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 [ 3/57]

Part 9: GMM Estimation [ 4/57]

The NYU No Action Letter

Part 9: GMM Estimation [ 5/57]

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 [ 24/57]

Part 9: GMM Estimation [ 25/57]

Part 9: GMM Estimation [ 26/57]

Part 9: GMM Estimation [ 27/57]

Part 9: GMM Estimation [ 28/57]

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 [ 52/57]

Application: Maquiladora

Part 9: GMM Estimation [ 53/57]

Maquiladora

Part 9: GMM Estimation [ 54/57]

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?