# 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,

- E

y

4

< 1:

- E kzk

4 < 1:

= 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 Qzx1

QxzQ1

zz

Q1

zz Qzx QxzQ1

zz Qzx1

:

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 Qzx1

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 Qzx1 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

- kk

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