ECON 837 - Econometrics
Assignment #3
due in class March 22nd
Provide detailed calculations and justications to get full credit.
Partial credit may be given.
1 Theoretical exercises
Exercise 1:
Prove that under the presence of conditional heteroskedasticity, GLS is asymptotically
more ecient than OLS (i.e., both are consistent but the asymptotic variance of GLS
is smaller than for OLS).
Exercise 2:
Consider the linear model yi = β1 x1,i + β2 x2,i + ϵi for i = 1, · · · , n and the usual
assumptions are valid including ϵ having a normal distribution and the spherical error
variance. We are interested in testing H0 : β2 = 0 versus Ha : β2 ̸= 0. To this end, we
estimate β = [β1 , β2 ]′ by OLS, β̂ = [β̂1 , β̂2 ]′ and we consider the following statistic:
t⋆ =
σ̂ ⋆
√
β̂2
[(X ′ X)−1 ]2,2
where σ̂ ⋆ = ϵ⋆′ ϵ⋆ /(n − 1) and ϵ⋆ = Mc y are the residuals of the regression where we
impose the restriction β2 = 0.
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(a) Explain why in general the test statistic t⋆ does not have an exact Student-t
distribution under the null hypothesis that β2 = 0 for a xed sample size n.
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(b) What is the asymptotic distribution of t⋆ under the null hypothesis?
Exercise 3:
Let X1 , · · · , Xn be an i.i.d. random sample generated according to the pdf
f (x|θ) = θxθ−1 , 0 ≤ x ≤ 1 , 0 < θ < ∞ .
(a) What is the MLE of θ?
(b) What is the variance of the MLE.
(c) Show that the MLE is consistent.
2 Computational exercises
Exercise 4:
Consider the model
y1t = βy2t + u1t
y2t = αxt + u2t
([
ut ∼ iid N
0
0
] [
,
1 0.99
0.99 1
])
in which y1t and y2t are dependent variables, xt ∼ iid U(0, 1) (the uniform distribution
over (0,1)) is the exogenous variable (held xed in repeated samples) and the parameters
are β and α. The sample size is 5 and 10,000 replications are used to generate the
sampling distribution of the estimator.
(a) Generate the sapling distribution of the IV estimator for the parameter values
β = 0 and α = {0.5, 1, 10}. Discuss the sampling properties of the IV estimator
in each case.
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(b) Generate the sampling distribution of the IV estimator for the parameters values
β = 0 and α = 0. Compare this sampling distribution to the three sampling
distribution obtained in part (a). Also compute the sampling distribution of the
OLS estimator for this case, and discuss your results.
(c) Repeat parts (a) and (b) for samples of size T = 50 and 500. Discuss whether
the results in parts (a) and (b) are aected by asymptotic arguments.
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