Econometrics Midterm Exam, Due June 8th, 2015

June 1, 2015

Instructions. This midterm consists of ten questions. Each question is worth 10 points. Some of these questions will require you to use Stata.

• Name it. Write your Student ID at the start of your midterm submission. Write your Name at the end of your midterm submission.

• Type it. Use Microsoft Word, or LYX, or any other program that allows you to type mathematics. Make sure your submission is neat. I will not accept handwritten work. If you put Stata output in your submission, change the font of the Stata output to monospace (Courier).

• Submit it. Submit your exam online by midnight on June 8th.

• Do your own work. Do not talk to each other about these questions, and especially do not share answers.

• Be concise. I will deduct points if you write things that are wrong or not directly related to answering the question. To avoid this, minimize the amount you write subject to answering the question.

A parent comments: �I think that younger siblings do better than older siblings in primary school because while younger siblings are still out of school (prior to Pre-K or Kindergarten), their older siblings (who are attending school) bring sicknesses from the school into the home. The younger siblings then get sick while they are still not yet in school, building up an immunity to school sickness. Then when the younger siblings �nally attend school, they don't get sick. Older siblings don't have this advantage�they live in a clean home until they attend school, and then they get bombarded by sickness when they start going to school. Older siblings then have to stay home sick more, which hurts their school grades.�

Question 1. Propose three hypotheses suggested by the parent's comment.

Question 2. Pick one of the hypotheses that you wrote in the previous question. Propose three observational datasets, one of which is cross-sectional, one of which is panel, one of which is time series, that you could use to study that hypothesis. Make sure to describe the important variables each dataset would contain.

Question 3. Pick one of the hypotheses that you wrote in the previous question. Propose a randomized experiment that could test the hypothesis. (The experiment does not have to be realistic or ethical.)

Question 4 [Requires Stata]. Consider the descriptive question based on the parent's claim from the previous problem: �in two-child families, does the older child do better in school than the younger child?�

On the course website �data� section, there is a �le, �NHES2007_twosiblings.dta�, which is an excerpt from the National Household Education Survey of 2007 (NHES2007�another product of the NCES). This is a survey of US households with children. The unit of observation is the child. I have restricted the dataset to children with one other sibling, and for which the parent reports �typical grades� of the child. The variable descriptions in the data should be self-explanatory.

Inspect the data. Is the data cross-sectional, panel, or time-series? Explain. Once you've answered that, answer the descriptive question presented at the beginning of this question using a

formal hypothesis test for the di�erence between two means. Present your answer in the four equivalent ways we've learned: acceptance region, con�dence interval, t-statistic, and p-value. Interpret your �ndings.

Question 5 [Requires Stata]. In the hypothesis test that you conducted in the previous question, mom's age was omitted from the analysis. Could this omission bias your hypothesis test? In your explanation, be sure to state clearly the requirements for omitted variable bias (including a picture if it helps). Then propose a multivariate

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regression model that contains a parameter which identi�es the e�ects of being the younger sibling on child grades, holding constant the e�ect of mom's age. Run your proposed regression using the provided data, put the coe�cients and standard errors in a neat table, and interpret the results. Does this alternative model provide any evidence for the parent's claim?

Question 6. Consider the following multivariate regression output explaining whether a child attends private school based on mother's and father's education, and household income. (This regression uses the same NHES2007 as the previous part of this exam.) In the NHES2007, household income is reported in 14 categories, so I include a dummy variable for each category (category 1, which is the lowest income, is omitted). The variables mom_college and dad_college are dummies for whether that parent has a college degree (2 or 4 year degrees are both treated as a 1).

. reg private mom_college dad_college i.householdincome , robust

Linear regression Number of obs = 3,428

F(15, 3412) = 33.65

Prob > F = 0.0000

R-squared = 0.0402

Root MSE = .33349

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

| Robust

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

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

mom_college | .0226086 .0133853 1.69 0.091 -.0036355 .0488526

dad_college | .0307699 .0141919 2.17 0.030 .0029444 .0585954

|

householdincome |

2 | .017251 .0160739 1.07 0.283 -.0142643 .0487664

3 | .062733 .0257021 2.44 0.015 .0123399 .113126

4 | .0930607 .0282164 3.30 0.001 .037738 .1483834

5 | .0468672 .0196209 2.39 0.017 .0083974 .085337

6 | .0037826 .0098874 0.38 0.702 -.0156031 .0231684

7 | .0654431 .0221277 2.96 0.003 .0220582 .108828

8 | .0438635 .0189764 2.31 0.021 .0066573 .0810698

9 | .0810275 .0272702 2.97 0.003 .02756 .1344951

10 | .0597434 .02136 2.80 0.005 .0178638 .1016231

11 | .0915305 .0168414 5.43 0.000 .0585103 .1245507

12 | .1107406 .0171696 6.45 0.000 .0770768 .1444044

13 | .1240095 .0163086 7.60 0.000 .0920339 .1559851

14 | .1836544 .0163583 11.23 0.000 .1515813 .2157276

|

_cons | -.0049965 .0025131 -1.99 0.047 -.0099238 -.0000691

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

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Suppose you are an analyst working for a private school system. Your goal is to determine who is more likely to attend private school, so that the school system can target advertisements to those who would not. Do you think this is a good model for forecasting the private school attendance of individual families? Explain.

Is father's college attainment signi�cantly related to private school attendance? Write down a formal hypothesis test and present your �ndings.

Do you think the coe�cients on mom_college and dad_college are signi�cantly di�erent? Interpret the following Stata output in your answer to this part.

. test mom_college=dad_college

( 1) mom_college - dad_college = 0

F( 1, 3412) = 0.13 Prob > F = 0.7165

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Question 7. Because this NHES2007 data contains pairs of mothers and fathers, we can compare within household di�erences in father's and mother's education. Let i index couples (mom + dad), and let mi be the mom's education, and fi be the father's education. Then mi −fi is the di�erence in education between the mom and dad within the same household. If it is positive, the mom is more educated than the dad; if it is negative, the dad is more educated than the mom.

In the population, there is an average di�erence µm−f = E[mi −fi]. Because E[mi −fi] = E[mi] − E[fi] there are two natural estimators for µm−f:

• (�two-sample estimator�) I could use m−f = ∑

i mi

# of moms −

∑ i fi

# of dads � this estimator is calculated by taking

the sample average for mothers and the sample average for fathers in my data, and then taking the di�erence. In order to calculate the standard error of this estimator I would have to worry about the fact that my sample of mothers and sample of fathers are not independent (because I sampled families, not mothers and fathers separately).

• (�paired estimator�) I could use m−f = ∑

i (mi−fi) N

. This estimator is calculated by taking within-family mother's education and subtracting o� within-family father's education, then averaging over families. Because each (mi − fi) is independent of (mj − fj) for j 6= i (random sampling occurred at the family level!), the standard variance formula applies.

Consider the hypothesis test: H0 : µm−f = 0 versus HA : µm−f 6= 0. Because there are two natural estimators for µm−f, there are two natural ways to conduct this hypothesis test. However, it turns out that the �paired estimator� is more e�cient. So that's the estimator we should use. You should read about this at http://blog.minitab.com/blog/statistics-and-quality-data-analysis/t-for-2-should-i-use-a-paired-t-or-a-2-sample-t.

The resulting paired t-test is easily calculated in Stata. The code and output are displayed below. (You might want to try this for yourself in Stata. Note that I don't have to specify the unequal option�there is only one sample here, so that option just doesn't make sense.)

Interpret the output. Do moms have signi�cantly more education than dads? How large is the di�erence? (For this, remember that you can interpret these expectations as probabilities.)

. ttest mom_college == dad_college

Paired t test

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

Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]

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

mom_co~e | 3,428 .4962077 .0085408 .5000586 .4794621 .5129533

dad_co~e | 3,428 .4276546 .0084512 .4948106 .4110847 .4442245

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

diff | 3,428 .0685531 .0090628 .5306179 .0507841 .0863221

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

mean(diff) = mean(mom_college - dad_college) t = 7.5642

Ho: mean(diff) = 0 degrees of freedom = 3427

Ha: mean(diff) < 0 Ha: mean(diff) != 0 Ha: mean(diff) > 0

Pr(T < t) = 1.0000 Pr(|T| > |t|) = 0.0000 Pr(T > t) = 0.0000

Question 8. Suppose I have an economic model that contains a parameter θ. I propose an estimator for θ using data on a variable X. Call my estimator θ̂. I am able to show that in large samples, θ̂ is uniformly distributed on the interval θ ± σX√

N .1

Suppose in a sample of size 100 I estimate σX ≈ 1 (by using the sample standard deviation for X, sX). My point estimate for θ̂ is 0.

Consider the hypothesis: H0 : θ = 0.1, versus HA : θ 6= 0.1. Construct a 90% acceptance region for this hypothesis test. Then construct a 90% con�dence interval for θ̂. Can you reject the null hypothesis at the 10% level?

1While I've never seen an estimator that is asymptotically uniformly distributed, so the situation in this problem is completely imaginary, we will talk about statistics which are not asymptotically normal. These statistics are generally asymptotically chi-square or F.

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Question 9. This and the next question are about regression weighting. Say instead of observing individual students, we only observe schools. That is, I have a cross-sectional dataset of

school average test score and school average attendance rate. I'm happy to estimate the relationship between school average attendance rate and school average test score, but I'm concerned that these variables might be noisier for smaller schools. Some schools have very few students. (In practice we may conduct such analysis on school-grades, and then we would have cells with very few students!)

De�ne Y i to be the average test score of school i and Xi to be the average attendance rate for school i. I assume the univariate regression model holds,

E[Y i|Xi] = β0 + β1Xi and my goal is to estimate and conduct inference on β1.

Suppose the standard deviation of test scores among students in school i was σyi. This could vary across schools, of course�some schools are more homogeneous than others. If school i has Ni students, what's the standard error of Y i? Explain.

Question 10. For this question, for ease of notation, call school average test score Yi. Note that this variable was called Y i (with a bar) in the previous question�for this question I'm going to be calling it Yi (without a bar). Same thing for Xi: Xi is now notation for Xi.

Now let's assume that σyi = σy (a constant) for all schools i. Then your answer to the previous part of this question simpli�es to a constant (in i) times 1/

√ Ni.

Write the conditional expectation model above as

Yi = β0 + β1Xi + �i

Recall that the assumption Var(�i|Xi) = σ2� (a constant) is an assumption of homoskedasticity, and under this assumption OLS is e�cient. E�ciency means that β1 will be more precisely estimated.

In our case, we don't know the form of Var(�i|Xi), but we do know that Var(Yi|Ni) = σ2y/Ni. So as Ni goes up, this variance goes down.

I believe that attendance rates are probably lower in larger schools, so Cov(Xi,Ni) < 0. If this is true, then what does this suggest about Var(�i|Xi)? Assume that the variance of Yi can only depend on Xi through Ni, that is, Var(Yi|Xi,Ni) = Var(Yi|Ni) (and by the univariate regression assumptions, Ni is redundant, meaning that E[Yi|Xi,Ni] = E[Yi|Xi] = β0 + β1Xi.).2

Now imagine I weight the regression by 1 1/Ni

= Ni. So instead of running the OLS regression

Yi = β0 + β1Xi + �i

I run the weighted least squares (WLS) regression

Yi √ Ni = β0

√ Ni + β1Xi

√ Ni + �i

√ Ni

(As discussed in class, this is weighting by Ni, not by √ Ni. I was wrong in assignment 2! We will discuss this

more later in the course.) De�ne the new error to be ui = �

√ Ni. Now what is Var(ui|Xi)? Provide a proof.3

To summarize, if you were given a dataset of averages for individual schools, or stores, or countries, and you knew the number of observations underlying each of those averages, why might you try to weight your OLS regression by the number of observations behind each average?

2 Hint: You can answer this question intuitively in a single sentence (but I cannot promise full credit for such a lazy answer). A

formal way to proceed is to expand Var(�i|Xi) by rewriting Yi = β0 + β1Xi + �i, and then thinking about how Var(Yi|Xi) will behave. If you go the formal route, you will want to use the conditional variance formula

Var(Y |X) = E[Var(Y |X,N)|X] + Var(E[Y |X,N]|X)

which because of our assumptions simpli�es to

Var(Y |X) = E[Var(Y |N)|X] + Var(E[Y |X]|X)

3 Hint: For this, I expect you will have to use the formula from the previous part,

Var(Y |X) = E[Var(Y |X,N)|X] + Var(E[Y |X,N]|X)

though I think you'll have to put other things in this formula besides Y,X, and N. Even with this hint, proving this is challenging�try your best!

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Assignment 2

The goal of the next few questions is to help you intuitively understand omitted variable bias and too-many variables bias, multicollinearity, and heteroskedasticity using a simulation in Stata.

Introduction. There's an important debate over how we can get more children to be able to read by 3rd grade. I have heard anecdotal evidence that the state government plans the number of prisons to build based on regressions that use current-year 3rd grade reading scores on the RHS, which might be suggestive of the importance of this goal. But we won't worry about the causal e�ect of literacy on crime in this assignment.

As we saw in class, children from poor families are already behind in terms of reading ability in fall of Kinder- garten. This is a problem. Local governments have two main policies to try to solve this problem: free full-day preschool, and family income supports.

Omitted variable & too-many variables bias.

Question 1. Suppose the process that determines child test score in Kindergarten is given by

test scorei = β0 + β1preschooli + β2incomei + �i

where β1,β2 > 0 and preschooli is a continuous measure of �preschool quality.� Preschool quality can be purchased with cash, and is purchased in cash according to the linear model

preschooli = α0 + α1incomei + ηi

You should interpret both of these linear models as structural models of human behavior. In other words, if you gave a random family another dollar, the family would indeed purchase α1 more units of preschool quality, and the family would purchase other children's stu� that has β2 additional e�ect on test scores. This means that the total e�ect on test scores of giving the family another dollar is:

(a) α0β1 (b) α1 (c) β2 + β1α1 (d) α1β1 + β2 Question 2. Let's try to simulate this model in Stata. Run Part I in the �le �assignment2.do.� This will

generate a fake dataset of 1000 students for this problem assuming that α0 = β0 = 0, α1 = 0.5, and β1 = β2 = 1. (You can easily play around by modifying these parameter assumptions.)

Regress testscore on preschool. As an estimator for β1, this regression's coe�cient is (a) biased upward (b) biased downward Question 3. A big city government is thinking about implementing a program that will raise a family's

preschool quality by 1 unit. An analyst uses the regression you ran in the previous question to make the case that this program is a highly e�ective way to improve student test scores�in particular it is better than direct income supports.

You counter that students who go to good preschools probably come from wealthier backgrounds, so at the very least the analyst should be controlling for family income. The analyst retorts: �these kinds of families wouldn't spend a dime of their own income on preschool!�

The analyst is proposing the testable null hypothesis that at least for the families he is considering, (a) β2 = 0 (b) β1 = 0 (c) α1 = 0 (d) α0 = 0 Question 4. Were the analyst's claim true, his estimator for β1 would be (a) unbiased and consistent (b) biased and inconsistent Question 5. You gather appropriate data and show that even in the analyst's population, α1 is signi�cantly

positive. The analyst then suggests that now that we have the nice data you collected, why don't we run the regression

test scorei = β0 + β1preschooli + β2incomei + �i

and then check whether β1 exceeds β2c, where c is the cost of a unit of government-provided preschool in dollars? He reasons that if β1 > β2c, the government should provide preschool since the test score return per dollar for preschool exceeds that for income supports; otherwise the government should provide income supports.

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Why might the analyst be wrong? (a) The estimator for β1 is inconsistent because of the inclusion of income on the RHS of the regression (b) One e�ect of income supports may be to increase family preschool purchases, thus the e�ect of income

supports is probably greater than β2 (too-many variables bias). Multicollinearity.

Question 6. Now we want to know whether income supports during preschool or before preschool are better. We consider the model

test scorei = β0 + β1income_beforei + β2income_duringi + �i

Try part II of the simulation. There, income_during is income_before plus a very small income change. Regress test score on income_before and income_during. You'll notice that at N = 100 observations, the

con�dence intervals for these parameters are very loose. Try N = 1000 observations by changing the -set obs 100- code to -set obs 1000-. Then try N = 10000. Notice that you need a ton of observations before the con�dence intervals really start to tighten. Why?

(a) Hard to distinguish income_before from income_during. (b) The OLS estimators for β1 and β2 are ine�cient. (c) The OLS estimators for β1 and β2 are biased. (d) Hard to distinguish income_during from the constant. Heteroskedasticity and weighted least squares.

The above analysis assumed that you could get individual-level data on income, preschool enrollment, and Kindergarten test scores. You might be able to do this in a survey dataset like the ECLS-K, but more generally you might �nd yourself working with averages at the school district level for instance.

Run part III of assignment2.do. This part simulates data for school districts of various sizes. The simulated data includes �population,� a variable that gives you the total number of kids in the district. There's one really big district�this is supposed to help you have some empathy for me since oftentimes I'm studying New York State and, you know, New York City is a single school district...

Try the regression reg avgtestscore avgincomebefore avgincomeduring, robust And then try the weighted regressions reg avgtestscore avgincomebefore avgincomeduring [w=population], robust and reg avgtestscore avgincomebefore avgincomeduring [w=sqrtpop], robust Make sure you understand what these weighted regressions are doing�review the lecture slides if need be. For me, all three approaches work reasonably well. Weighting by the square root of population seems to have

the tightest con�dence intervals around the true parameter values (because this is a simulation we all know the true parameter values�inspect the code to �nd them). I think there are theoretical reasons to justify the square root of population as the �best� weights in this case. Maybe I'll give you the chance to explore this deeper on your midterm exam.

For now let's make this simulation more realistic. Imagine that each district has some measurement error for average test scores. Thus run the regressions

reg avgtestscorewithdistrictshock avgincomebefore avgincomeduring, robust reg avgtestscorewithdistrictshock avgincomebefore avgincomeduring [w=pop], robust and reg avgtestscorewithdistrictshock avgincomebefore avgincomeduring [w=sqrtpop], robust I generally �nd that with district-level shocks regular unweighted OLS works better, followed by weighting by

the square root of population. My intuition was that if there are district-level errors in measuring the dependent variable, then we don't want to put a ton of weight on New York City, because then our OLS regression will be driven mostly by whether New York City happens to have a high or low error and high or low RHS variables. (In some analysis I was undertaking last fall, I was using teacher employment data and New York City had the unfortunate problem of �losing� all of its guidance counselors for a few years... wouldn't have been such a big deal if I wasn't weighting by district enrollments! It's tough when your analysis rests on a some 9-to-5er not making a coding mistake.)

Question 7. We might consider weighting by total district population/enrollment in the previous regressions because

(a) larger school districts are more important than smaller school districts

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(b) the dependent and independent variables, which are averages, will be estimated more precisely for larger school districts

(c) population is an omitted variable (d) all of the above

Question 8. In the regression following this question, r_1 is fall K reading test score, incthous_1 is household income in thousands in fall K, and age_1 is the child's age in months at the time of taking the test.

Richard's friend Kyle is 3 months younger than him, but attended the same school and was in the same cohort, and their families have about the same total household income. If they took this exam on the same day in Kindergarten, what do you expect Kyle's score would be relative to Richard's? (For your information, a standard deviation of r_1 is about 10.)

(a) about 1.14 scaled points more (or 11.4% of a standard deviation) (b) about 0.38 scaled points less (or 3.8% of a standard deviation) (c) about 0.38 scaled points more (or 3.8% of a standard deviation) (d) about 1.14 scaled points less (or 11.4% of a standard deviation)

. reg r_1 incthous_1 age_1

Source | SS df MS Number of obs = 16 ,747 −−−−−−−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− F(2 , 16744) = 794.64

Model | 152935.97 2 76467.9852 Prob > F = 0.0000 Residual | 1611276.4 16 ,744 96.2300764 R−squared = 0.0867

−−−−−−−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Adj R−squared = 0.0866 Total | 1764212.37 16 ,746 105.35127 Root MSE = 9.8097

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− r_1 | Coef . Std . Err . t P>|t | [95% Conf . Interval ]

−−−−−−−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− incthous_1 | .0451514 .0013383 33.74 0.000 .0425283 .0477745

age_1 | .3830728 .017546 21.83 0.000 .3486807 .4174649 _cons | 6.671546 1.208196 5.52 0.000 4.303354 9.039738

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

Question 9. Download the dataset kindergarten_version2.dta from the course website. Generate a new variable being a child's growth on the math exam score from Fall K to Spring K, m_2 minus

m_1. Generate a new variable equal to the child's age growth in month, age_2 minus age_1. Regress math growth on age growth and enroll_1. Use robust standard errors.

Let βE be the coe�cient for enroll_1 in this regression. Consider the hypothesis test

H0 : βE = 0

HA : βE < 0

What's the smallest signi�cance level at which you can reject the null hypothesis? (Careful�this is a one-tailed test!)

(a) about 1.2% (b) about 11.8% (c) about 5.9% (d) about 4.1% Question 10. The ECLS-K uses a complicated sampling scheme, and to account for this the National Center for

Education Statistics (NCES) includes sampling weights sample_weight which they recommend we use in estimation. Re-run your previous regression using these sample weights (put �[w=sample_weight]� before the comma in your regression.)

With this new speci�cation, what's the smallest signi�cance level at which you can reject the null hypothesis? (a) about 0.1% (b) about 10.4% (c) about 5.2% (d) about 3.4%

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Question 11. Finally, the ECLS-K is a clustered sample. This means that the NCES �rst samples schools and then samples students within schools. This sampling approach violates OLS assumption 2: simple random sample, since the NCES is not �shaking up the whole country� and drawing children at random. Because child outcomes are probably positively correlated within school, the standard errors are likely overstated.

One very general (and in many ways �hands-free�) way to control for this is to use �cluster-robust� standard errors. As the name implies, these standard errors are robust to heteroskedasticity, and also take into account within-cluster correlation.

Try it: replace the �robust� option in your current regression with �vce(cluster schlid)�. This will tell Stata to calculate cluster-robust standard errors, where the clusters are school IDs.

With this new speci�cation, what's the smallest signi�cance level at which you can reject the null hypothesis? (a) about 100% (b) about 15% (c) about 30% (d) about 10% Question 12. Open the kindergarten_version2.dta dataset, and plot a histogram of income_1. There are

some crazily large incomes. We know that OLS and other expectation-based analyses do not behave well when there are very large outliers. What to do?

One approach is to log very right-skewed variables like this. Apparently the income_1 variable is never less than 1, so this will work in this case: gen logincome_1 = log(income_1). A histogram of logincome_1 is much closer to normal, especially in the upper tail (you can assess this using -qnorm-, as you learned in the last assignment.)

Recall that the test scores were also very right-skewed. Log the math test score, creating a new variable logm_1. Then regress log reading score on log income. In other words, �t the model

log(math score) = β0 + β1log(income) + �

The standard approach to interpret this regression is to di�erentiate both sides w.r.t. income, treating math score as a function of income:

d

dincome log(math score) = β

d

dincome log(income)

I'm guessing it makes sense to assume � is not a function of income under OLS assumption 1. If you play around with this expression you'll get

%∆math score = β1%∆income

or %∆math score

%∆income = β1

thus β1 is interpreted as an elasticity. Which is lovely and very economic. This approach (using di�erentiation) has for some reason never confronted me as intuitive, because I cannot

see it with discrete changes in income using the original conditional expectations model. Nevertheless, you should remember that in a log-log regression like this, we give β1 the interpretation of an elasticity: it's the % change in the outcome variable expected from a 1% increase in the RHS variable. For example if β1 = 3, then a 3% increase in math score is expected from a 1% increase in income.

According to your estimates, a 1% increase in income is associated with about a (a) 0.12% increase in math test score (b) 1.2% increase in math test score (c) 12% increase in math test score (d) 1.9% increase in math test score

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