(Solved):     A.1 You wish to measure the effect of hiring more teachers on student performan ...

 

 

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: Bo + Bistr + B2ability + B3 (str x 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. score = (a) You do not have data on ability and so decide to estimate B1 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 x ability) > 0, B2 > 0 and 33 <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 Elability str) = E (ability). (2) Interpret the restriction in (2). Do you think it is likely to hold? (a) Assuming that (2) holds, demonstrate that the probability limit of the OLS estimator in (a) equals B1+B3E[ability). Interpret the probability limit. In particular, is it a meaningful measure of the effect of elasg 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 e, the following two conditions hold: E [scorestr, ability, mark] = E (score|str, ability), (3) ECON0019 2 CONTINUED E?ability str, mark] = E (ability mark] = Omark. Interpret the two conditions and compare them to (2). Do they seem reasonable? (f) You run the following regression, score = Bo + Bistr + Bemark + B3 (str x mark) What is the probability limit of B1 under (3)–(4)? Explain.