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THESIS

Illllllllllll‘IIIHIHIIIIIIIIHIIUlllllf‘illlHlllllHlllll

31293 01417 2666

This is to certify that the

dissertation entitled

IMPROVED GENERALIZED METHOD
OF MOMENTS ESTIMATORS

presented by

HAILONG QIAN

has been accepted towards fulfillment
of the requirements for

PhD degree in _Ecgnnmics_

 

1&3er

Major professor

Date July 22, 1995

MSU is an Affirmative Action/Equal Opportunity Institution 0- 12771

 

 

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MSU le An Nflrmetlve Action/Equal Opponunlty Inetltulon
W

”39.1

 

INIPROVED GENERALIZED METHOD
OF MOMENTS ESTIMATORS

HAILONG QIAN

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Economics

1995

ABSTRACT

Improved Generalized Method of Moments Estimators

By

Hailong Qian

This thesis introduces a new method to improve Generalized Method of Moments
estimators, given extra observable information. Monte Carlo simulation for a simple
model with intercept only confirms the accuracy of the asymptotic results obtained in this
thesis even when the sample size is quite small The three-stage least squares estimator of
a system of equations is shown to be asymptotically equivalent to an iterative two-stage
least squares estimator applied to each equation, augmented with the residuals fiom the

other equations.

ACKNOWLEDGENIENTS

It is impossible for me to express sufficient gratitude to my major professor, Peter
Schmidt, who masterly taught me econometrics, suggested this dissertation topic, carefully
guided me through each step of my research, meticulously read and corrected my drafts
time after time, and kindly offered me research assistantships for Summer, 1993, and the
1994-95 academic year. Without Professor Schmidt's valuable help, it would have been
impossible for me to complete this dissertation.

I am also extremely gratefiil to Professors Richard Baillie, Ching-Fan Chung,
Robert Rasche and Jeffrey Wooldridge for their valuable help with this dissertation. I
consider myself extremely fortunate to have the opportunity to associate with such an
excellent group of econometricians. Their intelligence, diligence and success have inspired
me so much.

I am heavily indebted to Professor Daniel Suits and Mrs. Adelaide Suits. It was
Professor Suits who helped get me admitted into this program, and most of all helped me
to get a scholarship from the Ford Foundation for my first year here. Without their help, I
would not have been here in the first place, and everything else would not be possible
either.

I also owe a debt of special gratitude to Mrs. Dottie Schmidt both for her help
with my spoken English and for her extreme kindness. Her generous help given to me and
numerous foreign students and scholars attending MSU makes our otherwise homesick
lives more enjoyable.

It is also my great pleasure to express my appreciation to Dan Hansen, Bill
Horrace, Young min Kwon, Hyung- Seung Lee, Paul Loetscher, Jess Reaser, Leslie

Schenk and Jen Tracey for all the help and entertainment they have given me in the last
five years. Their freindship has made my life here enjoyable.

I would also like to express my thanks to Ms. Ann F eldman and Terie Snyder for
their help in many ways in the last five years.

I owe my greatest debt to my family. Even though my parents are uneducated
farmers, through example they taught me the virtues of diligence, self-sufliciency, decency
and dignity. Though they often could not afford to buy what they needed, their sacrifice
for my education and their wishes for me to have a better life were unconditional I also
want to thank my four older brothers from my heart for all the things they have done for
me.

Finally, I gratefully acknowledge the scholarship for my first year here from the
US. Committee on Economics Education and Research in China sponsored by the Ford

Foundation.

TABLE OF CONTENTS

LIST OF TABLES ...................................................................................................... viii
CHAPTER 1: INTRODUCTION ................................................................................. 1
CHAPTER 2: IMPROVED GMM ESTIMATORS FOR THE LINEAR REGRESSION
MODEL ................................................................................................. 4
2.1. Introduction ............................................................................................... 4
2.2. GM with Moment Conditions Not Containing Unknown Parameters ....... 6
2.3. The Linear Model ....................................................................................... 8
2.4. Monte Carlo Results ................................................................................. 16
2.5. Concluding Remarks ................................................................................ 20
CHAPTER 3: IMPROVED GMM AND 3SLS ESTIMATORS FOR SYSTEM OF
EQUATIONS ....................................................................................... 23
3.1. Introduction .............................................................................................. 23
3.2. Model and Notation .................................................................................. 24
3.3. Improved GMM estimators ....................................................................... 25
3.4. Concluding Remarks ................................................................................. 42
CHAPTER 4: IMPROVED GMM ESTIMATORS FOR SYSTEM OF NONLINEAR
EQUATIONS ...................................................................................... 44
4.1. Introduction ............................................................................................. 44
4.2. Improved GMM Estimators ..................................................................... 44
4.3. Conchrding Remarks ................................................................................ 57

CHAPTER 5: THE ASYMPTOTIC EQUIVALENCE BETWEEN THE ITERATED

IMPROVED ZSLS ESTIMATOR AND THE BSLS ESTIMATOR ..... 58
5.1. Introduction ............................................................................................ 58
5.2. Improved ZSLS Estimator ...................................................................... 59
5.3. Iterated 12SLS Estimator ......................................................................... 63
5.4. The Convergence and Asymptotic Efliciency of the Iterated IZSLS
Estimators ............................................................................................... 71
CHAPTER 6: CONCLUSION ................................................................................... 97
REFERENCES ........................................................................................................... 99

LIST OF TABLES

TABLE 1 ..................................................................................................................... 21

TABLE 2 ..................................................................................................................... 22

CHAPTER]

INTRODUCTION

Suppose that we have a set of moment conditions EN)l (y: ,60 )] = 0 which
identify the unknown parameter 90, so that the generalized method of moments (GMM)
estimation of 60 is feasible. However, suppose that we also have available a set of
additional moment conditions E[¢2(y: )] = 0, where 62 is observable because it depends
only on the observed data y: . Then the question is how to utilize these additional moment
conditions in a simple way to improve the estimation of 60. This is possible when (I), is
correlated with m.

Problems of this type have been considered previously by Imbens (1992, 1993) and
Imbens and Lancaster ( 1994). Imbens (1992, footnote 3) considered estimation of
no = E(yt ). The sample mean, based on the moment condition E(yt — po) = O, is less
efficient than the GM estimate based on the moment conditions E[(yt — 110 ), ut ]' = 0, if
ut is observable, with E(ut ) = 0 and cov[(yt — Ho ), 11,] ¢ 0. Imbens (1992) and Imbens
and Lancaster (1994) analyze some other specific problems that lead to GMM estimation
with additional moment conditions that do not depend on the parameters of interest.

In this dissertation, we prove that the usual GMM estimator of 90, say 6 , using
the moment conditions E[¢1(y: ,60 )] = 0 and weighting matrix
0:: = aim... EIT'W 2.1.4». o: emu-“2 2.1m (yI.eo)1'}“. can be improved by
using the observed extra moment conditions E[¢2(y: )] = 0. Specifically, we prove that
the usual GMM estimator 6 is no more efficient than the augmented GMM (AGMM)

estimator, say 6 , defined as the GM estimator of 90 using the moment conditions

E[¢(Y:,Oo)] = E[¢l(Y:a90)'s¢2(Y:)'I= O and weighting matrix

2

C c ‘1 . ..
c“ = L” C12] ={1im-r_... EIT'W 2.1m. ,eomi‘” 21m. ,eon'r‘.
21 22

We further show that the AGMM estimator 6 is numerically the same as the improved
GMM (IGMM) estimator, say 6 , defined as the GMM estimator using the moment
conditions E[¢.(y:.eo)— Cncii‘l’z (y: )1 = o and weighting matrix

C“ =(C11— C12C2;C21)-1-

The structure of the dissertation is as follows. In chapter 2, we first provide a brief
general treatment of GMM estimation with additional moment conditions not containing
unknown parameters. We then give some more detailed results for the linear regression
model In the case of the linear regression model with conditional homoskedasticity and
uncorrelatedness, we show that the IGMM estimate is an improved ZSLS (IV) estimate
using as a new set of instruments the part of the original instruments that is orthogonal to
the observed extra variables, whereas the usual GMM estimate is just an ordinary ZSLS
(IV) estimate using the original set of instruments. We also provide some other estimators
that can be written in closed form and that are asymptotically equivalent to the IGMM
estimator. For the special case of a simple regression model with intercept only, we
provide some Monte Carlo evidence on the finite sample performance of some specific
improved estimators. For this simple model, the eficiency gains predicted by asymptotic
theory are realized even for quite small sample sizes.

In chapter 3, we extend the general results on improved GM to the case of a
system of linear equations. Under the assumptions of conditional homoskedasticity and
uncorrelatedness, we obtain explicit expressions for several asymptotically equally efficient
improved GMM estimators. While the usual GMM estimator is just an ordinary 3SLS
estimator, we prove that the IGMM is an improved 3SLS estimator. The improved 3SLS
estimator differs from the usual 3SLS estimator in two ways. First, the covariance matrix

of the residuals of the projection of the original model disturbances onto the observed

3

extra variables is used as the relevant error covariance matrix. Second, it uses as its
instruments the part of original instruments orthogonal to the observed extra variables.

In chapter 4, we extend the IGMM results from chapter 3 to the case of a system
of nonlinear equations. Under suitable regularity conditions and some "high-level"
assumptions, we show that essentially the same results as those in chapter 3 still hold for
this case.

In chapter 5, we further extend the improved GMM idea of previous chapters to
the case where the extra variables are not observed but consistently estimated. We
investigate this problem in the context of a system of linear equations. We show that
3SLS applied to the entire equation system is asymptotically equivalent to iterated ZSLS
applied to each equation, augmented by the residuals from the other equations. This result
generalizes a result ofTelser (1964) for the case of seemingly unrelated regressions. It
also provides an interesting example of a setting in which the improved GMM estimator
arises natually as an efficient estimator.

The final chapter concludes the dissertation with some brief comments on firrther

possible work in this line of research.

CHAPTER2

INIPROVED GMM ESTIMATORS
FOR THE LINEAR REGRESSION MODEL

2.1. Introduction

In this chapter, we provide (in section 2.2) a brief general treatment of GMM
estimation with additional moment conditions not containing unknown parameters. We
also give (in section 2.3) some more detailed results for the linear regression model.

Specifically, we consider the standard regression model

(2.1) yt=xt'B+8t, t=1,2, ...... , T,

with instruments zt satisfying E(z,et ) = 0. These moment conditions are the basis of
GM estimation of B ; under a conditional homoscedasticity assumption for at , the GM
estimator is the usual instrumental variables (IV) estimator. If we also have available a
vector of observable variables 111 that are uncorrelated with 2.1 but correlated with a, , the
additional moment conditions E(ut 69 2t ) = 0 will improve the efficiency of estimation of
[3. This principle applies in linear or nonlinear models, but in the linear case we obtain
very simple explicit results for the improved estimators.

We believe that these results are empirically relevant, notably in the estimation of
rational expectations models. In many empirical rational expectations models, the
orthogonath conditions used in estimation assert that a forecast error, written as a

fimction of data and parameters, is uncorrelated with variables in the information set at the

5
time the forecast was made. Thus a, is the error made in forecasting some variable at
time t, based on information available at time t- l, and zt consists of information available
at time t-l, so that it is uncorrelated with at. In this setting, ut can be the observable (ex
post) error in the forecast of a set of variables at time t based on information available at
time t- 1.
As a specific example, suppose that s is a spot exchange rate at time t and ft is the

one period forward rate. Many papers have tested the unbiasedness hypothesis that
f,_1 = E( st | OH ), where OH is the information set at time t-l. Thus we should have

a = 0 and B = l in the regression model

(2.2) st =or+13fH +st.

When SI and f,_] contain unit roots but are cointegrated, the above regression is often

replaced by a regression in stationary variables:

(23) (St - St—1)= ‘1 +B(ft—l _ St—1)+8t

where again or = O and B = 1 under the unbiasedness hypothesis. Because the forecast
error at is uncorrelated with variables in QM, (2.2) or (2.3) can be estimated by GM or
IV, where the instruments zt are variables in OH. This is a standard applied econometric
excercise. However, the esthate can be improved by using other observable variables 0.,
that are correlated with the forecast error a, but uncorrelated with 2,. Such variables will
typically be forecast errors in other related variables. An obvious example would be the
change in a security price from time t-l to t. We might reasonably expect a, and ut to be
correlated if spot exchange rates and security prices respond to the same unforecastable
economic shocks.

For Imbens's model of the estimation of the sample mean, we provide (in section

6

2.4) some Monte Carlo evidence on the finite sample performance of some specific
improved estimators. For this simple model, the eficiency gains predicted by asymptotic
theory are realized even for quite small sample sizes. The final section concludes the

chapter with some comments.

2.2. GMM With Moment Conditions Not Containing Unknown Parameters

Let 90 be a K x1 vector ofparameters to be estimated, and y:, t = 1, 2, T, be

observed data. Suppose that the following moment conditions hold:

(2.4) E¢(y:,90) = E[¢‘(Y:’?°)] = o, t = 1, T,
¢2(Yt)

where (bl is N X], with N 2 K, and 62 is Hx 1. We want to compare GMM based on 4)]
only with GMM based on d) =(¢1',¢2')'. Note that 4), does not depend on 60.

Define the following notation:
9 ..
(25A) Me) = [M )] = 1 Emma)
¢T2 Tt=1
(2 SB) C= C“ C" = limlT-Etb (9 )o (e r]
. C21 C22 T"°° T O T O
(2.50) D=[D‘]= lim M.
0 T—roo 60'

(The block "zero" in D arises because ¢T2 does not depend on 9 .) For identification of 90

7

we require D1 to be of fill] column rank. In the case that the y: are iid, C = V[¢(y: ,60)]
and D = E[a¢(y:,90)/ae'].

Let 6 denote the GM estimator of 90 based on the moment conditions (1)1 only,
using weighting matrix C; 11; and let 6 denote the augmented GMM (AGMM) estimator of
90 based on the moment conditions ¢ = (¢1',¢2')', using weighting matrix C"1. Under
suitable regularity conditions, standard GMM results indicate that these estimators are
consistent, with .AV[JT(é —eo)] = (ole-:1)l )-1 and Avwia‘) — 90)] = (lye-11))-l
=(D1'C11Dl)"1, where C11 = (CH - CHCZCZIYl is the block of C"1 corresponding to 4)]
(i e., the upper left block). For discussions of regularity conditions, see, e. g., Hansen
(1982) or Gallant and White (1988). The AGMM estimator is efficient relative to the
GMM estimator, since (Dl'C,"1‘D1)’l —(D1'C“D1)'1 is positive semidefinite. In fact, a
little algebra reveals that we‘ll)1 - lapel-1‘1), = (0,1031), )'(c22 - encgcu)‘l -
(C21C{11D1), so that the condition for no gain ill efficiency is Clel'I'D1 = 0. There is no
efficiency gain when C2] = 0 (t1)l and 92 are uncorrelated); when C21 at O, the AGMM
estimator is generally (but not necessarily) strictly better than the GM estimator.

We can also write the augmented GMM estimator as follows. Consider the

moment conditions

(2.6) Baronet)—Cocaoztytn=o,

which essentially deal with the residuals fiom a regression of ti), on (1)2. If
V[¢(yi,90)] = C. then Vim: .90) - Cacaottyt )1 = Cl. — 0.20302. = (6‘)“. With
this motivation, we define the improved GMM (IGMM) estimator, say 6, as the GM

estimator using the moment conditions (2.6) and weighting matrix
C11 = (C11 — c,,c;;c,,)“. It is then not difficult to show that the IGMM estimator 6 and

the AGMM estimator 6 are the same. This can be seen by noting that 6 satisfies the first

order condition

(2-7A) DTl(é).Cll[¢Tl (é) “ Clzcii‘i’rzl = 0

where DT1(6) = 64)“ (9) / 69'; 6 satisfies the first order condition

(2713) Datéwlon(é)+Drr(é)'c”om = 0.

But (2.7A) and (2.7B) are seen to be the same with the substitution C12 = —C”C12C;21 in
(2.7B).

The above discussion treats the weighting matrix C as known. Assuming suitable
regularity conditions, the superiority of the AGMM or IGMM estimator to the GMM
estimator will still hold asymptotically if C is replaced by a consistent estimate C. The
numerical equivalence of the AGMM and IGMM estimators would require that the same

estimate C be used for both estimators.

2.3. The Linear Regression Model

In this section we will apply the general results of the previous section to the case
of the linear regression model For this case we can give an explicit formula for the
IGMM estimator. When the errors are conditionally homoskedastic, further simplications
are possible and the IGMM estimator is related to some previous results.

The model considered in this section is as given in equation (2.1) above, which we
rewrite slightly as

(2.8) yt = xt'00+8,, t= l, 2, ...... , T,

where yt is the dependent variable , xt is a K x1 vector of explanatory variables, at is the
disturbance term, and 60 is the parameter vector to be estimated. Suppose that we have
available an M x1 vector of instruments zt satisfying M .2 K and E(z,et ) = 0, so that the
GM estimation of 90 based on the moment conditions E(z,e,) = 0 is feasible. However,
suppose that we also have available an L x] observable vector ut satisfying

E(ut ®z,)= 0 and E(utet) at 0. The observable data vector is y: = (yt,x,',zt',u,')', and

in our previous notation we have moment conditions E¢( y: ,90) = 0, with

(29A) ¢1(Y:’90)= no. — x390)
(2.913) My?) = u. ®z. = (It emut.

Asamatter ofnotation, let Z=(zl,---,zr)'; X=(x,,---,xT)'; a =(81,--°,8T)';
U=(u1,---,uT)'; y=(Y1’”'SYT)'; u(j) =(llJ-1,"',llij-)'f01‘j= 1’ m, L; and

n" =(u(1)',~--,u(L)')'= vec(U). Then straightforward calculation yields

(2.10A) on = T"z'(y - x9)
(2.1013) on = T“(IL ®Z')n‘ = T“veo(z' U)
(2.100) DT,(0) = —T“Z' x

Using the first order condition (2.7A) above, with these substitutions we arrive at the

IGMM estimator

(2.11) 6 = (X'ZCHZ'X)’1X'ZC”[Z'y — c,,c;;vee(z U)].

To proceed further, we need to put more structure on C. This is possible under

the assumption of no conditional heteroskedasticity or autocorrelation: suppose that,

10
conditional on Q, = {zt ;s,_1,u,_1,z,_,;...}, the (s,,u,')' are mutually uncorrelated, zt is

stationary, and that

(2.12) V([8‘]|z,)sz=[6: 2w].
u, 2

£118 uu
Then
(2.13) c = Holy:.eo)o(yt.eo)'= 2®E(ztzt')

for which a consistent estimate is

(2.14) f: = i cor-lz'z,

where E is any consistent estimate on. Then C11 = 65"‘(T’1Z'Z)’1 with
as = (63 — 2 12-12,, )-1, cl, = $3,, oar-122,6” = $3,, er'z'z,
C120}; = (Emil-1 )®IM. With these substitutions in (2.11), and noting that 6“ cancels,

we obtain
(2.15) 6 = (x' szy‘ X'Z(Z'Z)" {Z'y - [(233) ® IM]vec(Z' U)},

where PZ = Z(Z'Z)’1 Z'. More generally, if A is any matrix, we will define PA as the
projection onto A, so that PA = A(A'A)'1A' if A has firll cohrmn rank. Similarly, we
define M A = 1— PA. Obviously the first term in this expression is just (X' PZX)‘l X'sz,
the IV (ZSLS) estimator, which is GMM based on 411, given the assumption of no

conditional heteroskedasticity.

Using the matrix fact vec(BC) = (C'®I)vec(B), (2.15) can be rewritten as

(2.16) 6 = (x' PZX)’1 x' 2(z'Z)"[z'y — z' U232“, ].

11

It is reasonable to consider Em, = T'IU' U, in, = T“1U‘(y — X6), where 6 is any

consistent estimate of 9. Then (2.16) becomes
(2.17) 6 = (X'PZX)'1X' Pz[y — PU(y - xé)].

Finally, while (2.17) is defined for any consistent estimate 6, we may as well
consider 6 = 6. Then (2.17) implies (X' PZX)6 = x' sz — x' PZPUy + x' PZPUxé; solving

for 6 , we obtain
(2.18) 6 = (x'PZMnyl x' PZMUy.

The IGMM estimator 6 is very similar to an estimator considered by Schmidt
(1986, 1988):

(2-19) 6 = (X'P[MUZ]X)-1X'P[Mul]y'

This is IV of the regression equation (2.8), using as instruments MUZ, the part of Z
orthogonal to U. Schmidt also notes that '6. can be derived as IV of the augmented

equation
(2.20) yt = xt'Oo +ut'g +vt

using (Z, U) as instruments. Equation (2.20) is instructive because, speaking loosely, the

effect of adding the variable n1 is to reduce the relevant variance fiom o: to 03 = 0'2 -

elu —

2‘1 2‘. This result is closely related to the result of Wooldridge (1993), who

2-
as 2w an 118'

essentially considers the case xt = zt (in our notation).

12

To be more precise about the sense in which 6 and of). dominate the simple IV
estimator, and to exhibit some other asymptotically efficient estimators, we make some
more explicit assumptions. To make the asymptotic theory as simple as possible, we will
make the following "high level" assumptions.

 

 

F-A)O( X7 AXE Axu-
. 1 A2" A22 0 0 .
(A2.1) phm :r—[X,Z,8,U]‘[X,Z,8,U] = A“ 0 of: Zen exrsts.
_A,,x 0 2m Emu
(A2.2) Axx , Au and 2‘.“ are nonsingular; AZ, is of full column rank.
(A2.3a) T‘1’2vec[Z'(s,U)]—> N[O,‘I’].

(A2.3b) ‘11 = zeAu.

These high-level assumptions are derivable from various sets of more basic
assumptions. For example, in the rational expectations context, define et = (a, ,u,‘)' and
let Q, be the information (sub)set Q, = {z,;z,_l,e,_,;z,_2,et_2; ...... }. Then (A2.1)-(A2.3)
follow from the assumptions that E(et |Q,) = 0, V(et IQ, ) = Z, and xt and z, are
covariance stationary.

Let 6 be the usual IV estimator using Z as instruments: 6 =(X'PZX)’1X'PZy.
Under (A2. l)-(A2.3) we have the standard result:

(2.21) mil—90)»N10.oZ(A..zA;z‘A.,.)“l;
this is consistent with the general GMM result AV(6) = (Dl'Cl’llD1 )’I presented earlier.
We now turn to the IGMM estimator 6 , Schmidt's estimator '6', and the following

additional estimators

(2.22A) 6 = (X'MUPZMUX)" X'MUPZMUy

13

(2.22B) 6 = (X' PZX)” X' My — Uh)

where A = £32m. In practice, (2.22B) will require a consistent estimate of A.

We wish to show that the estimators 6 , '6', 6 and 6 are asymptotically equivalent,
with asymptotic variance matrix oilu(szA;zlAzx)‘l, where as above
oiln = 0': — 28,12,121”. This is consistent with the result of Schmidt (1988, Appendix C.4)
for .9". Comparing to the asymptotic variance matrix of 6 in (2.21) above, the inequality

0;“ S 0‘: establishes the asymptotic efficiency of 6 , '6', 6 and 6 relative to 6.

LEMMA 2.1: plim T’1X'MUZ = plim T'1x'z = An
plim T‘lz'MUz = plim T‘1Z'Z = A,z
Proof: plim T‘1x' MUZ = plim [T‘1X'Z — T“x' U(T"1U'U)“T"U'Z]
= A,z —A,,,2;;-o = An,

and similarly for plim T'IZ'MUZ.

LEMMA 2.2: plim T'IX'PIMUZIX = plim T‘IX'MUPZMUX
= plim T'IX' PZMUX
= plim r‘x' P2X
= A,,A;,‘A,x
Proof. plim T’1X'PIMUZ,X = plim Tlx'MUzqtlim T"z'MUZ)“plim T‘IZ'MUX
= szAgl‘A,x using Lemma 2.1.
The proofs for plim T’1X'MUPZMUX and plim T'IX'PZMUX are similar.

LEMMA 2.3: plim T‘IX' lemme = plim T’IX'MUPZMUe

= plimT‘IX'PZMUa

14
= plimT“x'PZ(y—- Uh)
= plimT'lX'PZs
= 0
Proof: plimT’IX'PlMUZIS = szAgzl-plimT'T MUS using Lemma 2.1.
But plim T‘IZ'MUs = plimr“z'e — plimT’IZ‘ U(plimT’1U'U)'lplim T'IU's
= 0—0-1232“, = 0
since plim T'IZ'a = O and plim T’IZ' U = O. The proofs for the other cases are similar.

Lemmas 2.2 and 2.3 imply that the estimators 6 , '6', B and 6 are consistent. For

example,

(2.23) plimBO=90+[AXZA;AZX]’1»O =90

using Lemmas 2.2 and 2.3; similar simple arguments apply to the other estimators. It is
interesting in Lemma 2.3 that the orthogonality of 8 with (MUZ) occurs because a is

orthogonal to Z Ed Z is orthogonal to U.

LEMMA 2.4: T'I’ZX'PIMUZIS,T‘1/2X'MUPZMU8,TWX'PZMUS and
T"“2X'Pz(y — U7») each converge in distribution to N[0,o:rquzA;zlAzx ].

Proof: We will give the proof for T‘“2 X' PlMoZIS . The other proofs are quite similar.

(2.24) T‘l/ZX'PIMUZIe = T’1/2X'MUZ(Z'MUZ)"Z'MU8
= (T’lx'MUZ)(T-‘z'MUZ)"T'mz'MUe
= AXZA2(T‘”2Z'MU8)+OP(1).

So we consider

T"’z' MUe = T‘1’2Z'[I—U(U'U)'l U']e

15

= T‘mZ‘e—(T‘ml‘ U)(T“U' U)"(T‘1U'e)
— ——r'1’1z'(e 112,3,2 ,,)+o (l)

Combining expressions, we have

(2.25) T‘1’1x'PIM ZIn: A A‘1'r'1’1z'(e— U2;u2 ue)+o (1)
But
(2.26) T‘1’1z'(s - U232“)

= T‘mvec[Z'(e — U232“, )]

= T’mvec{Z'(s,U)|:_ 2_ _12 :|}

0.11118

={[1,— —2 Z"]®IM}T"/2vec[Z'(s,U)].

w 1.111

But according to assumption (A2.3) above, T"mvec[Z‘ (e, U)] —) N[O,2 ®An].

Therefore

(2.27) T-1’1z'(e- 112,112 us)—al~l(o,13)

where

(2.28) B={[l,—2,u 2‘1]®IM}(£®AZZ ){[1,—2,,2;,1,]®1M)'

={[1 -2 su2,1,,1]2[1, -2 2;,12Z,I}®A
= (as _ 2"'tarzt—rllrzur: )‘Azz= Azz

O.slu

Using (2.27)-(2.28) in (2.25), we conclude

(2.29) T’WX'PlM 218 —+ N[0, AHA; zz.A;,1ltt,,]— — N[0, oi‘quzA‘zzlAzx].

OeluA

THEOREM 2.1: JT(6 -90)a s/T(6.—90), JT(6 -60) and «Edi-90) each converge in
distribution to N[0,(r:hl (AXZAEAZXYl ].

16
Proof. Ji(°t'1°— 90) = (T'IX'PIMUZIXYI T‘1’1x'PlMUZIe.
Then using Lemmas 2.2 and 2.4, fi(.9"-90) —) N[O, A], with
A = (A...A;.'A...)“1 -oit(A..A;;Aa)-(A..A;Aar1
= 0:111 (sz‘AE‘qzx).l -

The proofs for the other estimators are essentially identical. I

Thus the asymptotic variance matrix of each of the above estimators is
cit“ (p lim T'IX' P2X)'l. As noted above, this is less than the corresponding asymptotic
variance matrix for the ordinary IV estimator, 02(plim T'1X'PZX)'1, so long as 28“ at 0.
To achieve an efficiency gain, the additional variables 11 must be uncorrelated with the
instruments z an_d correlated with the errors a.

The estimator 6 in (2.22B) is infeasible because it depends on A = 2'12 We

“11 118'

can define a feasible version of it, say
(2.30) (i = (X'PZX)‘1X'PZ(y - Ui),
where it is a consistent estimate of A. Specifically, it = (U'U)'1U'§ with é = y - X6,

where 6 is any consistent estimate of 60. It is easy to show that 6 is consistent and has

the same asymptotic distribution as 6 (and, therefore, the same asymptotic distribution as
6, '6 and 6 ).

2.4. Monte Carlo Results

In the previous section we have considered four improved IV (UV) or improved

GMM estimators. Each is consistent and asymptotically more efficient than the usual

17

IV/GMM estimator 6 = (X'PZX)"1X' sz. The asymptotic efficiency gain for each of the
HV estimators over the usual IV estimator is ZNEQEW -(plim T "IX'PZXY‘, which
obviously depends on the strength of the correlation between 8 and u.

A natural question to ask is whether our IIV/IGMM estimators are still more
efficient than the usual IV or ordinary GMM estimator in finite samples. In order to
answer this question, we performed a Monte Carlo simulation on a very simple model. In
the simulation we considered our IIV estimators and also some estimators of Imbens
(1993) that are similar to GMM estimators. Our simulation plan is as follows. The

assumed regression model is:
(2.31) y, =90+e,, t= 1,2, ...... , T,

where 60 is a scalar parameter and a, is iid N(O,1). Thus we are estimating 60 = E(y, ).
Further we assume that we observe a random variable u,, which is also iid N(O,1). Let p
denote the correlation between a, and 11,. This simple model has also been considered by

Imbens (1993). An efficiency gain is possible here because the mean of u, is known to be

zero. Our DGP is therefore as follows:

(2.32A) y, = 1+8,,
(2323) “t = pet + l-p’nr,

where a, is iid N(O,l), n, is also iid N(O,1) and a, is independent of 11,. Thus 60 = 1.
Our results do not depend on this choice of 60, nor do they depend on the choice of the
variance of a, and 11, equal to one.

The following six estimators of 60 are considered ill our simulation:

(1) Sample mean (61): 61 = y. This is the GMM estimator based on E(y, —60) = 0.

18

(2) Infeasible GMM (6,): 62 = argmin,{¢T(e)'C'1¢T(e)} = y — pii, where

_ _ l p e,
¢r(9)=(y-9, u)',andC=[ ]=V([ ]).
p 1 at

(3) Feasible GM (6,): 63 = argnnine {¢T(9)'6‘1¢T(9)) = y — on, where

-—- — A 1 T A A. e 6 . A _é
¢T(9)=(y-9, u)';C=¥Ze,e, =[.” .12]wrthe,=[yt 1]; and
1:1 C21 C22 11,

6: 6:21 / 6:22-
(4) 11v estimator (6,): 64 = (i'Mui)“i'Muy, where i a (1, l)'.,,,.
(5) Imbens's first estimator (6 5): 6 5 is the pseudo maximum likelihood (PML) estimator

defined by Imbens (1993) as the first part of the solution to

g(926) = 2?:1 p(Yt ,u,,6,5) = 0’

 

where p(y,u,6,8) =(1y+—s:’ 1:511) and 5 is an artificial parameter.

(6) Imbens's third estimator (66): 66 is defined by Imbens (1993) as the first part of the

sohrtion to
g(e,6.tt) = 2.1.16(y.,u..e,6,n) = o.
where "156.11.935.10 = ((y-9)eXP(u -5u),u-eXP(11-8u),1- eXP(11-5u))'-

Notice that in our special case of a regression model with only intercept, some

estimators that are different in general become identical. The infeasible GMM estimator

l9
(6,) is the same as the infeasible IIV/IGMM estimator 6 = (x' sz)“l x' P,(y — uh) = y — pfi
defined in (2.22B) above. The feasible GMM estimator (63) is the same as the feasrhle
IIV/IGMM estimator 6 =(x'P,x)-1x'P,(y — ui.) defined in (2.30) above, where
71: 2,32,, = (U'U)’1[U'(y - 61)] = 6,, /6,,, provided that the initial consistent
estimator for e, is 6, in both cases. The three IIV/IGMM estimators
6 =(x'P,MUx)‘1x'P,MUy defined in (2.18), '6' = (x'nMU,,x)'1x'P,MU,,y defined in (2.19)
and 6 =(x'MUPzMUx)“1x'MUPzMUy defined in (2.22A) are the same and equal to
6, = (i'MUi)"i'MUy, when x = z = i = (l, m, l)'Tx,. The second estimator ofImbens
(1993), defined as the first part ofthe solution to g(6,8) = Z,=,p(y,,u,,6,6) = 0 with
p(y,u,6,8) = ((y -6)(1-5u), u(1— 8u))', is also the same as the first three HV/IGMM
estimators (equal to 6,). This leaves us with the six distinct estimators listed above.

6, = y is unbiased and var(6,) = UT. 6, = 7— on is unbiased and var(6,) =
(1 — p2 ) / T. For the remaining four estimators, finite sample properties are unknown, but
the estimators are consistent and their asymptotic variance is (l - p2 ) / T.

Our simulation results are based on 20,000 replications. The simulations were
performed in GAUSS 2.0 and used its random number generator.

Table 1 gives the means of the six distinct estimators, while Table 2 gives their
mean squared errors (MSE). In each case the estimators are nearly unbiased and MSE is
nearly the same as variance. For convenience we actually present MSE multiplied by
sample size (T), and the asymptotic variance of «E (6 —6) is given as the value for T = 00.
For the sample mean 6,, T-MSEl should equal 1.0 apart fiom sampling error for all
values of T and p , and deviations from unity in the first cohrmrl labelled T-MSE1 give an
indication of the sample variability in the experiment. Similarly, for the infeasible GMM
estimator 6,, T-MSE2 should equal (1- p2) apart from sampling error for all T and p.
For the other estimators T-MSE should converge to (l — p2) for large T.

The result in Table 2 are in close agreement with the asymptotic theory, and the
agreement is very close for T .>_ 50. T-MSE is nearly equal to its asymptotic value

20

(1— p2) for all estimators, all values of p, and all sample sizes except T = 25 and
occasionally T = 50. The IIV/IGMM estimators are better than the sample mean in all
cases except p =.l and T = 25 or T = 50; as expected, the size of the efliciency gain
depends on p.

For this simple model, at least, the difl‘erences among the various IIV/IGMM
estimators are quite small. As might be expected, the infeasible GMM estimator (6,) is
usually the best. The IIV estimator 6 4 (also equal to Imbens's second estimator) is
somewhat better than Imbens's first and third estimators (6 5 and 66). The feasible GMM
estimator (63) seems to be slightly better than the IIV estimator when p is small, and
slightly worse when p is larger. However, we repeat that the finite sample differences
among the asymptotically equivalent estimators are quite small The main message of the
simulations is that we can indeed improve on the usual IV estimator in finite samples, and
asymptotic theory is a reliable guide to the variability of these improved estimators. At

least this is so in the simple model we have considered.

2.5. Concluding Remarks

In this chapter we have shown how to improve on ordinary GMM (IV or ZSLS)
estimators, given observable extra variables which are uncorrelated with the instruments
but correlated with the error in the equation being estimated. The difference between the
improved ZSLS (IV) estimators and the ordinary ZSLS (IV) estimators is that the
projection matrix PZ in ordinary ZSLS (IV) is replaced by PIMUZI’ MUPZMU, or PZMU, so
that the 2SLS "fitted values" are constructed differently. For example, 6. uses MUZ, the
part of Z orthogonal to U, as the regressors in the "first stage" regression, whereas the

ordinary 2SLS estimator 6 just uses Z.

21

TABLE 1

Means of Alternative Estimators

.1

i o. o. o. o. o. o.
25 .9998 .9998 .9992 .9992 .9992 .9992
50 .9996 .9996 .9994 .9994 .9995 .9994
100 .9999 1.0000 1.0001 1.0001 1.0001 1.0000
200 .9996 .9997 .9997 .9997 .9997 .9997
500 .9999 .9999 .9999 .9999 .9999 .9999
00 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
25 .9998 .9998 .9993 .9993 .9942 .9993
50 .9996 .9996 .9995 .9995 .9996 .9995
100 .9999 1.0002 1.0002 1.0002 1.0003 1.0003
200 .9996 .9998 .9998 .9998 .9998 .9998
500 .9999 .9999 .9999 .9999 .9999 .9999
“3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
25 .9998 .9998 .9995 .9996 .9975 .9996
50 .9996 .9996 .9997 .9997 .9998 .9997
100 .9999 1.0003 1.0003 1.0004 1.0004 1.0004
200 .9996 1.0000 .9999 .9999 .9999 .9999
500 .9999 1.0000 1 .0000 1.0000 1 .0000 l .0000
00 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
25 .9998 .9998 .9997 .9999 .9993 .9999
50 .9996 .9997 .9998 .9999 .9998 .9998
100 .9999 1.0004 1.0004 1.0004 1.0004 1.0004
200 .9996 1.0001 1.0000 1.0000 1.0000 1.0000
500 .9999 1.0000 1.0000 1.0000 1.0000 1.0000
00 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
25 .9998 .9999 .9997 1.0000 1.0001 1.0000
50 .9996 .9999 .9999 1.0000 .9995 1.0000
100 .9999 1.0003 1.0003 1.0003 1.0003 1.0003
200 .9996 1.0001 1.0001 1.0001 1.0001 1.0001
500 .9999 1.0000 1.0000 1.0000 1.0000 1.0000
00 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

T‘

25

50
100
200
500

00

25

50
100
200
500

25

50
100
200
500

25

50
100
200
500

25

50
100
200
500

22

TYUBLIEZ

Mean Square Errors of Alternative Estimators

.9965
.9985
1.0073
‘L0008
L0120
L0000

.9965
.9985
1.0073
1.0008
L0120
1.0000

.9965
.9985
1.0073
1.0008
L0120
LOOOO

.9965
.9985
L0073
L0008
L0120
L0000

.9965
.9985
1.0073
L0008
L0120
L0000

T-MSE, T-MSE,

.9970
.9945
.9953
.9888
L0003
.9900

.9350
.9249
.9111
.9051
.9163
.9100

.7831
.7699
.7478
.7433
.7524
.7500

.5370
.5266
.5068
.5043
.5099
.5100

.1983
.1955
.1887
.1881
.1896
.1900

L0345
1.0123
L0047
.9937
L0027
.9900

.9684
.9406
.9196
.9094
.9180
.9100

.8113
.7829
.7556
.7468
.7535
.7500

.5615
.5372
.5130
.5068
.5109
.5100

.2199
.2035
.1918
.1893
.1902
.1900

L0447
L0147
1.0052
.9938
L0027
.9900

.9775
.9432
.9198
.9094
.9180
.9100

.8156
.7846
.7556
.7468
.7535
.7500

.5580
.5365
.5130
.5068
.5108
.5100

.2066
.1995
.1911
.1891
.1901
.1900

T-MSE, T-MSE5

L0586
L0181
1,0063
.9940
1.0025
.9900

.9963
.9469
.9207
.9096
.9180
.9100

.8188
.7873
.7564
.7468
.7535
.7500

.5600
.5377
.5136
.5068
.5110
.5100

.2192
.2003
.1913
.1892
.1900
.1900

T-MSE6

L0470
L0157
L0055
.9938
1.0025
.9900

.9781
.9439
.9200
.9098
.9180
.9100

.8159
.7849
.7558
.7468
.7535
.7500

.5592
.5367
.5132
.5068
.5110
.5100

.2069
.1998
.1912
.1890
.1900
.1900

CHAPTER3

IMPROVED GMM AND 3SLS ESTMATORS
FOR SYSTEM OF EQUATIONS

3.1. Introduction

In section 2.2 of Chapter 2 we defined the improved GMM (IGMM) estimator as
the GM estimator using moment conditions E[¢l(y: ,60) — €1,034), (y: )] = 0 and
weighting matrix C11 = (Cll — CHCQC21 )"1. In the definition, we intentionally did not
specify the functional forms of d), and (112 , nor did we require the observations
{o(y,‘,e) = (o,(y:,e)',o,(y;')' )', t = 1,2,...) to be conditionally homoskedastic or serially
uncorrelated so long as they satisfy suitable regularity conditions. In section 2.3 of
Chapter 2 we applied the general results of IGMM estimator to the case of the linear
regression model. Assuming conditional homoskedasticity and serial uncorrelation, and
imposing the regularity conditions (A.2.1)-(A.2.3), we obtained an explicit formula for the
IGMM estimator and related it to other previously known estimators, such as the
estimator of Schmidt (1988).

In this chapter we will provide a similar analysis for a system of linear equations.
We will first set up the model and make ' 'gh-level" assumptions of regularity conditions.
Under these assumptions we derive an explicit formula for the IGMM estimator and
several other asymptotically equivalent estimators, and demonstrate the efficiency of these

estimators relative to the usual three-stage least sqares (3 SLS) estimator.

23

 

 

Ill.

24

3.2. Model and Notation

The model considered in this chaper is

(3.1) ytg = x,,,'60g +8,8, g =1, 2, ...... , G; t =1, 2, ...... , T,

where yt8 is the dependent variable of equation g at observation t, x,g is the R, X] vector
of explanatory variables of equation g at observation t, 90,, is the K8 x1 unknown
parameter vector of equation g, and at, is the model disturbance of equation g at

observation t. We assume that in general cov(x,g,s,g) at O for g = 1, 2, ..., G.

We define the following notation:

ytl x11. 91 811
(3.2A) y,= 5 ,X,= ,6: 3 ,8,= 3 ,t=1,2,...,T;

 

YIg xlg
(3.2B) ya) = : , X“) =
YTg ng.
Y(l) X0) 8(1)
(3.2C) ytt- = E , Xe = .'. , as = E ,
y(G) X(G) 8(G)
a,‘
(3.2D) 8 = ‘
sT'

Then (3.1) can be rewritten as

25

(3.3A) y, = x,9, +e,, t = 1, 2, ...... , T,
01' as
(3.3B) y. = x.e, +e.

3.3. Improved GMM Estimators

Suppose that we have available an M x 1 vector of instruments z, satisfying the
moment conditions E(e, 8) z,) = O, with E[(IG ® 2, )X,] having full column rank.
Suppose that we also have available an L x 1 vector of observable variables 11, satisfying
E(u, ® z,) = 0 and E(u,e,') = 2“, ¢ 0. Then the additional moment conditions
E(u, ® 2,) = 0 will help us to improve the efficiency of estimation of 60. Using the

notation of Chapter 2, we have

(My: 13)] _ [(16 8’ Zr )(yt - X60]

3.4 La: , _
( ) "y ) [tum (heme.

where the observed data vectoris y:=(yr';Xr,',...,x,G';u,';z,1)'. Then

111(9)]=12o(yt,9)=[
on

r106 e Z'Xy. — Xe)
T t=l

(3'5) ¢r(9) =[ T‘1(IL ®Z')u.

where Z=(z,,...,zr)' and u. =vec(U)E(u(,)',...,u(L)‘)' with U=(u,,...,uT)'. Then

_ C11 C12 _ . . ,
(36A) C‘lca €22}ng Baronet-(9.11

26

T—>oo

= m E r10, ®Z')8.8.'(IG ®Z) T"(IG ®Z')8..ua'(1L ®Z)
T"(IL o'oz')n.e..'(1G 8 Z) T“(IL ® Z)u..u.'(1L ® Z)

(3.68) 1),, = 544$? = —%(1G ®Z')X. .

Substituting (3.5) and (3.6) into the first order condition (2.7A) in Chapter 2 for
the IGMM estimator 6 , and replacing C“, C12 and 0;,1 by consistent estimates C", C,,

and C; respectively, we arrive at:

(3.7) [x.'(1G ®Z)]C“[(IG 82')(y. — x.6)- 6,,6;,1(lL ®Z')u...] = 0.
Solving for 6 , we obtain
(3.8) 6 = [X.'(IG o3>Z)611(IG ®Z')X.]" x.'(lG ®Z)C”

{(1. e2'1y. - 6.26310. ®Z')u.1.

In order to simplify the above expression further, we need to put more structure on
C. This is possible under the assumption of conditional homoskedasticity and
lmcorrelatedness. Suppose that conditional on Q, = {z, ;e,_, ,u,_1 ,z,_, ;...} , the (s,',u,')'
are mutually uncorrelated and that:

2.

39 V 81 _ 2&8 2w
(. ) {JIM—[2 2 ]

“8 uu

27
Then C = £®E(z,z,') and C = E®T‘1Z'Z is a consistent estimate ofC, where E is any

consistent estimate of E. For the moment we will treat 2 as known, for simplicity.

Therefore we have

(3.10A) 611 = 2.. ®(T"Z'Z)'l with 288 = (2,, — 2,,2;,1,2,,,)-1
(3.1013) 6,, = 2,,,, ®T"Z‘Z

(310(3) 6;; = 2,1,1, ®(T"‘Z'Z)"

(3.10D) 6,,63 = 2,2,3, 81 1M.

Substituting (3.10A) and (3. 10D) into (3.8), we get

(3.11) 6 = [x.'(2“ ®Pz)X.]" x.'[2“ ®Z(Z'Z)"1]
{(IG @2')Y* ..(Zsuzl-Rll ® IM )(IL @Z')u.j.

Noticing that
(3.12) (2,,2;,1, ® 1,, )(1L ®Z')u. =(2,,2;,1, ®Z')vec(U)
= vec(z'U2,‘,,1,2,,)
= (1G 8z'U)vec(2;,1,2,, )

= (IG 82')(IG ®U)ve0(2{;l£aa)a

and substituting into (3.11), we obtain an explicit formula for the IGMM estimator 6:

(3.13) 6 = [x.'(2'i ®PZ)X.]‘1X.'(E“ 8P,)[y. -(lG sum

where A = vec(Eme) a (A,',A,',-~-,AG')'. Thus, for i = 1, 2, ..., G, A, = 2,1,1, times the
ith column of 2“,; equivalently, A, = (p1imT’1U'U)‘1plimT'lU's(,).

28

We can compare the IGMM estimator in (3.13) with the usual GM (3 SLS)

estimator
(3.14) 6 = [x.'(2;,1 8P,)x.]‘1x.'(2;,1 8P,)y.

based on moment conditions E[¢t,(y: ,60)] = 0. We see that the only difference between
the 3SLS estimator and the IGMM estimator is that 2;: and y. in (3.14) are replaced by
2" and [ya - (1G 18 U))t] respectively. It is interesting to notice that

[Ye — (16 ® U)},] = [(y(,) — UA)’ ,:--,(y(G) - UA)‘ ]' is just a vector of residuals from the
linear projection of ya) onto U. Thus the IGMM estimator 6 in (3.13) can also be
regarded as a purged GMM (PGMM) estimator.

We will now consider a specific form for a consistent estimate of 1.. Define

A

(3.15) 5. (h,',~--,5.G')'

with it, =(T'1U'U)‘1T"U'(y(g) — X(g)6g), where 6g is any consistent estimate of 68, for
g =1,..., G. Then
(U'U)_1U'(Y(l) - x(1)é(l) )

(3.16) (1G 8 0))". = (1G 8 U) .
(U.U)_1U.(Y(G) ’ X((i)9(o))

= (I. e was ®(U'U)"U')(ya — x6)
:08 ®PUXY--Xté)

where 6 = (6,',...,6G')'. Substituting the above expression into (3.13), we get

(3.17) 6 =lx.'(2°‘ elem-8:12“ eerie—(Is ePa)(y.-x.é)l.

29

This expression still depends on 2“, and we will discuss its consistent estimation later.
While (3.17) is defined for any consistent estimate 6, we may as well consider the

special case that 6 = 6. Then (3.17) implies

(3.18) [8'02“ mantle = xstzs many. -(I. ePUXy. - x611.
Solving for 6 , we obtain
(3.19) 6 =[x.'(2*=1 ®PZMU)X.]" x.'(2°i ®PZMU )y..

This is an obvious generalization of the single-equation IGMM estimator 6 of Chapter 2.
In order to be more precise about the sense in which the IGMM estimator 6

dominates the usual GMM (3 SLS) estimator 6, and to introduce some other equally

asymptotically eflicient estimators, we make some more explicit assumptions. To make

the asymptotics as simple as possible, we will make the following "high level"

assumptions:
X0).
. 1X '
(A3.l) phm-T— (2‘31 [x,,, xm, z e U]
8|
_ U. A

 

 

 

 

PAll A16 A12 A18 Alu 1
= AG] ' A66 A62 A68 AGu -
Azl . A26 Au 0 0 exrsts.
A81 ° A86 0 288 281.1
_Au1 ° AuG 0 2118 2‘ou
288 2w 0
(A32) All, 2,“, and 2 = [Z 2 :l are nonsmgular; Azg hasfull column rank
for g =1, 2, ..., G.
(A3 3) 1 (1 ®Z‘) 8‘ —>N(0 28A )
' fl G+L ll. 9 22 '

As was the case in Chapter 2, these high-level assumptions are derivable from

various sets of more basic assumptions. For example, let e, = (8,',u, ' )' and
Q, = {z,;z,_1 ,e,_,;z,_2 ,e,_,; ...... } ; then (A3.1)-(A3.3) follow from the assumptions that
E(e,|Q, ) = O, V(e,|Q,) = )2, and X, and z, are covariance stationary.

It is well known that under (A3. 1)-(A3.3), the usual GMM (3 SLS) estimator 6

defined in (3.14) has the following asymptotic variance:
" - 1 . -1 1—1 . —1 -l —1
(3.20) AV[Ji(e-0,)]=[phm¥x. (2,, ®PZ)X., = [A (2,, 8A,, )A]

A
where A = plim%(1G ®Z')X. =

21

Anti

We now wish to show that several estimators are asymptotically equally efficient,
and that they are efficient relative to the 3SLS estimator. One such estimator is the

IGMM estimator 6 defined in (3.19). The other such estimator is the PGMM estimator

31

defined in (3.13) with A known. In order to distinguish the PGMM estimator from the
IGMM estimator, we now denote the PGMM estimator by 6. We will also consider the

following two additional estimators

(3-21A) .9. = [Xa'Ole‘ ® PIMpzl)an_1 Xe'aw ® PlMuzl )Yt

(3.2113) 6=[x.'(2°18M P M )X.]'1X.'(2“®M P M )y..
U Z U U Z U

We will show that 6 , 6 , '6' and 6 are asymptotically equivalent, with asymptotic variance

matrix equal to

(3.22) [plim%X.'(£as ®PZ)X.]“ =[A'(2'1a ®A;,‘)A]‘1 a B“ .

Comparing to the asymptotic variance matrix of 6 in (3.20) above, the fact that the matrix
{[A'(2;,1 ®A;z‘ )A]" — [A'(2“ 8A,,1 )A]'1) is positive semidefinite (shown later in
Theorem 3.3) establishes the asymptotic efficiency of 6 , 6 , .6. and 6 relative to 6. We

now turn to a rigorous proof of these results.

 

LEMMA 3.1: plim T‘1z'MUx,,, = phm'r‘1z'x,,, = A,,, for g = 1, 2, G.
plimT'IZ'MUZ = plimT“1z'z = A,z
Proof: The proof is similar to the proof of Lemma 2.1 of Chapter 2. For example,
plim T"1z'M,,x,,, = plim[r-1z'x,,, -(r‘1z'U)(r"1U'U)'1(T‘1U'x,,,)1
= A, -O-2"A =A,,,

uuug

using (A3. 1) and (A32).

LEMMA 3.2: plim T'1x,,,,' PZMUX“, = plimT’IXm' P[Mnle(8)

32
- -l l
__ - -1 r
_ phmT xm sz

= A,,A;,1A,,

(a)

forh, g =1, 2, ..., G.
Proof. The proof is essentially the same as the proof of Lemma 2.2 of Chapter 2. For

example,

plim T‘1x,,,, 'PZMUX,,, = plim[r‘1x(,,,'z](r“1z'2)‘1[T‘1z'MUx,,,]
=A,,,A;1A,g

using (A31), (A32) and Lemma 3.1.

LEMMA 3.3: plim[T'1X.'(Z“ 8 P,MU )x.] = plim[r’1x.'(2i*= 8 Pmuz, )x.]
= plim[T‘1X.'(Z“ 8M,P,MU)X.]
= plim[r-1x.'(2“ 8P,)x.]
= A12“ 8A;)A a B,

and B isnonsingular.

Proof: Let 288 =(611')G,G. Then

phm[T'1x.'(2i‘ 8 PIMUZ, )X,]

I 11 16
x0) 0 PIMUZI G PIMUZJ X0)
0 —-1 O C O
=phmT
Gl GG
X(G). ‘3 PIMUZI ‘7 PIMUZJ X(G)
x '611P x x '61GP x
(1) 1Mu21 <1) (i) [Moll (G)
O -1 O O C
=phmT ' °

G1 I GG
xtG)'° PlMuZlXU) X(G)O PlMule(G)

33

OllAlen-zlAzl GIGAlegAnG
= E E 5 (usingLemma3.2)
OGIAGzAgAzl CGGAGzAgAnG
A,, 611A;,1 61%;,1 A,,
AGz (rGlAg,1 060A; A,G
=A’(Z“®A;)A

using the definition of A in (3.20). The probability limits for the other cases involve
essentially the same arguments. Finally, B = AXE“ ®A;z1)A is nonsingular bacause 2‘. ,
Em, and A2, are nonsingular, which implies 2“ ®A;,l nonsingular, and because A has fill]
column rank (see (A3.2)).

LEMMA 3.4: plim T'1x(,,,'P,MUe,,) = plimT"1X(h) 'P,MUZ,e,,,
= plimT‘1x,,,,'MUP,M,,e,,)
= phmT'Xm'P,(e,,, - Uh)
= plimT‘1x,,,,'Pze,,,
= 0
where h, g =1, 2, ..., G.
Proof. The proof is similar to the proof of Lemma 2.3 of Chapter 2. For example,
plim T“X(,,) 'MUPZMUe,,, = plim[T’1X(h)'MUZ](T"'Z'Z)" [T'1z'MUe,,,]
= A,, ;,1-plimT‘1z'MUe,,,
using Lemma 3.1. But
plim T‘1z'MUe(,, = plim'r'1z'e,,) - plim('r‘1z'U)(T"1U'U)‘1[T‘1U'e,,, ]
= 0—0-2,;,1,2,,,,, = 0

using (A3. 1) and (A32), where 2“,, is the g-th column of 2“,. Therefore

34
p lim T‘1x,,,,' MUPZMUe,,, = A,,,A;,1 -0 = 0.

The proofs for the other cases are similar.

LEMMA 3.5: plim[T’1X.'(X“ 8P,MU )e.)
= plim[T-1x.'(>:“ ® P[M,,Z])891
= phm[r‘1x.'(2“ 8 MUPZMU)8.]
= plim{T"1x.'(2“ 8 PZ )[e. -(1G 8U)7.]}
= plim[r'1x.'(2;,1 8P,)e.]
= 0
Proof. Let 2"" = (oi’l)GxG as above. Then

plim[r'1x.'(2** 811%,, )e.]

11 16
xtn' 0 PIMUZI 0 PIMUZI 8(1)
. -1 . . . .
=phmT
GI GG ‘
xtel' 5 PIMUZ1 0 PIMUZI 8(9)
G 1 lg
Zs=lx(1) 0' PiMUZ18(8)
o -1 e
=phmT -

G I Gs
Zs=lx(G) 1’ PIMUZISm

G 1 ° 1 r
EPIC gth'r x0) PIMUZIEm)

G G3 ' -1
28=10 PMT X(9)'P1Mo213(s)

23:1618 '0
= 5 (using Lemma 3.4)
26 10,63 .0
g:

35
= 0.
The proofs for the other cases are similar.

THEOREM 3.1: The improved GMM estimators 6 , 6 , '6. and 6 are consistent.
Proof: We will give the proof for 6. The other proofs are quite similar.
plim6 = plim[x.'(2'as 8 P,M,,)X.]'1 x.'(2“ 8P,M,,)y.
= plim[x.'(2“ 8P,M,,)X.]'1x.'(21° 8P,M,,)(x..6O +e.)
= e, +[plimx.'(2' ®PZMU)X.]'1[PlimX.'(2“ 8P,M,,)e.]
= 0, + B" -0 (using Lemmas 3.3 and 3.5)
= 0,.

LEMMA 3.6: T190, ®Z'MU)8. -> N[O, (2...)-1 8A,]

where 2“ = (2,, - 2,2;2mr1.

Proof: T‘1’1z'MUe,,, = T“1’1z'e,,, - (T-1’1z'U)(T‘1U' U)“ (T-1U'e,,, )
= T-1’1Ze,,, - (T'1’1z'U)h, +o,(1)
= T""2Z'(8(g) — UA8)+op(l)

for g =1, 2, ..., G, where A, = 2,1,1, -E(u,e,g). Then

T—1’2z' MUS“)
T‘1’1(IG ®Z'MU)8. = I
r'1’1z'MUe,G,

T'mZ'(8(,) — U711)
= 5 +0p (1)

36

= "1‘1”(1, 8z')[e. —(1G ®UE,‘,,1,)vec(2,,8)]+op(l)
(using the definition of A8 = 2,1,1, -E(u,8,,,))

= T'1’1 (1G 8 Z')[vec(8) -vec(U2;,§2,,, )] + op (l)

= T"1’2(1, 8 Z')vec[8 - U2;,1,2,,,]+ op (1)

= T'mvec[Z'(8 — U2;,1,2,, )] +op (1)

= T"”2vec{Z'(8,U)[_21_?z ]} + 0p (1)

11111.18

= T1” (11.. ,—2..£;.1.lein )veo12'(e.U)l +o, (1)
=T‘1”([Ie.—£...2‘11®IM)(IG..el)veo(e.U)

+o,,(1)

= (11. ,—2..2;.11®IM ){T‘1’1tiou e Z')[:‘ ]}

#

+op(1).

8e

But according to assumption (A33) above, T‘”2 (IG+L 81 Z')[ ]-> N(O, 2 ®A,z ).

6

Therefore
T‘1’2(IG ® Z'M,J )8. —> N(O, W)

w=(11..,—2..2;.11®Iu)(2®A..)([_2{?2 191M)

=03... -E.a2.12a.)®Azz
= (2"1'1)‘l 8A,,

LEMMA 3.7: T‘1’1x.'(2“ 8 P,M,,)e., T‘1’2X.'(2“ 8 11mm»...
T’mX.'(Z“ <8MUPZMU )8. and T'mX.'(£“ ® PZ )[8. —(IG ® U)A] each converge in
distribution to l~l[o,A'(2as 8A;,1)A] with A = diag(A,,, ...... , A,,,) as defined in (3.20).

37
Proof. We give the proof for T‘WX.'(E“ ® MUPZMU )8.. The proofs for the other

cases are quite similar.
T'1’1x.'(21‘ 8 MUPZMU )e.

X(,)' 611MUPZMU 61‘1MUPZMU em
=T'"2 ' E I E E
X(G)' OGIMUpzMU “° OGGMUpzMU 8(G)

G 1

___ T—1/2 E
G r G

Zs=1X(G) 5 BMUPZM06®

1g -l/2
3:10" T x,,)'MU P ,Mue,,,

G -1/2
86-,6 8T x,G,'MUP,MUe,,,

11-,6 68,,(T‘1xm,'M Z)(T"Z'Z)"[T"’ZZ'MUS(8)]

lg —1 -1/2

+o,,(1) (using Lemma 3.1)

[:z§-,618(T*1x,,,'MUZ)(T‘1z'Z)‘1[T‘1’12'Mue,,,]
L: 21-, 11-,6GBAG,A;,1 T'1’1z'MUe,,)

‘Alz’qul
(2“ 81M )[T‘1’1(lG 8z'MU)e.]+op (1)

AGzA-z-zi

= A'(1G 8A,,1 (2“ ®IM)[T“’2(IG 8z'MU)e.]+o,(1)
= A'(2°‘ ®A;,‘)[T‘"2(IG ®Z'MU)8..]+0,,(1).

38
But according to Lemma 3.6: T‘1/2(IG ®Z'MU)8. —> N[O, (2111)”1 8A,,]. Then
T‘1’2x.'(2“ ®MUPZMU)8. -) N[O, V]
where
v = A12“ 8A; )[(2* )-1 8A,,1[A'(2i‘ 8A; )]'= A'(2“ @AZ )A

which is the same as the matrix B defined ill (3.22) above.

THEOREM 3.2: JT(6 —60), JT(6 —60), «fl—"(6660) and JT(6 —60) each converge in
distribution to N(O, 13-1) with B = A12“ 8A;)A = plimT’IX.'(2“ ®PZ)X..
Proof: We will give the proof for JT (6 — 60). The proofs for the other cases are
essentially identical.

JT(6 -e,)=[T‘1x.'(2°‘ 8 PZMU )x.]‘l T‘1’1x.'(2* 8 PZMU )e.

= B‘1-T‘1’2X.'(2“ 8 P,M,,)e. +op (1)

using Lemma 3.3. But according to Lemma 3.7:

T‘1’2X.'(2“ 8 PZMU )e. —> N(O, 13).
Therefore

JT(6—00)-)N(O, A)

A = B“B(B’1)'=(B")'= B“.

THEOREM 3.3: The improved GMM estimators: 6 , 6 , .6. and 6 are asymptotically

 

efficient relative to the 3 SLS estimator 6. They are strictly more efficient than the 3 SLS
estimator if 2,, at 0.

Proof: From Theorem 3.2, the asymptotic variance matrix of each of the IGMM
estimators is B“1 = [A'(2“ ®A;zl )A]'1. The asymptotic variance matrix of the 3SLS
estimator is Q’1 = [A'(§.‘.,;,,1 (8 A; )A]'l. We wish to show that (Q'1 — B") is positive

39
semidefinite (psd), and positive definite (pd) when 2,, at O. This is equivalent to showing
that (B - Q) ispsd, and pd when 2,, at 0. But

B—Q=A'[(E“ — 2;,1 )8A;,1]A.

When 2,, = 0, 25‘ = 2;: and B - Q = 0; there is no efficiency gain for IGMM relative to
3SLS. But when 2,, at 0, 2“ - 2;: ispd;A;z1 ispd; this implies that (22';8 — 23;,,‘)®A”ul
is pd; and B - Q is pd because (2“ - 2;)8A; is pd and A has full cohlmn rank.

Theorem 3.3 is best understood by the intuitive explanation of the following

theorem

THEOREM 3.4: Consider the augmented system
(3.23) y,g = x,g'60g +u,'Ag +v,g, g =1, 2, ..., G; t = l, 2, ..., T,

where A, = [E(u,u,')]’l E(u,8,g) is the linear projection coefficient of 8,1, onto u,, and
v,g = 8,8 - u, ' A3 is the error in the linear projection. Define 6m as the estimator of
60 = (001',...,6,,G')' when the system (3.23) is estimated by 3SLS, using (z,',u,')' as
instruments. Then 6 m is numerically the same as the IGMM estimator 6. defined in

(3.21A) above.
Proof: Let v, = (v,,,...,v,G )'. Then

811 "ut'll ;‘l' E(Stlut ')[E(ut.ut )1"
v,= 5 =8,- 5 u,=8,- :

. u,
816 ’ut.7‘G AG. E(StGut')[E(ut 'ut ”—1

40
= e, — 2,,2;,,1,u,.
Therefore

V(v,)=2,,-2 2‘12 =(2i‘)-1.

“W118

As before, equation (3.23) can be rewritten as

9
(3.24) y. = x.e, +(lG 8 U)A+v. = [x.,(IG 8U)][ flaw. ,

where A = (A,',...,AG')' and V(v.) = (2%)-1 ®IT.
Applying 3SLS formula to (3.24) using (Z, U) as instruments, we obtain:

é3SLS _ x‘. as —l
(3.25) [ism] - {[06 My}: eiiz,u,)lx.,as 9U»)

x.' ,,
. (IG®U)' (2 @Plzmblc.

Because P1211] = PU +PIMUZ] and PIZJJIU = U, (3.25) becomes

(2“®U')X. X“®U'U

Aasrs

(3.25) [6m ] = [x4281 8(PU +P,,,UZ, )]x. x.'(2“ 8U)]‘

.1 X512“ ®(PU +P[M,,z])]yti
(2“ 8U')y. '

Using the partitioned inverse rule, we get

(3.26) 6,8,, = E"~X..'[21"®(PU +P,MUZ, )]y. — E‘1BD‘1-(2i‘ 8U')y.

41

where B = x.'(2as 80), D = 2‘11 ®U'U, and E = A—BD’IC with
A = x42“ 8(PU +P,MU,,)]X. and c = (2“ 8U')x..

But
(3.27) E = A — BD'IC

= x.'[2“ 8(PU +PIMUZ] )]x.

-x.'(2as ®U)(£“ 8U'U)‘1(2i‘ 8U')x.

= X362“ ®(Pn +P[M,,Z] )1-(2“ ®PU)}X*

= x.'(2“ ®1’[MUZ,)X. ,
and
(3.28) BD"1 = X.'(2“ 8U)(2i‘ ®U'U)“ = x..'[IG ®U(U'U)“].

Substituting (3.27) and (3.28) into (3.26), we obtain:

(3.30) 6,8,, = [x.'(2“ 811W, )x.1’1
-{x.'[2“ ®(Pn +P[MUZ] )]y. - Xe'llo ®U(U'U)'l 1(2"8 69 U')ya}
= [x.'(2“ ®P[M,,z] )x.]‘1x.'{[2‘*‘ ®(Pn +P[M,,z] )1-(2“ ® PU)}Y*
= [x.'(2'i's 811W, )x.]“1 x.'(2“ 811mg)»

=6.

Equation (3.23) is instructive because, speaking loosely, the effect of adding the

variable 11, is to reduce the relevant variance of 8, from V(a,) = 2,, to

V(8, lu, ) = 28,, - 2 2“2 Obviously, this result is a direct extension of the similar

8111111118.

result of Schmidt (1986), and is also closely related to the result of Wooldridge (1993).

42

2 2
Our discussion so far has assumed that the covariance matrix 2 = [2m 8"] is

118

known. Since we generally do not know 2 , the estimators 6 , 6 , .6. and 6 are infeasible.

However we can define feasible versions of them, say

(3.31A) 6,. = [x.‘(2'as 8 P,M,,)x..]‘1x.'(2as ®PZMU)y.
(3.318) 6,. = [x.'(2“ 8P,)x.]‘1x.'(2as 8P,)[y. —(1G 8 U)A]
(3.31C) '6'F =[x.'(21*= 811MHz, )x.]'1 x.'(2"=a 811mm)»
(3.31D) 6, = [)(.'(2118 8MUPZMU)X.]‘1X.'(2“ 8MUPZMU)y.

where E“ and A are consistent estimates of 2‘.“ and A respectively. Specifically,

2“ = (E - 28,232,, )‘1 and A = vec(EEEm) with 2,, = T‘lé'é, 2,,'= 28,, = T'E'U
and 2,, = T-1U'U, where 6 = (€,,---,€,)'= (y, — x,6, yT —x,6)' with 6 being any
consistent estimate of 60; for example, the usual ZSLS estimator. Then it is not difficult to

show that 61., 61., 6F and 6F have the same asymptotic distribution as 6 , 6, '6' and 6.

3.4. Concluding Remarks

This chapter provided some improved versions of 3SLS, and extended the results
in Chapter 2 on improved versions of IV (ZSLS). The improved 3SLS estimators differ
fiom the usual 3SLS estimator in two ways. The first difl‘erence is that the projection
matrix PZ in 3SLS is replaced by PZMU, MUPZMU, or PIMUZI’ so that the 3SLS "fitted
values" are constructed differently. For example, '6' uses MUZ, the part of Z orthogonal
to U, as the regressors in the "first stage" regressions, whereas the 3SLS estimator 6 just

uses Z. This is exactly the same as the difference between the improved and ordinary

43

2SLS estimators in Chapter 2. However, there is a second difference between the usual
and improved 3SLS estimators that did not arise in the case of 2SLS. Where 3SLS
estimator uses 2;, the improved 3SLS estimator uses 2“ = (2,, - 28,232,, )’1. Thus
3SLS estimator uses the inverse of V(8, ), while the improved 3SLS estimtors use the

inverse of V(8, lu, ), in the final "stage" of estimation.

CHAPTER4

INIPROVED GMM ESTIMATORS
FOR SYSTEM OF NONLINEAR EQUATIONS

4.1. Introduction

In this chapter we will extend the results of Chapter 3 to the case of a system of
nonlinear equations. The structure of the chapter is as follows. In the next section, we
will first define several improved GMM (IGMM) estimators, and then show that these
IGMM estimators are asymptotically equally eflicient and eflicient relative to the usual

GMM estimator. The final section gives some concluding remarks.

4.2. Improved GMM Estimators

The model considered in this chaper is

(4.1) f(y:8,60g) = 8,8, g =1, 2, ..., G; t =1, 2, ..., T,

with y; = (y,g,Y,g',x,8' )', where yt8 is the dependent variable of equation g at
observation t, Ytg is the t"1 observation on the Mg x 1 vector of other endogenous
variables included in equation g, xt8 is the N8 x 1 vector of exogenous variables of
equation g at observation t, 608 is the K8 x1 unknown parameter vector of equation g,

and at, is the model disturbance of equation g at observation t.

44

45

Suppose that we have available an M x 1 vector of instruments z, satisfying the
moment conditions E[(lG 8 z, )(e,,,...,etG )'1 = 0, and E[z,af(y:,,6,,)/60,'] has full
column rank, g = 1, 2, ..., G. Then the usual GMM estimation of(60,',...,606')' ill (4.1)
is feasible. However, suppose that we also have available an L x1 vector of observable
variables 11, satisfying E[(IL ® 2, )u,] = O and E(u,8,') = 2us at 0. Then the additional
moment conditions E[(IL ® z, )u,] = 0 will help us to improve the efficiency of the
estimation of(60,',...,606')'.

We define the following notation:

91
(42A) 6: 5 ;
0G
Yil “3’11 :91) 311
(4.28) y,‘= E , f(y,‘,e)= E , e, = 5 , t= l, ...,T;
311.6 “Yio 966 1' 8to
YIg f(YIgaeg) 813
(4.2C) y(8) = E , f(8)(y(g),68)= 5 , 8(8): 5 , g= 1, ..., G;
YTg f(YTgreg) 8T3 '
8: , U: : , Z: 3
ST' uT' zT'
f‘(1)(Y(ml)o91) 8(1)
(4.2D) f.(6)= E , u..=vec(U), 86=V60(8)= E ;

f(G) ()’(o) :96) 8(o)

 

8f ‘ ,0
(4.2E) l),(8,) = “11%” 11), l, ., G;
608'
D1(91)
(4.2F) D(6) = ,
06(96)
(4.2G) P,( = X(X'X)“ x', M,( = I— Px for any matrix x with full column rank.
Then (4.1) can be rewritten as
(43A) f(y.’,90) = 8., t = l, 2, T,
01‘
(4.38) f,,,(y;,,,8,,) = 8(8), g = 1, 2, G.

Using the notation of Chapter 2, we have

. ¢1(Y:’e) (IG®zt)f(Y:’e)
4.4 ,,e = , = ,
( ) My ) [4.0.11 [ (1.92mi. 1
Then
_ 96(9) ___1_T . _ T"(IG®Z')f.(6)
(4-5) ¢r(9)-[ 4m :I-Tt>=21¢(yt,9)-[ T_,(IL®Z,)u‘ ]
Therefore

(46A) c= C“ C” =1im1T-Bo (9)1» (9 )'1
. C21 C22 T"°° T 0 T O

47

T—eoo

- Hm E T“(Io ®Z‘)etet'(lo <8 Z) T‘1(IG 82')e..u..'(1L 82)
T_1(IL ®Z')u.8.'(lG ® Z) T‘1(IL ® Z')u,,u,,'(1L ® 2)

_ acute) = 1 6MB) _ 1

(4.68) 1),, _ 69, $0,, 82')—-—(367 - TOG ®Z')D(6).

According to the general results of Chapter 2, we know that the augmented GMM
(AGMM) estimator (6 ) using moment conditions E{(IG+L <8 2, )[f(y: ,60)' ,u,']'} = 0 and
weighting matrix C“ is numerically equivalent to the IGMM estimator (6 ) using the
moment conditions E[(IG ® 2, )f(y,' ,60 ) - CuC; (IL ® 2, )u,] = 0 and weighting
matrix 611 = (6,, — 6,,6;;6,, )'1, since both 6 and 6 satisfy the same first order
condition (2.7A) ill Chapter 2.

Under suitable regularity conditions, standard GMM results also tell us that both 6
and 6 are consistent. For discussions of regularity conditions, see, e. g., Hansen (1982),
Gallant and White (1988) or Amemiya (1985).

Substituting (4.5) and (4.6) into the first order condition (2.7A) in Chapter 2 for
the IGMM estimator 6 , and replacing C“, C12 and C3; by consistent estimates C”, 612

and 6;,1 respectively, we arrive at:
(4.7) 8(6)'(1G 8 2)611[(lG 8 Z')f.(6) — 6,,6;,1(1L 8 z')u.] = 0.

In order to simplify (4.7) further, we need to put more structure on C (or C). To
do so, and to allow us to investigate the asymptotic properties of our estimators, we make

the following "high-level" assumptions:

(A4. 1)

(A42)

(A43)

(4.8A)

48

 

 

 

PD1(901)'
l D 6 '
phm; 6‘21“) [Drteah Date...) 2 e U]
8|
_ U. .1
PAH A16 A12 A18 Aluw
= AG] A66 A62 A68 AGu '
All . A29 Au 0 0 670818.
A81 ' A86 0 2% 2w
__Aul AuG 0 2us: zuuj

 

Z 2
A2,, 2,,“ and 2 = [2“ 2“] are nonsingular; AZ, has full column rank

118 1111

for g =1, 2, ..., G.

1 .
‘/—T-(lG+L ®Z‘)[:] —> N(O, 28A,,).

C= 2®E(z,z,')= 2®AE

Note that (A43), and hence (4.8A), implicitly reflect an assumption of no conditional

heteroskedasticity. Furthermore,

(4.88)

6

6,, 6,, 2,,8T'1z'z 2,,8T‘12'2

49

is a consistent estimate of C, where i is any consistent estimate of 2. For the moment

we will treat 2‘. as known, for simplicity. Therefore we have

(4.9A) (“3“ = (Cl, —é,,é;§é,, )“ = z68 ®(T"‘Z'Z)“
with )3“ = (2,, - 28,23,sz
(4.9B) 6126;; = 2&2; 09111-

Substituting (4.9A) and (4.9B) into (4.7), we get

(4.10) D(é)'(1G ®Z)[E“ ®(T“Z'Z)“]~
{(16 @2')f.(é)—(2w2;3, «81,, )(IL ®Z')u.] = o.

Noticing that
(4.11) (2&2; 49 1M )(IL @2011. = (2&2; ®Z')vec(U)

= vec(Z' US$21“)

=(1G <8 z'U)vec(>:;3,2u,)

= (IG ®Z‘)(IG ®U)vec(2;},2,,),
and substituting this expression into (4.10), we get
(4.12) D(é)'(1G azure <2a>(z'2)"1 1(1G @Z')[f.(é) —(1G 49 um = 0
or
(4.13) D(5)'(2“ ®PZ)[f.(é)—(IG ®U)A]=O

where A = vec(BfiZm) a (Al',kz',---,AG')'. Thus, fori= l, 2, ..., G, Xi = 2;; times the
i‘h column of 2“,; equivalently, x, = (plimT’lU' U)’1p1imT'1U'em.

50

It is well known that the first order condition for the usual GMM estimator ((3)
using moment conditions E[¢1(y: .90 )] = 13((1G a 2, my: ,eo)] = o and weighting

matrix 2;: is
(4.14) 1)(é)'(2:;,1 a Pz)f.(é) = 0.

Comparing (4.14) with (4.13), we see that the only difi‘erence between the first order
condition of the usual GMM estimator and the first order condition of the IGMM
estimator is that E: and f.(9) in (4.14) are replaced by E“ and [f.(9)- (IG ®U))t]
respectively. It is interesting to notice that [f. (90) - (IG ® UM] =
[(r(1,(y;,, ,eo, ) - mt),- - -,(f(G) (yzg, .90G ) - van is just a vector of residuals from the
linear projection of fm)(y:g) ,Oog) onto U. Thus the IGMM estimator é implicitly
determined by (4. 13) can also be regarded as a purged GMM (PGMM) estimator, as in
Chapter 3.

Under assumptions (A4.1)-(A4.3), it is not difficult to show that the usual GMM

estimator é implicitly determined by (4. 14) has the following asymptotic variance:

(4.15) Av1fi(é-eo)1 =1pM%D(eo)'(2: ®Pz mm. )1"
= 14112;: ®A2>A1“
where
AZ]
(4.16) A = phm—;(IG ez')1)(90) =
Ad}

has full column rank because of assumption (A4.2).
We now wish to consider the IGMM estimator 6 and the following three more

estimators

51

(4.17A) ’6' = atgmin, {f.(0)'(2“ a qMUZ,)f.(e)}
(4.1713) 6 = argmine{f.(9)'(2“ @MUPZMU)f.(9)}
(417(3) 6' = otgmine {f.(9)'(2“ ®PZMU)f.(O)}.

We will show that these estimators are asymptotically equally eficient, with asymptotic

variance matrix equal to

(4.13) [plim%D(90)'(Z“ tsp, )D(90)]“ = [A'()3"' 69A; )A]" e B".

We will presume that the estimators are consistent, because each can be written as
a GM estimator that exploits valid moment conditions.

Comparing (4.18) with the asymptotic variance matrix of 6 in (4.15) above, the
fact that the matrix {[A'(z;,1 em;l )A]" — [A'(2“ eAg )A]“} is positive semidefinite
(shown later in Theorem 4.2) establishes the asymptotic eficiency of 6 , '9', 5 and 6

relative to 5. We now turn to a proof of these results.

LEMMA 4.1: plimT’IZ'MUZ= phmT“Z'Z=A,,;
plimr‘12'MU1),(90i ) = plimr’12'D,(90i ) = A,, for i = 1, 2, G.
Proof. The proof is similar to the proof of Lemma 3.1 of Chapter 3. For example,
plimT’IZ'MUDi (90,)
= pfim[T-IZ'D1(90i )-(T"Z'U)(T"'U'U)"(T"1U'Di(9ot ))1
= A, - 0.2;},Au,
= A”,
where the second equality is implied by (A4.]).

52

LEMMA 4.2: plim T’lDi (90i )'I’IMUmDJ- (90].) = plim T“D,(9Oi )'MUPZMUDJ-(Ooj )
= pMT—lDi(901 )'PZMUDj(90j)
= pMT—lDi(90i)'PZDj(90j)
: AizA—ulAq'
for i,j=1, 2, ..., G.
Proof: We will give the proof for plimT'lDi(90i )'MUPZMUDjwoj) = AizA ;Azj. The
proofs for the other cases are quite similar
P lim T-lDi (9m )' MUPZMUDj (901')
= pfimlT—1D1(901)'MUZ](T—1Z.Z)-l[T—IZ.MUDj(90j )1
= AgAgA,

using (A4.1), (A42) and Lemma 4.1.

LEMMA 4.3: plim[T”lD(90)'(Z‘8 ® EMUZ,)D(60)]
= pIimIT"D(Go)'(E“ ® MonMo)D(Oo)]
= pIimIT"D(Go)'(2“ ® PzMU)D(90)]
= plimlT“D(90)'(Z“ 8’ Pz)D(Bo)1
= A12“ ®A;z‘)A a B,
and B is nonsingular.
Proof: Let 2” = (oii)G,(,. Then

P limITIDwo )'(Zw ® P[MUZ] )D(90 )]

D1 (901 )' GIIP[MUZ] ' ‘ ' GIGP[MUZ]
= plimT-l '. : :
Do (906 )' O'Glpmuzl ' ° ’ OGGP[MUZ]

53

Dt(9m)
Do(9oo)
D1(901)'0“P|MUZJD1(901) D1(901).61GP[MUZ]DG(BOG)
=phmT-1 s s s
Dowoo )'OmleUlei (901) Do (900 )'GGGPlMUZJDG (906)
011A12A2A21 GIGAlegAi}
= 5 E E (usingLemma4.2)
OGlAGzAgAzl GGGAGzAZAzG
Alz oI‘A; o‘GA; AZ,
AGZ 061A; "' 066A; ’ Ad}

A12“ ®A;)A a B.

using the definition of A in (4.16). The probability limits for the other cases involve
essentially the same arguments. Finally, 13 = AXE“ (SA; )A is nonsingular bacause Z ,
Em, and Azz are nonsingular, which implies 2‘.“ ®A;z1 nonsingular, and because A has full
cohrmn rank (see (A4.2)).

LEMMA 4.4: plimr"(1G ®Z'MU)D(90) = plimT“ (1G @Z')D(00) = A.
Proof: plimT‘1(IG ®Z'MU )D(90)

54
Z'MU D1(901)
= plimT’1 .
Z'Mu Do (900)

plimT—IZ'MUD1(901)

plim T—IZ'MUDG (90G)

plimT"Z'Dt(9ot)
= (using Lemma 4.1)
plimT'IZ'DG (GOG)

21
= (using (A4.1))
A 26

using the definition of A in (4.16).

LEMMA 4.5: T'1/2(IG ®Z')[e. —(1G e um and T‘1’2(IG ® z'MU )e. each converge
in distribution to N[O, (2“ )-1 ®Au] with 2“ = (2,, — zwzggzue)".

Proof. See the proof of Lemma 3.6 in Chapter 3.

THEOREM 3.1: JT(§—90)r flay—90), «E(é—Bo) and JT(é—90) each convergein
distribution to N(O,B‘1), where B = A'(2‘.“ am; )A with A = ding(A,, ,...,A,G) as
defined in (4.16).

55

Proof: We will give the proof for $11.6. - 90 ). The proofs for the other cases are
essentially identical. Using the definition in (4. 17A), we know that '9' satisfies the

following first order condition:

(4.19) 1)('é')'(2“ e qMUZ,)f.('é') = 0.
Using the first order Taylor expansion formula, we have
(4.20) r('é') = f(90 ) +D(e‘ )(‘é’ -90) = e. +D(e‘ )(‘é’-90)

where 9' is between 60 and ‘9‘. Substituting (4.20) into (4.19), we get

(4.21) D('é')(z“ e PIMUZ] )D(9‘)(é‘-90)+D(é')'(2“ a PM,] )e. = o.
(4.21) is equivalent to
(4.22) T1/2('é'- 90)

= —1T-ID(6')'(2“ ® IiM.n>D(e')1-‘ -T-“D('é')'(2“ ®P1Mo21)8'-

Because .9. is consistent, (4.22) can be rewritten as

(4-23) T"2 (ii—Go) = -[p1imT"D(eo)'(>:“ e PIMUZ1)D(90)]—l -
.T-1/2D(9o “2% ® PIMUZ] )e, +0p (1)
= —B—1 .'1“—1/21)(9o “Zea <8 PIMUZI )3, +0p (1)

using Lemma 4.3. But

(4.24) T‘V’me0 )'(>:88 a PIMUZ, )e.

56
= r“‘”D(eo)'[2“ ®MUZ(Z'MUZ)‘1Z'MU]8.
= T"”D(90)'(IG ca MUZ)[Z“ <22>(Z'MUZ)“](IG ®z'MU)c.
=[T‘1D(90)'(IG ®MUZ)][2“ ®(T"Z'MUZ)“]-
~T‘1’2(IG ® z'MU )e.
= A12“ 49A; )-T'“2(IG ®Z'MU)8. +op(1)

using Lemmas 4.4 and 4.1.

Substituting (4.24) into (4.23), we arrive at

(4.25) T"2('é'-eo) = —B“A'(E“ ®A;ZI)-T‘”2(IG ®Z'MU)8. +01, (1)
—+ N(0,W)

using Lemma 4.5, where

(4.26) w = B“A'(Z§“ ®A;Z')-[(2“ )"1 ®An]-[B"A'(2“ @A; )]'

= 13“A'(2m ®A;)-[(£“)“ ®Au]-(£“ @A; )AB"
= 13‘1 -A'(Z“ ®A;z‘)A-B"‘

= B’1 ~B-B'1 (using the definition ofB in (4.18))

= B“.

THEOREM 4.2: The improved GMM estimators: é , '9', 5 and 6 are asymptotically

emcient relative to the usual GMM estimator 6. They are strictly more efiicient than the
usual GMM estimator (3 if 2“,, at 0.

Proof: See the proof of Theorem 3.3 in Chapter 3.

57

2 2
Our discussion so far has assumed that the covariance matrix 2 = [£68 8"] is

‘18 Z uu

known. Since we generally do not know 2‘. , the estimators é , '6', 5 and ii are infeasible.

However we can define feasible versions of them, say

(4.27A) 6,. = argmin, {[f.(9)— (1G a U)X]'(i“ <8 1),)[f..(9)-(1G e Um}
(4.2713) '6', = argmin, {f..(e)'(2as a PlMUZ] mm»

(4.270) 6, = urgmine {f.(9)'(i“ ® MUPZMU )f.(e)}

(4.27D) 6,. = urgmine {f.(9)'(2“ ® PZMU )f.(e)}

where if.“ and it are consistent estimates of 25‘ and A respectively. Specifically,

2“ = (2,, — 2,52,,12,)-l and i: vec(2;3,2,,) with 2,, = r—‘e'e, 2,: 2,, = r—‘é'u
and 2,, = T‘lU'U, where E = (a, ,---,éT )'= [f(y;,é), my; ,é)]' withé being any
consistent estimate of 60; for example, the usual GMM estimator. Then it is not difficult

to show that 61;, 6}, 6r and 6,, have the same asymptotic distribution as (3 , '6', 5 and 5.

4.3. Concluding Remarks

In this chapter, we have generalized the results of the previous chapter for 3SLS-
type estimators to the case of a nonlinear model We have simplified the analysis by
making high-level assumptions, and by not giving a rigorous proof of consistency. Given
these simplifications, the extension from linear to nonlinear BSLS is fairly straightforward.
As in the linear case, the improved estimators use a difiermt projection matrix than the
usual nonlinear 3SLS (e. g. the projection onto MUZ instead of onto Z), and they use the

inverse of V(st |u,) instead of V(e,) in weighting equations.

CHAPTER 5

THE ASYMPTOTIC EQUIVALENCE BETWEEN THE ITERATED
INIPROVED ZSLS ESTMATOR AND THE 3SLS ESTINIATOR

5.1. Introduction

In Chapters 2, 3 and 4, we showed that we can improve on the usual IV (ZSLS)
and 3SLS estimators provided that we have available an extra vector of observable
variables 111 which is uncorrelated with the instruments and correlated with the
disturbances of the model being estimated. In this chapter, we will extend the improved
IV (ZSLS) idea to the case when the extra information ut is consistenthi estimated instead
of observed. Because the asymptotic distribution of the estimated 111 depends on the the
model structure from which it is estimated, we choose to consider our extension in the
context of a system of linear equations.

It is well known that the only diflerence between equation-by-equation ZSLS and
BSLS is that the 3SLS estimator utilizes information about the relationships among the
disturbances of difi‘erent equations, but 2SLS does not. Then a natural question one wants
to ask is whether it is possible for us, on one hand, to still keep the simplicity of the
equation-by—equation 2SLS estimator; on the other hand, to also utilize the information
contained in the covariance structure of the model disturbances, such that the new
equation-by-equation estimator has the same asymptotic efficiency as the BSLS estimator
applied to the entire equation system

Telser (1964) has addressed essentially the same question in the context of a

seemingly uncorrelated equation (SURE) system:

58

59

(5.1) ytg = x,,'eo, +e,,, g = 1, 2, G; t = 1, 2, T,

where ytg is the dependent variable of equation g at observation t, xtg is the K8 x1 vector
of explanatory variables of equation g at observation t, at, is the model disturbance of
equation g at observation t, 6,8 is the K8 x1 unknown parameter vector of equation g,

and xt8 is strictly exogenous. Telser proved that the iterated LS estimators of the

augmented equations

(5.2) y, = xtg'eo, +e,(,)'7.o, +v,,, g = 1, 2, G; t = 1, 2, T,

where at“) = (8,1,...,a,,8_1,a,_g,l,...,etG)', log = [E(e,(g)s,(g)')]‘1E(s,(g)atg) and
vtg = at8 - “9'20? are asymptotically as efficient as the SURE estimator of the system
(5.1). It turns out that the analogous result is also true for a simultaneous equation
system

The rest of the chapter is organized as follows. Section 5.2 defines the improved
ZSLS (IZSLS) estimator. Section 5.3 describes the iterated IZSLS estimators. Section
5.4 proves that the iterated IZSLS estimators converge to a limit, and that their limit is

asymptotically as eflicient as the 3SLS estimator.

5.2. Improved ZSIS Estimator

The model considered in this chapter is

(5.3) ytg = x,'e,, +stg, g = l, 2, ..., G; t = l, 2, ..., T,

60

where ytg is the dependent variable of equation g at observation t, xtg is the K8 x1 vector
of explanatory variables of equation g at observation t, 8,8 is the model disturbance of
equation g at observation t, and 903 is the Kg x1 unknown parameter vector of equation
g. We assume that in general cov(x,g,etg) at O for all g. Suppose that we have available
an M x1 vector ofinstruments zt satisfying E(ztstg) = 0, with E(ztxtg') having fiJll
column rank for g = l, 2, ..., G.

We define the following notation:

YIg xlg 813 zl
(5.4A) y,= : ,Xg= : ,88= : ,Z= : ;
YTg ng' 8T3 ZT.
91
(5.4B) 9 = E , a = [81,...,EG], a. = vec(e);
9o
(5.4C) 8“,) = (8,1,...,et’g_1,e,,g+1,...,e,G)';
31(3).
(5.4D) 8(8) = : =(81,...,88_] 388+1’”"SG);
8m).
(5.4E) PZ = Z(Z'Z)“Z‘ provided that 22 is nonsingular;
(SAP) )2, = sz8 for g = 1, 2, G;
(5.4G) L(h|W) = W20 = the linear projection of h onto W, with

20 = [E(W'W)]'1E(W'h) defined for any s x 1 vector h and any sx m

matrix W as long as E(W‘W) is nonsingular.

61

It is important to note the distinction between a, (T observations for the error of equation

g) and 8(8) (T observations for the errors of all the equations except equation g).

With this notation, (5.3) can be rewritten as

(5.5)

yg = x8908 +88, g :1, 2, ..., G.

We make the following "high level" assumptions.

(A5.1)

(As.3)

plim—

 

 

Xl'

 

[X1,...,XG ,Z,81,...,SG]

A16 A12 Axe.“ AXEJG

AGG AGz Axe,Gl Axa,GG

A2G A22 0 0 exists.
AflJG 0 511 0' lG

Asx,GG 0 CG] GGG j

 

T'”2 (1G ez')e. —+ N(o,2eA,,).

62
(ASA) T‘1/28(i)'vi —) N(O,Qi)
With Vi = 8i - L(8i

 

8(3)) for i = 1, 2, ..., G.

0'“ Ola—1 51,1“ 016
q._ 0-._ ._ o’._. c._ .
Define 2,, = ‘ I" ' I" l ‘ 1"“ ‘ LG 1= 1, 2, ..., G.
01+” ' ” o'i+1.i—1 °i+1,i +1 "' 0i+1,G
_ CG] oGJ—l oG,i+1 6G6 _

 

 

Assumption (A5.2) implies that 2n is nonsingular for all i

These high-level assumptions are derivable from various sets of more basic
assumptions. For example, in a time series context, define et = (8,1,...,e,G)' and
‘I’t = {z,;z,_1,e,_l;z,_2 ,e,_2 ;...}. Then (A5.1)-(A5.4) follow from the assumptions that
E(e,|‘l’,) = 0, V(e,|‘l’,) = E, (xu',...,x,G',z,')' is covariance stationary, and the fourth
moments of et exist.

We now define our IZSLS estimator as the IV estimator of the augmented

equation
(5.6) yg = X890g +s(g)kog +vg, g = l, 2, ..., G,

using (Eyew) as instruments, where

(5.7A) 103 =[E(8(8)18(8))]‘1E(8(8)'88)_=_(kogl,n,,)\,og’g_l,xog,g+1,o.., 1086 )1

(5.7B) v8 = a, — L(ag|e(g)) = as -—s(g))tog.

63

If we knew 8(8), then as pointed out in Chapter 3, adding 8(8) to (5.6) reduces the
variance of the model disturbance in (5.6) from V(sg) to V(88|8(8) ), and IV applied to

(5.6) would be more eficient than IV applied to (5.3). However, the IV estimation of
(5.6) is infeasible because we do not observe 8“,). In the next section we will define the

iterated feasible IZSLS estimator.

5.3. Iterated IZSLS Estimator

Our iterated feasrble IZSLS estimator of 90 = (601',...,OOG’)' in (5.3) is defined as
follows.
TKO—mill): Apply the usual IV (ZSLS) estimation method to (5.3) equation by equation,
using Z as instruments, to get the initial consistent estimate 6(0) = [6A)1(0)',...,éG (0)']' of
90. Then we estimate the model disturbance a, by

(5.8) e,(0) = y, -x,é,(0)

for g =1, 2, ..., G.

Round 1: For equation 1, apply the usual IV estimation method to
(5-9) Y1 = X1901+8(1)(1)}~01+V1(1)

using pt, ,8(,) (1)] as instruments, where 2.0, is defined in (5.7A), and

(5-10A) 3(1)“) = [32(0)a---a50(0)]
(5-103) V10) = Y1 ‘ X1901— 3(1)(1)7~01 = 81" 8(1)(l))~01~

64

Denote the IV estimate of (5.9) by [61(1)',il(1)']'. We then update the estimate of a, by
(5.11) e,(1)=y,—x,é,(1).
For myation 2: Apply the usual IV estimation method to
(5-12) Y2 = X2902 +3(2)(1)}~02 +V2(1)
using [222 ,e(2)(1)] as instruments, where Am is defined in (5.7A), and

(5-13A) 8(2)“) = [81(l)a33(0)a---aeo(0)]
(5.13B) v2(1)= 82 —8(2)(1))toz.

Denote the resulting IV estimate of (5. 12) by [62(1) , 22(1)]2 We then update the

estimate of 82 by

(5.14) 32(1)=Y2 4962(1).

Generally, in Round 1 for equa_tiqn_g: We apply the usual IV estimation method to
(5.15) y8 = X8908 +8(g)(1)7cog +vg(l)

using [x,,e(,)(1)] as instruments, where 2.0, is defined in (5.7A), and

(5.16A) 8(g)(l) = [81(1),...,88_l(1),8g+1(0),...,80(0)]
(5.16B) vg(1) = 3:; -e(g)(l)7tog.

65

Denote the resulting IV estimate of(5.15) by [68(1), 28(1)}. We then update the

estimate of as by

(5.17) e,(1) = y8 — x8680).

Round 2: For equation 1, apply the usual IV estimation method to
(5-18) Y1 = x1901+3(1)(2)7~01+"1(2)

using [22, ,8“) (2)] as instruments, where 9.0, is defined in (5.7A), and

(5-19) 8(1)(2)=[82(1)a---a80(1)]
(5-20) V1(2) = Y1 — X1901 " 5(1)(2))~01 = 81 ‘ 8(1) (2)101-

Denote the IV estimate of(5. 18) by [61(2)',il(2)']'. We then update the estimate of 81

 

by
(5.21) 81(2) = y1 — x,é,(2).

For equation 2: Apply the usual IV estimation method to
(5-22) Y2 = X2902 +8(2)(:’3)7~02 +"2(2)

using [39 ,s(2)(2)] as instruments, where 202 is defined in (5.7A), and

(523) 8(2)(2)=[81(2)’83(1)r°"80(1)]

66

(5.24) v2(2) = 82 — 8(2)(2)7\.02.

Denote the resulting IV estimate of (5.22) by [02(2), 712(2)}. We then update the

estimate of a, by

(5.25) 82(2) = y, — x,é,(2).

Generally, in Round 2 for equation g: We apply the usual IV estimation method to
(5.26) yg = Xg90g +i»:(g)(2)?to8 +vg(2)

using [233(c)(2)] as instruments, where 208 is defined in (5.7A), and

(5-27) 8(9(2) = [81(2)a---rag—1(2)a8g+1(1)a-~-936(1)]
(5.28) vg(2) = a, - 8(g)(2))tog.

Denote the resulting IV estimate of (5.26) by [08(2) , 28(2)']'. We then update the

estimate of 88 by

(5.29) e,(2) = y, —x,é,(2).

Further rounds continue in the same fashion. Generally, in Rormd n for equgtion g,

we apply the usual IV estimation method to
(5.30) y8 = X3903 +e(g)(n)}t0g +vg(n)

using [5(8,s(g)(n)] as instruments, where 208 is defined in (5.7A), and

67

(5.3 1A) 8(g)(n) = [81(n),...,eg_1(n),ag+1(n — l),...,sG(n — 1)]
(5.31B) vg(n) = as — 8(g)(n)}tog.

Denote the resulting IV estimate by [03(n)',ig(n)']'. We then update the estimate of a,

by

(5.32) e,(n) = y8 — xgé,(n).

Thus, we have finished describing the iterative procedure of the feasible IZSLS
estimator. This process is essentially the same as defined by Telser (1964) for the SURE
model, except that IV is used here where OLS was used in the SURE model We may
note that other similar iterative processes are also possible. For example, for equation g in

rormd n, we could augmented the equation and instruments set by
8Z8)(n) = [81(11 — 1),...,88_1(n —1),8g+1(n — 1),...,8G(n - 1)]

instead of 8(8) (11) as in (5.31A); this amounts to using the estimates of a from round n-l
to estimate all equations in rormd n. This would change our algebra but not any of our
conclusions since, if the interative processes based on 8(g)(n) and 8:901) both converge,
they obviously converge to the same limit.

We now show that the iterated IZSLS estimators defined above are consistent and

asymptotically normal in every rormd of iteration.

LEMMA 5.1: T"2[é,(n) - 90,1» N[o,n,(n)], where Q,(n) is a finite positive definite

 

(pd) matrix, forn = 0,1, 2, ..., and g =1, 2, ..., G.

68

Proof: We will only prove the case of G = 2. The proof for general G is essentially the

same.

When n = o, é,(0) is just the usual 2SLS estimate ciao, in (5.3). Then it is well

known that

(5.33) T1/2[é,(0)—90,] —> N[o,o,,(A,,A-,}A,,)-l]

using (A5.1)-(A5.3), for every g.
When n = 1, using the definition that [08(l)',5tg(l)']' is the usual IV estimate of

(5.9) using [3(1,a(,)( 1)] as instruments, we have

(5 34) T1/2[él(1)-601] _ T‘lil'xl T'lil'8(1)(l) 4 T‘mil'vlfl)
. T1/2[i1(1)_}\'01] — T_]8(1)(1)'X1 T_18(1)(1)'8(l)(1) T-1/28(1)(1)'v1(1) '

But
(5.35) Tfig'xl = (T-‘x,'Z)(T"'z' Z)"(T”‘Z'X, )
: Aleir:z1Azl +0p(l)
(5.36) T"x,'e(l,(1) = T“x,'e,(0) (using the definition of 8(1)(1) in (5.10A))

= T92. 'iya - 2962(0)]

= T-lx, '[(y, — x,90, ) + x, (9,, - é,(0))]

= Tame, — x,(é‘),(0)-eo2 )]

= T‘1il'ez -(T-1x,'x,)[é,(0)-e,,]

= (T'IXI'ZXT'll' Z)'1(T"Z' e,)
—(T-lx,'Z)(T-1z' Z)-1(T-1z' x2 )[é,(0)—90,]

:Al' ;.0p(1)—AIZA; 22‘0p(1)+0p(1)

69

(using (A5. 1) and the consistency of 02 (0))
= 0p( 1)

(5.37) T'1em(l)'x1 = T“‘e,(0)'xl
= T“[ya — Xaéamn'X.
= T“[82 — Xa(éz(9)-eo)1'xr
= T-le,'x1 -(é,(0)-902 )'(T-lx2'x1)
= A8X,21—0p(1).A21+0p(1)
(using (A5. 1) and the consistency of 02(0))
= Aam +op(1)

(5-38) T_]8(1)(1)"3(1)(1) = T_1[32 " X2 (62(0)‘902 )“52 " X2(éz(0)"902 )]

= 022 +op(1)

using (A5. 1) and the consistency of 02(0). Combining (5.35)-(5.38), we have

T-lx 'x T-lx' 1 A A-IA o ,
(539) —1 1' l -1 l 8(l)'( ) =I: lz zz zl +0p(1)-
T 8(0(1) X1 T 8(1)(1)3(1)(1) Aex,21 022
Because
(5-40) V1“) = Y1 _X1901 _8(1)(1)x01

3 Y1 ‘ X1901 " 82(0))”01 (using (5'10A»
= 81 — [Y2 " Xzézmnxm
= 31_[82 — X2(éz(0) _902)])\v019

70

(5.41) T‘mit'v. (1) = T_1/2X1'{31"32}V01 autism—902110.}
= Twigs1 —T"’2x,'e,xm
+T“’25‘<.'Xzié. ((1)—9.211..
= A A“ .T'mz'e, —A A" ~(T‘1/2Z'82 )lm

lzzz lzzz

+A12A221Az2 “Tm [62(0) “9021' 7V01"' 0p (1)

using (A5.1)-(A5.2). Combining (A53), (5.33) and (5.41), we see that T‘1’2x,'v1 (1) is

asymptotically normal with mean zero.
Similarly

(5.42) T‘1/28(D(1)'v,(1)

= T‘1’282(0)' {e., 4.22.0, +x,[é,(0)-e,,]lm}

= T-1/2{g2 _x2[(),(o)—902 ]}'{e1 —t:2)tm +X2 [92(0)_902]7L01}

= T-me,t(a, —82101)-{Tm[éz(0)-9os]}'[T-1X2'(81 - an...”
+(T-‘e,'x,)-T"’[€1,(0)—e,,]7t01
— {Tm 162(0)-9a1}'(T“X.'X. )ié.(9)-9a1>~u

= T'mez'e. mi»...)—{T"2162(0)—9..1}'(A..,.. "'Axeazxm)

+A,,,22 -T‘”[é,(0)-90,]i.01 +o,(1).

Then combining (A5.4), (5.33) and (5.42), we see that T_1/28(1)(1)'V1(1) is

asymptotically normal with mean zero.

Substituting (5.39), (5.41) and (5.42) into (5.34), we have proved that
T1’2[01(1)- 001] is asymptotically normal with zero mean. Thus

T”2 [01(1) — 001] —) N[0,Q1 (1)], where 91(1) is the corresponding asymptotic

covariance matrix. (For our present purposes we do not need to evaluate 01(1).)

71
The proof that T"2 [02 (1) — 002 ] —> N[0,Q2 (1)] is essentially the same as the

proof given above. Then, using the inductive method, we can prove that
T"2 [6,(n) - 90,1» N[o,o,(n)] for all g and n.

5.4. The Convergence and Asymptotic Efficiency of the Iterated IZSLS Estimators

In this section, we give a convergence result for the iterated IZSLS estimator, and

show that it has the same asymptotic efficiency as 3SLS.
Since [08(n)',5tg(n)']' is the usual IV estimate of (5.30) using [5(8,e(g)(n)] as

instruments, we have

(5.43) X8.X8 x8'8(8)(n) 9801) = XS'YS .
3<g)(n)'xu 3(s>(n)'8(s)(n) 7‘s“) 8(e)(n)'ys
Because
(5.44) y8 = X390g +88
2 X8908 +8(3)A'08 +Vg (USing (5.7B))
= [Xg,8(g)](903'3x0g.).+vgs
then

)2 'y )2 ' 903]
5.45 8 3 = 8 x,
( > [WW8] [,(g)(n).]{t ,e,,1[,08 e.,}

72

= Xs'xa xs'em) [9%]4. xs'vs .
8(8)(n)'x8 8(11.)(11)'8(11) A'08 8(3)(n)'vs

Substituting (5.45) into (5.43), we get

(5.46) [ x,'x, x,'e(,,(n) [64m]

8(s)(n)'xg 8(a)(n)'8(s)(n) 9:801)

: xs'xs xa'em) [9%]+ xs'va .
8(a)(n)'xg 8(a)(n)'8(s) )‘Os 8(g>(n)"’s

In order to examine the convergence of[08(n)',ig(n)']' as n —> 00, we define the

following notation:
x 'x x '
(5.47A) 130 =[ 8. 8 8.803)]
8(11) X2. 8(8) 8(a)
(5.4713) B = XB'XS X8.8(8)(n)
8(s)(n)'xs 8(8)(n).8(8)(n)
(5470 C =[ XS'XB x,'a(,) ]
s(g)(n)'X, 8(s)(n)'8<s)
(5.47D) d,(n) = é,(n)—9,,, for g = 1,2, G and n = 1, 2,
(5.47E) Dg(n) = [X1d1(n),...,X8_1dg_,(n),Xg,,dg,1(n “1),...,XGd6(n —1)]f01‘

allnandg

73

(5.47F) AB, = 0 48.138“)
—Dg(n)"Xg —8(8)'D8(n)— D8(n)'e(g) +Dg(n)'Dg(n)

0 0
a...) A..-[_,gm,xg _,8,),,®].

Then (5.46) can be rewritten as
(5.48) B, [93m] = c,[e°“]+[ x,'v', ].
A11.01) 2'08 8(8)“) V8

Because

(5.49) e,(n) = y, —x,é,(n)
= (y, - X,90,)-1X,é,(n)- X,9.,1
= e, — x,[é,(n)—9,,]
= 8s - ngg(n)

using (5.47D), then substituting (5.49) into the definition of 8(8) (11) in (5.31A), we get

(5.50) 8(g)(n)=[81(n),...,88_1(n),88+1(n —1),...,eG(n — 1)]
= [81,...,88_1,88+1,...,SG]
-[X1d1(n),...,Xg_,d8_1(n),X8+]dg+l (n — 1),...,XGdG (n — 1)]

= 8(8) _ D801)

using the definition of D8(n) in (5.47E). Substituting (5.50) into (5.47B), we get

(5.51) B, =B0+ABn

74

using the definitions of B0 ill (5.47A) and ABn in (5.47F). Substituting (5.50) into
(5.47C), we also get

(5.52) C, = B, +AC,

using the definitions of B0 in (5.47A) and ACn in (5.47G). Substituting (5.51) and (5.52)

into (5.48), we obtain

(5.53) (B,+AB,)[98(n)]=(B,+AC,)[9°8]+[ XS'VB ].

is“) )‘08 8(8)“qu

This can also be rewritten as

(5.54) (B0+ABn)[eg(n)_9°3]=(AC,-AB,)[9°B:I+[ x,'v, ],

i,(n) ‘ ;‘08 7‘08 3(g)(n)'vg

01'

1/2 “ _
1 1 AB,)[T 19,01) 9.,1]

(555) (;B0 +_vf T1/2[ig(n)—)\rog]

e -l/2 A o
= T-m (ACn ' ABu )[ 08 ] + :1172 xg v.8 '
)‘08 T iany“) Vs

Premultiplying (5.55) by (31,-B0 +%AB,)'1, we arrive at

(5.56)

75

Tm[0,(n)-00,]
T1/2[7~g(11)- log]
9 -l/2 A 1
= (113, +-1—AB, )-1 {T‘1’2(AC, - AB, )[ ”]+ 3,, x, v', }.
T T log T 8(,)(n) v,
Now we wish to show that plimT_,,, T‘1AB, = 0, and that

T‘“2 (AC, - AB, )(008',}tog')', T'mig'v, and T‘1/28(,)(n)'v, each are asymptotically

multivariate normal with mean zero.

LEMMA 5.2: For any i,j = l, 2, ..., G, and n, m = 0, l, 2, ...,

 

(1)
(2)
(3)
(4)
(5)
(6)
(7)
(3)
(9)

plim T’lii'em =0;
plim
plim
plim
plimT_,,, T’ld,(n)'Xi'XJ-dj(n) = 0;
plim
plim
plim
plim

T—)ao

-l I _ .
T Xi dej(n)—0,

T—boo

-1A 1 _ .
T xixjdj(u)_o,

T—Mo

T—)ao

-l l _ .
T a, dej(n)—0,

-l i _ .
T xi Dj(n)—0,

T—No

—lAr _ .
T x,Dj(n)_o,

T—)oo

T“D,(n)'Dj(m) = o;

T—)oc

T—)oo

T'lam'DJ-(n) = 0.

Proof. (1) Because

plimT‘1iri'ej = plim(T“‘x,'Z)(T"z'Z)‘l (T“z'ej)

=A,A;,‘.o
=0

using (A5.1)-(A5.2), then

plimT'lii'em = plim[T’15(,'al,...,T"1$(i'aj_1,T‘lii'ew,...,T"'5(i'eG]

=0.

76
(2) plimT‘IXi'XJ-dj(n) = plim(T"Xi'Xj )dj(n) = A,- -0 = 0 where the second
equality is gotten from the definition of dj(n) = éj(n) —00j in (5.47D) and Lemma 5.1.
(3) plimT‘lii'deJ-(n) = p1im(T“xi 'Z)(z'Z)-‘(T-‘z'xj )dj(n)
= A,A;,‘A, .0 (using (A5.1) and Lemma 5.1)
= 0.

(4) plimT—lei'deJ-(n) = plim(T"18i'Xj )dj(n) = A9,,- -0 = 0, where the second
equahty is gotten fi'om (A5. 1) and Lemma 5.1.
(5) plimT"‘di (n)'x,'xjdj(n) = plimdi(n)'(T'1Xi'XJ-)dj(n) = o-Aij -o = o,
where the second equality is gotten from (A51) and Lemma 5.1.
(6) PlimT'IXr'an)

= plimT'IX,'[X1d1(n),...,Xj_1dj_1(n),Xj,1dj+l (n — l),...,XGdG(n — 1)]

(using the definition of Dj (11) in (5.47E))

= 0
using part (2) of this lemma.
(7) plimT"f<.'D,-(n)

= plimT‘1xi '[x,d,(n),...,xj_,dj_l (n),xj,,dj,1 (n —1),...,XGdG(n - 1)]

(using the definition of D J- (n) in (5.47E))

= 0
using part (3) of this lemma.
(8) Plim T_1Di(n)'Dj(m)

F dl(n)'xl'

di—l (n)' Xi-l'

= limT’l . .
P di+1 (n — 1)'Xi+1

 

 

_ dG(n—l)'XG' _

77
-[X1d1(m),...,Xj_1dj_1(m),Xj+ldJ-+1(m-1),...,XGdG(m- 1)]
(using the definition of Di (n) and Dj(m) in (5.47E))
= (plimT'ld8(n. ).Xg.xhdh (m' ))(G-l)x(G-l)
= 0

using part (5) ofthis lemma, where n’ = n or n—1, and m' = m or m-l.
(9) PlimT—18(t)'DJ-(n)

=p1imT'l 8H [x1d,(n),...,xj_,dj_,(n),xj,1dj,,(n—1),...,deG(n—1)]

 

 

Z (plim'r-leg'xhdh(n.))(G-l)x(G—l)
= 0
using part (4) ofthis lemma, where 11‘ = n or n-l.

A E

A “A o
LEMMA 5.3: (1) plimT_,,,T"Bo=[ “A.” ’3 JEB,where
8x88 88

A8888 =[A ' ,A

“’18 ,ooc

“(84),',A,x,(,+1),',...,A,X,G,']', and B is nonsingular;

(2) plimT—bco T_1ABn = 0'
Proof: (1) We must evaluate the four elements of T“‘B0. First,

—1A I _ —l I -l —1 -l I
(5.57) T xgxg—(T xgsz Z‘Z) (T zx,)

= AyagA, +o,(1)

78

using (A5. l)-(A5.2). Second,

-1A 1 _
(5.58) T X, 8“,) — o,(l)

using part (1) ofLemma 5.2. Third,

P -

T'lel'X, F ARI, -
T“e 'x A
5.59 T‘1e 'x = 8“ 8 = “(8“)“ +0 1=A’ +0 1
( ) (g) g T-183+1.x8 A8X.(g+l)g p( ) sx,gg p( )
L T"‘eG'x, - _ Ashe, _

 

 

 

 

where the second equality is gotten from (A5. 1). Fourth,

(5.60) T“e(,)'e(,) = T"1 3", ,[81,...,8,_,,8,,1,...,SG]= 2,, +o,(1)

 

 

using (A5. 1) and the definition of 2,, on page 62. So, substituting (5.57)-(5.60) into
T‘1BO with B0 defined in (5 .47A), we obtain

)2 'x x ' A A‘IA 0
(5.61) plimT‘1B0=plimT"[ 8' 8 3.8(8):|=[ 82 .u 2% :|=B'
8(ta) X81.) 8(8) 8(a) Asxes 2,,

But, according to (A52), A 2, has firll column rank, and 2,, is nonsingular. Thus B is

nonsingular.

79

(2) Using the definition of AB, in (5.47F), we have
(5.62) plim T‘1AB,

= plim[ o -T"x,'D,(n)

—T’1D,(n)'X, -T—18(,) 'D,(n) — T"1D,(n)'a(,J + T"1D,(n)'D,(n)

using Lemma 5.2, parts (6)-(9).

LEMMA 5.4: Foranyi,j=l, 2, ,G, andn= 1,2, ...,asthe samplesize T—)oo,
(1) T'IIZX-'XJ--d J(n)—)N[0, A,Q J.(n)A,];

(2) T'mX 'x J.d J,(n)—)N[0 A ,Agz‘A ,0 JJ.(n)A zA‘,,‘A,];

(n)A
Proof: (1) mei'deJ.(u) =(T'1xi XJ-)[deJ-(n)]

(3) T’mei'XJ-dJ-(n)—>N[0, Amp J. ,, ,1.

— A,[T“2dJ -(n)]+o, (1) (using (A5.1))
—>N[0, A- J.er.(n)A, ]
using Lemma 5.1.
(2) T-“zx,'xJ.dJ.(n) = (T‘lx,'2)(T-‘z' Z)’1(T"Z‘XJ.)[T1/2dJ-(n)]
= A,A;,‘A ,.T“’dJ.(n)+oJ, (1) (using (A5.1))
—> N[O, A,A;,1A,J.QJ(n)AJ.,A;,‘A,.]
using Lemma 5.1.
(3) T‘mai'XJ-dJ-(n) = (T-‘e.'xJ.)[T”2d.(n)]
=A,, ,de. J(n)+o, (1) (using (A5.1))
—+ N[O, A,,J.Q J(n)A,, ,]

8O

usingLemma 5.1.

LEMMA 5.5: Each element of T‘1/2(AC, - AB,) converges in distribution to a normal

distribution with mean zero as the sample size T—) co.
Proof. Using the expressions for AB, in (5.47F) and AC, ill (5.47G), we get

--l/2A l
(5.63) T—U2(AC,-AB,)= 0 -1/2 . T Xg 2,201) . J
0 T 8(,)D,(n)—T D,(n) D,(n)
But
-l/2A I
(5.64) T X, D,(n)

= T'l’zi,'[xld1(n),...,X,_1d,_,(n),X,,1d,,l(n —1),...,deG(n —1)].

Then using Lemma 5.4, part (2), we see that each column of T’mi,'D,(n) converges ill

distribution to a multivariate normal distribution with mean zero as T —> 90. Similarly,

(5.65) T'1/28(,J'D,(n)

= T'“2 8-1 [X1d1(n),...,X,_1d,_1(n),X,+ld,+l(n—1),...,XGdG(n-1)]

 

 

= (Pr—“2%'dej(n.))(G—l)x(G-l)

81

where i,j=1,...,(g-1),(g+l),..., G, and 11‘ = n ifj < g otherwise n' = n - 1. Then, using
Lemma 5.4, part (3), we see that each element of T'1/28(,J 'D,(n) is asymptotically

normal with mean zero. Finally,
(5.66) T'mD,(n)'D,(n) = (T'mdtfllt )'Xt'delea ))(G-1)x(G—l)

using the definition of D,(n) in (5.47B), where n], ll2 = n or n - 1. But, as T—> 60,

(5-67) dethlt )'Xt 'dejflla ) = di(n1)'(T—lxi 'Xj)[de,-(n2 )1

= 0p(1)

using Lemma 5.1 and (A5. 1). Substituting (5.67) into (5.66), we obtain

(5.68) T'mD,(n)'D,(n) = o,(l)

as T —) 00. Therefore, substituting (5.64), (5.65) and (5.68) into (5.63), we prove that
each element of T’m(AC, — AB,) converges in distribution to a normal distribution with

mean zero asthe sample size T—)oo.

LEMMA 5.6: Both Twig“, and T“1/28(,J(n)'v, converge ill distribution to

multivariate normal distributions with mean zero as the sample size T —> 00.
Proof: Using the definition of v, in (5.7B), we have

(5.69) T“’2x,'v8 = T‘1’2x,'[e, - 2,80,]
= (T"X,'Z)(T"Z' Z)"‘ ~T’U2Z'[a, — 8(,))to,]
= —A,,A;,l -T“’2Z'e)to, +o,(1),

82

where we define
(5,70) 71,, = (8,1,...,l,,,_,,—1,>.,,,,,,...,>.,,G )'

with 7.0, for j = 1, g - 1, g + 1, Gdefined in (5.7A). (Note that 5.0, is not an

estimate.) But

—l/2 " __ —l/2 . "
(5.71) T 282,, — vec(T Z 820,)
= r“2 (51,691,, )vec(Z'i-:)
= T“2 ('22,, '81,, )(1G 8 Z')vec(8)
= (i,,'<8IM)-T‘“2(IG ®Z')vec(8)

-> N(O,Q.)

using (A53), where a. = (i0,'®1M)(2®A,,)(ioJJ ®IM) = (i,,'2i,,)-A,,.

Combining (5.69) and (5.71), we see that

-l/2A I
(5.72) T x, v,J —>N(0,Q)

° -1 --1 ~ I ~ -1
WIth Q = A,,An ll. -A,,A,, = (1.0, 220,)-(A,,A,,A,, ).

Similarly, using the expresion for 8“,) (n) in (5.50), we have

(5.73) T'ms(,J(n)'v, = T‘"2 [e(,) —D,(n)]‘v,

_ 1/2 -l/2 I

But

(5.74) T‘1’2a(,'v, —> N(0,Q,)

83

using (A5.4).
T-Uzd] (11)le 1v8

T-l/2dg_1 (nyxg—l ng

(5.75) T'mD (n)'v = _ J J
8 g T ll2dg+1(n_l) xg+l V

8

 

 

_ T'de(n —1)'XG'V, J

using the definition of D,(n) in (5.47E). The expression ill (5.75) converges to a

multivariate normal distribution with mean zero because, for any i and 11‘ = n or n- 1,

T'1’2d,(n")'x,'v,J = [T"2d,(n‘)]'-T“x,'[e, — 2,30,]
(using the definition of v, in (5.7B))
= - 1T"2d. (n‘ )1'-T"X: '81.,
(using the definition of X0, in (5.70))
= 1T‘”d.(n‘>1'iA.... WA 1%, +0, (1)

using (A5.l). Then, according to Lemma 5.1, T‘mdi (n')'X, 'v, converges to a normal

distribution with mean zero.
Substituting (5.74) and (5.75) into (5.73), we prove that T“ma(,)(n)'v, converges

ill distribution to a multivariate normal distribution with mean zero.

THEOREM 5.1: (i) For any g = l, 2, G, both Tm[0,(n)—00,] and
T”2 [i,(n) — 20,] converge ill distribution to multivariate normal distributions with mean

zero.

(ii) T“’[é,(n)-e,,]=(A,,A;,‘A,,)-‘[T*“2x,'1),(n).i.,, +T—“2x,'v,]+o,(1).

84

Proof: (i) Applying Lemmas 5.3, 5.5 and 5.6 to (5.56), the result follows immediately.
(ii) Substituting Lemma 5.3 and (5.63) into (5.56), we obtain

1/2 " _
(5.76) Tmfligm 908]
T [xg(n)—A'Og]

_A,,A“Az, 0 "{o T“”i,'D,(n) [90,]
AM, 2 o T’1/28(,J'D,(n)-T'mD,(n)'D,(n) 2.0,

+[ T-1/2X' ,v, 8]}+°p(1)-

T-1/28(8)(n)lv

Then, using the partitioned inverse rule, we obtain

(5.77) T1’2[0,(n)—00,]=(A,,A;21A,,)‘1[T‘1/2i,'D,(n)-Ao, +T-1’2i,'v,]+o, (1).

We will rewrite (5.77) slightly, as

(5.78) T1/2[é,(n)-90,]
= (T-lx,'x,)-1[T-V2x,'D,(n)20, +T-1’2)“(,'v,]+op (l).

Prermlltiplying (5.78) by (vigor, ), we obtain

(5.79) (T-li,'x,)-TW[é,(n)—eo,]
= T-1/2x,'D,(n)-l,, +T-1’2x8'v, +oJ,(1)

85

01'

(5.80) (T-lxg'x,).T1/2d,(n) = T-1/2x,'D,(n).7.0, +T‘1’25(,'v, +oJ, (1)
using the definition of d,(n) in (5.47D). Substituting (5.47E) into (5.80), we obtain

(5.81) (T-lx,'x,)-T1/2d,(n)
= T-lnig'lxldl(n):---axg—ldg—l(n):xg+1dg+l(n _ 1))-":XGdG (ll —1)]A‘Og
+T'1’25(,'v, +o,(l)

. 8-1 G a
= T-1/2X,'{J§1x,d,(n)x,, + JZJXJ-dJ-(n—1)}uo,J-}+T‘1’2X,'v, +o,(l)

J=8+

using the definition of 20, in (5.7A). This can also be rewritten as

(5.82) (T-lx,')"(,)-T1I2d,(n)
g—l A G A
= 2(T‘1X,'XJ-)T"2dJ-(n))t0,- + z (T-lx,'xJ.)T1/2dJ.(n—1)>.O,

j=1 j=g+l
+T—l/2xgovg+op(l)
,_1 A A G a ..

= E(T-txgtxjfltndjwno, +j=§+J(T—IXB'XJ )Tl/sz.(n— 1))to,
+T-l/2Xg'v8 +01) (1)

using the fact that )2ng = (sz,)'xJ. = (PZX,)'(PZXJ.) = 52352,.

Define
' 0 0 0 0
7,02,52,66, 0 0 0
(5.83A) L=T-l 2.03,)? '22, 7.0,,x3'x, 0 0

 

y
.8.
x)
Q-
y
§
X
G.
tax,
>’
§
>4
Q-
cox)
>2
.8
‘3
...x’
o-
>4)
‘1’

 

86

 

 

 

 

(5.83B)
0 Aolle'xz 7*0132‘1'7‘3 A01,ci—1Z(1'Xo-l )‘01le 9ft;
0 0 2.023X2 'X3 A02,6-1X2'Xc—1 Amoxz 'XG
U = '1‘.1 E . i '
0 0 0 0 )‘0,G—I,GXG—l '26
_0 0 0 0 0 -
Pi] -
a )2, _J ..J ..
(5.83C)X= ,D=T XX
- XG .1
(1101) v1
(5.83D) d(n) = E , V: ‘
dG(n) VG
With this notation, (5.82) for g = 1, ..., G can be expressed in matrix form as
(5.84) D - T1’2d(n) = L - T1’2d(n) + U-T1’2d(n — 1)+ T'l’zi'v + 0p (1).
Solving for T1’2d(n), we obtain
(5.85) T1’2d(n) = (D — L)‘1U-T1’2d(n — 1) +(D — L)‘1 ~T‘1/2X'v + o, (1)

forn =1, 2,
The iteration procedure defined in (5.85), apart from the 0p (1) term, is just the

Gauss- Seidel iteration method (see Varga, 1962). We now wish to show that this iterative
process converges to a limit, say d', and that AV(de') equals the asymptotic variance

87

of the usual BSLS estimator of 00 in (5.3). In order to prove these results, we first need

to establish Lemmas 5.7 and 5.8. Define

(5.86A) V = (v1,...,vG) with v,, i = 1, 2, ..., G defined in (5.7B)
(5.86B) c = E(T’18' V) a (c, )G)“,
(5.86C) A = (i,,,...,i,,) with 720,, i = 1, 2, G defined in (5.70).

LEMMA 5.7: (l) —V=8A;

011
(2) C = ispd;

cGG
(3) —C = 2A;
(4) L— D+U = T-lx'm'el, )2.
Proof: (1) Using (5.70) and the definitions of a and 8(,) in (5.4B) and (5.4D), equation

(5.7B) can be rewritten as

(5.87) —v = 810,.

Because (5.87) holds for all g, we can stack the equations together as

(5.88) —V = 8A.

(2) and (3): Because v, = a, — L(8,

 

a“) ), then E(sm 'v, ) = 0 for any i. Therefore

c,- = T‘1E(8,'vJ-) = 0 for i ¢ j. Premultiplying (5.88) by T'le', we get

88

(5.89) —T“e'v = T‘le'eA.

Taking expectations of both sides of ( 5.89), we obtain

(5.90) -c = 2A

using (5.86B) and (A5. 1). Equation (5.90) can be rewritten as

(5.91) —2“C = A.

Denoting 2‘1 = (oij )GxG, then oii > 0 for allibecause 2"1 = (oij )6,6 is pd. Comparing

the diagonal elements on both sides of (5.91), we get

(5.92) --o“cii = —1

using (5.70), (5.86C) and the diagonality ofC. Then cii = 1/ oii > 0 for all i.

(4) Using the definitions of L, D, U, and )2 in (5.83) and A in (5.86C), we can easily
verify that L— D+U = T-lx'(A'eIT)x.

LEMMA 5.8: All the eigenvalues of(D — L)"1 U equal zero.

Proof.

A

x1. cllIT
(5.93) X'(C®I,) =

89

CIIIK, x1.

cGGIKO XG.

III
01
)5)

where Ki is the number ofcolumns in X, for i = l, 2, ..., G. According to Lemma 5.7,
part (2), c, > 0 for all i. So we can define

cillzlx,
(5.94) C. =
cgélxo
Then
(5.95) C = C3 .
Using Lemma 5.7, part (4),
(5.96) L - D + U = T-lx'm'el, )i

= T-lx'((—2-IC)'8>IT))‘( (using Lemma 5.7, part (3))
= 44221081, )(2-1 81,)22

= 446212-181”)? (using (5.93))

= -T-lcz)"('(2-l elm?

90

using (5.95). Then

(5.97) C:‘(D — L— U)C. = C.[T")('(2'1®IT)5(]C. a M

using (9.96). Because both C. and T“)“('(2‘l ® IT))( are pd, then the matrix M defined

ill (5.97) is also pd. Therefore there exists a nonsingular matrix, say P, such that
(5.98) P‘IMP = lK

where K = 20:, K,.
Suppose x is any eigenvalue of the matrix (D — L)”1 U, then it satisfies

(5.99) [xiK —(D—- L)“ U| = 0

where |A| = det(A). Using the facts that |A ~B| = |A| |B| and (D - L) is nonsingular, (5.99)

is equivalent to

(5.100) |x(D—L)- U| = 0,
which is also equivalent to
(5.101) :‘[x(D—L)—U]C. =0

 

 

C.

 

 

because ¢ 0. Substituting (5.97) into (5. 101), we get

(5.102) ‘xM+(x-1)C:1UC.. = 0.

 

91

Because |P| =1: 0, (5. 102) is also equivalent to

(5.103) |P“[xM +(x—1)CI1UC.]PI = 0.

Substituting (5.98) into (103), we obtain

(5.104) IxIK +(x— 1)(c.1>)‘l U(C.P)| = 0.

Using the fact that all the eigenvalues of a strict upper triangular matrix are zero, we
conlude that all the eigenvalues of U are zero, since U is a strict upper triangular matrx.
Because (C.P)‘1U(C.P) conjugates with U, then all the eigenvahles of (C..P)’1 U(C.P)
are zero. Next we use the facts that ifan H x H matrix Q has eigenvalues p1, ..., on, then
(1) for any scalar a, the eigenvalues ofmatrix orQ are up], ..., con; and (2) for any scalar
or, the eigenvalues ofmatrix (011H +Q) are (or +p1), ..., (a +pH). From this we can
conclude that all the eigenvalues of the matrix [xIK +(x —1)(C.P)'1U(C.P)] are equal to

x. Then (5.104) is equivalent to
(5.105) xK = 0
because the determinant of a matrix equals the product of its eigenvalues. Solving for x,

we get x = 0 with multiplicity K. Therefore we have proved that all the eigenvalues of the

matrix (D— L)‘1U are equal to zero.

Define d’ to be the limit of the iterative process

(5.106) Tl/2d(n) = (D— L)-1 U-T1/2d(u — 1)+(1)- L)-1.T-1/2x'v,

92

which is the same as (5.85) except for an 0p (1) term Because all of the eigenvalues of

(D — L)‘1 U equal zero, d' exists and the iterative process (5. 106) reaches (1" in no more
than K iterations. Furthermore, since (5.85) and (5.106) difi‘er only by an o,(1) term, the

probability that the process (5.85) has a limit (in n) approaches one as T —> co; and the
limit of this iterated 12SLS estimator has the same asymptotic distribution as d’. We now

proceed to show that d‘ (and hence the iterated IZSLS estimator) has the same asymptotic

distribution as the 3SLS estimator.

THEOREM 5.2: de‘ —> N(0,W), with w = [A'(2"' 8A; )A]", where
A: diag(A,1,...,A,fi).

Proof. Since (1' is the limit of the process (5.106), it satisfies

(5.107) TWd‘ = (D— L)‘1U-T"2d‘ +(D— L)--1 .T-Wx'v.
Solving for de‘, we get
(5.108) TWd' = —(L+U— D)-1-T-1/2x'v

= —[T-15('(A'<8>I,))“(]-1-T-1/25('v,

using Lemma 5.7, part (4). But

(5.109) T-‘x'(A'®1,)5‘( = T1 (A'®IT)

93
= [A'®(T"Z'Z)“]-
T"‘xG'z

T"z'xl

T"z'xG

=A'(A'®A;,1)A+o,(1) (using (A5.1))
= —A'(C2“ 8A;)A+o,(1)

(using Lemma 5.7, part (3) and the diagonality of C)
= —A'(C®IM)(2" 8A;)A +oJ,(l).

Similarly

(5.110) T‘mx'v = T"“2

T'mil'vl 1

-l/2A r
T XG vG

(T—lxl rZ)(T-lZlZ)-l . T—l/2 Z'V]

(T-IXG IZ)(T—lZlZ)—l . T—l/ZZIVG

94
A —1 ‘T-1/2Z.V1

1 22
= 5 + 0p ( 1) (using (A5.1))
AGZA'Z',‘ -T‘“’z'vG

= A'(IG ®A;Z‘)-T‘“2 (IG ®Z‘)vec(V)+o,(l)
(using the definition of v in (5.83D) and V in (5.86A))
= —A'(1G 8A;_,1).T-1/2(1G eZ')vec(sA)+op(l)
(using Lemma 5.7, part (1))
= —A'(IG <8Ag.l )-T“”(Io ®Z')(A'®Ir )vec(8)+°r(1)
= —A'(IG ®A221 )(A'®IM ) 'T-W (16 ® Z')Ve°(8) + 013(1)
—> N(0,Wt)
using (A5.3), where

(5.111) w1 = A'(IG <8A;,§)(A'erM)-(28»A,,).[A'(1G ®A;z‘ )(A'®IM)]'
= A'(A'2A®A;,‘)A
= A'[(—2"C)'2(—2“C)<8>A',1 ]A (using Lemma 5.7, part (3))
= A'(C2-IC®A;,1)A
= A'(C®IM)(2" ®A;,1)(C®IM)A.

Combining (5.108)-(5.111), we obtain
(5.112) de‘ = —[T")“<'(A'®l,)ik)‘l me'v

= [A'((:<81M)(2'1 8>A;,‘)A]‘l .T‘1’2x'v+o,(1)

—> N(0,w‘)

(5.113) w’ =[A'(C®IM)(2‘1<8A‘,‘z‘)A]’l -wl .

95

°{[A'(C ®IM)(E_1 ®AZ )1‘11—I }'

= [A'(C®Iu)(2" 94;).«1‘1 -

-A'(C®IM)(2“ 8A;)(ceIM)A.

-{[A'(C GNMXY1 @142 )A1'1}'

= [A'(C®IM)(2_1®A;)A]—l -

~A'(C®IM)(2“ 8A;)(C®IM)A.

-[A'(2" 8A; )(C ®IM )Ar‘.

But
cllIM A21
(5.114) (C®IM )A = '
cGGIM
cllAzl
CGGAZG
A21 c111K
A26
5 A .("3

where Ki = the number ofcolumns ill X,, i = l, 2, ..., G. Substituting (5.114) into

(5.113), we get

(5.115) w‘ =[(AC)'(2'1®A;Z')A]’l ~(AC)'(2“ ®A;)AC-[A'(2“®A;)AC]"1

=[A'(2"@AQMT‘C"-CA'(2“(29A;)AC-C"1[A'(2'1®A;,')A]’1

96
=[A'(2“<8>A;21)A]‘1
=w.

Because W = [A'(2‘l (8 A; )A]‘1 is just the asymptotic variance of the the usual
3SLS estimator of 00 ill (5.3), theorem 5.2 tells us that the iterated 12SLS estimator

defined in section 5.3 of this chapter does indeed have the same asymptotic eficiency as
the usual 3SLS estimator.

CHAPTER 6

CONCLUSION

In this dissertation we have shown how to improve on standard GMM estimators,
given observable extra information. For linear models, and for certain kinds of nonlinear
models, the additional information consists of variables that are uncorrelated with the
instruments but correlated with the error(s) of the equation(s) being estimated. We
believe that these results are empirically relevant, notably in the estimation of rational
expectations models. An obvious further research question is the size of the efficiency
gain that can be obtained in actual empirical work. Here the relevant issue is the strength
of the correlation between the forecast errors in related series.

We have also considered the case that the extra moment conditions involve
parameters that need to be estimated, and we discussed the 3SLS problem in detail More

generally, we could consider GMM based on the moment conditions
',0 ,0
(6.1) 9=E10(y:,9.)1=E[¢‘(y: °‘ ”)1,
(1’2 (Yt £01,902)

and using the weighting matrix C‘l, where

C11 C12]

(6.2) C = “HIT—>00 Eli'l"°d’T(9017902)(1>T(901’902 )'1 E [C C
21 22

T .. .. ...
= T—l Z¢(Y:ael :92 ) Let 9 = (01.99;). be the

t=l

and where (1).,(91’92) = [¢Tl(91r92 )]

¢r2(91.92 )
corresponding GMM estimates. An alternative to this (standard) GMM treatment is to
consider an iterative procedure, and this is feasible if 4), identifies 001 with 002 given, and

97

98

6, identifies 9,, with 0,1 given. Then, iii), and 0, are any initial estimates, we can

consider the GM problems

(6.3A)
(6.3B)

where

(6.4A)
(6.4B)
(6.4C)
(6.4D)

6l = ”Emilie, {¢112(91262 )'C11¢1|2(91562 )}
62 = argmine,{¢2|1(ét.92)'C22¢2|1(9t.92)}

¢1|2(91’éz ) = 1911(91 aéz)" C12C2i¢T2 (61 ’62 )1
¢2|1(éla92)=1¢T2(éla92)"C21C1i¢rl(élaéz)1
C“ =(C11_C12C221C21)—1

C22 = (C22 - C21C1‘11C12 )'1.

This yields new estimates 0,, 0,, and the iterative process can be continued ill an obvious

fashion. We conjecture that the limit of this iterative process is asymptotically equivalent

to the GM estimate 0. This result, if true, is a considerable generalization of our results

of Chapter 5 on 3SLS and iterated improved ZSLS.

REFERENCES

REFERENCES

[1] Amemiya, T. (1985), Advanced Econometrics, Cambridge, Mass: Harvard
University Press.

[2] Gallant, AR. and H. White (1988), A Unified Theory of Estimation and Inference
for Nonlinear Dynamic Models, Oxford: Basil Blackwell

[3] Hansen, LP. (1982), "Large Sample Properties of Generalized Method of Moments
Estimators," Econometrica, 50, 1029-1054.

[4] Imbens, G.W. (1992), "An Efiicient Method of Moments Estimator for Discrete
Choice Models with Choice-Based Sampling," Econometrica, 60, 1187- 12 14.

[5] Imbens, G.W. (1993 ), "A New Approach to Generalized Method of Moments
Estimation," unpublished manuscript, Harvard University.

[6] Imbens, G.W. and T. Lancaster (1994), "Combining Micro and Macro Data in
Microeconometric Models," Review of Economic Studies, 61, 65 5-680.

[7] Schmidt, P. (1986), "A Curious IV Result," unpublished manuscript, Michigan State
University.

[8] Schmidt, P. (1988), "Estimation of A Fixed-Effect Cobb-Douglas System Using
Panel Data, " Journal of Econometrics, 37, 36 1-3 80.

[9] Telser, LG. (1964), "Iterative Estimation of a Set of Linear Regression Equations,"
American Statistical Association Journal, September, 845-862.

[10] Varga, R S. (1962), Matrix Iterative Analysis, Englewood Clifi’s, New Jersey:
Prentice-Hall, Inc.

[1 l] Wooldridge, J.M. (1993), "Efficient Estimation with Orthogonal Regressors,"
Econometric Theory, 9, 687.

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