ECON 837 - Econometrics
Assignment #4
due in class April 7th
Provide detailed calculations and justications to get full credit.
Partial credit may be given.
• Exercise 1:
Consider the linear model yt = θ0 zt + ϵt where θ0 and zt are scalars. The associated
orthogonality conditions are,
E[xt (yt − θ0 zt )] = 0
and the associated GMM objective function is denoted Qn (θ).
Assume that θ0 > 0 (which means Θ = (0, ∞)) and consider the reparametrization
λ = 1/θ with orthogonality conditions,
E[xt (zt − λ0 yt )] = 0
and the associated GMM objective function is denoted Q̃n (λ).
(a) Write the 2 GMM objective functions Qn (θ) and Q̃n (λ) (using the same weighting
matrix W ).
(b) Is it the case that Qn (θ) = Q̃n (1/θ)? that Q̃n (λ) = Qn (1/λ)?
(c) What can you conclude from the previous question?
• Exercise 2:
Consider the ecient GMM estimator that presumes the moment functions, {f (zt , θ)},
to be serially uncorrelated. We are interested in studying asymptotic properties of the
associated 2S-GMM estimator when {f (zt , θ)} is in fact correlated.
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(a) Is the 2S-GMM estimator consistent?
(b) Is the 2S-GMM estimator ecient?
(c) What would you recommend to a practitioner?
• Exercise 3:
Consider the model
i = 1, · · · , n,
yi = g(xi ; θ0 ) + ui
where g is a known function and θ0 an unknown vector of coecients. Suppose further
that potentially
P [E(u1 |x1 ) = 0] < 1 ,
but that there is an observable vector zi for which E[z1 u1 ] = 0.
Answer the following questions using GMM.
(a) What are the moment conditions in this case?
(b) Determine the optimal weight matrix.
(c) Determine the asymptotic distribution using the optimal weight matrix.
• Exercise 4:
Suppose that,
[
xi
yi
]
([
∼N
γ0
β0 γ0
] [
,
δ02
δ02 β0
δ02 β0 σ02 + δ02 β02
])
Suppose further that β0 is the parameter of interest.
(a) Derive the limit distribution of the MLE β̂ of β0 .
(b) Derive the conditional distribution of yi given xi = x.
(c) Show that the MLE β̂ in (a) is numerically identical to the conditional MLE of
β0 .
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