Econometrics I 21

7/25/2019 Econometrics I 21

1/67

Part 21: Generalized Method of Moments1-1/67

Econometrics I

Professor William Greene

Stern School of Business

Department of Economics


2/67


Econometrics I

Part 21 Generalized

Method of Moments


3/67


I also have a questions about nonlinear GMM - which is more or less nonlinear IV technique

I suppose.

I am running a panel non-linear regression (non-linear in the parameters) and I have L

parameters and K eogenous variables with L!K.

In particular m" model loo#s #ind o$ li#e this% & ' b*+b, e and so I am tr"ing to

estimate the etra b,that don/t usuall" appear in a regression.

0rom what I am reading to run nonlinear GMM I can use the K eogenous variables toconstruct the orthogonalit" conditions but what should I use $or the etra b,coe$$icients1

2ust some more possible IVs (li#e lags) o$ the eogenous variables1

I agree that b" adding more IVs "ou will get a more e$$icient estimation but isn/t it onl" the

case when "ou believe the IVs are trul" uncorrelated with the error term1

3o b" adding more 4instruments4 "ou are more or less imposing more and more restrictive

assumptions about the model (which might not actuall" be true).

I am as#ing because I have not $ound sources comparing nonlinear GMM5IV to nonlinear

least squares. I$ there is no homoscadesticit"5serial correlation what is more e$$icient5give

tighter estimates1


4/67


Im trin! to minimize a nonlinear pro!ram "ith theleast s#uare under nonlinear constraints$ Its firstintroduced % &n' ( Geman )2***+$ It consisted onthe minimization of the sum of s#uared difference%et"een the moment !eneratin! function and thetheoretical moment !eneratin! function


5/67


, ,,

, ,,

Method o$ Moment Generating 0unctions

0or the normal distribution the MG0 is

M(t6 )'78ep(t)9'ep8t 9

Moment 7quations% ep( ) ep8t 9 : ,.

;hoose two values o$ t

n

j i j ji

t

t x tn

=

= =

, , , ,

, ,

and solve the two moment equations $or and .

Miture o$ se the method o$ moment generating $unctions with ? values o$ t.

M(t6 )'7

N N

= +

, , , , , ,, ,

8ep(t)9' ep8t 9 ( )ep8t 9t t +


6/67


( )

,

, ,

? , , , , , ,, ,

0inding the solutions to the moment equations% Least squares@M(t ) ep( ) and li#ewise $or t ...

MinimiAe( )

@M(t ) ep8t 9 ( )ep8t 9

Bltern

n

ii

jj

t xn

t t

=

=

=

+

{ } , , , ,

ative estimator% Maimum Li#elihood

L( ) log 8 6 9 ( ) 8 6 9

N

i ii N N == +


7/67Part 21: Generalized Method of Moments1-7/67

,G-S

,easi%le G-S is %ased on findin! an estimator

"hich has the same properties as the true G-S$

E.ample /ar0i 20E.p)zi+2$

3rue G-S "ould re!ress 40E.p)zi+on the same transformation of xi$

With a consistent estimator of 055 sa 0s5c5 "e dothe same computation "ith our estimates$

So lon! as plim 0s5c 055 ,G-S is as 6!ood7 astrue G-S$ 8onsistent Same &smptotic /ariance Same &smptotic 9ormal Distri%ution



3he Method of Moments

1 2 ;

9 91 1 i 1 i i 1 i9 9

Estimatin! Parameters of Distri%utions Moment E#uation> ) 15$$$5;+

m . f ) 5 5$$$5 +

Method of Moments

? ! )m 5$$$5m +5 15$$$5;

=

= =

= =

=


9/67Part 21: Generalized Method of Moments1-/67

Estimatin! a Parameter Mean of Poisson

p)+e.p)=@+ @4 A

E0 @$plim )149+ii @$

3his is the estimator

Mean of E.ponential

p)+ @ e.p)= @+ E0 14 @$

plim )149+ii 14@


10/67Part 21: Generalized Method of Moments1-1!/67

Mean and /ariance of a

9ormal Distri%ution

= =

= =

=

= = +

= = +

= =

2

2

2 2 2

9 9 2 2 21 1i 1 i i 1 i9 9

2 9 2 2 9 21 1i 1 i i 1 i9 n

1 ) +p)+ e.p

22

Population MomentsE0 5 E0

Moment E#uations

5

Method of Moments Estimators

?5 ? ) + ) +



Gamma Distri%utionP P 1

2

2

e.p) +p)+

)P+

PE0

P)P 1+E0

E014 P 1

E0lo! )P+ lo! 5 )P+dln )P+4dP

)Each pair !iCes a different ans"er$ Is there a >%est> pairD es5

the ones that are >sufficient> statistics$

=

=

+=

=

=

E0 and E0lo!$ ,or a

different course$$$$+



3he -inear Fe!ression Modeli i i

i i

9

i i1 1 i2 2 i; ; i1i 1

9

i i1 1 i2 2 i; ; i2i 1

9

i i1 1 i2 2 i; ; i;i 1

Population

.

Population E.pectation

E0 . *

Moment E#uations1

) . . $$$ . +. *9

1) . . $$$ . +. *

9

$$$

1) . . $$$ . +. *

9

Solution : -inea

=

=

=

= +

=

=

=

=

r sstem of ; e#uations in ; unno"ns$

-east S#uares



Instrumental /aria%les

i i i

i i 1 ;

9

i i1 1 i2 2 i; ; i1i 1

9

i i1 1 i2 2 i; ; i2i 1

i i1 1i 1

Population

.


E0 z * for instrumental Caria%les z $$$ z $

Moment E#uations

1) . . $$$ . +z *

9

1) . . $$$ . +z *

9

$$$1

) .9

=

=

=

= +

=

=

=

9

i2 2 i; ; i;

=1

I/

. $$$ . +z *

Solution : &lso a linear sstem of ; e#uations in ; unno"ns$

% ) +

=

Z'X Z'y/n ( /n)



Ma.imum -ielihood91i 1 i i 1 ;9

9 i i 1 ;

i 1

-o! lielihood function5 lo!- lo!f) G . 5 5$$$5 +

Population E.pectations

lo!-E *5 15$$$5;

Sample Moments

lo!f) G . 5 5$$$5 +1*

9

Solution : ; nonlinear e#uations in ; un

=

=

=

=

9 i i 15M-E ;5M-E

i 15M-E

no"ns$

? ?lo!f) G . 5 5$$$5 +1*

?9 =

=



BehaCioral &pplication

+

+

+

+ = + =

=

=

=

+ =

t 1t t

t

t

t

tH1 t 1 t

t t 1 t

-ife 8cle 8onsumption )te.t5 pa!e JJ+

c1E )1 r+ 1 *

1 c

discount rate

c consumption

information at time t

-et 14)1H +5 F c 4 c 5 =

E 0 )1 r+F 1 *

What is in the information set Each piece of >information>

proCides a moment e#uation for estimation of the t"o parameters$



Identification

8an the parameters %e estimated

9ot a sample Kpropert

&ssume an infinite sample Is there sufficient information in a sample to reCealconsistent estimators of the parameters

8an the Kmoment e#uations %e solCed for the

population parameters



Identification

E.actl Identified 8ase: ; population moment e#uations

in ; unno"n parameters$ Lur familiar cases5 L-S5 I/5 M-5 the MLM estimators

Is the countin! rule sufficient

What else is needed

LCeridentified 8ase Instrumental /aria%les

& coCariance structures model



LCeridentification

i i i 1 ;

i i 1 M

9

i i1 1 i2 2 i; ; i1i 1

Population

. 5 5$$$5


E0 z * for instrumental Caria%les z $$$ z M ;$

3here are M ; Moment E#uations = more than necessar1

) . . $$$ . +z *9

1

=

= +

=

=9

i i1 1 i2 2 i; ; i2i 1

9

i i1 1 i2 2 i; ; iMi 1

) . . $$$ . +z *9

$$$1

) . . $$$ . +z *9

Solution: & linear sstem of M e#uations in ; unno"ns$ DDDDD

=

=

=

=



LCeridentification

= +

= +

=

=

1 1 1

2 2 2

1 1 1

2 2 2

3"o E#uation 8oCariance Structures Model

8ountr 1:

8ountr 2:

3"o Population Moment 8onditions:E0)143+ >) +

E0)143+ >) +

)1+ No" do "e com%ine the t"o sets of e#

" #

" #

# " # !

# " # !

=

1 2

1 2 2

uations

)2+ GiCen t"o L-S estimates5 and 5 ho" do "e

reconcile them

9ote: 3here are eCen more$ E0)143+ >) + $

$ $

# " # !


20/67Part 21: Generalized Method of Moments1-2!/67



= +

=

z

x

i i i

i i 0 i0

i

Consider the Mover - Stayer Model

Binary choice for whether an individual 'moves' or 'stays'

d ( u 0)

!utcome e"uation for the individual# conditional on the state$

y % (d 0) &

y % (d ) +

xi i' '

i0 i 0 0

&

( # ) *(0#0)#( # # )+

,n individual either moves or stays# ut not oth (or neither).he arameter cannot e estimated with the oserved data

re1ardless of the samle si2e. 3t is unidenti4ed.


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23/67



24/67


9onlinear -east S#uares--> NAMELIST ; x = one,age,educ,female,hhkids,married --> !alc ; k=col"x# --> NLS$ ; Lhs = hhninc ; %cn = & ' ex&"()*x#

; la(els = k+(,& ; sar = k+,) ; maxi = .Moment matrix has become nonpositive definite.

Switching to BFGS algorithm

Normal exit: 16 iterations. Stats!". F! #$1.1"%$

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

'ser (efined )ptimi*ation.........................

Nonlinear least s+ares regression ............

,-S!--NN/ Mean ! .#0%"$

Standard deviation ! .1621

Nmber of observs. ! %#%6Model si*e 3arameters !

(egrees of freedom ! %#12

4esidals Sm of s+ares ! 6%.%"001

Standard error of e ! .16"1

&&&&&&&&5&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

ariable7 /oefficient Standard 8rror b9St.8r. 37;7% B#7 &."00%@@@ .""1"0 &0%.$"2 .""""

B>7 &."1$>#@@@ .""0$" .1$" .""10

B07 ."0>>0@@@ .""660 $.1$> .""""

B67 &.%6>%>@@@ .""$%# %.1"2 .""""

37 .6#%#2 2"00.>2# .""" .2222 ?!!!!!!!

&&&&&&&&5&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Nonlinear least s+ares did not worA. hat is the implication of the

infinite standard errors for B1 Cthe constantD and 3.


25/67


Ma.imum -ielihood&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&Gamma C,oglinearD 4egression Model

(ependent variable --NN/

,og liAelihood fnction 1>%2#.""%1>4estricted log liAelihood 1120."620#/hi s+ared 6 d.f.= %6120.$60%%

Significance level ."""""

McFadden 3sedo 4&s+ared &1".20220# C> observations with income ! "8stimation based on N ! %#%%E ! were deleted so log, was comptable.D

&&&&&&&&5&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

ariable7 /oefficient Standard 8rror b9St.8r. 37;7"$>1@@@ ."%10> 10$.%1# .""""

HG87 .""%"0@@@ ."""%$ .>1# ."""" >#.0%%

8('/7 &."00%@@@ .""1%" &>6.>26 ."""" 11.#%"% F8MH,87 &.""0>% .""0>0 &.220 .#12$ .>$$1

--(S7 ."601%@@@ .""61$ 1".0>% ."""" .>"%%MH448(7 &.%6#>1@@@ .""62% $.">1 ."""" .0$62

7Scale parameter for gamma model3Iscale7 0.1%>$6@@@ .">%0" 1%".02> .""""&&&&&&&&5&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

M,8 apparentlJ worAed fine. KhJ did one method CnlsD fail and anotherconsistent estimator worA withot difficltJL


26/67


Moment E#uations: 9-S

( ) ,

8 6 9 5 ep( )

/ 5 ep( )

,/E

ep( )

,/

ep( );onsider the term $or the constant in the model .


27/67


Moment E#uations M-E

( )

Fhe log li#elihood $unction and li#elihood equations are

logL' log log ( ) ( ) log

log log ( )log ( ) log E ( )

log using .

Hecall

N

i i i ii

N

i ii

N ii i i i ii

i i

P P y P y

L d PP y P

P dP

L Py

=

=

=

+

= + = =

= =

x

the epected values o$ the derivatives o$ the log li#elihood equal

Aero. 3o a loo# at the $irst equation reveals that the moment equation in

use $or estimating = is 78log" 6 9 ( ) log and anothei i iP= x r K moment

equations 7 " E are also in use. 3o the ML7 uses K

$unctionall" independent moment equations $or K parameters while


28/67


&!enda

3he Method of Moments$ SolCin! the moment e#uationsE.actl identified cases

LCeridentified cases

8onsistenc$ No" do "e no" the method of moments is

consistent&smptotic coCariance matri.$

8onsistent Cs$ Efficient estimation

& "ei!htin! matri.

3he minimum distance estimator

What is the efficient "ei!htin! matri.

Estimatin! the "ei!htin! matri.$3he Generalized method of moments estimator = ho" it is computed$

8omputin! the appropriate asmptotic coCariance matri.


29/67


3he Method of Moments

Moment E#uation: Defines a sample statistic that

mimics a population e.pectation:

%he &o&'lation ex&ectation ortho(onalit"

condition:

E0 mi)

+ !$ Su%script i indicates it dependson data Cector inde.ed % >i> )or >t> for a time

series settin!+


30/67

Part 21: Generalized Method of Moments1-3!/67

3he Method of Moments = E.ample

P P 1

i ii

i i

9

1 i1 i

9

2 i1 i

Gamma Distri%ution Parameters

e.p) +p) +

)P+Population Moment 8onditions

PE0 5 E0lo! )P+ lo!

Moment E#uations:

E0m ) 5P+ E0Q)14n+ R P 4 *

E0m ) 5P+ E0Q)14n+ lo! R )

=

= =

=

)P+ lo! + * =


31/67


&pplication

7lot of 7si(7)8unction

P

-10

-8

-6

-4

-2

0

2

-12

1 2 3 4 5 60

PSI

2 21 1 2 2

2 2

1 2

1

2

SolCin! the moment e#uations


32/67


Method of Moments Solution

creae ; /)=/ ; /.=log"/#calc ; m)=x(r"/)# ; ms=x(r"/.#minimi0e; sar = .1, 12 ; la(els = &,l ; fcn = "l3m)-.

5 "ms - &si"log"l## 4. 5---------------------------------------------5

6 7ser 8efined 9&imi0aion 66 8e&enden :aria(le %uncion 66 Num(er of o(ser:aions ) 66 Ieraions com&leed 2 66 Log likelihood funcion 12.


33/67


9onlinear Instrumental /aria%les

3here are ; parameters5

i f)xi5+ H i$3here e.ists a set of ; instrumental Caria%les5 zisuch that

E0zii *$3he sample counterpart is the moment e)'ation

)14n+izii )14n+i zi0i= f)xi5+ )14n+imi)+ )+ !$

3he method of moments estimatoris the solution to themoment e#uation)s+$

)No" the solution is o%tained is not al"as o%Cious5 andCaries from pro%lem to pro%lem$+

m


34/67


3he MLM Solution

3here are ; e#uations in ; unno"ns in ) + If there is a solution5 there is an e.act solution

&t the solution5 ) + and 0 ) +>0 ) + *

Since 0 ) +>0 ) + *5 the solution can %e fou

m !

m !* m m

m m

nd

% solCin! the pro!rammin! pro%lem Minimize "rt : 0 ) +>0 ) +

,or this pro%lem5

0 ) +>0 ) + 0)14n+ 0)14n+

3he solution is defined %

0 ) +>0 ) + 0)14n+

m m

m m

m m

'Z Z'

'Z 0)14n+

Z'


35/67


MLM Solution

[ ] =

=

=

i

i

i i

0)14n+ 0)14n+ 2 )14 n+ 0)14n+

f) 5 +

n ; matri. "ith ro" i e#ual to

,or the classical linear re!ression model5

f) 5 + > 5 5 and the ,L8 are =20)1

x

G (

x x + # G #

'Z Z' G'Z Z'

4n+) + 0)14n+ >

?

#,# #

#,# #,"

-

+ &0

which has uni"ue solution &( )


36/67


/ariance of the Method

of Moments Estimator

n

ii 1

i i

=1 =1

3he MLM estimator solCes m) +

1 1) + ) + so the Cariance is for some

n 9Generall5 E0 ) + ) +

3he asmptotic coCariance matri. of the estimator is

1&s$/ar0 ) + ) + "9

=

!

MOM

m m

m m '

G G') +

here

m

G= '


37/67


E.ample 1: Gamma Distri%ution

=

=

=

=

= +

=

=

2

n1 P1 i 1 in

n12 i 1 in

i i i

i i i

1 P9

i 1 1

m ) +

m )lo! )P+ lo! +

/ar) + 8oC) 5lo! +1 1

8oC) 5lo! + /ar)lo! +n n

1

n >)P+

G


38/67


E.ample 2: 9onlinear I/ -east S#uares

=

= +

= =

=

=

=

i i i

2i

i i i

2

i

n2 2 2 2

i 1

*

i i

f) 5 + 5 the set of ; instrumental Caria%les

/ar0

/ar0

With independent o%serCations5 o%serCations are uncorrelated

/ar0 ) +)14n + ) 4n + >

)14n+ >

i

i i

i i

x z

m z

m zz'

m zz' Z Z

G zx=

=

=

=

n *ii 1* ii i

*

1 1 * 1 2 2 * 1

2

"here is the Cector of >pseudo=re!ressors5>

f) 5 +$ In the linear model5 this is Vust $

)14n+ > $

) + ) +> 0 )14n+ > 0) 4 n + > 0 )14n+ >

0

x

xx x

G Z X

G V G Z X Z Z X Z

Z * 1 * 1> 0 > 0 > X Z Z X Z


39/67


/ariance of the Moments

=

i

n

i i i ii1

n

i ii1

No" to estimate )14n+ /ar0 ) +

/ar0 ) +)14n+/ar0 ) + )14n+

Estimate /ar0 ) + "ith Est$/ar0 ) + )14n+ ) + ) +>

3hen5

? ? ?)14 n+ )14 n+ ) + ) +>

,or the linear re!ression

m

m m

m m m m

m m

=

= =

=

=

i i i

n n 2

i i i i i i ii1 i1

n=1 2 =1

MLM i i ii1

n=1 2 =1

i i ii1

model5

5

? )14 n+ )14 n+ e e )14 n+ )14 n+ e

)14 n+

Est$/ar0% 0)14 n+ 0)14 n+ )14 n+ e 0)14 n+

0 0 e 0

m x

V x x' x x'

G X'X

X'X x x' X'X

X'X x x' X'X )familiar+


40/67


Properties of the MLM Estimator

8onsistent 3he --9 implies that the moments are consistent estimators of

their population counterparts )zero+


41/67


Generalizin! the Method

of Moments Estimator

More moments than parameters

the oCeridentified case E.ample: Instrumental Caria%le

case5 M ; instruments


42/67


3"o Sta!e -east S#uares

No" to use an 6e.cess7 of instrumental Caria%les

)1+ #is ; Caria%les$ Some )at least one+ of the ;

Caria%les in #are correlated "ith .$

)2+ +is M ; Caria%les$ Some of the Caria%les in

+are also in #5 some are not$ 9one of the

Caria%les in +are correlated "ith .

)+ Which ; Caria%les to use to compute +0#and

+0"


43/67


8hoosin! the Instruments

8hoose ; randoml

8hoose the included s and the remainder randoml


44/67


2S-S &l!e%ra

9

9 9 9( )

But# & ( ) and ( ) is idemotent.9 9 ( )( ) ( ) so

9 9( ) & a real 3: estimator y the de4nition.

9ote# lim( /n) &

=

=

=

-1

2SLS

-1

Z Z

Z Z Z

2SLS

X Z(Z'Z Z'X

! X'X X'y

Z(Z'Z Z'X I -M X I -MX'X= X' I - M I - M X= X' I - M X

! X'X X'y

X' 0

-

9since columns of are linear cominations

of the columns of # all of which are uncorrelated with

( ) + ( )

2SLS Z Z

X

Z

! X' I -M X X' I -M y

= [


45/67


Method of Moments Estimation

m 0

m 0"

Same Moment ;"uation

( )&

ow# M moment e"uations# < arameters. here is no

uni"ue solution. here is also no e6act solution to

( )&

We 1et as close as we can.

=ow to cho

m 'm 'Z Z' 'ZZ'

ose the estimator? >east s"uares is an ovious choice.

Minimi2e wrt $ ( ) ( );.1.# Minimi2e wrt $ *(/n) ( ) +*(/n) ( )+&(/n ) ( ) ( )


46/67


,L8 for MLM

m 'm = G 'm

'ZZ' = - X Z Z' y - X

0

8irst order conditions

() eneral

( ) ( )/ ( ) ( ) & 0

() he 3nstrumental :ariales 7rolem(/n ) ( ) ( )/ (/n )( ' )* ( )+

&

!r#

X Z Z' y - X 0

0

X Z Z' y - X 0

Z' y - X 0

( ' )* ( )+ &

(< M) (M )( ) &

,t the solution# ( ' )* ( )+ &

But# * ( )+ as it was efore.


47/67


8omputin! the Estimator

Pro!rammin! Pro!ram

9o all purpose solution

9onlinear optimization pro%lem

solution Caries from settin! to settin!$


48/67


&smptotic 8oCariance Matri.

-1

m 0

G 'm 0 G '

G ' V G V= m-

eneral @esult for Method of Moments when M


49/67


More Efficient Estimation

m 'm

Mi#im$m %i&(#)* +&

We have used least s"uares#

Minimi2e wrt $ ( ) ( )

to 4nd the estimator of . 3s this the most eAcient

way to roceed?enerally not$ We consider a more 1eneral aroach

M%

im(i,#

. / = m ' m

>et e any ositive de4nite matri6$

9>et & the solution to Minimi2e wrt( ) ( )

/his is a minimum distance (in the metric of ) estimator.


50/67


Minimum Distance Estimation

M%

.

/ = m ' m

m

>et e any ositive de4nite matri6$

9>et & the solution to Minimi2e wrt

( ) ( )

where ;* ( )+ & 0 (the usual moment conditions).

his is a minimum distance (in th

M%

M%

e metric of ) estimator.

9 is consistent

9 is asymtotically normally distriuted.

Same ar1uments as for the MM estimator. ;Aciency of

the estimator deends on the choice of .


51/67


MDE Estimation: &pplication

= + =i iy X X

!

i i i

i

units# oservations er unit# S country y country# roduces estimators of

() =ow to comine the estimators?

We hav

=

2

!

!0

"""!

e 'moment' e"uation$ ;

=ow can 3 comine the estimators of


52/67


-east S#uares

=

=

=

= = =

2 2

2

! !

! ! 0 m

""" """

! !

m m ! ' !

!

!m m I I I ! 0

"""!

!

i ii

ii

i

; . ( )&

o minimi2e ( )' ( ) & ( ) ( )

( )' ( )* # #...# + ( ) .

he solution is ( )= =

= = 0 = ! ! ii i

or


53/67


Generalized -east S#uares

X X

X X

he recedin1 used !>S - simle unwei1hted least s"uares.

3 0 ... 00 3 ... 0

,lso# it uses & . Suose we use wei1hted# >S.... ... ... ...

0 0 ... 3

* ( ) + 0 ... 0

0 * (hen# &

( )

X X

m m

XX !

= XX

i i i ii&

i i i ii&

) + ... 0

... ... ... ...

0 0 ... * ( ) +

he 4rst order condition for minimi2in1 ( )' ( ) is

* ( ) + D( ) & 0

or * ( ) + D * (

XX !

!

i i ii&

i ii&

) + D

& & a matri6 wei1hted avera1e.


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Minimum Distance Estimation

he minimum distance estimator minimi2es

( ) ( )

he estimator is() Consistent

() ,symtotically normally distriuted

(E) =as asymtotic covariance matri6

9,sy.:ar* + * ( )=M%

/ = m ' m

G

( )+ * ( ) ( )+* ( ) ( )+

'G G 'VG G 'G


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Lptimal Wei!htin! Matri.

is the Wei1htin1 Matri6 of the minimum distance estimator.

,re some 's etter than others? (Fes)

3s there a est choice for ? Fes

he variance of the MG; is minimi2ed when

& ,sy.

3*#*4(5iz*6 m*7,6 ,8 m,m*#& *&im(,4

m

V

-

-

:ar* ( )+D

his de4nes the .

&


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GMM Estimation

=

=

= =

=

i

i i

i i

GM

m m 9x 9

m 2 m 9x 9 m 9x 9

m 9x 9 '2 m 9x 9

i i i

i i i i i

i i i i i i

( )& (y )

,sy.:ar* ( )+ estimated with & (y ) (y )

he MM estimator of then minimi2es

" (y ) (y ) .

9;st.,sy.:ar*

=

-1

M

mG'2 G G=

( )+ * + #


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GMM Estimation

=

= =

=

=

i

i

m m 9x 9 0

m 9x 9 '2 m 9x

i i i

i i i i i i

;6actly identi4ed MM rolems

When ( ) & (y ) is < e"uations in

< un5nown arameters (the e6actly identi4ed case)#the wei1htin1 matri6 in

" (y ) (y

=

=

i

-1 -1 -1

9

m 0

G'2 G G 2G'

)

is irrelevant to the solution# since we can set e6actly( ) so " & 0. ,nd# the asymtotic covariance matri6

(estimator) is the roduct of E s"uare matrices and ecomes

* +


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& Practical Pro%lem

i i i i i

i i i i i i

,sy.:ar* ( )+ estimated with

& (y ) (y )

he MM estimator of then minimi2es

" (y ) (y ) .

3n order to comute # you need to 5now

=

= =

=

i i

i i

m

2 m 9x 9 m 9x 9

m 9x 9 '2 m 9x 9

2 # and you are

tryin1 to estimate . =ow to roceed?

yically two stes$

() Hse & Simle least s"uares# to 1et a reliminary

estimator of . his is consistent# thou1h not eAcient.

(

I"

) Comute the wei1htin1 matri6# then use MM.


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Inference

3estin! hpotheses a%out the parameters:

Wald test

& counterpart to the lielihood ratio test

3estin! the oCeridentifin! restrictions


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3estin! Npotheses

m '2 m

restricted unrestrict

() Wald ests in the usual fashion.

() , counterart to li5elihood ratio tests

MM criterion is " & ( ) ( ) when restrictions are imosed on

" increases.

" " d

ed chi s"uared*I+(he wei1htin1 matri6 must e the same for oth.)


61/67


&pplication: Dnamic Panel Data Model

= + + +xi#t i#t i#t i#t i

(,rellano/Bond/Bover# I ournal of ;conometrics# JJK)

y y u

Gynamic random eLects model for anel data.

Can't use least s"uares to estimate consistently. Can't use 8>S without

esti

+ =

+ =

x

x

i# i i#

i# i i#

mates of arameters.Many moment conditions$ What is ortho1onal to the eriod disturance?

;*( u) + 0 & < ortho1onality conditions#


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&&lication nnoation


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&&lication nnoation


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&pplication: MultiCariate Pro%it Model

? J ,

? , J ?

? - variate =robit Model

" " 8" E9

log 8K(, ) ...? 9

Hequires ? dimensional integration o$ the :oint no

i i i i i

it it it it it

i it it i i i i iL y s t ds ds ds ds ds

= + = >

= = x x x x x

x

, , ,

J J J

rmal densit". Ver" hard

Nut 78" 6 9 ( ).

Orthogonalit" ;onditions% 78K" - ( )

K" - ( )

K" - ( )

Moment 7quations% K" - ( )K" - ( )

it it it

it it it

i i i

i i in

i i ii

i i

n =

=

=

x x

x x 0

x x

x x

x x

x

? ? ?

I? equations in P parameters.

K" - ( )

i

i i i

=

0

x

x x


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Pooled Pro$it (norin( orrelation


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andom ffects 9:;1-


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?nrestricted orrelation Matrix

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Econometrics I 21