Midterm Exam in Econometrics
ECON 837
February 23rd, 2015
Exercise 1
Consider the following two models:
Y
= Xβ + ϵ
(1)
Y
= Xβ + Zγ + ϵ
(2)
where X is (N, KX ) and Z is (N, KZ ). Denote the OLS estimator of β in Equation (1) by
β̂ (1) . Denote the OLS estimator of β in Equation (2) by β̂ (2) .
(a) What are E[β̂ (1) |X, Z] and V ar[β̂ (1) |X, Z] equal to if in fact the true model is (2)?
(b) What are E[β̂ (2) |X, Z] and V ar[β̂ (2) |X, Z] equal to if in fact the true model is (1)?
(c) For the special case KX = KZ = 1, compare V ar[β̂ (1) |X, Z] and V ar[β̂ (2) |X, Z] if the
true model is (1).
(d) For the special case KX = KZ = 1, compare V ar[β̂ (1) |X, Z] and V ar[β̂ (2) |X, Z] if the
true model is (2).
(e) Discuss and interpret your ndings.
Exercise 2
Consider the two regressions,
yi = β1 x1,i + β2 x2,i + β3 x3,i + ui
(3)
yi = α1 z1,i + α2 z2,i + α3 z3,i + vi
(4)
where z1,i = x1,i − 2x2,i , z2,i = x2,i + 4x3,i , and z3,i = 2x1,i − 3x2,i + 5x3,i .
Let X = [x1 x2 x3 ] and Z = [z1 z2 z3 ], where x1 = [x1,1 x1,2 · · · x1,N ]′ (a column-vector).
We use similar denitions for the other components of X and Z .
(a) Find the elements of the (3, 3) matrix A such that Z = XA. Show that A is invertible.
(b) How is the OLS estimator β̂1 related to the OLS estimators α̂1 , α̂2 , and α̂3 ?
1
(c) Show that the two regressions give the same predicted values and residuals. Provide
some intuition for this result.
Exercise 3
Consider again the linear regression model
yi = x′i θ0 + ui ,
i = 1, · · · , n
(5)
d1
d2
where xi ∈ Rd can be partitioned into 2 subvectors x(1)
and x(2)
(d = d1 + d2 )
i ∈ R
i ∈ R
with corresponding coecient vectors θ0,1 and θ0,2 . Suppose that the object of principal
interest is θ0,1 . Suppose that xi is endogenous and that a vector of mean zero instruments
(2)′
zi ∈ Rdz is available such that E[zi ui ] = 0 and for which also E[zi xi ] = 0. Assume iid
data.
(a) If d1 ≤ dz < d then E[zi x′i ] cannot possibly have full column rank because the linear
system is underidentied. Moreover, zi is uncorrelated with the regressors x(2)
i .
Show that θ0,1 can nevertheless be consistently estimated by proposing a consistent estimator
θ̂1 of θ0,1 and derive its limit distribution.
(b) Now suppose that dz ≥ d and that zi is independent of both ui and x(2)
i .
Under what circumstances would you use the standard 2SLS estimator of θ0,1 (i.e. the subvector of the 2SLS estimator corresponding to θ0,1 ) and under what circumstances would it
be better to use the estimator you proposed in your answer to question (a)? Show your work.
Exercise 4
Consider the linear regression model
yi = x′i θ0 + ui ,
i = 1, · · · , n
Assume iid data, and suppose that there is homoskedasticity.
(a) Derive the test statistic that you would use to test
2
2
2
2
H0 : θ01
− θ02
= 5 and θ02 + θ03 = 1 vs H1 : θ01
− θ02
̸= 5 or θ02 + θ03 ̸= 1
(b) What is the limit distribution of your test statistic?
2
(6)