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Answer ALL TWO questions from Part A and answer ONE question from Part B.
Questions in Part A carry 60 per cent of the total mark and questions in Part B carry
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ECON0019 1 TURN OVER
be the ones that count (not the best answers). All remaining answers will be ignored.
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PART A
Answer all questions from this section.
A.1 You wish to measure the effect of hiring more teachers on student performance. To this end
you randomly select 300 schools; for each school you collect data on the number of students per
teacher (str) and the average exam score (score) for final year students in 2021. Suppose that
in the population the following equation holds:
score = β0 + β1str + β2ability + β3 (str × ability) + u, (1)
where ability is the average level of student ability in a given school. This equation statisfies
MLR.3–MLR.4 in Wooldridge’s text book.
(a) You do not have data on ability and so decide to estimate β1 by regressing score on str.
Derive the probability limit of the estimator.
(b) A colleague of yours conjectures that Cov (str, ability) < 0, Cov (str, str × ability) > 0,
β2 > 0 and β3 < 0. Do these sign restrictions seem plausible to you Explain. Supposing they hold, is it possible to determine the sign of the asymptotic bias of the estimator in (a) (c) You speculate that the joint population distribution of ability and str is such that E [ability|str] = E [ability] . (2) Interpret the restriction in (4). Do you think it is likely to hold (d) Assuming that (4) holds, demonstrate that the probability limit of the OLS estimator in (a) equals β1+β3E [ability]. Interpret the probability limit. In particular, is it a meaningful measure of the effect of class size on student performance (e) You decide to collect additional data on the average mark that the final–year students earned in their first year at each school. With mark denoting this new variable, you hypothesise that, for some unknown coefficients θ, the following two conditions hold: E [score|str, ability, mark] = E [score|str, ability] , (3) ECON0019 2 CONTINUED E [ability|str, mark] = E [ability|mark] = θmark. (4) Interpret the two conditions and compare them to (4). Do they seem reasonable (f) You run the following regression, s ccore = β 0 + β 1str + β 2mark + β 3 (str × mark) What is the probability limit of β 1 under (5)–(6) Explain. A.2 You are interested in the relationship between sales, profits and research & development (R&D). For that purpose you obtain the following regression based on data collected from a sample of 45 firms in the UK concrete industry in 2016, b rd = .42 (1.369) + .21 (.116) log (sales) + .07 (.046) prof it, ˉR 2 = .079, (5) where rd is expenditures on R&D of a firm as percentage of its annual sales, sales is the firm’s annual sales (in millions GBP) and prof it is its annual profits as percentage of sales. Robust standard errors are reported in parentheses. (a) Interpret the coefficient on log (sales). If sales increases by 10% what is the exact estimated percentage point change in rd Is this an economically large effect (b) Test the hypothesis that rd does not change with sales against the alternative that it does increase with sales. Perform the test at the 5% and 10% level. What is the p-value of the test Conclude. (c) You compute the F-test statistic of the hypothesis that sales and prof it are jointly in significant and obtain F = 4.12. Do you accept or reject the null at the 5% level Explain. (d) Do you trust the critical values that you used in (b) and (c) and the the p-value that you computed in (b) Are they valid What do you conclude about the reported test results (e) You estimate the following alternative regression model for rd, b rd = .35 + .030sales .0000070sales2 + 0.048prof it, ˉR 2 = .099. (6) (1.245) (.014) (.00000038) (.047) At what point does the estimated marginal effect of sales on rd become negative in this model (f) Write up a composite model that would allow you to test (7) and (9), respectively, against the composite model. Would the outcomes of these two tests be able to determine which of the two models, (7) and (9), is the preferred one Explain. (g) You collect data on annual R&D, sales and profits of the same 45 firms in 2017 and re estimate (9) by running a pooled regression across the two years, 2016 and 2017. A colleague tells you that you should rather estimate the model using the first-difference estimator. Is your colleague right Explain. ECON0019 3 TURN OVER PART B Answer ONE question from this section. B.1 In “Rainfall and Conflict: A Cautionary Tale” (Journal of Development Economics, 2015), Heather Sarsons studies whether lower income can lead to more violent conflict among religious groups in India. She studies a sample of 142 districts in the country’s 28 states. Simplifying things a bit, the baseline equation of interest is: Ci = βYi + 28 X s=1 γs1 [Si = s] + εi , where Ci (“conflict”) is the number of riots in district i in a particular year, Yi (“income”) is income per capita, Si is the state in which district i is located, 1 [Si = s] indicates a dummy variable which takes the value of one when Si = s, and εi is the error term. (a) Why may the OLS estimate of β be inconsistent Provide at least one economic justification. Sarsons proceeds to use an instrumental variable strategy: she instruments income with two measures of rainfall in the district. The first one, R1i , is the amount of rainfall in district i in the year of study minus its typical value (across many years) for the district. The second measure, R2i , is a dummy variable that R1i is below its 20th percentile. The idea is that agricultural production is a key source of income in much of India and it relies on sufficient rainfall. (b) Explain in detail (step by step) how her instrumental variable estimate β is constructed from data on (Ci , Yi , R1i , R2i , Si). Then write down the formal conditions under which β is consistent for β. Which of them can be tested For those which can, describe the testing procedure. For those which cannot, explain why not. (c) Why does Sarsons subtract the typical rainfall in the district when constructing R1i Which condition or conditions from part (b) would be more likely violated if she did not do this Give an economic justification. For simplicity, drop R2i and keep R1i as the single instrument in the final parts of the question. Sarsons observes that in some Indian districts there are dams on local rivers, and thus reservoirs of water which do not dry out even in low-rainfall years. This should make income less dependent on weather. Another issue is that the causal effects of income on conflict can be heterogeneous across regions. (d) With these complications, can the instrumental variable estimate β still be interpreted as some average of causal effects of income on conflict and, if so, what kind of average Which ECON0019 4 CONTINUED condition or conditions would have to be added, compared to part (b), for such an interpretation to be valid Is it (or are they) plausible (e) Sarsons finds that in the districts with a dam the reduced-form coefficient is significantly different from zero, but the first-stage coefficient is not. She concludes that rainfall may not be an exogenous instrument. Explain intuitively and formally how she makes this conclusion. B.2 In “Does Hospital Crowding Matter Evidence from Trauma and Orthopedics in England” (American Economic Journal: Economic Policy, forthcoming), Thomas Hoe examines the im pact of hospital crowding on medical treatment outcomes exploiting variation in emergency admissions. For simplicity, we abstract from some of the details examined in the paper. One possible outcome of interest is the length of the illness (measured in days) for a particular patient i. Let this be denoted by yi . Suppose one focusses on the following model relating this outcome for an individual i admitted to a particular hospital: ln(yi) = β0 + x 0iβx + i , (7) where xi comprises variables such as the individual’s age, race and disease stage at the time of admission. As indicated in the article, other elements that might affect the outcome of interest relate to patient composition and hospital operation details (such as capacity constraints and utilisation) at the hospital where the individual is admitted, among other factors. Note that the unit of observation in items (a)-(c) below are the individual whereas in items (d)-(e) relates to a time period (day) for a particular hospital. (a) Suppose that one is interested in estimating (11) with data from a particular hospital. Let ci > 0 denote the number of days an individual is at the hospital. This variable and xi
is
observed for every patient in the hospital. If individual i is discharged after recovering from
the illness (i.e., ci ≥ yi), yi
is known, but otherwise we only know ci and that yi > ci
. What
additional assumptions would one need to estimate (11) by maximum likelihood Write
down the log-likelihood function for this regression and explain your answer.
(b) Suppose that individual i opts to go to a hospital according to the following choice model:
hi = 1(γ0 + γddi + vi ≥ 0) (8)
where hi = 1 if individual i goes to the hospital and = 0, otherwise. The variable di
records i’s distance to the closest hospital and vi marks idiosyncratic unobservable factors
informing this decision. Assume that vi follows a standard normal distribution. One is
interested in estimating (11) and information on yi
is only available for those who go to the
hospital. Assume that OLS estimates are obtained for those observations. First, if i and vi
are not necessarily independent, but γd = 0, would the OLS estimator above be consistent
Explain. What if γd is not necessarily zero, but di and xi are independent Explain.
ECON0019 5 TURN OVER
(c) Suppose that going to the hospital depends not only on di but also on xi
. In other words,
consider now the following extended version of equation (12):
hi = 1(γ0 + γddi + γ
0xxi + vi ≥ 0) (9)
How would you estimate the parameters in (11) consistently Explain your answer.
(d) Suppose you have time series data on daily admissions to a particular hospital, denoted by
qt where t is a particular date, and consider for simplicity a linear regression of qt on week of-the-year dummy variables st recording which week of the year t pertains to. Using the
number of admissions per hospital, Hoe notes that there is no evidence of serial correlation
in the residuals of a regression of (emergency) admissions on seasonal dummies once one
examines the estimated AR(1) coefficients for a regression of residuals on lagged residuals.
Under what conditions does this residual regression offer a valid test for the absence of
serial correlation Describe the test in detail.
(e) For the regression above, under what conditions would the OLS estimator be consistent
Explain your answer.