Write down a gmm estimator for beta beta0 beta1 there are

Question

Economics 706 - Problem Set 3

Problem 1 - Suppose that

E (Y_i|X_i) =β₀e^β_1X_i

where X_i is a scalor random variable

1. Write down a GMM estimator for β = (β₀, β₁). (There are many that you can use. You can make it simple as long as it works.)

2. Calculate the asymptotic variance for your estimator, that is

√N(β^{^} - β₀) ∼ N (0, V )

What is V?

3. Now suppose we want to use the model to estimate the derivative of E (Y_i|X_i) with respect to X_i evaluated at zero. Use the Delta method to derive a standard error of your estimator of this derivative.

Problem 2 - Choose any panel data set you would like.

Choose some regression specification and run the following regression models. Before running anything in stata type "xtset id yr" for whatever variables are your unique person identifiier (id) and time identifier (yr) or the corresponding alternatives if you are not using person year data.

a) Straight regression with robust standard errors

b) Regression clustering standard errors (reg Y X, cluster(id))

c) Random Effect Regression (xtreg Y X, re)

d) Fixed Effect Regression (xtreg Y X, fe)

e) Fixed Effect Regression clustering standard errors

f) First Difference Regression (reg d.y  d.x, nocons)

g) A regression with full individual dummy variables (xi: reg Y X i.id, cluster(id))

h) Construct Fixed effect yourself. In stata you can construct Ybari using (sort id;by id: egen ybari=mean(Y)). Then you can construct Y-Ybari using (gen Ytil=Yybari). Do the same thing for all of your X's and regression Ytil on Xtil.

i) Restrict your data to at most two periods per person. Show that in that case Fixed effects is the same as first differencing.