Econometrics 718 - Problem Set 3

Problem 1 - Consider a model in which

Y_i = β₀ + β₁T_i + µ_i + ξ_i

Assume that

- ξ_i is mean 0 and independent of everything else in the model

- µ_i is 0 with probability 0.5 and 1 with probability 0.5.

- There is a further variable X_i which is

X_i = ½ µ_i + ν_i

where ν_i is independent of everything else in the model and uniformly distributed on [0, 1]

- T_i is binary and determined according to the following rule

• When X_i < x∗ and µ_i = 0 then Pr(T_i = 1) = 0
• When X_i ≥ x∗ and µ_i = 0 then Pr(T_i = 1) =p₀
• When X_i < x∗ and µ_i = 1 then Pr(T_i = 1) = 0
• When X_i ≥ x∗ and µ_i = 1 then Pr(T_i = 1) =p₁

- The econometricial observes X_i, T_i, and Y_i but not the other variables.

a) Suppose you run a regression of Y_i onto T_i. What is the plim of β^{^}1 (where β^{^}1 is the OLS estimator of the coefficient on T_i)?

b) Suggest a methodology to get a consistent estimate of β₁ and sketch how identification works.

c) Now suppose the type 1 people (i.e. the people for whom µ_i = 1) can manipulate their X_i. In particular they can change their X_i to X_i+δ at some cost, and anyone with X_i < x∗ < X_i + δ pay this cost and manipulate there X_i. How does this affect your estimates from you estimation method above?

d) Now suppose they cannot manipulate it exactly but in reality

X_i =Z_i + ω_i

and they can manipulate Z_i but not ω_i where ω_i is independent of everything else and uniform. Sketch why this form of manipulation is not a problem. For simplicity (if you want) assume that Z_i only takes on two values, but that the distribution of Z_i depends on µ_i.

Problem 2 - Take a real data set with a real X and a real Y that are related somehow. Construct a placebo treatment T_i by choosing some rule for so that T_i = 1 when X_i > x∗ for some x∗. Now try estimating the model

Y_i =β₀ + αT_i + u_i

by regression discontinuity in several different ways (i.e. kernel regression, local linear regression, using polynomials etc.) Compare the results (given that you should get an effect of zero)

Problem 3 - Construct a dummy data set as I did in class in the rd.do file (which is on my website). Modify the simulated model and the estimation method for a fuzzy design rather than a sharp design and show what you get.

Problem 4 - Take any data set you would like and consider using some instrument for some treatment. Control for a bunch of regressors.

a) First run IV and get estimates.

b) Now calculate the bias assuming that "selection on the observables is the same as selection on unobservables." In doing so you can assume that the unobservables are uncorrelated with the observables and that your answer in a) gave consistent estimates of γ. Given the formula for the bias of IV given in the lecture notes in discussing the bias using "Catholic as an instrument" calculate the bias.

c) Now construct for yourself a perfect instrument. That is construct a model with

T_i =β₀ + β₁Z_i + X'_iβ₂ + u_i

Y_i =γ₀ + γ₁T_i + X'_iγ₂ + ε_i

where Z_i is independent of X_i and ε_i, but u_i is correlated with ε_i. That will involve using a random number generator to generate new values of u_i, ε_i, Z_i, T_i, and Y_i, but using the X's you used in a). Estimate the IV model and also estimate the bias in this case.

Attachment:- Data File.rar