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• There are 4 questions worth 70 points in total.
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Question 1. (25 points in total, each part is worth 5 points) You wish to estimate the causal effect β1 of X on Y :
Yi = β0 + β1Xi + ui . (1)
You are concerned endogeneity bias might lead to inconsistency of the OLS estimate of β1. You have a control variable Ci, which is not binary. The control variable satisfies conditional mean independence:
E[ui|Xi, Ci] = E[ui|Ci] . (2)
However, the conditional mean of ui depends on Ci in a nonlinear fashion:
E[ui|Ci] = δ0 + δ1Ci + δ2C2 , (3)
where each of the δ coefficients is non-zero.
You have data on Xi, Yi and Ci drawn i.i.d. from their joint distribution. You also know that each of Yi and Xi has finite nonzero fourth moments and Ci has finite nonzero eighth moment.
Hint: conditioning on Ci is the same as conditioning on Ci and C2. This is because C2 contains no
extra information beyond that contained in Ci. Therefore, E[ui|Ci] = E[ui|Ci, C2] and similarly for
other conditional expectations.
(a) Propose an approach for consistently estimating β1 from data on Xi, Yi and Ci.
Be sure to clearly describe the model you would estimate. You should state what the dependent and explanatory variable/s are and the method you would use to estimate β1.
(b) Write the model from (a) in BLP form. In answering, clearly relate the BLP coefficients to your parameter of interest β1. Show your working to receive full credit.
(c) Show that the procedure you describe in part (a) will produce a consistent and unbiased estimate of β1.
You do not need to provide a formal proof of consistency and unbiasedness, but you should be able to show whether or not the relevant key assumption is satisfied.
(d) Briefly explain and distinguish the concepts of consistency and unbiasedness. In answering, give an example of an estimator we’ve used this semester which is consistent but not unbiased.
(e) How, if at all, would your answer to (a) change if Ci was binary? Explain.
Question 2. (15 points in total, each part is worth 5 points)
You wish to investigate whether an individual’s previous union membership status influences their current status. You have panel data on individuals’ union membership over 4 years (t = 1, 2, 3, 4) on the variable Mit, which takes the value 1 if individual i was a union member in year t and 0 otherwise. You model individual i’s utility from choosing to be a union member (U1) or not (U0) in year t as a function of previous membership status Mit−1, a fixed effect αi, and random components εit,1 and εit,0:
U1(Mit−1, αi, εit,1) = u1(Mit−1, αi) + εit,1 , (4)
U0(Mit−1, αi, εit,0) = u0(Mit−1, αi) + εit,0 . (5)
The εit,0 and εit,1 terms represent the parts of individual i’s utility from each choice in year t that are not explained by previous membership status and the fixed effect. These are drawn randomly each year whereas the fixed effect is constant over time. You assume
u1(Mit−1, αi) − u0(Mit−1, αi) = β1Mit−1 + αi . (6)
You also assume that, for each year t, the conditional distribution of εit,1 − εit,0 given Mit−1 and
αi is a logistic distribution:
(εit,1 − εit,0)|Mit−1, αi has cdf Λ, where Λ(u) = 1 + e−u . (7)
(a) Derive an expression for Pr(Mit = 1|Mit−1, αi).
(b) Explain the role of the individual fixed effects in this model. What is it that we are attempting to control for by the inclusion of individual fixed effects?
(c) Unlike panel regression models, here there is no obvious way to difference out the individual fixed-effect αi from the expression you obtained in (a). After some algebra, you deduce
Pr(Mi2 = 1|Mi4, Mi2 + Mi3 = 1, Mi1, αi) = 1 + e−β1(Mi1−Mi4) , (8)
e−β1(Mi1−Mi4)
Pr(Mi2 = 0|Mi4, Mi2 + Mi3 = 1, Mi1, αi) = 1 + e−β1(Mi1−Mi4) . (9)
Describe how you could use these expressions to estimate β1. Be sure to clearly describe the model you would estimate. You should state what the dependent and explanatory variable/s are, the (subset of) data you would use, and the method you would use to estimate β1.Hint: You might want to consider only “switchers”: these are individuals who change union membership status between dates 2 and 3 (i.e., for whom Mi2 + Mi3 = 1).
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