paneldatanotes-9

7/23/2019 PanelDataNotes-9

http://slidepdf.com/reader/full/paneldatanotes-9 1/57

Part 9: GMM Estimation [ 1/57]

Econometric Analysis of Panel Data

William Greene

Department of Economics

Stern School of Business




http://people.stern.nyu.edu/wgreene/CumulantInstruments-Racicot-AE(201!"#(10!.pd$




The NYU

No ActionLetter




GMM Estimation for One Equation

= =

= =

=

′Σ − = Σ

′ ′Σ σ Σ= ÷ ÷

′= Σ − ÷

N Ni 1 i i i 1 i i

N 2 N 2

i 1 i i 1 i

i

N

i 1 i i

1 1( )= (y )ε

N N

e1 1Asy.Var[ ( )] , estimated withN N N N

based on 2SLS residuas e. !he "## estimator then minimi$es

1 1

% (y )N

i

i i i i

i

gβ z xβ z

zz zzgβ

z xβ '

−

=

=

′Σ

′Σ − ÷ ÷

1N 2Ni 1 i

i 1 i i

e 1

(y ) .N N N

i i

i

zz

z xβ




GMM for a System of Equations

h h

w w

Simutaneous e%uations

Labor su&&y

hours = '(wae, ) =

wae = '(hours, ) =rodu*t mar+et e%uiibrium

uantity demanded = '(ri*e,...)

ri*e = '(mar+et demand,

′+ ε + ε

′+ ε + ε

h h h

w w w

g xβ

g xβ

1 1

2 2

# #

...)

"enera 'ormat-

y =

y =

...

y =

′ + ε′ + ε

′ + ε

1 1

2 2

M M

xβ

xβ

xβ




SUR Model with Endogenous

RHS aria!les

1 1 1

2 2

# "

S/ System

y = , 0[ , ,... ]

y = ,...

...

y = ,...

0a*h e%uation has a set o' L 3 instruments,0a*h e%uation *an be 4t by 2SLS, 5V, "##, as be'ore.

′ ′ ′ ′+ ε ε ≠

′ + ε

′ + ε

≥

1 1 1 2 G

2 2

G G

xβ x x x

xβ

xβ

z




GMM for the System " #otation

′ ′ ′ ÷ ′ ′ ′ ÷= = ÷ ÷ ′ ′

i1 i1

i2 i2i

i"

5nde6- i = 1,...,N 'or indi7iduas

= 1,...," 'or e%uations (this woud be t=1,...! 'or a &ane)

8ata matri*es- " rows,

y

y,

... ...

y

i

x 0 ... 0

0 x ... 0y X

... ... ...

0 0 ...

ε ÷ ÷ ÷ε ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷′ ε

=

i"

1 2 "

i

, ,

3 3 ... 3 *oumns

1 i1

2 i2

i

G iG

i i

β

ββ= ε =

... ...

xβ

y Xβ+ε




$nstruments

′ ′ ′ ÷′ ′ ′ ÷= ÷ ÷

′ ′ ′

ε

ε ε = =

ε 1

i1

i2i

i"

1 2 "

i1,1 i1

i1,2 i1i1 i1

i1,L i1

, " rows (1 'or ea*h e%uation)...

L L ... L *oumns

Su*h that

$

$ 0 0......

$

z 0 ... 0

0 z ... 0Z

... ... ...

0 0 ... x

z

÷÷

÷ ÷

ε

1

i2 i2

'or L instrumenta 7ariabes

Same 'or , ...z




Moment Equations

i1 1

i2 2

i" "

i1

i2

L rows

L rows0[ ] 0 , 'or obser7ation i

... ...L rows

Summin o7er i i7es the orthoonaity *ondition,

1 10 0

...N N

ε ÷ε ÷′ = = ÷ ÷ ÷ε

εε ′Σ = Σ

i1

i2

i

iG

i1

i2N N

i=1 i i=1

z 0

z 0Zε

...z 0

z

zZε

z

1

2

i" "

L rows

L rows

...

L rows

÷ ÷ ÷ ÷= ÷ ÷ ÷ ÷ ÷ ÷ε iG

0

0

...

0




Estimation"%

( )

( )

′ε = −

′Σ − =

′Σ −

∑

∑

i i

2# N

i=1 i,m i im=1

" Ni=1 i,m i i=1

y

9or one e%uation,

: :the minimi$er o' (1;N) $ (y ) ( ) ( )

Leads to 2SLS9or a e%uations at the same time

: :the minimi$er o' (1;N) $ (y )

ig g

g g g g g

xβ

β = x β g β 'g β

β = x β

=

∑

∑

2#

m=1

"

=1

( ) ( )

5' the s are a di<erent, sti e%uation by e%uation 2SLS

g g g g

g

gβ 'g β

β




Estimation"&

−

== =

ε

′Σ ′ ′= Σ − Σ − ÷ ÷ ÷ ÷ ÷ ÷

Σ

i

1N 2i 1 iN N

i 1 i i i 1 i i

"=1

Assumin are a un*orreated, e%uation by e%uation "##

e1 1 1% (y ) (y ) .

N N N N

9or the system,

% = %

ases to *onsider

ig ig

ig g ig g

z zz xβ ' z x β

-

(1) oe>*ient 7e*tors ha7e eements in *ommon or are

restri*ted

(2) 8isturban*es are *orreated.




Estimation"'

−

= = ==

=

=

=

′ ′ ′Σ − Σ ε Σ − = ÷ ÷ ÷ ÷

′Σ − ÷′Σ − ÷= ÷ ÷ ÷′Σ −

∑1

" N N 2 Ni 1 i i i 1 i i 1 i i 1

Ni 1 i1 i1

Ni 1 i2 i2

Ni 1 i" i"

ombinin "## *riteria

1 1 1 1(y ) (y ):

N N N N

(y )

(y )% ?

...

(y )

ig g ig ig ig g

i1 1

i2 2

iG G

z xβ ' z z z x β

z xβ

z xβ

z xβ

−=

=

=

=

=

=

′Σ ε ÷′Σ ε ÷ ÷ ÷ ÷′Σ ε

′Σ − ′Σ −×

′Σ −

1N 2i 1 i1

N 2i 1 i2

N 2i 1 i"

Ni 1 i1 i1

Ni 1 i2 i2

Ni 1 i" i"

:

:

:

(y )

(y )

...

(y )

i1 i1

i2 i2

iG iG

i1 1

i2 2

iG G

z z 0 ... 0

0 z z ... 0

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

0 0 ... z z

z xβ

z xβ

z xβ

÷÷÷

÷ ÷




Estimation"(

Ni 1 i1 i1

Ni 1 i2 i2

Ni 1 i" i"

N 2 N Ni 1 i1 i 1 i1 i2 i 1 i1 i"

Ni 1

2

5' disturban*es are *orreated a*ross e%uations,

(y )

(y )1% ?

N ...

(y )

: : : : :

:1

N

=

=

=

= = =

=

′Σ − ÷′Σ − ÷= ÷

÷ ÷′Σ −

′ ′ ′Σ ε Σ ε ε Σ ε εΣ ε

i1 1

i2 2

iG G

i1 i1 i1 i2 i1 iG

z xβ

z xβ

z xβ

z z z z ... z z1

N 2 Ni2 i1 i 1 i2 i 1 i2 i"

N N N 2

i 1 i" i1 i 1 i" i1 i 1 i"

Ni 1 i1 i1

: : : :

: : : : :(y

1

N

−

= =

= = =

=

÷′ ′ ′ε Σ ε Σ ε ε ÷ ÷ ÷

÷′ ′ ′Σ ε ε Σ ε ε Σ ε ′Σ −

×

i2 i1 i2 i2 i2 iG

iG i1 iG i1 iG iG

i1 1

z z z z ... z z

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

z z z z ... z zz xβ

Ni 1 i2 i2

Ni 1 i" i"

)

(y )

...

(y )

=

=

÷′Σ − ÷ ÷ ÷ ÷′Σ −

i2 2

iG G

z xβ

z xβ




Estimation")

" " N Ni 1 i i i 1 ih ih 1 h 1

N 2 Ni 1 i1 i 1 i1

2

5' disturban*es are *orreated a*ross e%uations,

:% (1; N) (y ) (1;N) (y )

:where = the h bo*+ o' the in7erse matri6

: : :

1

N

= == =

= =

′ ′= Σ − Σ −

′Σ ε Σ ε ε

∑ ∑ ig g ih h

i1 i1

gh

gh

z xβ W z x β

W

z z1N

i2 i 1 i1 i"

N N 2 Ni 1 i2 i1 i 1 i2 i 1 i2 i"

N N N 2i 1 i" i1 i 1 i" i1 i 1 i"

: :

: : : : :

: : : : :

−

=

= = =

= = =

′ ′Σ ε ε ÷′ ′ ′Σ ε ε Σ ε Σ ε ε ÷

÷ ÷ ÷′ ′ ′Σ ε ε Σ ε ε Σ ε

i1 i2 i1 iG

i2 i1 i2 i2 i2 iG

iG i1 iG i1 iG iG

z z ... z z

z z z z ... z z

... ... ... ...z z z z ... z z




*he Panel Data +ase

×

ε =it it

is the same in e7ery e%uation.

!he number o' moment e%uations is ! L

i' ea*h moment e%uation is L &er &eriod,

0[ ] ,5' e7ery disturban*e at time t is aso orthoona

to e7ery set o' instruments i

β

z

ε = 2it is

n e7ery other &eriod, s,

!hen

0[ ] , !L &er &eriod, 'or ! &eriods, or ! L

0.., L=1 instruments, !=@ &eriods, 3=@ &arameters,

2@ moment e%uations () 'or 4ttin @ &arameters.

z




Hausman and *aylor ,E-RE Model

it it i

i i

i i

2i i i u

i i

2i i

i

y u

0[u ]

0[u ]

Var[u ]

0[ ]=

Var[ ]=

o7[ ,u

ε

′ ′ ′ ′= + + + + ε +

=

≠ ⇒

= σ

ε

ε σ

ε

it 1 it 2 i 1 i 2

it

it

it it

it it it

it it it

it

x1β x2 β z1 α z2 α

x1 ,z1

x2 ,z2 OLS an GLS a!" in#$n%i%t"nt

x1 ,x2 ,z1 ,z2

x1 ,x2 ,z1 ,z2

x1 ,x2 ,z1 ,z2

x i i

2 2i i i u

2i i i i u

]=

Var[ u ]=

o7[ u , u ]=

εε + σ + σ

ε + ε + σ

it it

it it it

it i% it it

1 ,x2 ,z1 ,z2

x1 ,x2 ,z1 ,z2

x1 ,x2 ,z1 ,z2




Useful Result. /SD is an $ Estimator

( ) ( ) ( )

= + ++

′ ≠

=′ ′ ′ ′= + = +

′=

8

8 8 8

8

=

1&im , so is endoenous. orreated with be*ause o' .

N!

B = 6?s in rou& mean de7iations.1 1 1 1

B? B? C =N! N! N! N!

1

N!

y X &

X w

Xw 0 X w &

M X X

X w X & XM & XM X0 XM

XM

ε

ε ε ε

( ) −

′= =

= = ≠

′

8

8

1

1 1, so &im B? &im

N! N!

1 1&im B? &im ? within rou&s sums o' s%uares .N! N!B is a 7aid instrument.

&im B=&im B ? B

X w XM 0

X X XM X 0

X

X X X y=

ε ε




Hausman and *aylor

′ ′ ′ ′= + + + + ε +

− = + + ε

it it i

it i it

y u

8e7iations 'rom rou& means remo7es a time in7ariant 7ariabes

y y ( ) ( )

5m&i*ation- , are *onsistenty estimated by LS8V.

(

it 1 it 2 i 1 i 2

i iit 1 it 2

1 2

i

x1β x2 β z1α z2 α

x1 ( x1 'β x2 ( x2 'β

β β

x1 1 1

2 2

1

2

) = = 3 instrumenta 7ariabes

( ) = = 3 instrumenta 7ariabes

= L instrumenta 7ariabes (un*orreated with u)

= L instrumenta 7ariabes (wher

it &

iit &

i

( x1 M X

x2 ( x2 M X

z1

)

≥1 1 1 2

e do we et themD)

EF!- = ( G ) = 3 additiona instrumenta 7ariabes. Needs 3 L .i &x1 * M X




H0*1s ,G/S Estimator

21 2

1 1 1 2 2 2 N N N

Ni=1 i

i1 i2

i1 i2 i

i1 i2

(1) LS8V estimates o' , ,

(2) ( ) (e ,e ,..., e ),(e ,e ,..., e ),...,(e ,e ,..., e )

( ! obser7ations).

! rows, re&eat in7ariant 7ariabe B

εσ

Σ

′ ′ ′ ′ = = ′ ′

i

β β

" '=

z z

z zZ

z z

# #

i

1 2

i1 i1,1

i i1 i1,ti1 i1,2

1 1

i1 i1,!

s

L L *oumns

! rows, re&eat , time 7aryin

L 3 *oumns

+

′ ′ ′ ′′ ′ = = +

′ ′

i

z x

z xz xW

z x

# #




H0*1s ,G/S Estimator 2cont34

ε

ε

σ + σ

σ + σ

1 2

2 2u

2 2u

(2 *ont.) 5V reression o' on with instruments

*onsistenty estimates and .

(H) Iith 46ed !, residua 7arian*e in (2) estimates ; !

Iith unbaan*ed &ane, it estimates ;! or s

i

" Z

W α α

ε

ε

ε ε

σ

σ σ

θ = − σ σ + σ

2

2 2u

2 2 2

i i u

omethin

resembin this. (1) &ro7ided an estimate o' so use the two

to obtain estimates o' and . 9or ea*h rou&, *om&ute

: 1 ; ( ! ): : :(J) !rans'orm [ ] toit1 it2 i1 i2x ,x ,z ,z

θ

θi

it it it i i

: [ ] G [ ]

: and y to y B = y G y.

i it1 it2 i1 i2 i1 i2 i1 i2W = x ,x ,z ,z x ,x ,z ,z




H0*1s ( S*EP $ Estimator

=

1

2

1

1

5nstrumenta Variabes

( ) = 3 instrumenta 7ariabes

( ) = 3 instrumenta 7ariabes = L instrumenta 7ariabes (un*orreated with u)

= 3 additiona in

i

iit

iit

i

i

x1 ( x1

x2 ( x2z1

x1

′ ′G1

strumenta 7ariabes.

Now do 2SLS o' on with instruments to estimate

a &arameters. 5.e.,

: : :[ , , , ]=( )1 2 1 2

y W

β β α α W W W y .




Dynamic 2/inear4 PanelData 2DPD4 Models

A&&i*ation Kias in on7entiona 0stimation 8e7eo&ment o' onsistent 0stimators 0>*ient "## 0stimators




Dynamic /inear Model

Bi,t i,t i,t 1

Bi,t 1 2 i,t H i,t J i,t @ i,t i,t i,t

KaestraGNero7e (1M), H States, 11 ears

8emand 'or Natura "as

Stru*ture

New 8emand- " " (1 )"

8emand 9un*tion " N N

"=as demand

N

−= − − δ= β + β + β ∆ + β + β ∆ + β + ε

i,t 1 2 i,t H i,t J i,t @ i,t i,t O i,t 1 i i,t

= &o&uation

= &ri*e = &er *a&ita in*ome

/edu*ed 9orm

" N N " −= β + β + β ∆ + β + β ∆ + β + β + α + ε




A General DPD model

i,t i,t 1 i i,t

i,t i2 2i,t i i,t i,s i

i

y y *

0[ ,* ] 0[ ,* ] , 0[ ,* ] i' t s.

0[* ] ( )

No *orreation a*ross indi7iduas

−

ε

′= + δ + + ε

ε =ε = σ ε ε = ≠

=

i,t

i

i i

i i

xβ

XX X

X X




O/S and G/S are inconsistent

i,t i,t 1 i i,t

i,t 1 i i,t

2* i,t 2 i i,t

2*

y y *

o7[y ,(* )]

o7[y ,(* )]

5' ! were are and G1P P1,

this woud a&&roa*h 1

−

−

−

′= + δ + + ε

+ ε =

σ + δ + εδ

σ

− δ

i,txβ

*-/i#ati$n $th OLS an GLS a!"

in#$n%i%t"nt.




/SD is $nconsistent

52Ste6en4 #ic7ell 8ias9

i,t i i,t i i,t 1 i i,t i

2 !

i,t 1 i i,t i 2 2

y y ( ) C (y y ) ( )

(! 1) !o7[(y y ),( )] ! (1 )

Lare when ! is moderate or sma.

ro&ortiona bias 'or *on7entiona ! (@ G 1@), is

on the order o' 1@Q G Q

−

ε−

− = − δ − + ε − ε

−σ − − δ + δ− ε − ε ≈ − δ

x x 'β

.




Anderson Hsiao $ Estimator

− − − − −− = − δ − + ε − ε

− = − δ − + ε − ε

− = − δ −

i,t i,t 1 i,t i,t 1 i,t 1 i,t 2 i,t i,t 1

i,H i,2 i,H i,2 i,2 i,1 i,H i,2

i1

i,J i,H i,J i,H i,H i

Kase on 4rst di<eren*es

y y ( ) C (y y ) ( )

5nstrumenta 7ariabes

y y ( ) C (y y ) ( )

an use y

y y ( ) C (y y

x x 'β

x x 'β

x x 'β + ε − ε

−,2 i,J i,H

i2 i,2 i,1

) ( )

an use y or (y y )

And so on.

Le7es or aed di<eren*esD

Le7es aow you to use more data

Asym&toti* 7arian*e o' the estimator is smaer with e7es.




Arellano and 8ond Estimator " %


i,H i,2 i,H i,2 i,2 i,1 i,H i,2

i1

i,J i,H i,J i,H i,H i

Kase on 4rst di<eren*es

y y ( ) C (y y ) ( )

5nstrumenta 7ariabes

y y ( ) C (y y ) ( )

an use y

y y ( ) C (y y

− − − − −− = − δ − + ε − ε

− = − δ − + ε − ε

− = − δ −

x x 'β

x x 'β

x x 'β ,2 i,J i,H

i,1 i2

i,@ i,J i,@ i,J i,J i,H i,@ i,J

i,1 i2 i,H

) ( )

an use y and y

y y ( ) C (y y ) ( )

an use y and y and y

+ ε − ε

− = − δ − + ε − εx x 'β




Arellano and 8ond Estimator " &

− = − δ − + ε − ε

− = − δ − + ε − ε

i,H i,2 i,H i,2 i,2 i,1 i,H i,2

i1 i,1 i,2

i,J i,H i,J i,H i,H i,2 i,J i,H

i,1 i2 i,1 i,2

#ore instrumenta 7ariabes G redetermined

y y ( ) C (y y ) ( )

an use y and ,

y y ( ) C (y y ) ( )

an use y , y , ,

X

x x 'β

x x

x x 'β

x x

− = − δ − + ε − ε

i,H


i,1 i2 i,H i,1 i,2 i,H i,J

,

y y ( ) C (y y ) ( ) an use y , y ,y , , , ,

x

x x 'βx x x x




Arellano and 8ond Estimator " '

− = − δ − + ε − ε

− = − δ − + ε − ε

i,H i,2 i,H i,2 i,2 i,1 i,H i,2

i1 i,1 i,2 i,!

i,J i,H i,J i,H i,H i,2 i,J i,

07en more instrumenta 7ariabes G Stri*ty e6oenous

y y ( ) C (y y ) ( )

an use y and , ,..., (a &eriods)

y y ( ) C (y y ) (

X

x x 'β

x x x

x x 'β

− = − δ − + ε − ε

H

i,1 i2 i,1 i,2 i,!


i,1 i2 i,H i,1 i,2 i,!

) an use y , y , , ,...,

y y ( ) C (y y ) ( )

an use y , y ,y , , ,...,

!he number o' &otentia instruments is hue.

!hese de4ne the rows

x x x

x x 'β

x x x

o' . !hese *an be used 'or

sim&e instrumenta 7ariabe estimation.

iZ




$nstrumental aria!les

− −

′ ′ ′ ′ ′ =

′ ′ ′

′

=

i,1 i,1 i,2

i,1 i,2 i,1 i,2 i,H

i,1 i,2 i,! 2 i,1 i,2 i,! 1

i,1

y , , ...

y ,y , , , ... (! rows)

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

... y ,y ,..., y , , ,...

y ,

i

i

!""t"!-in" 3a!ia/"%

x x

x x xZ

x x x

St!i#t/y 4x$g"n$5% 3a!ia/"%

Z

−

−

− −

′ ′ ′ ′ ′

′ ′ ′

i,1 i,2 i,! 1

i,1 i,2 i,1 i,2 i,! 1

i,1 i,2 i,! 2 i,1 i,2 i,! 1

, ,... ...

y ,y , , ,... ... (! rows)

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

... y ,y ,..., y , , ,...

x x x

x x x

x x x




Sim:le $ Estimation( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

−−

−

−−

∆ε

= =∆ε

′ ′ ′Σ Σ Σ × ′ ′ ′Σ Σ Σ

′ ′ ′σ Σ Σ Σ

Σ Σσ =

11

1

11

2

N !i 1 t H2

:

!his is two stae east s%uares.

:0st.Asy.Var[ ]=:

[(:

N N N

i=1 i i i=1 i i i=1 i i

N N N

i=1 i i i=1 i i i=1 ii i

N N N

i=1 i i i=1 i i i=1 i i

6= X Z ZZ ZX

XZ ZZ Zy

6 XZ ZZ ZX

− − −

=

−

− − − − δ −

Σ −

−

2i,t i,t 1 i,t i,t 1 i,t i,t 1

Ni 1 i

i,t i,t 1

: :y y ) ( ) (y y )]

(! 2)

Note that this 7arian*e estimator understates the true asym&toti*

7arian*e be*ause obser7ations are auto*orreated 'or one &eriod.

(y y )

x x 'β

( ) ( ) ( )

−

− + ε

−−

= + ε − ε = += = −σ

′ ′ ′ ′Σ Σ Σ

i,t i,t 1 i, t

2i,t i,t 1 i,t i,t 1

11

... ( ) ... 7o7[7 , 7 ] [7 , 7 ] ( 'or oner as, and eads)

se a RIhiteR robust estimator

: : :0st.Asy.Var[ ]= N N N

i=1 i i i=1 i i i i i=1 i i6 XZ Z3 3 Z ZX




Arellano-8ond

,irst Difference ,ormulation

−′∆ = ∆ + δ∆ + ∆ε

δ

′∆ − ∆ ′∆ ∆ − = =

∆ ′∆ −

#

i

it i,t 1 it

i,2 i,1iH

iJ i,H i,2

i

i! i,! i,!1

y y

= [ , ]

y yy

y y y, , ! G2 rows

...

y y y

3

it

i7

i8

i i

i9

xβ

a!a-"t"!% 6 β

9h" atax

xy X

x

1 *oumns




Arellano-8ond " G/S

i,t i,t 1 i,t i,t 1 i,t 1 i,t 2 i,t i

i,H i,2

i,J i,H

2 2

i,@ i,J

i,! i,! 1

y y ( ) C (y y ) ( )

2 1 ...

1 2 1 ...

o7 1 2 ...

... ... 1 ... 1...

... 1 2

− − − −

ε ε

−

− = − δ − + ε − ε

ε − ε −

÷ ε − ε − − ÷ ÷ = σ = σε − ε − ÷ ÷ − − ÷ ÷ −ε − ε

i

x x 'β

:




Arellano-8ond G/S Estimator

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

−−

−

−− −

′ ′ ′Σ Σ Σ ×

′ ′ ′Σ Σ Σ

′ ′ ′ ′ ′ ′

11

1

11 1

: N N N

i=1 i i i=1 i i i i=1 i i

N N Ni=1 i i i=1 i i i i=1 i i

6= XZ Z:Z ZX

XZ Z:Z Zy

= XZ Z:Z ZX XZ Z:Z Zy




GMM Estimator

( ) ( )

i,t i,t i,t 1 i,t


1

y C y

Ie ma+e no assum&tions about the disturban*e. 5n 4rst di<eren*es

y y ( ) C (y y ) ( )

(1) !wo stae east s%uares

:

−

− − − − −

−

= δ + ε

− = − δ − + ε − ε

′ ′Σ ΣN N

i=1 i i i=1 i i

x 'β

x x 'β

6= XZ ZZ ( ) ( ) ( ) ( )1 1

2

1: : :(2) 9orm the weihtin matri6 'or "##-N

!he *riterion 'or "## estimation is

1 1:%=N N

− − ′ ′ ′ ′Σ Σ Σ Σ ′ ′= Σ ÷

′ ′Σ Σ ÷

N N N N

i=1 i i i=1 i i i=1 i i i=1 i i

N

i=1 i i i i

N (1 N

i=1 i i i=1 i i

ZX XZ ZZ Zy

W Z3 3Z

3Z W Z3

( ) ( ) ( ) ( )

( ) ( )

11 1

11

: : :

: :0st.Asy.Var[ ]

−− −

−−

÷

′ ′ ′ ′Σ Σ Σ Σ

′ ′= Σ Σ

N N N NGMM i=1 i i i=1 i i i=1 i i i=1 i i

N N

GMM i=1 i i i=1 i i

6 = XZ W ZX XZ W Zy

6 XZ W ZX




Arellano-8ond-8o6er1s ,ormulation

Start with H0*

it it i

1

2

1

y u

5nstrumenta 7ariabes 'or &eriod t

( ) = 3 instrumenta 7ariabes( ) = 3 instrumenta 7ariabes

= L instrumenta 7ariabes (un*o

′ ′ ′ ′= + + + + ε +it 1 it 2 i 1 i 2

iit

iit

i


x1 ( x1x2 ( x2

z1

1 1 2

it it i

it

i

rreated with u)

= 3 additiona instrumenta 7ariabes. 3 L .

Let 7 u

Let [( ) ,( ) , , ]

!hen 0[ 7 ]

Ie 'ormuate this 'or the ! obser7ations in rou

≥

= ε +′ ′=

=

i

i iit it it i

it

x1

z x1 ( x1 ' x2 ( x2 ' z1 x1'

z 0

& i.





Dynamic Model

− ′ ′ ′ ′= δ + + + + ε +

′ ′ ′ ′δ

′ ′ ′ ′

′ ′ ′ ′ = =

′ ′ ′

#

i

it i,t 1 it i

i,2 i,1

i,H i,2

i,! i,!G1

y y C u

= [ , , , , ]

y y

y y,

y y i i

it 1 it 2 i 1 i 2

1 2 1 2

i2 i2 i i

i7 i7 i i

i i

i9 i9 i


a!a-"t"!% 6 β β α α '

9h" atax1 x2 z1 z2

x1 x2 z1 z2y X

x1 x2 z1

′

i, !G1 rows

1 31 32 L1 L2 *oumns

iz2





it i,t 1 it i

i,1 i,2

y y u

5nstrumenta 7ariabes 'or &eriod t as de7eo&ed abo7e

Let [y ,y ,...,( ) ,( ) , , ]

ombine EF! treatment with 88 "## estima

− ′ ′ ′ ′= δ + + + + ε +

′ ′=

it 1 it 2 i 1 i 2

i iit it it i

+x1β x2 β z1α z2 α

z x1 ( x1 ' x2 ( x2 ' z1 x1'

tor.

5nstrumenta 7ariabe *reation is based on rou& mean

de7iation rather than 4rst di<eren*es.





−′ ′= − −

=

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′

′

=

i,1 i i,t 1 i

it

i

i,1

[y y ,...,y y ,( ) ,( ) , , ]

!hen 0[ 7 ]

Ie 'ormuate this 'or the ast !G1 obser7ations in rou& i.

(y , , ) ( ,,) ( ,,) ... ( , ,)

( ,

i iit it it i

it

i2 i2 i

i

z x1 ( x1 ' x2 ( x2 ' z1 x1'

z 0

,x1 x2 z1

Z

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′

i,1 i,2

i,1 i,2 i,H

i,1 i,!G2

,) (y y , , ) ( ,,) ... ( , ,)

( , ,) ( ,,) (y y y , , ) ... ( , ,)

( , ,) ( ,,) ( ,,) ... (y ,...,y , ,

i7 i7 i

i8 i,8 i

i,;9(1< i,;9(1< i

, ,x1 x2 z1

, , ,x1 x2 z1

,x1 x2 z1

′ ′ ′

′ ′ ′ ′ ′ ′

′′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ −

− = =

−

#

i,1 i,!G1

i i

(,,)

(,,)

(,,)

)(y ,...,y )( , ,) ( ,,) ( ,,) ... ( , ,)

1;(! 1)

1;(! 1), where with

...

1;(! 1)

&,;9 (1< &,;9 (1<i i

i

i

i i

i &

i

,z1,x1

' M M M out the ast *oumn.

!hese bo*+s may *ontain a&re7ious e6oenous 7ariabes, or ae6oenous 7ariabes 'or a &eriods.

!his may *ontain the a &eriods o' data on 61 rather than ust the rou& mean. (Amemiya and

#aurdy).





For unbalanced panels the number of

columns for %i varies. Given the form of

%i, the number of columns depends on T i.

We need all %i to have the same number

of columns. For matrices with lesscolumns than the larest one, e!tra

columns of "eros are added.





2

2 2u

i i

!he *o7arian*e matri6 de4nes the mode-

= G assi*a (&ooed) reression mode (no e<e*ts)= C G /andom e<e*ts mode

= A &ositi7e de4nite !6! matri6 G "/ mode

ε

ε

σσ σ

i

i

i

: *: * ii'

:




Arellano-8ond-8o6er Estimator

( ) ( ) ( )

( ) ( ) ( )

11

1

: :

:

!wo ste& ("##) estimation

:: :(1) se = . om&ute residuas

−−

−

′ ′ ′ ′ ′Σ Σ Σ × ′ ′ ′ ′ ′Σ Σ Σ

= −

N N N

i=1 i i i i=1 i i i i i i=1 i i i

N N N


i i i i

>= X Z Z : Z Z X

X Z Z : Z Z y

: * 3 y X

( ) ( ) ( )

Ni i i 1 i i

11

1: : : !hen =N

:(2) /e*om&ute .

: : 0st.Asy.Var[ ]=

=

−−

′ ′ ′Σ

′ ′ ′ ′ ′Σ Σ Σ

i i i

N N N


>

: 3 3

>

> X Z Z : Z Z X




GMM +riterion

( )1

N

i 1 i i

2

!he "## *riterion whi*h &rodu*es this estimator is

:: :

ost estimation, use this as [89] to test the o7eridenti'yin

restri*tions. !he derees o' 'reedom

−

= ′ ′ ′ ′ ′Σ Σ

χ

N

i i i=1 i i i i i i i?= 3Z Z : Z Z3

is the tota number o'

moment *onditions (*oumns in T) minus the number o'

&arameters in .>




A::lication. Maquiladora




Maquiladora




#ide $ssue

%ow does &'t( ) *.++*- &'t/*( / .+0+*12 &'t/+( 3 a behave4

&'t( ) *.++*- &'t/*( 3 a is obviousl& e!plosive.

*.++*- .+0+*12%ow to tell5 )

*

−

A

#mallest 'possibl& comple!( root must be reater than *..




Postscri:t

!here is no theoreti*a uidan*e on theinstrument set

!here is no theoreti*a uidan*e on the 'orm o'the *o7arian*e matri6

!here is no theoreti*a uidan*e on thenumber o' as at any e7e o' the mode

!here is no theoreti*a uidan*e on the 'orm o'

the e6oeneity U and it is not testabe. /esuts 7ary widy with sma 7ariations in the

assum&tions.




Ahn and Schmidt

it i,t 1 it i

i,

i,t i,

y y C u

!here are (hue numbers o') additiona moments.

(1) 5nitia *ondition, y

0[ y ] im&ies ! more estimatin e%uations

(2) n*orreatedn

−

′ ′ ′ ′= δ + + + + ε +

′= ε

ε =

it 1 it 2 i 1 i 2

i,0 i,0

x1β x2 β z1α z2 α

x@+

is it i,t 1

i! it i,t 1

ess with di<eren*es,

0[y ( )] , t 2,..., !,s ,..., ! 2 is

!(!G1);2 *onditions

(H) (Noninear)

0[ ( )] im&ies !G2 restri*tions.

And so on.07en moderatey si$ed modes embed &otentia

−

−

ε − ε = = = −

ε ε − ε =

y

thousands o' su*h estimatin e%uations 'or usuay

7ery sma numbers (say @ or 1) &arameters.

Eow mu*h e>*ien*y *an be ainedD 5s there a *ostD

paneldatanotes-9

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