Problem Set #8 Answer Key
Economics 435: Quantitative Methods
Fall 2011
Please use WebCT to turn in both the Word document and R code for question (1) and turn in your answer
to question (2) in class.
1
Public sector unionization and the size of government: Part V
Please see the file http://www.sfu.ca/~bkrauth/econ435/restricted/psu5.pdf. The R code used to generate these results is available at http://www.sfu.ca/~bkrauth/econ435/restricted/qunionization5.r
2
Fixed effects with measurement error
a)
plim β̂ OLS
=
=
=
=
=
cov(y
ˆ it , x̃it )
var(x̃
ˆ it )
cov(yit , x̃it )
var(x̃it )
cov(ai + βxi + βvit + uit , xi + vit + it )
var(xi + vit + it )
cov(ai , xi ) + βvar(xi ) + βvar(vit )
var(xi ) + var(vit ) + var(it )
β(σx2 + σv2 ) + σx σa ρa,x
(σx2 + σv2 ) + σ2
plim
b)
plim β̂ F D
=
=
=
=
=
cov(∆y
ˆ
it , ∆x̃it )
var(∆x̃
ˆ
it )
cov(∆yit , ∆x̃it )
var(∆x̃it )
cov(β∆vit + ∆uit , ∆vit + ∆it )
var(∆vit + ∆it )
βvar(∆vit )
var(∆vit ) + var(∆it )
σ2
β 2 v 2
σv + σ
plim
1
ECON 435, Fall 2011
2
c) If there is no fixed effect, then the probability limit of the OLS estimator is:
plim β̂ OLS = β
Since
2
(σx
+σv2 )
2 +σ 2 )+σ 2
(σx
v
is closer to one than
σv2
σv2 +σ2
(σx2 + σv2 )
+ σv2 ) + σ2
(σx2
is, the OLS estimator has smaller asymptotic bias.
d) If the fixed effect is uncorrelated with xi , then the probability limit of the OLS estimator is:
plim β̂ OLS = β
Since
2
(σx
+σv2 )
2 +σ 2 )+σ 2
(σx
v
is closer to one than
σv2
σv2 +σ2
(σx2 + σv2 )
(σx2 + σv2 ) + σ2
is, the OLS estimator has smaller asymptotic bias.
e) If there is no measurement error, then the probability limit of the OLS estimator is:
plim β̂ OLS =
β(σx2 + σv2 ) + σx σa ρa,x
(σx2 + σv2 )
and the probability limit of the FD estimator is
plim β̂ F D = β
Since the FD estimator is consistent and the OLS estimator is not, the FD estimator obviously has smaller
asymptotic bias.
f) The first statement is the correct one.