part 9: gmm estimation [ 1/57] econometric analysis of panel data william greene department of...
TRANSCRIPT
Part 9: GMM Estimation [ 1/57]
Econometric Analysis of Panel Data
William Greene
Department of Economics
Stern School of Business
Part 9: GMM Estimation [ 2/57]
http://people.stern.nyu.edu/wgreene/CumulantInstruments-Racicot-AE(2014)_46(10).pdf
Part 9: GMM Estimation [ 6/57]
GMM Estimation for One Equation
N Ni 1 i i i 1 i i
N 2 N 2i 1 i i 1 i
i
Ni 1 i i
1 1( )= (y ) ε
N N
e1 1Asy.Var[ ( )] , estimated with
N N N N
based on 2SLS residuals e. The GMM estimator then minimizes
1 1q (y )
N
i
i i i i
i
g β z xβ z
zz zzg β
z xβ '
1N 2Ni 1 ii 1 i i
e 1(y ) .
N N Ni i
i
zzz xβ
Part 9: GMM Estimation [ 7/57]
GMM for a System of Equations
h h
w w
Simultaneous equations
Labor supply
hours = f(wage, ) =
wage = f(hours, ) =
Product market equilibrium
Quantity demanded = f(Price,...)
Price = f(market demand,
h h h
w w w
g x β
g x β
1 1
2 2
M M
...)
General format:
y =
y =
...
y =
1 1
2 2
M M
x β
x β
x β
Part 9: GMM Estimation [ 8/57]
SUR Model with Endogenous RHS Variables
1 1 1
2 2
M G
g g g
SUR System
y = , E[ | , ,... ] 0
y = ,...
...
y = ,...
Each equation has a set of L K instruments,
Each equation can be fit by 2SLS, IV, GMM, as before.
1 1 1 2 G
2 2
G G
x β x x x
x β
x β
z
Part 9: GMM Estimation [ 9/57]
GMM for the System - Notation
i1 i1
i2 i2i
iG
Index: i = 1,...,N for individuals
g = 1,...,G for equations (this would be t=1,...T for a panel)
Data matrices: G rows,
y
y,
... ...
y
i
x 0 ... 0
0 x ... 0y X
... ... ...
0 0 ...
iG
1 2 G
i
, ,
K K ... K columns
1 i1
2 i2i
G iG
i i
β
ββ= ε =
... ...
x β
y Xβ+ε
Part 9: GMM Estimation [ 10/57]
Instruments
1
i1
i2i
iG
1 2 G
i1,1 i1
i1,2 i1i1 i1
i1,L i1
, G rows (1 for each equation)...
L L ... L columns
Such that
z 0z 0
E E......0z
z 0 ... 0
0 z ... 0Z
... ... ...
0 0 ... x
z
1
i2 i2
for L instrumental variables
Same for , ...z
Part 9: GMM Estimation [ 11/57]
Moment Equations
i1 1
i2 2
iG G
i1
i2
L rows
L rowsE[ ] E , for observation i
... ...
L rows
Summing over i gives the orthogonality condition,
1 1E E
...N N
i1
i2i
iG
i1
i2N Ni=1 i i=1
z 0
z 0Zε
...
z 0
z
zZε
z
1
2
iG G
L rows
L rows
...
L rows
iG
0
0
...
0
Part 9: GMM Estimation [ 12/57]
Estimation-1
ig ig
2M Ni=1 ig,m i im=1
G Ni=1 ig,m i ig=1
y
For one equation,
ˆ ˆthe minimizer of (1/N) z (y ) ( ) ( )
Leads to 2SLS
For all equations at the same time
ˆ ˆthe minimizer of (1/N) z (y )
ig g
g g g g g
x β
β = x β g β 'g β
β = x β
2M
m=1
G
g=1( ) ( )
If the s are all different, still equation by equation 2SLS
g g g g
g
g β 'g β
β
Part 9: GMM Estimation [ 13/57]
Estimation-2
ig
1N 2i 1 igN N
g i 1 ig ig i 1 ig ig
Gg=1 g
Assuming are all uncorrelated, equation by equation GMM
e1 1 1q (y ) (y ) .
N N N N
For the system,
q = q
Cases to consider
ig igig g ig g
z zz x β ' z x β
:
(1) Coefficient vectors have elements in common or are
restricted
(2) Disturbances are correlated.
Part 9: GMM Estimation [ 14/57]
Estimation-3
1
G N N 2 Ni 1 ig ig i 1 ig i 1 ig igg 1
Ni 1 i1 i1Ni 1 i2 i2
Ni 1 iG iG
Combining GMM criteria
1 1 1 1(y ) (y )ˆ
N N N N
(y )
(y )q '
...
(y )
ig g ig ig ig g
i1 1
i2 2
iG G
z x β ' z z z x β
z x β
z x β
z x β
1N 2i 1 i1
N 2i 1 i2
N 2i 1 iG
Ni 1 i1 i1Ni 1 i2 i2
Ni 1 iG iG
ˆ
ˆ
ˆ
(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 x β
z x β
z x β
Part 9: GMM Estimation [ 15/57]
Estimation-4
Ni 1 i1 i1Ni 1 i2 i2
Ni 1 iG iG
N 2 N Ni 1 i1 i 1 i1 i2 i 1 i1 iG
Ni 1
2
If disturbances are correlated across equations,
(y )
(y )1q '
N ...
(y )
ˆ ˆ ˆ ˆ ˆ
ˆ1N
i1 1
i2 2
iG G
i1 i1 i1 i2 i1 iG
z x β
z x β
z x β
z z z z ... z z1
N 2 Ni2 i1 i 1 i2 i 1 i2 iG
N N N 2i 1 iG i1 i 1 iG i1 i 1 iG
Ni 1 i1 i1
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ
(y
1
N
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 z
z x βNi 1 i2 i2
Ni 1 iG iG
)
(y )
...
(y )
i2 2
iG G
z x β
z x β
Part 9: GMM Estimation [ 16/57]
Estimation-5
G G N Ni 1 ig ig i 1 ih ihg 1 h 1
N 2 Ni 1 i1 i 1 i1
2
If disturbances are correlated across equations,
ˆq (1/ N) (y ) (1/ N) (y )
ˆwhere = the gh block of the inverse matrix
ˆ ˆ ˆ
1N
ig g ih h
i1 i1
gh
gh
z x β W z x β
W
z z1N
i2 i 1 i1 iGN N 2 Ni 1 i2 i1 i 1 i2 i 1 i2 iG
N N N 2i 1 iG i1 i 1 iG i1 i 1 iG
ˆ ˆ
ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ
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
Part 9: GMM Estimation [ 17/57]
The Panel Data Case
it it
is the same in every equation.
The number of moment equations is T L
if each moment equation is L per period,
E[ ] 0,
If every disturbance at time t is also orthogonal
to every set of instruments i
β
z
2it is
n every other period, s,
Then
E[ ] 0, TL per period, for T periods, or T L
E.g., L=10 instruments, T=5 periods, K=5 parameters,
250 moment equations (!) for fitting 5 parameters.
z
Part 9: GMM Estimation [ 18/57]
Hausman and Taylor FE/RE Model
it it i
i i
i i
2i i i u
i i
2i i
i
y u
E[u | ] 0
E[u | ] 0
Var[u | ]
E[ | ]=0
Var[ | ]=
Cov[ ,u |
it 1 it 2 i 1 i 2
it
it
it it
it it it
it it it
it
x1 β x2 β z1 α z2 α
x1 ,z1
x2 ,z2 OLS and GLS are inconsistent
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
]=0
Var[ u | ]=
Cov[ u , u | ]=
it it
it it it
it is it it
1 ,x2 ,z1 ,z2
x1 ,x2 ,z1 ,z2
x1 ,x2 ,z1 ,z2
Part 9: GMM Estimation [ 19/57]
Useful Result: LSDV is an IV Estimator
D
D D D
D
=
1plim , so is endogenous. Correlated with because of .
NT * = x's in group mean deviations.
1 1 1 1*' *' + =
NT NT NT NT1
NT
y X D
X w
Xw 0 X w D
M X X
X w X D XM D XM X0 XM
XM
D
D
1
1 1, so plim *' plim
NT NT1 1
plim *' plim ' within groups sums of squares .NT NT
* is a valid instrument.
plim *=plim * ' *
X w XM 0
X X XM X 0
X
b X X X y=
Part 9: GMM Estimation [ 20/57]
Hausman and Taylor
it it i
it i it
y u
Deviations from group means removes all time invariant variables
y y ( ) ( )
Implication: , are consistently estimated by LSDV.
(
it 1 it 2 i 1 i 2
i iit 1 it 2
1 2
i
x1 β x2 β z1 α z2 α
x1 - x1 'β x2 - x2 'β
β β
x1 1 1
2 2
1
2
) = = K instrumental variables
( ) = = K instrumental variables
= L instrumental variables (uncorrelated with u)
= L instrumental variables (wher
it D
iit D
i
- x1 M X
x2 - x2 M X
z1
?
1 1 1 2
e do we get them?)
H&T: = ( - ) = K additional instrumental variables. Needs K L .i Dx1 I M X
Part 9: GMM Estimation [ 21/57]
H&T’s FGLS Estimator
21 2
1 1 1 2 2 2 N N N
Ni=1 i
i1 i2
i1 i2 i
i1 i2
(1) LSDV estimates of , ,
(2) ( ) (e ,e ,...,e ),(e ,e ,...,e ),...,(e ,e ,...,e )
( T observations).
T rows, repeat invariant variable *
i
β β
e* '=
z z
z zZ
z z
i
1 2
i1 i1,1
i i1 i1,ti1 i1,2
1 1
i1 i1,T
s
L L columns
T rows, repeat , time varying
L K columns
i
z x
z xz xW
z x
Part 9: GMM Estimation [ 22/57]
H&T’s FGLS Estimator (cont.)
1 2
2 2u
2 2u
(2 cont.) IV regression of on with instruments
consistently estimates and .
(3) With fixed T, residual variance in (2) estimates / T
With unbalanced panel, it estimates /T or s
i
e* Z*
W α α
2
2 2u
2 2 2i i u
omething
resembling this. (1) provided an estimate of so use the two
to obtain estimates of and . For each group, compute
ˆ 1 / ( T )ˆ ˆ ˆ
(4) Transform [ ] to
it1 it2 i1 i2x ,x ,z ,z
i
it it it i i
ˆ [ ] - [ ]
ˆ and y to y * = y - y .
i it1 it2 i1 i2 i1 i2 i1 i2W* = x ,x ,z ,z x ,x ,z ,z
Part 9: GMM Estimation [ 23/57]
H&T’s 4 STEP IV Estimator
1
2
1
1
Instrumental Variables
( ) = K instrumental variables
( ) = K instrumental variables
= L instrumental variables (uncorrelated with u)
= K additional in
i
iit
iit
i
i
V
x1 - x1
x2 - x2
z1
x1
-1
strumental variables.
Now do 2SLS of on with instruments to estimate
all parameters. I.e.,
ˆ ˆ ˆ[ , , , ]=( )1 2 1 2
y* W* V
β β α α W* W* W* y* .
Part 9: GMM Estimation [ 29/57]
Dynamic (Linear) PanelData (DPD) Models
Application Bias in Conventional Estimation Development of Consistent Estimators Efficient GMM Estimators
Part 9: GMM Estimation [ 30/57]
Dynamic Linear Model
*i,t i,t i,t 1
*i,t 1 2 i,t 3 i,t 4 i,t 5 i,t 6 i,t i,t
Balestra-Nerlove (1966), 36 States, 11 Years
Demand for Natural Gas
Structure
New Demand: G G (1 )G
Demand Function G P N N Y Y
G=gas demand
N
i,t 1 2 i,t 3 i,t 4 i,t 5 i,t 6 i,t 7 i,t 1 i i,t
= population
P = price
Y = per capita income
Reduced Form
G P N N Y Y G
Part 9: GMM Estimation [ 31/57]
A General DPD model
i,t i,t 1 i i,t
i,t i
2 2i,t i i,t i,s i
i
y y c
E[ | ,c ] 0
E[ | ,c ] , E[ | ,c ] 0 if t s.
E[c | ] g( )
No correlation across individuals
i,t
i
i i
i i
x β
X
X X
X X
Part 9: GMM Estimation [ 32/57]
OLS and GLS are inconsistent
i,t i,t 1 i i,t
i,t 1 i i,t
2c i,t 2 i i,t
2c
y y c
Cov[y ,(c )]
Cov[y ,(c )]
If T were large and -1< <1,
this would approach 1
i,tx β
Implication: Both OLS and GLS are
inconsistent.
Part 9: GMM Estimation [ 33/57]
LSDV is Inconsistent[(Steven) Nickell Bias]
i,t i i,t i i,t 1 i i,t i
2 T
i,t 1 i i,t i 2 2
y y ( ) + (y y ) ( )
(T 1) TCov[(y y ),( )]
T (1 )
Large when T is moderate or small.
Proportional bias for conventional T (5 - 15), is
on the order of 15% - 60%
x x 'β
.
Part 9: GMM Estimation [ 34/57]
Anderson Hsiao IV Estimator
i,t i,t 1 i,t i,t 1 i,t 1 i,t 2 i,t i,t 1
i,3 i,2 i,3 i,2 i,2 i,1 i,3 i,2
i1
i,4 i,3 i,4 i,3 i,3 i
Base on first differences
y y ( ) + (y y ) ( )
Instrumental variables
y y ( ) + (y y ) ( )
Can use y
y y ( ) + (y y
x x 'β
x x 'β
x x 'β
,2 i,4 i,3
i2 i,2 i,1
) ( )
Can use y or (y y )
And so on.
Levels or lagged differences?
Levels allow you to use more data
Asymptotic variance of the estimator is smaller with levels.
Part 9: GMM Estimation [ 35/57]
Arellano and Bond Estimator - 1
i,t i,t 1 i,t i,t 1 i,t 1 i,t 2 i,t i,t 1
i,3 i,2 i,3 i,2 i,2 i,1 i,3 i,2
i1
i,4 i,3 i,4 i,3 i,3 i
Base on first differences
y y ( ) + (y y ) ( )
Instrumental variables
y y ( ) + (y y ) ( )
Can use y
y y ( ) + (y y
x x 'β
x x 'β
x x 'β ,2 i,4 i,3
i,1 i2
i,5 i,4 i,5 i,4 i,4 i,3 i,5 i,4
i,1 i2 i,3
) ( )
Can use y and y
y y ( ) + (y y ) ( )
Can use y and y and y
x x 'β
Part 9: GMM Estimation [ 36/57]
Arellano and Bond Estimator - 2
i,3 i,2 i,3 i,2 i,2 i,1 i,3 i,2
i1 i,1 i,2
i,4 i,3 i,4 i,3 i,3 i,2 i,4 i,3
i,1 i2 i,1 i,2
More instrumental variables - Predetermined
y y ( ) + (y y ) ( )
Can use y and ,
y y ( ) + (y y ) ( )
Can use y , y , ,
X
x x 'β
x x
x x 'β
x x
i,3
i,5 i,4 i,5 i,4 i,4 i,3 i,5 i,4
i,1 i2 i,3 i,1 i,2 i,3 i,4
,
y y ( ) + (y y ) ( )
Can use y , y ,y , , , ,
x
x x 'β
x x x x
Part 9: GMM Estimation [ 37/57]
Arellano and Bond Estimator - 3
i,3 i,2 i,3 i,2 i,2 i,1 i,3 i,2
i1 i,1 i,2 i,T
i,4 i,3 i,4 i,3 i,3 i,2 i,4 i,
Even more instrumental variables - Strictly exogenous
y y ( ) + (y y ) ( )
Can use y and , ,..., (all periods)
y y ( ) + (y y ) (
X
x x 'β
x x x
x x 'β
3
i,1 i2 i,1 i,2 i,T
i,5 i,4 i,5 i,4 i,4 i,3 i,5 i,4
i,1 i2 i,3 i,1 i,2 i,T
)
Can use y , y , , ,...,
y y ( ) + (y y ) ( )
Can use y , y ,y , , ,...,
The number of potential instruments is huge.
These define the rows
x x x
x x 'β
x x x
of . These can be used for
simple instrumental variable estimation.iZ
Part 9: GMM Estimation [ 38/57]
Instrumental Variables
i,1 i,1 i,2
i,1 i,2 i,1 i,2 i,3
i,1 i,2 i,T 2 i,1 i,2 i,T 1
i,1
y , , 0 ... 0
0 y ,y , , , ... 0 (T rows)
... ... ... ...
0 0 ... y ,y ,..., y , , ,...
y ,
i
i
Predetermined variables
x x
x x xZ
x x x
Strictly Exogenous variables
Z
i,1 i,2 i,T 1
i,1 i,2 i,1 i,2 i,T 1
i,1 i,2 i,T 2 i,1 i,2 i,T 1
, ,... 0 ... 0
0 y ,y , , ,... ... 0 (T rows)
... ... ... ...
0 0 ... y ,y ,..., y , , ,...
x x x
x x x
x x x
Part 9: GMM Estimation [ 39/57]
Simple IV Estimation
11
1
112
N Ti 1 t 32
ˆ
This is two stage least squares.
ˆEst.Asy.Var[ ]=ˆ
[(ˆ
N N Ni=1 i i i=1 i i i=1 i i
N N Ni=1 i i i=1 i i i=1 ii i
N N Ni=1 i i i=1 i i i=1 i i
θ= X Z ZZ ZX
X Z ZZ Zy
θ X Z ZZ ZX
2i,t i,t 1 i,t i,t 1 i,t i,t 1
Ni 1 i
i,t i,t 1
ˆ ˆy y ) ( ) (y y )]
(T 2)
Note that this variance estimator understates the true asymptotic
variance because observations are autocorrelated for one period.
(y y )
x x 'β
i,t i,t 1 i,t
2i,t i,t 1 i,t i,t 1
11
... ( ) ... v
Cov[v ,v ] [v ,v ] (0 for longer lags, and leads)
Use a "White" robust estimator
ˆ ˆ ˆEst.Asy.Var[ ]= N N Ni=1 i i i=1 i i i i i=1 i iθ X Z Zv v Z ZX
Part 9: GMM Estimation [ 40/57]
Arellano/BondFirst Difference Formulation
i
it i,t 1 it
i,2 i,1i3
i4 i,3 i,2i
iT i,T i,T1
y y
= [ , ]
y yy
y y y, , T -2 rows
...y y y
K
it
i3
i4i i
iT
x β
Parameters : θ β
The data
x
xy X
x
1 columns
Part 9: GMM Estimation [ 41/57]
Arellano/Bond - GLS
i,t i,t 1 i,t i,t 1 i,t 1 i,t 2 i,t i
i,3 i,2
i,4 i,3
2 2i,5 i,4
i,T i,T 1
y y ( ) + (y y ) ( )
2 1 0 ... 0
1 2 1 ... 0
Cov 0 1 2 ... 0
... ... 1 ... 1...0 0 ... 1 2
i
x x 'β
Ω
Part 9: GMM Estimation [ 42/57]
Arellano/Bond GLS Estimator
11
1
11 1
ˆ
N N Ni=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
θ= XZ ZΩZ ZX
XZ ZΩZ Zy
= XZ ZΩZ ZX XZ ZΩZ Zy
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 + y
We make no assumptions about the disturbance. In first differences
y y ( ) + (y y ) ( )
(1) Two stage least squares
ˆ
N Ni=1 i i i=1 i i
x 'β
x x 'β
θ= XZ ZZ 1 1
2
1ˆ ˆ ˆ(2) Form the weighting matrix for GMM: N
The criterion for GMM estimation is
1 1ˆq=N N
N N N Ni=1 i i i=1 i i i=1 i i i=1 i i
Ni=1 i i i i
N -1 Ni=1 i i i=1 i i
ZX XZ ZZ Zy
W Zv vZ
vZ W Zv
11 1
11
ˆ ˆ ˆ
ˆ ˆEst.Asy.Var[ ]
N N N NGMM i=1 i i i=1 i i i=1 i i i=1 i i
N NGMM i=1 i i i=1 i i
θ = XZ W ZX XZ W Zy
θ XZ W ZX
Part 9: GMM Estimation [ 44/57]
Arellano/Bond/Bover’s FormulationStart with H&T
it it i
1
2
1
y u
Instrumental variables for period t
( ) = K instrumental variables
( ) = K instrumental variables
= L instrumental variables (unco
it 1 it 2 i 1 i 2
iit
iit
i
x1 β x2 β z1 α z2 α
x1 - x1
x2 - x2
z1
1 1 2
it it i
it
i
rrelated with u)
= K additional instrumental variables. K L .
Let v u
Let [( ) ,( ) , , ]
Then E[ v ]
We formulate this for the T observations in grou
i
i iit it it i
it
x1
z x1 - x1 ' x2 - x2 ' z1 x1'
z 0
p i.
Part 9: GMM Estimation [ 45/57]
Arellano/Bond/Bover’s FormulationDynamic Model
i
it i,t 1 it i
i,2 i,1
i,3 i,2
i,T i,T-1
y y + 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
i3 i3 i ii i
iT iT i
x1 β x2 β z1 α z2 α
Parameters : θ β β α α '
The data
x1 x2 z1 z2
x1 x2 z1 z2y X
x1 x2 z1
i, T-1 rows
1 K1 K2 L1 L2 columns
iz2
Part 9: GMM Estimation [ 46/57]
Arellano/Bond/Bover’s Formulation
it i,t 1 it i
i,1 i,2
y y u
Instrumental variables for period t as developed above
Let [y ,y ,...,( ) ,( ) , , ]
Combine H&T treatment with DPD GMM estima
it 1 it 2 i 1 i 2
i iit it it i
+x1 β x2 β z1 α z2 α
z x1 - x1 ' x2 - x2 ' z1 x1'
tor.
Instrumental variable creation is based on group mean
deviation rather than first differences.
Part 9: GMM Estimation [ 47/57]
Arellano/Bond/Bover’s Formulation
i,1 i i,t 1 i
it
i
i,1
[y y ,...,y y ,( ) ,( ) , , ]
Then E[ v ]
We formulate this for the last T-1 observations in group i.
(y , , ) (0 ,0 ,0) (0 ,0 ,0) ... (0 ,0 ,0)
(0 ,
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,3
i,1 i,T-2
0,0) (y y , , ) (0 ,0 ,0) ... (0 ,0 ,0)
(0 ,0 ,0) (0 ,0 ,0) (y y y , , ) ... (0 ,0 ,0)
(0 ,0 ,0) (0 ,0 ,0) (0 ,0 ,0) ... (y ,...,y , ,
i3 i3 i
i4 i,4 i
i,(T-1) i,(T-1) i
, ,x1 x2 z1
, , ,x1 x2 z1
,x1 x2 z1
i,1 i,T-1
i i
(0 ,0 ,0)
(0 ,0 ,0)
(0 ,0 ,0)
)(y ,...,y )(0 ,0 ,0) (0 ,0 ,0) (0 ,0 ,0) ... (0 ,0 ,0)
1/(T 1)
1/(T 1), where with
...
1/(T 1)
D,(T -1) D,(T -1)i i
i
i
i ii D
i
,z1 ,x1
H' M M M out the last column.
These blocks may contain all previous exogenous variables, or all exogenous variables for all periods.
This may contain the all periods of data on x1 rather than just the group mean. (Amemiya and MaCurdy).
Part 9: GMM Estimation [ 48/57]
Arellano/Bond/Bover’s Formulation
For unbalanced panels the number of columns for Zi varies. Given the form of Zi, the number of columns depends on Ti.
We need all Zi to have the same number of columns. For matrices with less columns than the largest one, extra columns of zeros are added.
Part 9: GMM Estimation [ 49/57]
Arellano/Bond/Bover’s Formulation
2
2 2u
i i
The covariance matrix defines the model:
= - Classical (pooled) regression model (no effects)
= + - Random effects model
= A positive definite TxT matrix - GR model
i
i
i
Ω I
Ω I ii'
Ω
Part 9: GMM Estimation [ 50/57]
Arellano/Bond/Bover Estimator
11
1
ˆ ˆ
ˆ
Two step (GMM) estimation
ˆˆ ˆ(1) Use = . Compute residuals
N N Ni=1 i i i i=1 i i i i i i=1 i i i
N N Ni=1 i i i i=1 i i i i i i=1 i i i
i i i i
δ= X H Z Z H Ω H Z Z H X
X H Z Z H Ω H Z Z H y
Ω I v y X
Ni i i 1 i i
11
1ˆ ˆ ˆ Then = N
ˆ(2) Recompute .
ˆ ˆ Est.Asy.Var[ ]=
i i i
N N Ni=1 i i i i=1 i i i i i i=1 i i i
δ
HΩH Hv vH
δ
δ X H Z Z H Ω H Z Z H X
Part 9: GMM Estimation [ 51/57]
GMM Criterion
1Ni 1 i i
2
The GMM criterion which produces this estimator is
ˆˆ ˆ
Post estimation, use this as [DF] to test the overidentifying
restrictions. The degrees of freedom
Ni i i=1 i i i i i i iq= vHZ Z H Ω H Z ZHv
is the total number of
moment conditions (columns in Z) minus the number of
parameters in .δ
Part 9: GMM Estimation [ 55/57]
Side Issue
How does y(t) = 1.220175 y(t-1) - 0.262198 y(t-2) + a behave?
y(t) = 1.220175 y(t-1) + a is obviously explosive.
1.220175 0.262198How to tell: =
1 0
A
Smallest (possibly complex) root must be greater than 1.0.
Part 9: GMM Estimation [ 56/57]
Postscript
There is no theoretical guidance on the instrument set
There is no theoretical guidance on the form of the covariance matrix
There is no theoretical guidance on the number of lags at any level of the model
There is no theoretical guidance on the form of the exogeneity – and it is not testable.
Results vary wildly with small variations in the assumptions.
Part 9: GMM Estimation [ 57/57]
Ahn and Schmidtit i,t 1 it i
i,0
i,t i,0
y y + u
There are (huge numbers of) additional moments.
(1) Initial condition, y
E[ y ] 0 implies T more estimating equations
(2) Uncorrelatedn
it 1 it 2 i 1 i 2
i,0 i,0
x1 β x2 β z1 α z2 α
x λ+
is it i,t 1
iT it i,t 1
ess with differences,
E[y ( )] 0,t 2,..., T,s 0,..., T 2 is
T(T-1)/2 conditions
(3) (Nonlinear)
E[ ( )] 0 implies T-2 restrictions.
And so on.
Even moderately sized models embed potentia
lly
thousands of such estimating equations for usually
very small numbers (say 5 or 10) parameters.
How much efficiency can be gained? Is there a cost?