Transcript The Hausman-Taylor Estimator, GMM Estimation

Part 8: IV and GMM Estimation [ 1/48]
Econometric Analysis of Panel Data
William Greene
Department of Economics
Stern School of Business
Part 8: IV and GMM Estimation [ 2/48]
Dear Professor Greene,
I have to apply multiplicative heteroscedastic models,
that I studied in your book, to the analysis of trade
data.
Since I have not found any Matlab implementations, I
am starting to write the method from scratch. I was
wondering if you are aware of reliable
implementations in Matlab or any other language,
which I can use as a reference.
Part 8: IV and GMM Estimation [ 3/48]
a “multi-level” modelling feature along the following lines? My data has a “two
level” hierarchical structure: I'd like to perform an ordered probit analysis such that
we allow for random effects pertaining to individuals and the organisations they
work for.
density  ordered probit = f ( y oit | x oit ,  oit ,  oi ,  o )
(Integrate  oit out directly - leads to norm al probability)
P rob(y oit  j | x oit ,  oi ,  o )  G j ( y oit | x oit ,  oi ,  o )
 G ( y oi t ,   x oit    w oi    h0 )
L ogL 

O
o 1

N
i 1
log 
T
t 1
G ( y oit ,   x oit    w oi    h 0 )
Part 8: IV and GMM Estimation [ 4/48]
L ogL 

O
o 1

log 
N
i 1
T
t 1
G ( y oit ,   x oit    w oi    h 0 )
N eed to integrate out w oi and h o .
L ogL 




O
o 1
O
o 1
N
i 1

log  
 oi
N
i 1
log
 L ogL (  ,   ,   )

o

 oi


G ( y oit ,   x oit    w oi    h 0 ) ( w oi ) dw oi 
t 1

T
G ( y oit ,   x oit    w oi    h 0 ) ( w oi ) dw oi   ( h o ) dh o
t 1

T
Part 8: IV and GMM Estimation [ 5/48]
L ogL (  ,   ,   )


O
o 1

N
i 1
log

o

 oi

G ( y oit ,   x oit    w oi    h 0 ) ( w oi ) dw oi   ( ho ) dho
t 1

T
H ow to do the integration? M onte C arlo sim ulation
L ogL (  ,   ,   )



O

O
o 1
o 1


N
i 1
N
i 1
log
log

o
1
R
 1
M


R
r 1

M
m 1
 1
M




G ( y oit ,   x oit    w oim    h 0 )   ( h o ) dh o
t 1

T
M
m 1


G ( y oit ,   x oit    w oim    h 0 r ) 
t 1

T
Part 8: IV and GMM Estimation [ 6/48]
L ogL (  ,   ,   )



O

O
o 1
o 1


N
i 1
N
i 1
log
1
R
log

1 1
R M
 1
M

R
r 1

R
r 1


M
m 1
M
m 1


G ( y oit ,   x oit    w oim    h 0 r ) 
t 1

T
T

G ( y oit ,   x oit    w oim    h 0 r ) 

t

1


(C om bine tw o sim ulations in one loop ove r tw o variables s im ulated at the sam e
tim e. h o , rm stays still w hile w oi , rm varies.)


O
o 1

N
i 1
log
1
RM

RM
rm  1

T
  t 1 G

(h 1 , w1 ), (h 1 , w 2 ), (h 1 , w 3 ), etc .

 y oit ,   x oit    

 w oi , rm
 
 h0 , rm
 
  
  
Part 8: IV and GMM Estimation [ 7/48]
----------------------------------------------------------------------------Random Coefficients OrdProbs Model
Dependent variable
HSAT
Log likelihood function
-1856.64320
Estimation based on N =
947, K = 14
Inf.Cr.AIC =
3741.3 AIC/N =
3.951
Unbalanced panel has
250 individuals
Ordered probit (normal) model
LHS variable = values 0,1,...,10
Simulation based on
200 Halton draws
--------+-------------------------------------------------------------------|
Standard
Prob.
95% Confidence
HSAT| Coefficient
Error
z
|z|>Z*
Interval
--------+-------------------------------------------------------------------|Nonrandom parameters
Constant|
3.94945***
.24610
16.05 .0000
3.46711
4.43179
AGE|
-.04201***
.00330
-12.72 .0000
-.04848
-.03553
EDUC|
.05835***
.01346
4.33 .0000
.03196
.08473
|Scale parameters for dists. of random parameters
Constant|
1.06631***
.03868
27.57 .0000
.99050
1.14213
|Standard Deviations of Random Effects
R.E.(01)|
.05759*
.03372
1.71 .0877
-.00851
.12369
|Threshold parameters for probabilities
Mu(01)|
.13522**
.05335
2.53 .0113
.03065
.23979
...
Mu(09)|
4.66195***
.11893
39.20 .0000
4.42884
4.89506
--------+--------------------------------------------------------------------
Part 8: IV and GMM Estimation [ 8/48]
Agenda

Single equation instrumental variable estimation





Exogeneity
Instrumental Variable (IV) Estimation
Two Stage Least Squares (2SLS)
Generalized Method of Moments (GMM)
Panel data




Fixed effects
Hausman and Taylor’s formulation
Application
Arellano/Bond/Bover framework
Part 8: IV and GMM Estimation [ 9/48]
Structure and Regression
E a rn in g s (s tru c tu ra l) e q u a tio n
y it  x it β   E it   it , i,t" = siblin g t in fa m ily i
E it  ' tru e ' e du ca tio n m e a su ra ble o n ly w ith e rro r
S it  m e a su re d 'sch o o lin g' = E it  w it , w = m e a su re m e n t e rro r
R e d u c e d fo rm
y it  x it β   (S it  w it )   it
= x it β   S it + (  it   w it )
E s tim a tio n p ro b le m fo r le a st squ a re s (O LS o r G LS )
2
C o v[S it , (  it   w it )]    w  0
C o n siste n cy re lie s o n th is co va ria n c e e qu a lin g 0 .
H o w to e stim a te β a n d  (co n siste n tly)?
Part 8: IV and GMM Estimation [ 10/48]
Exogeneity
S tru ctu re
y = Xβ + ε
E [ ε | X ] = g (X )  0
R e g re ssio n y = X β + g (X ) + [ ε - g (X ) ]
= E [y | X ] + u , E [ u | X ]= 0
P ro je ctio n ε
= Xθ + w
"R e g re ssio n o f y o n X "
y
= X (β + θ ) + w
T h e p ro b le m :
X is n o t e x o g e n o u s .
E x o g e n e ity:
E [  it | x it ]  0 (cu rre n t p e rio d )
S trict E x o g e n e ity: E [  it | x i1 , x i2 , ..., x iT ]  0 (a ll p e rio d s)
(W e a ssu m e n o co rre la tio n a cro ss in d ivid u a ls.)
Part 8: IV and GMM Estimation [ 11/48]
An Experimental Treatment Effect
H ealth O utcom e =
f(unobserved individu al characteristics, a
observed individual characteristics,
x
treatment (interventions),
T
random ness,
)
C ardiovascular D isease (C V D ) =  +  x +  T (H orm one R eplacem ent T herapy) + a + 
P roblem : H R T is associated w ith greater risk of cardiovascular disease.
E xperim ental evidence suggests  > 0. O b servational evidence suggests  < 0.
W hy? T = T (a). A lready healthy w om en w ith higher education and higher incom e initiated
the treatm ent to prevent heart disease. H R T users had low er C V D in spite of the bad effects
of the treatm ent T . T is endogenous in this m odel. (A pparently.)
Part 8: IV and GMM Estimation [ 12/48]
Instrumental Variables

Instrumental variable associated with changes in x, not with ε

dy/dx = β dx/dx + dε /dx = β + dε /dx. Second term is not 0.

dy/dz = β dx/dz + dε /dz. The second term is 0.

β =cov(y,z)/cov(x,z) This is the “IV estimator”

Example: Corporate earnings in year t
Earnings(t) = β R&D(t) + ε(t)
R&D(t) responds directly to Earnings(t) thus ε(t)
A likely valid instrumental variable would be R&D(t-1)
which probably does not respond to current year
shocks to earnings.
Part 8: IV and GMM Estimation [ 13/48]
Least Squares
y = Xβ + ε
E[ y | X ]  X β  E[ ε | X ]  X β
b  ( X X )
1
X y  ( X X )
1
X ( X β  ε )
1
= β  ( X X / N) ( X ε / N)
p lim b = β  p lim ( X X / N)
= β + Qγ
 β
1
p lim ( X ε / N)
Part 8: IV and GMM Estimation [ 14/48]
The IV Estimator
T h e v a ria b le s
X  [ x 1 , x 2 , ...x K -1 , x K ],
Z  [ x 1 , x 2 , ...x K -1 , z ]
T h e M o d e l A ssu m p tio n
E [  it | z it ]  0
E [ z it  it | z it ]  E[ z it ( y it  x it β ) | z it ]  0
n
(U sin g "n " to d e n ote  i= 1 T i )
E [(1 /n )  i,t z it  it | z it ]  E[( 1 /n )  i,t z it ( y it  x it β ) | z it ]  0
E [(1 /n ) Z 'y ] = E [(1 /n ) Z 'X β ]
T h e E stim a to r : M im ic th is con d ition (if p ossib le )
ˆ so (1 /n ) Z 'y = (1 /n ) Z 'X β
ˆ
F in d β
Part 8: IV and GMM Estimation [ 15/48]
A Moment Based Estimator
T h e E stim a to r
ˆ so (1 /n ) Z 'y = (1 /n ) Z 'X β
ˆ
F in d β
In stru m e n ta l V a ria b le E stim a to r
ˆ = ( Z 'X ) -1 Z 'y
β
(N ot eq u ivalen t to rep lacin g x K w ith z.)
Part 8: IV and GMM Estimation [ 16/48]
Cornwell and Rupert Data
Cornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 Years
Variables in the file are
EXP
WKS
OCC
IND
SOUTH
SMSA
MS
FEM
UNION
ED
LWAGE
=
=
=
=
=
=
=
=
=
=
=
work experience, EXPSQ = EXP2
weeks worked
occupation, 1 if blue collar,
1 if manufacturing industry
1 if resides in south
1 if resides in a city (SMSA)
1 if married
1 if female
1 if wage set by unioin contract
years of education
log of wage = dependent variable in regressions
These data were analyzed in Cornwell, C. and Rupert, P., "Efficient Estimation with Panel
Data: An Empirical Comparison of Instrumental Variable Estimators," Journal of Applied
Econometrics, 3, 1988, pp. 149-155. See Baltagi, page 122 for further analysis. The data
were downloaded from the website for Baltagi's text.
Part 8: IV and GMM Estimation [ 17/48]
Wage Equation with
Endogenous Weeks
logWage=β1+ β2 Exp + β3 ExpSq + β4OCC + β5 South + β6 SMSA + β7 WKS + ε
Weeks worked is believed to be endogenous in this equation.
We use the Marital Status dummy variable MS as an exogenous variable.
Wooldridge Condition (5.3) Cov[MS, ε] = 0 is assumed.
Auxiliary regression: For MS to be a ‘valid’ instrumental variable,
In the regression of WKS on
[1,EXP,EXPSQ,OCC,South,SMSA,MS, ]
MS significantly “explains” WKS.
A projection interpretation: In the projection
XitK =θ1 x1it + θ2 x2it + … + θK-1 xK-1,it + θK zit , θK ≠ 0.
(One normally doesn’t “check” the variables in this fashion.
Part 8: IV and GMM Estimation [ 18/48]
Auxiliary Projection
+----------------------------------------------------+
| Ordinary
least squares regression
|
| LHS=WKS
Mean
=
46.81152
|
+----------------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Constant
45.4842872
.36908158
123.236
.0000
EXP
.05354484
.03139904
1.705
.0881
19.8537815
EXPSQ
-.00169664
.00069138
-2.454
.0141
514.405042
OCC
.01294854
.16266435
.080
.9366
.51116447
SOUTH
.38537223
.17645815
2.184
.0290
.29027611
SMSA
.36777247
.17284574
2.128
.0334
.65378151
MS
.95530115
.20846241
4.583
.0000
.81440576
Part 8: IV and GMM Estimation [ 19/48]
Application: IV for WKS in Rupert
+----------------------------------------------------+
| Ordinary
least squares regression
|
| Residuals
Sum of squares
=
678.5643
|
| Fit
R-squared
=
.2349075
|
|
Adjusted R-squared
=
.2338035
|
+----------------------------------------------------+
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
Constant
6.07199231
.06252087
97.119
.0000
EXP
.04177020
.00247262
16.893
.0000
EXPSQ
-.00073626
.546183D-04
-13.480
.0000
OCC
-.27443035
.01285266
-21.352
.0000
SOUTH
-.14260124
.01394215
-10.228
.0000
SMSA
.13383636
.01358872
9.849
.0000
WKS
.00529710
.00122315
4.331
.0000
Part 8: IV and GMM Estimation [ 20/48]
Application: IV for wks in Rupert
+----------------------------------------------------+
| LHS=LWAGE
Mean
=
6.676346
|
|
Standard deviation
=
.4615122
|
| Residuals
Sum of squares
=
13853.55
|
|
Standard error of e =
1.825317
|
| Fit
R-squared
= -14.64641
|
|
Adjusted R-squared
= -14.66899
|
| Not using OLS or no constant. Rsqd & F may be < 0. |
+----------------------------------------------------+
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
Constant
-9.97734299
3.59921463
-2.772
.0056
EXP
.01833440
.01233989
1.486
.1373
EXPSQ
-.799491D-04
.00028711
-.278
.7807
OCC
-.28885529
.05816301
-4.966
.0000
SOUTH
-.26279891
.06848831
-3.837
.0001
SMSA
.03616514
.06516665
.555
.5789
WKS
.35314170
.07796292
4.530
.0000
OLS-----------------------------------------------------WKS
.00529710
.00122315
4.331
.0000
Part 8: IV and GMM Estimation [ 21/48]
Generalizing the IV Estimator-1
D efin e a p a rtition ed reg ression for n ob serv a tion s
y = X 1β 1 + X 2 β 2 + ε
K1
K 2 v a ria b les
su ch th a t p lim ( X 1 ε /N ) = 0 a n d p lim ( X 2 ε /N )  0 .
T h ere ex ists a set of M  K 2 v a ria b les W su ch th a t
p lim (1 /n ) W 'X 1  Q W 1  0
p lim (1 /n ) W 'X 2  Q W 2  0
p lim (1 /n ) W 'ε
 Q W ε = 0 , W is ex og en ou s
p lim (1 /n ) X 1 ε
 Q 1ε
 0 , X 1 is ex og en ou s
p lim (1 /n ) X 2 ε
 Q 2ε
 0 , X 2 is n ot ex og en ou s
Part 8: IV and GMM Estimation [ 22/48]
Generalizing the IV Estimator - 2
D e fin e th e se t o f in stru m e n ta l va ria ble s Z
Z1  X1
Z 2  K 2 lin e a r co m bin a tio n o f th e M W s
= WP
P = a n M xK 2 m a trix. W P is N xK 2
Z = [ Z 1 , Z 2 ]= [ X 1 , Z 2 ]= [ X 1 , W P ]
W h y m u st M be  K 2 ? S o Z 2 ca n h a ve fu ll co lu m n ra n k.
Part 8: IV and GMM Estimation [ 23/48]
Generalizing the IV Estimator
B y th e d e fin itio n s, Z is a se t o f in stru m e n ta l v a ria b le s.
ˆ = [ Z 'X ] -1 Z 'y
β
is co n siste n t a n d a sy m p to tica lly n o rm a lly d istrib u te d .
2

ˆ 
ˆ )'( y  X β
ˆ)
(y  Xβ
N o r (N -K )
A ssu m in g h o m o sce d a sticity a n d n o a u to co r e la tio n ,
2
-1
-1
ˆ]  
E st.A sy.V a r[ β
ˆ  [ Z 'X ] Z 'Z [ X 'Z ]
Part 8: IV and GMM Estimation [ 24/48]
The Best Set of Instruments
Z1  X1
Z 2  K 2 lin e a r co m b in a tio n o f th e M W s
= WP
P = a n M x K 2 m a trix . W P is N x K 2
Z = [ Z 1 , Z 2 ]= [ X 1 , Z 2 ]= [ X 1 , W P ]
W h a t is th e b e st P to u se (th e b e st w a y to
co m b in e th e e x o g e n o u s in stru m e n ts)?
(a ) If M = K 2 , it m a k e s n o d iffe re n ce .
(b ) If M < K 2 , th e re a re to o fe w in stru m e n ts to co n tin u e
(c) If M > K 2 , th e re is o n e b e st co m b in a tio n , 2 S L S .
Part 8: IV and GMM Estimation [ 25/48]
Two Stage Least Squares
A C la ss o f IV e stim a to rs by Z = [ Z 1 , Z 2 ] = [ X 1 , W ] P
2 S LS is de fin e d by
(1 ) R e gre ss ( X 1 a n d X 2 ) o n a ll o f ( X 1 a n d W ), c o lu m n by co lu m n ,
ˆ and X
ˆ ) = X
ˆ . X is re pro du ce d
a n d co m pu te pre dicte d va lu e s, ( X
1
2
1
I
ˆ
pe rfe ctly by re gre ssin g it o n itse lf so X 1 = [ X 1 , W ] P1 =[ X 1 , W ]  
0
ˆ  [ X , W ] P . Fo r 2 S LS , X
ˆ = Z (Z 'Z ) -1 Z 'X
X
2
1
2
ˆ to e stim a te β .
(2 ) R e gre ss y o n X
(D o e s it w o rk a s a n IV e stim a to r?
ˆ  is a lin e a r co m bin a tio n o f X a n d W , so ye s.)
X
2
1
Part 8: IV and GMM Estimation [ 26/48]
2SLS Estimator
B y th e d e fin itio n s, Z is a se t o f in stru m e n ta l v a ria b le s.
ˆ = [X
ˆ 'X ] -1 X
ˆ 'y
β
is co n siste n t a n d a sy m p to tica lly n o rm a lly d istrib u te d .
2

ˆ 
ˆ )'( y  X β
ˆ)
(y  Xβ
N o r (N -K )
A ssu m in g h o m o sce d a sticity a n d n o a u to co r e la t io n ,
2 ˆ
-1 ˆ ˆ
ˆ]  
ˆ ] -1
E st.A sy.V a r[ β
'X ] X
'X [ X 'X
ˆ  [X
Part 8: IV and GMM Estimation [ 27/48]
2SLS Algebra


ˆ 'X = Z (Z 'Z ) -1 Z 'X 'X
X

-1

= X ' Z (Z 'Z ) Z ' X
= X'

-1
Z (Z 'Z ) Z '

-1
Z (Z 'Z ) Z '

X
ˆ 'X
ˆ
= X
T h e re fo re
ˆ 'X ] -1 X
ˆ 'X
ˆ [ X 'X
ˆ ] -1  [ X
ˆ 'X
ˆ ]1
[X
ˆ]  
ˆ 'X
ˆ] ,
E st.A sy.V a r[β
ˆ  [X
ˆ 
2
-1
2
ˆ )'( y  X β
ˆ)
(y  Xβ
n o r (n -K )
Part 8: IV and GMM Estimation [ 28/48]
A General Result for IV

We defined a class of IV estimators by the set of
variables
Z1  X1
Z 2  K 2 lin e a r co m b in a tio n o f th e M W s
= WP
P = a n M x K 2 m a trix . W P is N x K 2
Z = [ Z 1 , Z 2 ]= [ X 1 , Z 2 ]= [ X 1 , W P ]

The minimum variance (most efficient) member in this
class is 2SLS (Brundy and Jorgenson(1971))
(rediscovered JW, 2000, p. 96-97)
Part 8: IV and GMM Estimation [ 29/48]
GMM Estimation –
Orthogonality Conditions
G e n e ra l M o d e l F o rm u la tio n :
y = X β + ε ; p lim [(1 /n ) X 'ε ]  0 (p o ssib ly ) K re g re sso rs in X
M  K In stru m e n ta l v a ria b le s Z ; p lim [(1 /n ) Z 'ε ] = 0 .
IV fo rm u la tio n im p lie s M o rth o g o n a lity c o n d itio n s
E [z m ( y  x 'β )]  0 .
2S L S o n ly K o f th e se in th e fo rm
ˆ m ( y  x 'β )]  0 w h e re
E [x
M
ˆ
x m =  l= 1  lm z m
ˆ = (X
ˆ 'X
ˆ ) -1 X
ˆ 'y
S o lu tio n is β
C o n sid e r a n e stim a to r th a t u se s a ll M e q u a tio n s w h e n M > K
T h e o rth o g o n a lity co n d itio n to m im ic is
n
E [ (1 /n )  i= 1 z im ( y i  x iβ )]= 0 , m = 1 ,...,M
T h is is M e q u a tio n s in K u n k n o w n s e a ch o f th e fo rm E [ g m ( β )]= 0 .
Part 8: IV and GMM Estimation [ 30/48]
GMM Estimation - 1
n
E [(1 /n )  i= 1 z im ( y i  x iβ )]= 0 , m = 1 ,...,M
E [g m ( β )]= 0 , m = 1 ,...,M .
S a m p le cou n te rp a rts - fin d in g th e e stim a tor:
N
ˆ )= 0
(1 /n )  i= 1 z im ( y i  x iβ
(a ) If M = K , th e e x a ct solu tion is 2 S LS
(b ) If M < K , th e re a re too f e w e q u a tion s. N o solu tion .
(c) If M > K , th e re a re e x ce ss e q u a tion s . H ow to re con cile th e m ?
F irst P a ss: "Le a st S q u a re s"
ˆ :
T ry M in im izin g w rt β

M
m=1

ˆ)
(1 /n )  i= 1 z im ( y i  x iβ
n

2
 g ( β ) 'g ( β )
Part 8: IV and GMM Estimation [ 31/48]
GMM Estimation - 2
ˆ = th e m in im ize r of
β

g ( β ) 'g ( β ) = g ( β ) '  I 
-1
M
m=1

ˆ)
(1 /n )  i= 1 z im ( y i  x iβ
n

2
 g ( β ) 'g ( β )
g(β )
D e fin e s a "M in im u m D ista n ce E stim a tor" w ith w e ig h t m a trix = I.
M ore g e n e ra lly : L e t
ˆ = th e m in im ize r of g ( β ) ' A g ( β )
β
R e su lts: F or a n y p ositiv e d e fin ite m a tr ix A ,
ˆ is con siste n t a n d a sy m p totica lly n orm a lly d istrib u te d .
β



)
β
(
g

)
β
(
g







ˆ ]=
A
)]
β
(
g
r[
a
sy.V
A
A
A sy .V a r[ β






'
β

'
β




 

(S e e JW , C h . 1 4 for a n a ly sis of a sy m p totic p rop e rtie s.)
Part 8: IV and GMM Estimation [ 32/48]
IV Estimation
ˆ = th e m in im izer of g ( β ) ' A g ( β )
β
R esu lts: F or an y p ositive d efin ite m atr ix A ,
ˆ is con sisten t.
β




g
(
β
)

g
(
β
)






ˆ ]=
A sy.V ar[ β
A
A
sy.V
ar[
g
(
β
)]
A







β
'

β
'



 

-1
F or IV estim ation , g ( β ) = (1 /n ) Z '( y - X β )
 g(β ) 

   (1 / n) Z 'X
 β ' 
2
n
2
A sy.V ar[ g ( β )]  (1 / n)  i  1   z i z i
a ssu m in g h o m o sce d a sticity a n d n o a u to co r re la tio n .
Part 8: IV and GMM Estimation [ 33/48]
An Optimal Weighting Matrix
ˆ = th e m in im ize r o f g ( β ) ' A g ( β )
β
Fo r a n y p o sitiv e d e fin ite m a trix A ,
ˆ is c o n siste n t a n d a sy m p to tic a lly n o rm a lly d istrib u te d .
β








g
(
β
)

g
(
β
)


ˆ ]=
A sy .V a r[ β
A
A
sy.V
a
r[
g
(
β
)]
A







β
'

β
'



 

1
Is th e re a 'b e st' A m a trix ? T h e m o st e ffic ie n t e stim a to r in th e G M M c la ss
has A =
 A sy.V a r[ g ( β )] 
1
. A  A sy.V a r[ g ( β )]  A =  A sy.V a r[ g ( β )] 
1
ˆ
β
=
th
e
m
in
im
ize
r
o
f
g
(
β
)
'
A
sy.V
a
r[
g
(
β
)]
g( β )


GMM
1
Part 8: IV and GMM Estimation [ 34/48]
The GMM Estimator
1
ˆ
β
=
th
e
m
in
im
ize
r
of
q
=
g
(
β
)
'
A
sy.V
a
r[
g
(
β
)]
g( β )


GMM



1   g(β )  
  g(β ) 
ˆ
A sy.V a r[ β G M M ]=  
  A sy.V a r[ g ( β )]  


β
'

β
'






F or IV e stim a tion , g ( β ) = (1 /n ) Z '( y - X β ),
2
N
2
2
1
 g(β ) 

   (1 / N) Z 'X
 β ' 
2
A sy.V a r[ g ( β )]  (1 / n)  i  1   z i z i =(   / n ) Z 'Z

2
2
-1
ˆ
A sy.V a r[ β
]= (  (1 / n) X 'Z )[(   / n ) Z 'Z ] (  (1 / n) Z 'X )
GMM

1
2

-1
=   X 'Z [ Z 'Z ] Z 'X )

1
!!!! !
IM P L IC A T IO N : 2 S L S is n o t ju st e fficie n t fo r IV e stim a to rs th a t u se a lin e a r
c o m b in a tio n o f th e co lu m n s o f Z . It is e ffici e n t a m o n g a ll e stim a to rs th a t u se
th e co lu m n s o f Z .
Part 8: IV and GMM Estimation [ 35/48]
GMM Estimation
g ( β )=
1
N
N
 i  1 z i ( y i  x iβ ) 
1
N
N
 i1 z i ε i
A ssu m in g h o m o sce d a sticity a n d n o a u to co r re la tio n 2 S L S
is th e e fficie n t G M M e stim a to r. W h a t if th e re is h e te ro sce d a sticity ?
1 1 n 2
 1 N 2




A sy.V a r[ g ( β )]   2  i  1  i z i z i  , e stim a te d w ith    i  1 e i z i z i  
n


nn
b a se d o n 2 S L S re sid u a ls e i . T h e G M M e stim a to r m in im ize s
1 n
 1
q    i  1 z i ( y i  x iβ )  ' 
n
 n
1 n 2

  i  1 e i z i z i  
n

1
1 n



z
(
y

x
β
)
i1 i
i
i

.
n

-1
T h is is n o t 2 S L S b e ca u se th e w e ig h tin g m a trix is n o t ( Z 'Z ) .
Part 8: IV and GMM Estimation [ 36/48]
Application - GMM
NAMELIST
NAMELIST
2SLS
NLSQ
; x = one,exp,expsq,occ,south,smsa,wks$
; z = one,exp,expsq,occ,south,smsa,ms,union,ed$
; lhs = lwage ; RHS = X ; INST = Z $
; fcn = lwage-b1'x
; labels = b1,b2,b3,b4,b5,b6,b7
; start = b
; inst = Z
; pds = 0$
Part 8: IV and GMM Estimation [ 37/48]
Application - 2SLS
Part 8: IV and GMM Estimation [ 38/48]
GMM Estimates
Part 8: IV and GMM Estimation [ 39/48]
2SLS
GMM with Heteroscedasticity
Part 8: IV and GMM Estimation [ 40/48]
Testing the Overidentifying Restrictions
q = g ( β ) '  A sy.V a r[ g ( β )] 
1
g( β )
U n de r th e h ypo th e sis th a t E [ g ( β )] = 0 ,
d
2
q    [M  K ]
M = n u m be r o f m o m e n t e qu a tio n s
K = n u m be r o f pa ra m e te rs e stim a te d
(In o u r e x a m ple , M = 9 , K = 7 .)
M - K = n u m be r o f 'e x tr a ' m o m e n t e qu a tio n s. M o re th a n a re
n e e de d to ide n tify th e pa ra m e te rs.
Fo r th e e x a m ple ,
| Value of the GMM criterion :
| e(b)tZ inv(ZtWZ) Zte(b) =
|
537.3916 |
Part 8: IV and GMM Estimation [ 41/48]
Inference About the Parameters

N
2
-1
ˆ
β
 (X 'Z )[  i= 1ˆ i z i z i ] (Z 'X )
GMM

1
  (X 'Z )[ 
N
i= 1
N
2
-1
ˆ

E st.A sy.V a r[ β
]=
(X
'Z
)
[


z
z
]
(Z 'X )
ˆ
GMM
i= 1 i
i i

2
-1
ˆ i z i z i ] (Z 'y )

1
R e strictio n s ca n be te ste d u sin g W a ld sta tistics;
H 0 : r ( β )= h
H 1 :N o t H 0
W a ld 
R 

ˆ
r (β
)- h
GMM

ˆ
 r (β
)- h
GMM

ˆ
'  R  E st.A sy.V a r[ β
]  R 
GMM


1
 r ( βˆ
GMM
)- h


ˆ
β
GMM
E .g., fo r a sim ple te st, H 0 :  k = 0 , th is is th e squ a re o f th e t-ra tio .
Part 8: IV and GMM Estimation [ 42/48]
Specification Test Based on the Criterion
C o n sid e r a n u ll h yp o th e sis H 0 th a t im p o se s re strictio n s o n a n
a lte rn a tive h yp o th e sis H 1 ,
U n d e r th e n u ll h yp o th e sis th a t E [ g ( β 0 )] = 0 ,
q 0 = g ( β 0 ) '  A sy.V a r[ g ( β 0 )] 
1
d
2
g ( β 0 )    [M  K 0 ]
U n d e r th e a lte rn a tive h yp o th e sis, H 1
q 1 = g ( β 1 ) '  A sy.V a r[ g ( β 1 )] 
d
1
d
2
g ( β 1 )    [M  K 1 ]
2
U n d e r th e n u ll, q 0  q 1    [K 1  K 0 ]
R e strictio n s ca n b e te ste d u sin g th e cri te rio n fu n ctio n s
sta tistic, q 0  q 1 .
(W e ig h tin g m a trix m u st b e th e sa m e fo r H 0 a n d H 1 . U se
th e u n re stricte d w e ig h tin g m a trix .)
Part 8: IV and GMM Estimation [ 43/48]
Extending the Form of the GMM
Estimator to Nonlinear Models
V e ry little ch a n ge s if th e re gre ssio n fu n ctio n is n o n lin e a r.
1 1 N 2
 1 N 2


A sy.V a r[ g ( β )]   2  i 1  i z i z i  , e stim a te d w ith    i 1 e i z i z i  
N


N N
ba se d o n n o n lin e a r 2 S LS re sidu a ls e i . T h e G M M e stim a to r m in im ize s
1 N
 1

q    i 1 z i ( y i  f ( x iβ ))  ' 
N
 N
1 N 2


  i 1 e i z i z i  
N

T h e pro ble m is e sse n tia lly th e sa m e .
1
1 N



z
(
y

f
(
x
β
))
i 1 i
i
i

.
N

Part 8: IV and GMM Estimation [ 44/48]
A Nonlinear Conditional Mean
f ( x iβ )  e x p( x β )
E[ z i ( y i  e x p( x iβ ))]  0
N o n lin e a r in stru m e n ta l va ria ble s (2 S LS ) m in im ize s

N
i= 1


1
N
( y i  e x p( x iβ )) z i '[ Z 'Z ]
 i= 1 z i ( y i  e x p( x iβ ))

N o n lin e a r G M M th e n m in im ize s
1 N

2
N
2
1


(
y

e
x
p(
x
β
))
z
'[(1 / N)  i 1ˆ i z i z i ]
i= 1
i
i
i 

N

1 N
 ˆ 1
   i= 1 ( y i  e x p( x iβ )) z i  ' W
N

1 N



z
(
y

e
x
p(
x
β
))
i= 1 i
i
i


N


1 N



z
(
y

e
x
p(
x
β
)
)
i= 1 i
i
i


N

Part 8: IV and GMM Estimation [ 45/48]
Nonlinear Regression/GMM
NAMELIST
; x = one,exp,expsq,occ,south,smsa,wks$
NAMELIST
; z = one,exp,expsq,occ,south,smsa,ms,union,ed$
? Get initial values to use for optimal weighting matrix
NLSQ
; lhs = lwage ; fcn=exp(b1'x) ; inst = z
; labels=b1,b2,b3,b4,b5,b6,b7 ; start=7_0$
? GMM using previous estimates to compute weighting matrix
NLSQ (GMM)
; fcn = lwage-exp(b1'x) ; inst = Z
; labels = b1,b2,b3,b4,b5,b6,b7 ; start = b
; pds = 0 $ (Means use White style estimator)
Part 8: IV and GMM Estimation [ 46/48]
Nonlinear Wage Equation Estimates
NLSQ Initial Values
Part 8: IV and GMM Estimation [ 47/48]
Nonlinear Wage Equation Estimates
2nd Step GMM
Part 8: IV and GMM Estimation [ 48/48]
IV for Panel Data
Fixed E ffects
b 2 sls ,lsdv    

N
i 1
X i M D Z i   
N
i 1
Z i M D Z i 
1

N
i 1
Z i M D X i  

1

1
N
N
  N X M Z




Z
M
Z

Z
M
y






i 1
i
D
i
i 1
i
D i 
 i  1 i D i

R andom E ffects




N
ˆ 1Z
b 2 sls , R E    i 1 X i 
i
i

  N X 
ˆ 1Z
 i  1 i i i
N
ˆ 1Z
 i 1 Z i 
i
i
N
i 1
ˆ Z
Z i 
i
1
i
 
1
 
1

N
ˆ 1 X 
 i 1 Z i 
i
i

N
i 1

ˆ 1y 
Z i 
i
i

1
