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# Residuals and Regression Fit

.
12.16 Asymptotic Distribution of 2SLS
We now show that the 2SLS estimator satisÖes a central limit theorem. We Örst state a set of
su¢ cient regularity conditions.
Assumption 12.2 In addition to Assumption 12.1,

1. E

y
4

< 1:

1. E kzk
4 < 1:

2. = E

zz0
e
2

is positive deÖnite.
Assumption 12.2 strengthens Assumption 12.1 by requiring that the dependent variable and
instruments have Önite fourth moments. This is used to establish the central limit theorem.
Theorem 12.2 Under Assumption 12.2, as n ! 1:
p
n

b
2sls

d ! N (0;V )
where
V =

QxzQ1
zz Qzx1
QxzQ1
zz
Q1
zz Qzx QxzQ1
zz Qzx1
:
This shows that the 2SLS estimator converges at a p
n rate to a normal random vector. It
shows as well the form of the covariance matrix. The latter takes a substantially more complicated
form than the least-squares estimator.
As in the case of least-squares estimation, the asymptotic variance simpliÖes under a conditional
homoskedasticity condition. For 2SLS the simpliÖcation occurs when E

e
2
i
j zi

= 
2
. This holds
when zi and ei are independent. It may be reasonable in some contexts to conceive that the error ei
is independent of the excluded instruments z2i
, since by assumption the impact of z2i on yi
is only
through xi
, but there is no reason to expect ei to be independent of the included exogenous variables
x1i
. Hence heteroskedasticity should be equally expected in 2SLS and le
CHAPTER 12. INSTRUMENTAL VARIABLES 425
Nevertheless, under the homoskedasticity condition then we have the simpliÖcations
= Qzz
2
and V = V
0

# def

QxzQ1
zz Qzx1

2
.
The derivation of the asymptotic distribution builds on the proof of consistency. Using equation
(12.41) we have
p
n

b
2sls

# 


1
n
X0Z
  1
n
Z
0Z
1 
1
n
Z
0X
!1


1
n
X0Z
  1
n
Z
0Z
1 
1
p
n
Z
0e

:
We apply the WLLN and CMT for the moment matrices involving X and Z the same as in the
proof of consistency. In addition, by the CLT for i.i.d. observations
1
p
n
Z
0e =
1
p
n
Xn
i=1
ziei
d ! N (0;
)
because the vector ziei
is i.i.d. and mean zero under Assumptions 12.1.1 and 12.1.7, and has a
Önite second moment as we verify below.
We obtain
p
n

b
2sls

# 


1
n
X0Z
  1
n
Z
0Z
1 
1
n
Z
0X
!1


1
n
X0Z
  1
n
Z
0Z
1 
1
p
n
Z
0e

d !
QxzQ1
zz Qzx1 QxzQ1
zzN (0;
) = N (0;V )
as stated.
For completeness, we demonstrate that ziei has a Önite second moment under Assumption 12.2.
To see this, note that by Minkowskiís inequality (B.33)

E

e
4

# 1=4


E

y x
0
4
1=4


E

y
4
1=4

• k k

E kxk
4
1=4
< 1
under Assumptions 12.2.1 and 12.2.2. Then by the Cauchy-Schwarz inequality (B.31)
E kzek
2 

E kzk
4
1=2

E

e
4
1=2
< 1
u