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Part 9: GMM Estimation [ 1/57]
Econometric Analysis of Panel Data
William Greene
Department of Economics
Stern School of Business
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Part 9: GMM Estimation [ 2/57]
http://people.stern.nyu.edu/wgreene/CumulantInstruments-Racicot-AE(201!"#(10!.pd$
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Part 9: GMM Estimation [ 3/57]
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Part 9: GMM Estimation [ 4/57]
The NYU
No ActionLetter
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Part 9: GMM Estimation [ 6/57]
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
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Part 9: GMM Estimation [ 7/57]
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
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Part 9: GMM Estimation [ 8/57]
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
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Part 9: GMM Estimation [ 9/57]
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
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Part 9: GMM Estimation [ 10/57]
$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
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Part 9: GMM Estimation [ 11/57]
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
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Part 9: GMM Estimation [ 12/57]
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
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Part 9: GMM Estimation [ 13/57]
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.
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Part 9: GMM Estimation [ 14/57]
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
÷÷÷
÷ ÷
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Part 9: GMM Estimation [ 15/57]
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
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Part 9: GMM Estimation [ 16/57]
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
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Part 9: GMM Estimation [ 17/57]
*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
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Part 9: GMM Estimation [ 18/57]
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
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Part 9: GMM Estimation [ 19/57]
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
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Part 9: GMM Estimation [ 20/57]
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
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Part 9: GMM Estimation [ 21/57]
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
# #
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Part 9: GMM Estimation [ 22/57]
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
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Part 9: GMM Estimation [ 23/57]
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 .
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Part 9: GMM Estimation [ 24/57]
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Part 9: GMM Estimation [ 25/57]
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Part 9: GMM Estimation [ 26/57]
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Part 9: GMM Estimation [ 27/57]
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Part 9: GMM Estimation [ 28/57]
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Part 9: GMM Estimation [ 29/57]
Dynamic 2/inear4 PanelData 2DPD4 Models
A&&i*ation Kias in on7entiona 0stimation 8e7eo&ment o' onsistent 0stimators 0>*ient "## 0stimators
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Part 9: GMM Estimation [ 30/57]
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
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Part 9: GMM Estimation [ 31/57]
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
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Part 9: GMM Estimation [ 32/57]
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.
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Part 9: GMM Estimation [ 33/57]
/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
.
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Part 9: GMM Estimation [ 34/57]
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.
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Part 9: GMM Estimation [ 35/57]
Arellano and 8ond Estimator " %
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
− − − − −− = − 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
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Part 9: GMM Estimation [ 36/57]
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,@ i,J i,@ i,J i,J i,H i,@ i,J
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
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Part 9: GMM Estimation [ 37/57]
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,@ i,J i,@ i,J i,J i,H i,@ i,J
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
7/23/2019 PanelDataNotes-9
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Part 9: GMM Estimation [ 38/57]
$nstrumental aria!les
− −
′ ′ ′ ′ ′ =
′ ′ ′
′
=
i,1 i,1 i,2
i,1 i,2 i,1 i,2 i,H
i,1 i,2 i,! 2 i,1 i,2 i,! 1
i,1
y , , ...
y ,y , , , ... (! rows)
... ... ... ...
... y ,y ,..., y , , ,...
y ,
i
i
!""t"!-in" 3a!ia/"%
x x
x x xZ
x x x
St!i#t/y 4x$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
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Part 9: GMM Estimation [ 39/57]
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
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Part 9: GMM Estimation [ 40/57]
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
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Part 9: GMM Estimation [ 41/57]
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
:
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Part 9: GMM Estimation [ 42/57]
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
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Part 9: GMM Estimation [ 43/57]
GMM Estimator
( ) ( )
i,t i,t i,t 1 i,t
i,t i,t 1 i,t i,t 1 i,t 1 i,t 2 i,t i,t 1
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
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Part 9: GMM Estimation [ 44/57]
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
x1β x2 β z1 α z2 α
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.
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Part 9: GMM Estimation [ 45/57]
Arellano-8ond-8o6er1s ,ormulation
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
x1β x2 β z1 α z2 α
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
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Part 9: GMM Estimation [ 46/57]
Arellano-8ond-8o6er1s ,ormulation
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.
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Part 9: GMM Estimation [ 47/57]
Arellano-8ond-8o6er1s ,ormulation
−′ ′= − −
=
′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′
′
=
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).
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Part 9: GMM Estimation [ 48/57]
Arellano-8ond-8o6er1s ,ormulation
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.
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Part 9: GMM Estimation [ 49/57]
Arellano-8ond-8o6er1s ,ormulation
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'
:
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Part 9: GMM Estimation [ 50/57]
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=1 i i i i=1 i i i i i i=1 i i i
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
i=1 i i i i=1 i i i i i i=1 i i i
>
: 3 3
>
> X Z Z : Z Z X
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Part 9: GMM Estimation [ 51/57]
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 .>
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Part 9: GMM Estimation [ 52/57]
A::lication. Maquiladora
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Part 9: GMM Estimation [ 53/57]
Maquiladora
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Part 9: GMM Estimation [ 54/57]
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Part 9: GMM Estimation [ 55/57]
#ide $ssue
%ow does &'t( ) *.++*- &'t/*( / .+0+*12 &'t/+( 3 a behave4
&'t( ) *.++*- &'t/*( 3 a is obviousl& e!plosive.
*.++*- .+0+*12%ow to tell5 )
*
−
A
#mallest 'possibl& comple!( root must be reater than *..
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Part 9: GMM Estimation [ 56/57]
Postscri:t
!here is no theoreti*a uidan*e on theinstrument set
!here is no theoreti*a uidan*e on the 'orm o'the *o7arian*e matri6
!here is no theoreti*a uidan*e on thenumber o' as at any e7e o' the mode
!here is no theoreti*a uidan*e on the 'orm o'
the e6oeneity U and it is not testabe. /esuts 7ary widy with sma 7ariations in the
assum&tions.
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Part 9: GMM Estimation [ 57/57]
Ahn and Schmidt
it i,t 1 it i
i,
i,t i,
y y C u
!here are (hue numbers o') additiona moments.
(1) 5nitia *ondition, y
0[ y ] im&ies ! more estimatin e%uations
(2) n*orreatedn
−
′ ′ ′ ′= δ + + + + ε +
′= ε
ε =
it 1 it 2 i 1 i 2
i,0 i,0
x1β x2 β z1α z2 α
x@+
is it i,t 1
i! it i,t 1
ess with di<eren*es,
0[y ( )] , t 2,..., !,s ,..., ! 2 is
!(!G1);2 *onditions
(H) (Noninear)
0[ ( )] im&ies !G2 restri*tions.
And so on.07en moderatey si$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
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