Econometrics 718 - problem set 1 two staged least squares

Question

Econometrics 718 - Problem Set 1

Problem 1 - Take any data set you would like and verify the two part regression results (if you google "econometric data sets" there are a lot to choose from). That is, think about a multiple regression which you can separate the independent variables into to groups.

Y_i = X'_1iβ₁ + X'_2iβ₂ + u_i

Verify that you get exactly the same results for bβ₁^{^} doing the two things:

a) A big multiple regression of Y_i on (X_1i, X_2i).

b) Two part regression where you first run Y_i and X_1i on X_2i and take residuals and then run the residuals from the Y regression on the residuals for the X regressions.

c) Now suppose that rather than the first set you just regressed Y_i on X^~_1i. (where Y_i is the original data and X^~_1i is the residuals). Does that give you the same result? Why or why not?

Problem 2 - Again with any data set you would like and with any software you would like think about the exactly identified IV problem. I would like you to produce the IV estimate in four different ways and show they are numerically equivalent (I don't care whether the instrument is really uncorrelated with the residual):

a) IV, that is (Z'X)^-1Z'Y (you can use ivregress with the gmm option in stata)

b) Two staged least squares. That is first run the treatment variable on the instrument and the X's, form the predicted value, then run a regression of the outcome on the predicted value and the other X's.

c) Ratio of reduced form coefficients. Run the two reduced forms and take the ratio of the coefficients on the Z.

d) Literally use

α^{^} = scov(Z^~, Y^~)/scov(Z^~, T^~)

where the tildes mean residuals after regressing on X, scov means sample covariance, and svar means sample variance.

Problem 3 - Verify the measurment error result. That is I want you to use your statistical package to contruct

T_i to have whatever distribution you want (a uniform might be easy)

Y_i = β₀ + β₁T_i + u_i

where u_i is N(0, σ_u²). You can choose these parameters to be anything you want.

τ_1i = T_i + ξ_i

where ξ_i is N(0, σ²_ξ)

τ_2i = T_i + η_i

where η_i is N(0, σ²_η)

Show that

a) If you run a regression of Y_i on T_i you get something close to β₁

b) If you run a regression of Y_i on τ_1i you get something close to

β₁(V ar(T_i)/V ar(T_i) + σ²_ξ)

c) Doing IV using τ_2i as an instrument for τ_1i gives an estimate close to β₁.