs i +β 6 t i +u i i=1,....,97 where q is the quantity (in pounds) purchased on a particular day p is the price on a particular day (per pound) t is a trend variable (1 for 1 st day, 2 for 2 nd day, etc.) Mon =1 if the day is Monday Tues =1 if the day is Tuesday Wed =1 if the day is Wednesday Thurs =1 if the day is Thursday The omitted variable is Friday; no fish are sold on the weekend (the market is closed). OLS Estimates Dependent variable is q Number of observations = 97 R 2 = 0.23 Coefficient Standard Error p Mon Tues Wed Thurs t Constant (a) (5 pts) Based on the OLS results above, is price an important determinant of quantity demanded? (Comment on both the magnitude and statistical significance.) (b) (5 pts) Do you have any concerns about the OLS estimate of β 1?

6 (c) (10 pts) Weather conditions can be assumed to affect supply while having a negligible effect on demand. Average wind speed (wind) and average wave height (wave) over the past two days are proposed as instrumental variable for price in the demand equation. What needs to be true for these variables to be valid instruments? (d) (10 pts) Given the attached regression results reported below, discuss the validity of wind and wave. You may need to use the provided F or chi-squared tables. (Note that not all of the information provided may be relevant.) (e) (5 pts) Give what you found in (d), explain what these results imply about the TSLS estimates.

7 OLS Model p i = π 0 + π 1 wind i + π 2 wave i + π 3Mon i + π 4 Tues i + π 5 Wed i + π 6Thurs i + π 7 t i +v i Dependent Variable is p Number of observations = 97 R 2 = 0.33 Coefficient Standard Error wind wave Mon Tues Wed Thur t Constant H0: π 1 = 0 H1: π 1 0 Test Statistic: F(1, 89) = 0.31 H0: π 2 = 0 H1: π 2 0 Test Statistic: F(1, 89) = H0: π 1 = π 2 = 0 H1: H0 is not true Test Statistic: F(2, 89) = H0: π 3 = π 4 = π 5 = π 6 =0 H1: H0 is not true Test Statistic: F(4, 89) = 0.63 TSLS Model qi 0 1 pˆ i 2Moni 3Tues i 4Wed i 5Thurs i 6ti ui p i = π 0 + π 1 wind i + π 2 wave i + π 3Mon i + π 4 Tues i + π 5 Wed i + π 6Thurs i + π 7 t i +v i TSLS uˆ i 0 1wind i 2Wave i 3Moni 4Tuesi 5Wed i 6Thurs i Dependent Variable = q Instruments = wind and wave Number of observations = 97 7 Coefficient Standard Error pˆ Mon Tues Wed Thurs t Constant H0: β 1 =0 H1: β 1 0 Test Statistic: F(1, 90) = 5.43 H0: β 2 = β 3 = β 4 = β 5 =0 H1: H0 is not true Test Statistic: F(4, 90) = 4.02 TSLS Dependent Variable = û Number of observations = 97 R 2 = 0.02 Coefficient Standard Error wind wave Mon Tues Wed Thurs t Constant H0: θ 1 =0 H1: θ 1 0 Test Statistic: F(1, 94) = 1.52 H0: θ 2 =0 H1: θ 2 0 Test Statistic: F(1, 94) = 0.29 H0: θ 1 =θ 2 =0 H1: H0 is not true Test Statistic: F(2, 94) = 0.77 t i e i

8 1. There are different ways to combine features of the Breusch-Pagan and White tests for heteroskedasticity. One possibility that we did not cover is to run the regression: 2 uˆi on x 1i, x 2i, x 3i,...., x ki, and 2 where the uˆi are the squares of the OLS residuals (estimated under homoskedasticty) and 2 the yˆi are the squares of the predicted values from the OLS regression. a) (6pts) How would you construct the LM statistic for this test? What are its degrees of freedom? 2 y ˆi b) (5pts) How will the R 2 from this regression compare with the R 2 from the usual BP test and the White test? c) (5pts) Does this imply that the new test statistic always delivers a smaller/larger p-value than the BP or White tests? Explain. d) (5pts) Suppose someone suggested also adding i ŷ to this regression. What do you think of this idea?

9 2. For one semester, you collect data on a random sample of college juniors and senior for each class taken. In other words, you have multiple observations for each student, with a different observation for each class the student takes. This information includes measures of a standardized final exam score, percentage of lectures attended, a dummy variable indicating whether the class is within the student s major, cumulative GPA prior to the semester, and SAT score. a) (6pts) Write a model for final exam performance in terms of attendance and other characteristics. Use subscripts for students and classes. b) (6pts) Suppose that your main interest is the effect of attendance on final exam performance. If you pool all of the data and use OLS, what are you assuming? Explain this in the context of the other variables in your model.

10 c) (10 pts) What other models (other than OLS) might you consider using? Carefully describe under what circumstances would you use those models?

11 3. Yet another attempt to identify the returns to education! Based on 1991 data from the NLSY, you run a regression of log wages (lwage) on the highest year of education completed (educ):. reg lwage educ Source SS df MS Number of obs = F( 1, 1228) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lwage Coef. Std. Err educ _cons a) (5pts) Interpret the coefficient on educ. Comment on its magnitude and statistical significance. b) (5pts) Of course, having taken ECON562, you know that this return may not represent the causal effect of education. Why not?

12 c) (6pts) You are considering using the change in college tuition for an individual from age 17 to age 18, ctuit, as a potential instrument. This variable is measured in thousands of dollars. What conditions should ctuit satisfy in order to be a valid instrument? d) You use ctuit as an instrument to predict education, and include use the predicted values of education (educhat1) to estimate the return to education. Your second stage results are the following:. reg lwage educhat1 Source SS df MS Number of obs = F( 1, 1228) = 3.72 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lwage Coef. Std. Err educhat _cons Assume for this part of the question that ctuit is a valid instrument. e) (3pts) Interpret the coefficient on educ. What is the 95% confidence interval for the return based on these results?

13 f) (8pts) What do the results thus far indicate about whether education is exogenous? (Your answer should include a test statistic and your interpretation of that test statistic.) g) (5pts) After puzzling over this, you remember that you were supposed to check your first stage results. Those results are Education Regression #1 on the attached sheet. Based on these, comment on the validity of ctuit as an instrument. h) (6pts) Does your answer in (g) affect your conclusions in (f) about the exogeneity of education? Explain.

14 i) (7pts) After thinking about this some more, you decide other control variables are warranted. You include a quadratic in years of experience (exper), dummy variables for region of the country where a person resides, and a dummy variable for whether or not the person lives in an urban area. You aren t exactly sure what to do here, so you estimate a bunch of regressions and make several predictions for education (See the attached sheet Education regressions #1 and #2 and Wage Regressions #1- #4). Which of the wage regressions gives you the best estimate of the return to education and why? What is that estimated return? j) (5pts) Comment on the validity of ctuit as an instrument based on the education regression you choose.

15 k) (7pts) Explain why your conclusions about (1) the validity of the instrument and (2) the return to education changed in the way that they did after adding control variables. What exactly changed that affected your conclusions (coefficients? Standard errors?) Can you propose an explanation for why those parameters changed in the way that they did?

16 Education Regression #1 and Prediction educhat1. reg educ ctuit Source SS df MS Number of obs = F( 1, 1228) = 0.35 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = educ Coef. Std. Err. t P> t [95% Conf. Interval] ctuit _cons predict educhat1, xb. test ctuit ( 1) ctuit = 0 F( 1, 1228) = 0.35 Prob > F = Education Regression #1 and Prediction educhat2. reg educ ctuit exper expersq ne nc west urban Source SS df MS Number of obs = F( 7, 1222) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = educ Coef. Std. Err. t P> t [95% Conf. Interval] ctuit exper expersq ne nc west urban _cons test ctuit ( 1) ctuit = 0 F( 1, 1222) = Prob > F = predict educhat2, xb

17 Wage Regression #1: reg lwage educ exper expersq ne nc west urban Source SS df MS Number of obs = F( 7, 1222) = Model Prob > F = Residual R-squared = lwage Coef. Std. Err. t P> t [95% Conf. Interval] educ exper expersq ne nc west urban _cons Wage Regression #2: reg lwage ctuit exper expersq ne nc west urban Source SS df MS Number of obs = F( 7, 1222) = Model Prob > F = Residual R-squared = lwage Coef. Std. Err. t P> t [95% Conf. Interval] ctuit exper expersq ne nc west urban _cons Wage Regression #3: reg lwage educhat1 exper expersq ne nc west urban Source SS df MS Number of obs = F( 7, 1222) = Model Prob > F = Residual R-squared = lwage Coef. Std. Err. t P> t [95% Conf. Interval] educhat exper expersq ne nc west urban _cons Wage Regression #4: reg lwage educhat2 exper expersq ne nc west urban Source SS df MS Number of obs = F( 7, 1222) = Model Prob > F = Residual R-squared = lwage Coef. Std. Err. t P> t [95% Conf. Interval] educhat exper expersq ne nc west urban _cons

18 1. For a large university, you are asked to estimate the demand for tickets to women s basketball games. You can collect time series data over 10 seasons, for a total of about 150 observations (assuming about 15 games a season. One possible model is lattend t = β0+ β1lprice t + β2 WINPERC t +β3 RIVAL t +β4weekend t + u t where lattend is the natural log of attendance at a game, lprice is the natural log of the real price of admissions, WINPERC is the team s winning percentage, RIVAL is a dummy variable indicating whether the game is with the major rival, and WEEKEND is a dummy variable for whether or not the game is on a weekend. a. Do you think a time trend should be included in the equation? Why or why not? b. The supply of tickets is fixed by stadium capacity which has not changed over this time period. Does this mean that price is necessarily exogenous (uncorrelated with u t ) in the equation? Explain. c. Suppose that the nominal price of admission changes slowly say at the beginning of each season. The athletic office chooses the price based partly on last season s average attendance, as well as the last season s team success. If lpricet is in fact endogenous, under what assumptions is last season s winning percentage a valid instrument for lpricet?

19 d. Suppose you are worried that some of the series, particularly lattend and lprice might have unit roots. Why would this be a concern? How might you change the estimated equation if these series did in fact have unit roots? e. If some games are sold out, what problems does this cause for estimating the demand function?

20 2. One type of partial adjustment model is y* t = x t +e t y t -y t-1 = (y* t -y t-1 ) + u t where y* t is the desired or optimal level of y and y t is the actual observed level. For example, y* t is the desired growth in firm inventories and x t is growth in firm sales. The parameter 1 measures the effect of x t on y* t. The second equation describes how the actual y depends on the relationship between the desired y in time t and the actual y in time t-1. The parameter measures the speed of adjustment and satisfies 0< <1. a. Show that we can rewrite the model as a regression of y t on x t and y t-1. Show how the coefficients in this model relate to the parameters in the two equation model above, and show how the error term relates to the error terms above. b. If E(e t x t, y t-1, x t-1, y t-2,....) = E(u t x t, y t-1, x t-1, y t-2,....) = 0 and all series are weakly dependent, if we estimate the model you specified in (a) by OLS will we get consistent estimates? Explain why or why not.

21 c. Will the errors in your model in (a) be serially correlated? Explain why or why not.

22 3. Let hy6 t denote the three-month holding yield (in percent) from buying a 6 month T-bill at time (t-1) and selling it at time t (three months later) as a three month T bill.let hy3 t-1 be the three month holding yield (in percent) from buying a 3 month T bill at time (t-1). At time (t-1), then, hy3 t-1 is known, whereas hy6 t is unknown because the price in time t of three month T-bills is unknown at time t-1. The expectations hypothesis says that, of course, these two different three month investments should be the same, on average. In other words, Suppose you were to estimate the model E(hy6 t all information up to time t-1) = hy3 t-1 hy6 t = hy3 t-1 + u t a. 4 pts How would you test the expectations hypothesis using this model? What is the null hypothesis? b. 4pts Suppose your estimates of that equation produced the following ^hy6 t = hy3 t-1 + u t (.070) (.039) N=123, R 2 =.866 Do you reject the test in (a) at the 5% significance level?

23 c. 4pts Another implication of the expectations hypothesis is that no other variables dated as t-1 or earlier should explain hy6 t after controlling for hy3 t-1. Suppose you were to test this implication by estimating the following equation, which includes the lagged spread between the 6 and 3 month T-bill rates: ^hy6 t = hy3 t (r6 t-1 r3 t-1 ) (.067) (.039) (.109) According to this, is the lagged spread term significant at the 5% level? What do the results imply about whether you should invest in 6-month or 3-month T bills if at time t-1, r6 is greater than r3? d. 3pts Conduct the test you specified in (a) using the results in (c). Do you conclude anything different from your results in (b)? e. 6pts The sample correlation between hy3 t and hy3 t-1 is.914. Does this raise any concerns with your previous analysis? Be specific.

24 4. For the US economy, let gprice be the monthly growth in prices and gwage be the monthly growth in wages (gprice = log(price); gwage= log(wage)) Using monthly data, suppose we estimate a distributed lag model: Dependent var: gprice Standard errors in parentheses Constant (.00057) gwage.119 (.052) gwage t (.039) gwage t (.039) gwage t (.039) gwage t (.039) gwage t (.039) gwage t (.039) gwage t (.039) gwage t (.039) gwage t (.039) gwage t (.039) gwage t (.039) gwage t (.052) N=273 R 2 =.317, Adj. R 2 =.283 a. 4 pts What is the long run propensity? Is it much different from one? Explain what the LRP tells us.

25 b. 6 pts What regression would you run to obtain the standard error of the LRP directly? c. 6pts How would you test the joint significance of 6 more lags of gwage? Be specific about the distribution of your test statistic (ie., if the distribution of the test statistic depends on degrees of freedom, be sure to state these.)

26 No. of Incidents 5. The Report of the Presidential Commission on the Space Shuttle Challenger Accident in 1986 shows a plot of the calculated joint temperature in Fahrenheit and the number of O-rings that had some thermal distress. You collect the data for the seven flights for which thermal distress was identified before the fatal flight and produce the accompanying plot Temperature a. 5 pts Do you see any problems other than sample size in fitting a linear model to these data? b. 6pts You look at all successful launches before Challenger, even those with no incidents. You simplify the problem by specifying a binary variable, Ofail, which equals 1 if there was some O-ring failure and is 0 otherwise. You fit a linear probability model: ^OFail = Temperature (0.496) (0.007) The numbers in parentheses are heteroskedasticity-robust standard errors. Interpret the equation. What is your prediction for some O-ring thermal distress when the temperature is 31 o, the temperature on January 28, 1986? Above which temperature do you predict values of less than zero? Below which temperature do you predict values of greater than one?

27 c. 5pts Why do you think that heteroskedasticity-robust standard errors were used? d. 5pts To fix the problem encountered in (b), you re-estimate the relationship using a logit regression: Pr(OFail=1 Temperature)=G ( Temperature) (7.329) (0.107) Recall that the logistic function is z e G( z) 1 e Calculate the effect of a decrease in temperature from 80 0 to 70 0, and from 60 0 to 500. Why is the change in probability not constant? How does this compare to the linear probability model? z e. 5pts You want to see how sensitive the results are to using the logit, rather than the probit estimation method. The probit regression is as follows: Pr(OFail=1 Temperature)=Φ ( Temperature) (3.983) (0.058) Why is the slope coefficient in the probit so different from the logit coefficient?

28 6. Suppose an author has data on outcomes (Y) for two periods, 1 and 2, for a number of cross-sectional units (e.g., firms). Suppose also that a subgroup of firms were hit with an intervention (e.g., minimum wage hike) sometime between period 1 and 2. a. 6 pts Write an equation for the difference-in-difference estimate (e.g., one that controls for group and time effects) that provides a consistent estimate of the intervention by using the first differences in outcomes between two periods. That is, the dependent variable would be ΔY i = Y 2i -Y 1i. b. 5 pts What is the advantage of the difference-in difference approach in this context? c. 5pts Are there potential problems with inference that the difference-indifference approach does not resolve?

29 7. Consider the bivariate regression model y i = α + βx* i + u i We do not observe the true x* i, but instead estimate y i = α + βx i + v i where the covariate of interest x i is subject to classical measurement error. In particular, assume that the measured value x i is related to the true (unobserved) value x i* according to the linear equation: x i = x i * + e i where e i is an iid random error, with E[e i ]=0 and var(e i )=σ 2 e and Cov(e i,x i *)=0. (Also assume X and u are uncorrelated in expectation.) We demonstrated in class that with measurement error, ˆ is biased and inconsistent. a. 5pts Suppose there is an instrument Z i where Cov(Z, v)=0, Cov(Z,e)=0 but Cov(Z,X) 0. In the presence of measurement error in X, is the IV estimate of ˆ consistent? Explain. b. 5pts Now consider this in a time series context: y t = α + βx* t + u t; x t = x t * + e t To simplify the algebra for the rest of this question, assume that the mean of x t * is zero. In addition to the previous assumptions, assume that u t and e t are uncorrelated with all past values of x t * and et. Show that E(x t-1, v t ) = 0, where v t is the error term in the model from part (b).

30 c. 5pts Are x t and x t-1 likely to be correlated? Explain. d. 6 pts What does all of this suggest as a useful strategy for consistently estimating β?

Lecture 8: Instrumental Variables Estimation

Lecture Notes on Advanced Econometrics Lecture 8: Instrumental Variables Estimation Endogenous Variables Consider a population model: y α y + β + β x + β x +... + β x + u i i i i k ik i Takashi Yamano