Reference no: EM131095942
Econometrics 742 - Problem Set 1
Problem 1 - Take any data set you would like and verify the two part regression results (if you google something like 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.
Yi = X'1iβ1 + X'2iβ2 + ui
Verify that you get exactly the same results for β^1 doing the two things:
a) A big multiple regression of Yi on (X1i, X2i).
b) Two part regression where you first run Yi and X1i on X2i 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 Yi on X~1i. (Where Yi 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 residua):
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~)/svar(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
- Ti to have whatever distribution you want (a uniform might be easy)
Yi = β0 + β1Ti + ui
where ui is N(0, σ2u). You can choose these parameters to be anything you want.
τ1i = Ti + ξi
where ξi is N(0, σ2ξ)
τ2i = Ti + ηi
where ηi is N(0, σ2η)
Show that
a) If you run a regression of Yi on Ti you get something close to β1
b) If you run a regression of Yi on τ1i you get something close to
β1(Var(Ti)/Var(Ti) + σ2ξ)
c) Doing IV using τ2i as an instrument for τ1i gives an estimate close to β1.
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