montana state university | montana state university - pa nj ...1. there are different ways to...

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 23.33 20.44

FTE Employment after 21.17 21.03

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 diffindiff

(b) Since minimum wages represent a price floor, did you expect diffindiff to be

positive or negative?

(c) The standard error for diffindiff is 1.36. 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. ( 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

Random

Xit

0.3201

(0.0842)

0.1201

(0.0542)

0.0543

(0.0321)

0.0271

(0.0242)

0.0735

(0.0223)

(a) What is the Hausman test statistic for the null hypothesis that ui and Xit 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. (20 points) A classmate is interested in estimating the variance of the error term in the

following equation

yi =β0 + β1xi + ui

where i denotes entities, y is the dependent variable, and x is an explanatory variable

for each entity and z is an instrument.

Suppose that she uses the following estimator for 2u from the second-stage

regression of TSLS:

210

2 )ˆˆˆ(2

1ˆ i

TSLSTSLSi xy

n

where ix is the fitted value from the first-stage regression. Is this a consistent

estimator for2u ?

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

collegei = β0 + β1CathHSi + ui

(i) (5 pts) Interpret the magnitudes of the OLS coefficient estimates of β0 and

β1 (I am looking for more substantive answers than “β0 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 β1, using this

variable as an IV for CathHS? Use the definition of the Wald estimator in

your answer.

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

Explain.

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:

qi =β0 +β1 pi +β2 Moni +β3 Tuesi +β4 Wedi +β5 Thursi +β6 ti +ui 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 1st 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 R2 = 0.23

Coefficient Standard Error

p -2484.70 742.05

Mon -1110.75 776.06

Tues -2326.37 763.37

Wed -1981.41 753.49

Thurs -39.46 754.01

t -4.24 9.07

Constant 7519.11 1030.33

(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 β1?

(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.

OLS Model

pi = π 0 + π 1 windi + π 2 wavei+ π 3Moni + π 4 Tuesi + π 5 Wedi + π 6Thursi + π 7 ti +vi

Dependent Variable is p Number of observations = 97 R

2 = 0.33


wind -0.005 0.009

wave 0.105 0.021

Mon -0.058 0.097

Tues -0.009 0.094

Wed 0.038 0.093

Thur 0.0877 0.093

t -0.002 0.001

Constant 0.450 0.147

H0: π1 = 0 H1: π1 0 Test Statistic: F(1, 89) = 0.31

H0: π2 = 0 H1: π2 0 Test Statistic: F(1, 89) = 25.76

H0: π1 = π2 = 0 H1: H0 is not true Test Statistic: F(2, 89) = 15.86

H0: π3 = π4 = π5 = π6 =0 H1: H0 is not true Test Statistic: F(4, 89) = 0.63

TSLS Model

iiiiiiii utThursWedTuesMonpq 6543210 ˆ

pi = π 0 + π 1 windi + π 2 wavei+ π 3Moni + π 4 Tuesi + π 5 Wedi + π 6Thursi + π 7 ti +vi

iiiiiiiiTSLSi etThursWedTuesMonWavewindu 76543210ˆ

Dependent Variable = q Instruments = wind and wave

Number of observations = 97 Coefficient Standard Error

p -3400.917 1459.99

Mon -1150.396 784.484

Tues -2349.607 770.464

Wed -1987.691 759.896

Thurs 0.203 762.305

t -7.565 10.211

Constant 8463.304 1657.892

H0: β1 =0 H1: β10 Test Statistic: F(1, 90) = 5.43

H0: β2 = β3 = β4 = β5 =0 H1: H0 is not true Test Statistic: F(4, 90) = 4.02

Dependent Variable = TSLSu

Number of observations = 97 R2 = 0.02


wind -94.234 76.397

wave 91.137 169.682

Mon 206.940 797.556

Tues 21.025 767.865

Wed 21.498 760.581

Thurs 0.629 760.174

t -0.411 9.102

Constant 637.768 1208.353

H0: θ1 =0 H1: θ 10 Test Statistic: F(1, 94) = 1.52

H0: θ2 =0 H1: θ 20 Test Statistic: F(1, 94) = 0.29

H0: θ1 =θ2 =0 H1: H0 is not true Test Statistic: F(2, 94) = 0.77

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:

2îu on x1i, x2i, x3i, . . . ., xki, and

2îy

where the 2îu are the squares of the OLS residuals (estimated under homoskedasticty) and

the 2îy 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?

b) (5pts) How will the R2 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 iy to this regression. What do

you think of this idea?

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.

c) (10 pts) What other models (other than OLS) might you consider using?

Carefully describe under what circumstances would you use those models?

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 = 1230

-------------+------------------------------ F( 1, 1228) = 236.62

Model | 69.9901106 1 69.9901106 Prob > F = 0.0000

Residual | 363.229151 1228 .295789211 R-squared = 0.1616

-------------+------------------------------ Adj R-squared = 0.1609

Total | 433.219262 1229 .352497365 Root MSE = .54387

------------------------------------------------------------------------------

lwage | Coef. Std. Err.

-------------+----------------------------------------------------------------

educ | .1013613 .0065894

_cons | 1.092319 .0872969

------------------------------------------------------------------------------

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?

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


-------------+------------------------------ F( 1, 1228) = 3.72

Model | 1.30777496 1 1.30777496 Prob > F = 0.0541


-------------+------------------------------ Adj R-squared = 0.0022

Total | 433.219262 1229 .352497365 Root MSE = .59306

------------------------------------------------------------------------------

lwage | Coef. Std. Err.

-------------+----------------------------------------------------------------

educhat1 | .8202563 .4253841

_cons | -8.280202 5.545928

------------------------------------------------------------------------------

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?

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.

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.

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?

Education Regression #1 and Prediction educhat1 . reg educ ctuit


-------------+------------------------------ F( 1, 1228) = 0.35

Model | 1.94372373 1 1.94372373 Prob > F = 0.5540

Residual | 6810.33595 1228 5.54587618 R-squared = 0.0003

-------------+------------------------------ Adj R-squared = -0.0005

Total | 6812.27967 1229 5.54294522 Root MSE = 2.355

------------------------------------------------------------------------------

educ | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

ctuit | -.0494466 .0835227

_cons | 13.03836 .0671676

------------------------------------------------------------------------------

. predict educhat1, xb

. test ctuit

( 1) ctuit = 0

F( 1, 1228) = 0.35

Prob > F = 0.5540

Education Regression #1 and Prediction educhat2 . reg educ ctuit exper expersq ne nc west urban


-------------+------------------------------ F( 7, 1222) = 163.72

Model | 3296.87961 7 470.982801 Prob > F = 0.0000

Residual | 3515.40006 1222 2.87675946 R-squared = 0.4840

-------------+------------------------------ Adj R-squared = 0.4810

Total | 6812.27967 1229 5.54294522 Root MSE = 1.6961

------------------------------------------------------------------------------

educ | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

ctuit | -.1952777 .0604338 -3.23 0.001 -.3138431 -.0767122

exper | -.8877036 .0816872 -10.87 0.000 -1.047966 -.727441

expersq | .0171023 .0037967 4.50 0.000 .0096535 .024551

ne | .1141177 .1458075 0.78 0.434 -.1719431 .4001784

nc | -.0081578 .1269651 -0.06 0.949 -.2572517 .240936

west | .2501201 .1558156 1.61 0.109 -.0555757 .555816

urban | -.2010627 .1320998 -1.52 0.128 -.4602303 .0581049

_cons | 20.53855 .4470007 45.95 0.000 19.66158 21.41552

------------------------------------------------------------------------------

. test ctuit

( 1) ctuit = 0

F( 1, 1222) = 10.44

Prob > F = 0.0013

. predict educhat2, xb

Wage Regression #1: reg lwage educ exper expersq ne nc west urban Source | SS df MS Number of obs = 1230

-------------+------------------------------ F( 7, 1222) = 46.79

Model | 91.5734225 7 13.0819175 Prob > F = 0.0000


lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

educ | .1348698 .0088801 15.19 0.000 .1174479 .1522917

exper | .111761 .026586 4.20 0.000 .0596018 .1639202

expersq | -.0032241 .0011917 -2.71 0.007 -.0055621 -.0008861

ne | .1278787 .045432 2.81 0.005 .0387454 .217012

nc | -.0120877 .0395779 -0.31 0.760 -.0897358 .0655605

west | -.0023592 .0486249 -0.05 0.961 -.0977568 .0930384

urban | .2230756 .04121 5.41 0.000 .1422253 .3039259

_cons | -.3462225 .2287505 -1.51 0.130 -.7950098 .1025648

Wage Regression #2: reg lwage ctuit exper expersq ne nc west urban Source | SS df MS Number of obs = 1230

-------------+------------------------------ F( 7, 1222) = 12.51

Model | 28.9618019 7 4.13740028 Prob > F = 0.0000



-------------+----------------------------------------------------------------

ctuit | -.0488464 .0204938 -2.38 0.017 -.0890533 -.0086396

exper | -.0095934 .027701 -0.35 0.729 -.0639402 .0447534

expersq | -.0008583 .0012875 -0.67 0.505 -.0033842 .0016676

ne | .1410388 .0494449 2.85 0.004 .0440325 .2380451

nc | -.0137673 .0430552 -0.32 0.749 -.0982377 .0707031

west | .0315301 .0528388 0.60 0.551 -.0721347 .1351948

urban | .1972751 .0447965 4.40 0.000 .1093886 .2851616

_cons | 2.433934 .1515828 16.06 0.000 2.136543 2.731325

Wage Regression #3: reg lwage educhat1 exper expersq ne nc west urban Source | SS df MS Number of obs = 1230

-------------+------------------------------ F( 7, 1222) = 12.51

Model | 28.9617815 7 4.13739736 Prob > F = 0.0000



-------------+----------------------------------------------------------------

educhat1 | .9878573 .4144624 2.38 0.017 .1747204 1.800994

exper | -.0095934 .027701 -0.35 0.729 -.0639402 .0447534

expersq | -.0008583 .0012875 -0.67 0.505 -.0033842 .0016676

ne | .1410389 .0494449 2.85 0.004 .0440326 .2380452

nc | -.0137673 .0430553 -0.32 0.749 -.0982377 .0707031

west | .03153 .0528388 0.60 0.551 -.0721347 .1351948

urban | .1972751 .0447965 4.40 0.000 .1093885 .2851616

_cons | -10.4461 5.396812 -1.94 0.053 -21.03415 .1419391

Wage Regression #4: reg lwage educhat2 exper expersq ne nc west urban Source | SS df MS Number of obs = 1230

-------------+------------------------------ F( 7, 1222) = 12.51

Model | 28.9618029 7 4.13740041 Prob > F = 0.0000



-------------+----------------------------------------------------------------

educhat2 | .2501384 .1049467 2.38 0.017 .0442427 .4560342

exper | .2124554 .0957597 2.22 0.027 .0245838 .4003271

expersq | -.0051362 .0021646 -2.37 0.018 -.009383 -.0008895

ne | .1124936 .0513506 2.19 0.029 .0117486 .2132386

nc | -.0117267 .0430533 -0.27 0.785 -.0961932 .0727398

west | -.0310346 .0589366 -0.53 0.599 -.1466627 .0845935

urban | .2475686 .0500256 4.95 0.000 .1494229 .3457143

_cons | -2.703547 2.15156 -1.26 0.209 -6.924708 1.517615

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

lATTENDt= β0+ β1lPRICEt+ β2 WINPERCt +β3 RIVALt +β4WEEKENDt + ut

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 ut) 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?

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?

2. One type of partial adjustment model is

y*t = 0 + 1xt+et

yt-yt-1 = (y*t-yt-1) + ut

where y*t is the desired or optimal level of y and yt is the actual observed level. For

example, y*t is the desired growth in firm inventories and xt is growth in firm sales. The

parameter 1 measures the effect of xt 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 yt on xt and yt-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(et|xt, yt-1, xt-1, yt-2, . . . .) = E(ut|xt, yt-1, xt-1, yt-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.

c. Will the errors in your model in (a) be serially correlated? Explain why or

why not.

3. Let hy6t 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 hy3t-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, hy3t-1 is known, whereas hy6t 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,

E(hy6t|all information up to time t-1) = hy3t-1

Suppose you were to estimate the model

hy6t = 0 + 1 hy3t-1 + ut

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

^hy6t = -.058 + 1.104 hy3t-1 + ut

(.070) (.039)

N=123, R2 = .866

Do you reject the test in (a) at the 5% significance level?

c. 4pts Another implication of the expectations hypothesis is that no other variables

dated as t-1 or earlier should explain hy6t after controlling for hy3t-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:

^hy6t = -.123+ 1.053 hy3t-1 + .480(r6t-1 – r3t-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 hy3t and hy3t-1 is .914. Does this raise any

concerns with your previous analysis? Be specific.

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 -.00093

(.00057)

gwage .119

(.052)

gwaget-1 .097

(.039)

gwaget-2 .040

(.039)

gwaget-3 .038

(.039)

gwaget-4 .081

(.039)

gwaget-5 .107

(.039)

gwaget-6 .095

(.039)

gwaget-7 .104

(.039)

gwaget-8 .103

(.039)

gwaget-9 .159

(.039)

gwaget-10 .110

(.039)

gwaget-11 .103

(.039)

gwaget-12 .016

(.052)

N=273 R2 =.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.

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.)

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.

0

0.5

1

1.5

2

2.5

3

3.5

50 55 60 65 70 75 80

Temperature

No

. o

f In

cid

en

ts

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 = 2.858 – 0.037×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 31o, 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?

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 (15.297 – 0.236× Temperature)

(7.329) (0.107)

Recall that the logistic function is z

z

e

ezG

1)(

Calculate the effect of a decrease in temperature from 800 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?

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 (8.900 – 0.137× Temperature)

(3.983) (0.058)

Why is the slope coefficient in the probit so different from the logit coefficient?

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 ΔYi = Y2i-Y1i.

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-in-

difference approach does not resolve?

7. Consider the bivariate regression model

yi = α + βx*i + ui

We do not observe the true x* i , but instead estimate

yi = α + βxi + vi

where the covariate of interest xi is subject to classical measurement error.

In particular, assume that the measured value xi is related to the true (unobserved) value

x i* according to the linear equation: xi = x i * + ei

where ei is an iid random error, with E[ei]=0 and var(ei)=σ2

e and Cov(ei,xi*)=0. (Also

assume X and u are uncorrelated in expectation.) We demonstrated in class that with

measurement error, is biased and inconsistent.

a. 5pts Suppose there is an instrument Zi 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

consistent? Explain.

b. 5pts Now consider this in a time series context:

yt = α + βx*t + ut;

xt = x t* + et

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 ut and et are uncorrelated with all

past values of x t* and et.

Show that E(x t-1, vt) = 0, where vt is the error term in the model from part (b).

c. 5pts Are xt and xt-1 likely to be correlated? Explain.

d. 6 pts What does all of this suggest as a useful strategy for consistently

estimating β?

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