Econometrics questions
wsyvcjwpxgs1. Consider the following equation that relates a primary student’s test score to per-student spending in his or her school:
score∗ =β0+β1spend∗+u Where score∗ and spend∗ represent the true values of the underlying variables, and score and spend
represent what you observe.
· (a) Suppose score is measured with measured with random error that is uncorrelated with the ex- planatory variables. Using a model of measurement error score = score∗ + e, show that the estimator for β1 will still be unbiased. Express the variance of βˆ1 under measurment error in score.
· (b) Now suppose spend is measured with random error that is uncorrelated with the other explana- tory variables. Using a model of measurement error spend = spend∗ +ν, show that the estimator for β1 will be biased or inconsistent. (It is sufficient to show that one of the regression assump- tions will be viloated.) What is the direction of the bias?
· (c) Now assume that both score and spend are measured without error, but you are worried that the zero conditional mean assumption is violated in the above equation because, for example, students in high-spending schools may have higher ability. That is, the true model should be score = β0 + β1spend + β2abil + u
You decide to use the student’s score in the previous year as a proxy variable and estimate
score = α0 + α1spend + α2 prescore + εt
i. Write down the auxiliary equation that decomposes a student’s ability into a portion related to the previous year’s test score and a portion that isn’t.
ii. Substitute the auxiliary equation into the true model and identify the new error term.
iii. Based on the result in (ii), write down the assumption required for the estimation using the proxy variable to be unbiased. Explain in words.
2. Suppose that you wish to estimate the effect of class attendance on college students’ performance:
f inal = β0 + β1ACT + β2attend + β3GPA + u
where ACT is the student’s ACT score, attend is the fraction of classes attended over the semester, and GPA is the student’s GPA.
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· (a) Why might attend be correlated with u?
· (b) Let dist be the distance from the student’s living quarters to the lecture hall. If living quarters are assigned by lottery, will dist be uncorrelated with u? Explain briefly.
· (c) Assuming that dist and u are uncorrelated, what other assumption must dist satisfy to be a valid IV for atndrte?
· (d) Which IV assumption can be tested? Explain how you would test it in this case.
· (e) Now suppose living quarters are not assigned by lottery; rather, they are chosen by the students. Based on this, how might the exclusion restriction be violated? Speculate on the direction of the resulting bias.
3. The data in fertil2.dta includes, for women in Botswana during 1988, information on number of children, years of education, age, and religious and economic status variables.
· (a) Estimate the model children = β0 + β1educ + β2age + β3age2 + u by OLS, and interpret the estimates. In particular, holding age fixed, what is the estimated effect of another year of education on fertility? If 100 women receive another year of education, how many fewer children are they expected to have?
· (b) The variable f rsthal f is a dummy variable equal to one if the woman was born during the first six months of the year. Assuming that f rsthal f is uncorrelated with the error term from part (a), show that f rsthal f is a reasonable IV candidate for educ. (Hint: you need to do a regression).
· (c) Consider a simplified bivariate regression of children on educ: children = β0 + β1educ + u Compute the IV estimates for this model, using f rsthal f as an instrument. Do this in two ways: i. By computing the relevant covariances between children, educ, and f rsthal f , and applying the IV formula. ii. By using the ivregress command.
· (d) Now, back to the full model: children = β0 + β1educ + β2age + β3age2 + u i. Compute the IV estimates using two-stage least squares, running each “stage” separately. ii. Confirm your answer using the ivreg command.
· (e) UsetheIVestimatesinpart(d)toproposeadirectionofbiasoftheOLSestimateforβ1.Propose an omitted variable that could plausibly explain this bias. (You can assume that the IV estimates are unbiased.)