Midterm Answer Key
Economics 435: Quantitative Methods
Fall 2011
1
Warmup
a) Yes. A proof is not required, but here it is:
!
n
1X
xi
E(x̄n ) = E
n i=1
n
=
1X
E (xi )
n i=1
n
=
=
=
=
b) As usual, var(x̄n ) =
σ2
n .
1X
(µE I (i is even) + µO I (i is odd))
n i=1
!
!
n
n
1X
1X
µE
I (i is even) + µO
I (i is odd)
n i=1
n i=1
1n
1n
µE
+ µO
n2
n2
µE + µO
2
A proof is not required, but here it is. First, note that since the observations
1
ECON 435, Fall 2011
2
are independent cov(xi , xj ) = 0 for all i 6= j. Then:
n
var (x̄n )
=
=
=
=
=
=
=
1X
xi
n i=1
var
!
!
n
X
1
var
xi
n2
i=1


n
n
1 X X
cov(xi , xj )
n2 i=1 j=1
!
n
1 X
var(xi )
n2 i=1
!
n
1 X 2
σ
n2 i=1
1
nσ 2
n2
σ2
n
c) Yes. A proof is not required, but here it is. Let yj =
x2j−1 +x2j
2
and let N = n/2. Then:
n
x̄n
=
1X
xi
n i=1
=
N
1 X
x2j−1 + x2j
2N j=1
=
N
1 X
2yj
2N j=1
=
N
1 X
yj
N j=1
= ȳN
Now, since the x’s are independent, so are the y’s. In addition, the y’s are identically distributed: y ∼
2
N (θ, σ2 ). Therefore we have a random sample of size N on y and can apply the law of large numbers to say
that ȳN →p θ. Since x̄n = ȳN , this implies x̄n →p θ as well.
ECON 435, Fall 2011
2
3
The Mincer human capital model
a) We use our standard procedure
plim β̂1
=
=
=
=
=
=
=
=
=
cov(LN
ˆ
W AGE, EDU C)
(usual OLS formula)
var(EDU
ˆ
C)
plim cov(LN
ˆ
W AGE, EDU C)
(by Slutsky)
plim var(EDU
ˆ
C)
cov(LN W AGE, EDU C)
var(EDU C)
cov(β0 + β1 EDU C + β2 EXP ER + u, EDU C)
(by substitution)
var(EDU C)
cov(EXP ER, EDU C)
β1 + β2
var(EDU C)
cov(AGE − EDU C − 6, EDU C)
β1 + β2
var(EDU C)
cov(AGE, EDU C) − var(EDU C)
β1 + β2
var(EDU C)
−var(EDU C)
β1 + β2
(since cov(AGE, EDU C) = 0)
var(EDU C)
β1 − β2
plim
b) We can substitute (AGE − EDU C − 6) for EXP ER in the original model and get:
LN W AGEi
= β0 + β1 EDU Ci + β2 EXP ERi + ui
= β0 + β1 EDU Ci + β2 (AGEi − EDU Ci − 6) + ui
=
(β0 − 6β2 ) + (β1 − β2 )EDU Ci + β2 AGEi + ui
Since E(u|EDU C, AGE) = 0:
plim β̂1
= β1 − β2
c) The coefficient β̂1 (and thus its probability limit) doesn’t even exist, because the explanatory variables
in this regression are perfectly collinear (EXP ER is an exact linear function of EDU C and AGE).
d) These regressions would tend to underestimate β1 .
3
Vector autoregressions
a) First we substitute to get:
mt
= b0 yt + b1 yt−1 + b2 mt−1 + vt
= b0 (a1 yt−1 + a2 mt−1 + ut ) + b1 yt−1 + b2 mt−1 + vt
=
(b0 a1 + b1 )yt−1 + (b0 a2 + b2 )mt−1 + (b0 ut + vt )
= π1 yt−1 + π2 mt−1 + t
ECON 435, Fall 2011
4
Then we note that:
E(t |yt−1 , mt−1 )
=
E(b0 ut + vt |yt−1 , mt−1 )
=
b0 E(ut |yt−1 , mt−1 ) + E(vt |yt−1 , mt−1 )
=
b0 0 + 0
=
0
b)
σu
= cov(ut , t )
= cov(ut , b0 ut + vt )
= b0 cov(ut , ut ) + cov(ut , vt )
= b0 σu2 + 0
= b0 σu2
c) Note that:
b1
σu
σu2
= π1 − b0 a1
b2
= π2 − b0 a2
b0
=
So applying the analog principle produces the estimators
σ̂u
σ̂u2
b̂0
=
b̂1
= π̂1 −
b̂2
σ̂u
â1
σ̂u2
σ̂u
= π̂2 − 2 â2
σ̂u
ECON 435, Fall 2011
5
d)
plim b̂0
plim b̂1
σ̂u
σ̂u2
plim σ̂u
=
(Slutsky)
plim σ̂u2
σu
=
(given)
σu2
b0 σu2
=
(previously shown)
σu2
= b0
σ̂u
= plim π̂1 − 2 â1
σ̂u
σ̂u
(Slutsky)
= plim π̂1 − plim 2 plim â1
σ̂u
σ̂u
= π1 − plim 2 a1
(given)
σ̂u
= π1 − b0 a1
(previously shown)
=
plim
=
(b0 a1 + b1 ) − b0 a1
=
b1
σu
= plim π̂2 − 2 â2
σu
σ̂u
(Slutsky)
= plim π̂2 − plim 2 plim â2
σ̂u
= π2 − b0 a2
(given/previously shown)
plim b̂2
=
(b0 a2 + b2 ) − b0 a2
= b2