Instrumental Variable Methods Questions and Exercises

Question 1: We have the classic ability bias (what is the likely sign of the bias?)

Question 2: How can we obtain an unbiased estimate of r if ability is unobserved?

Question 3: For instance: between 15 and 16 years old both will be in the same grade, but those born earlier can drop out of school because they already turn 16 while those born late, will need to go through one additional school year before being allowed to leave leaving. Do you think this IV satis.es the exclusion restriction?

Exercise 4 - Show that r = Cov(Yi , ˜zi )/Cov(Si , ˜zi ), where ˜zi is the residual from the regression of Zi on exogenous covariates X

Exercise 5 - A special case: the wald estimator - A dummy IV

Derive the IV estimator in the case where the instrument is a dummy variable which is as good as randomly assigned.

Exercise 6 - Classify in the draft lottery example individuals in these groups.

Question 7 - Is this the causal effect of Di on Yi?

Question 8 - Can we take expectation on both side of and identify the average causal effect of D on Y : E [Yi (1) Yi (0)]?

Exercise 9 - Show that comparing the outcome of those assigned and those not assigned is equal to the weighted difference between the average effects for compliers and the average effect for defiers.

Exercise 10 - The Causal Question: what is the wage premium for an individual of staying one additional year with an employer?

Does this question face the Fundamental Problem of Causal Inference?

Assume you have a dataset on two groups of individuals: moovers and stayers.

To what extent this will help you identifying the returns to tenure?

Derive the OLS estimator, discuss potential bias in the following two cases:

• Best individuals are more likely to moove.
• Individuals are more likely to select better jobs.

Assume you have a variable indicating whether individuals have changed employers because of plant closing.

How this variable can be useful, state formally all assumptions needed to obtain the LATE associated to one additional year of tenure.

Additional References: Topel (1991), Altonji and Williams (2005).