B9324 Introduction to
Econometrics and Statistical Inferece
Problem Set V
Due: December 3rd, 2021
1. Let X1, …, XN be an iid sample of Bernoulli random variables; that is, each Xi has
density f(x, θ) = θx(1 − θ)(1−x). Instead of using ML estimation, consider instead
estimation of θ0 using the generalized method of moments (GMM).
(a) Consider GMM estimation of θ using the single moment condition ψ(x, θ) =
x− θ. Show that E[ψ(Xi, θ)] = 0 for θ = θ0, and E[ψ(Xi, θ)] 6= 0 for θ 6= θ0.
(b) Compute the GMM estimator for θ0.
(c) Determine the asymptotic distribution of θ̂gmm. How would you estimate the
variance? Is this distribution the same as the asymptotic distribution of θ̂mle?
2. Use the data in the ascii file card wage 2008.txt that you also used in problem set
4. The data consists of observations on eleven years of average hourly wage data
for 1312 individuals. Again discard all individuals who in any period change wages
by more than a factor 10 in any one year, so that you have 1302 individual left. We
only use the first four years of data. Do everything in levels of wages. We consider
the following panel data model: Yit = ηi + θ · Yit−1 + �it, where �it has mean zero
given {Yit−1, Yit−2, …}. We have observations Yit for t = 1, …, T and i = 1, …, N ,
with N large relative to T . We use the moments
ψ1t(Yi1, …, YiT , θ) =
Yit−2
Yit−3
…
Yi1
· (Yit − Yit−1 − θ · (Yit−1 − Yit−2)).
This leads to t − 2 moment functions for each value of t = 3, …, T , leading to a
total of (T − 1) · (T − 2)/2 moments. In addition, under the assumption that the
initial condition is drawn from the stationary long-run distribution, the following
additional T − 2 moments are valid:
ψ2t(Yi1, …, YiT , θ) = (Yit−1 − Yit−2) · (Yit − θ · Yit−1).
(a) Do the discarding based on all 11 periods, even though we only use four periods
of data later.
(b) Given that we use T = 4, how many moments do we have, combining ψ1 and
ψ2? Write down all moment functions explicitly.
(c) Write down the derivatives of the moment function.
(d) Fix θ = 0.3. Calculate the value of the average moments at θ = 0.3.
(e) First we do standard two-step GMM. In the first step we use the identity
matrix. At θ = 0.3, what is the value of the objective function?
(f) Calculate the derivative of the objective function with respect to θ at θ = 0.3.
(g) Minimize the objective function and report the estimate
(h) Using the estimated value of θ, estimate the optimal weight matrix.
(i) Calculate the two-step GMM estimate by minimizing the objective function
with the estimated weight function, and estimate the standard error.