### Linear Regression with Panel Data

```Topics in Microeconometrics
William Greene
Department of Economics
Part 3: Basic Linear Panel Data Models
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
BLK
LWAGE
=
=
=
=
=
=
=
=
=
=
=
=
work experience
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 union contract
years of education
1 if individual is black
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
Balanced and Unbalanced Panels
•
•
Distinction: Balanced vs. Unbalanced Panels
A notation to help with mechanics
zi,t, i = 1,…,N; t = 1,…,Ti
•
The role of the assumption
•
Mathematical and notational convenience:


•
•
Balanced, n=NT
N
Unbalanced: n   i= 1 Ti
Is the fixed Ti assumption ever necessary? Almost
never.
Is unbalancedness due to nonrandom attrition
from an otherwise balanced panel? This would
require special considerations.
Application: Health Care Usage
German Health Care Usage Data, 7,293 Individuals, Varying Numbers of Periods
This is an unbalanced panel with 7,293 individuals. There are altogether 27,326 observations. The number of
observations ranges from 1 to 7.
(Frequencies are: 1=1525, 2=2158, 3=825, 4=926, 5=1051, 6=1000, 7=987).
Variables in the file are
DOCTOR = 1(Number of doctor visits > 0)
HOSPITAL = 1(Number of hospital visits > 0)
HSAT
= health satisfaction, coded 0 (low) - 10 (high)
DOCVIS
= number of doctor visits in last three months
HOSPVIS = number of hospital visits in last calendar year
PUBLIC
= insured in public health insurance = 1; otherwise = 0
= insured by add-on insurance = 1; otherswise = 0
HHNINC = household nominal monthly net income in German marks / 10000.
(4 observations with income=0 were dropped)
HHKIDS
= children under age 16 in the household = 1; otherwise = 0
EDUC
= years of schooling
AGE
= age in years
MARRIED = marital status
An Unbalanced Panel: RWM’s
GSOEP Data on Health Care
N = 7,293 Households
Fixed and Random Effects
•
Unobserved individual effects in regression: E[yit | xit, ci]
Notation: y it = x it  + c i +  it
 x i1 


x i2

Xi  





x
 iTi 
•
Ti ro w s, K co lu m n s
Linear specification:
Fixed Effects: E[ci | Xi ] = g(Xi). Cov[xit,ci] ≠0
effects are correlated with included variables.
Random Effects: E[ci | Xi ] = 0. Cov[xit,ci] = 0
Convenient Notation
•
Fixed Effects – the ‘dummy variable model’
y it =  i + x it  +  it
Individual specific constant terms.
•
Random Effects – the ‘error components model’
y it = x it  +  it + u i
Compound (“composed”) disturbance
Estimating β
•
β is the partial effect of interest
•
Can it be estimated (consistently) in
the presence of (unmeasured) ci?
•
•
Does pooled least squares “work?”
Strategies for “controlling for ci” using
the sample data
The Pooled Regression
•
Presence of omitted effects
y it = x it β + c i + ε it , o bse rva tio n fo r pe rso n i a t tim e t
y i = X iβ + c ii+ ε i , Ti o bse rva tio n s in gro u p i
= X iβ + c i + ε i , n o te c i  (c i , c i ,...,c i ) 
N
y = X β + c + ε ,  i= 1 Ti o bse rva tio n s in th e sa m ple
•
Potential bias/inconsistency of OLS – depends
on ‘fixed’ or ‘random’
OLS with Individual Effects
-1
b = ( X X ) X 'y
N
= β +  (1 /N )Σ i= 1 X i X i 


N
+  (1 /N )Σ i= 1 X i X i 
-1
-1
 (1 /N )Σ Ni= 1 X i c i  (p a rt d u e to th e o m itte d c i )


 (1 /N )Σ Ni= 1 X i ε i  (c o v a ria n c e o f X a n d ε w ill = 0 )


T h e th ird te rm v a n ish e s a sy m p to ti c a lly b y a ssu m p tio n
1 N

p lim b = β + p lim  Σ i= 1 X i X i 
N

So, w hat becom es of
-1
 N Ti

Σ
x
c
(le ft o u t v a ria b le fo rm u la )
i i 
 i= 1
N


 Σ Ni= 1 w i x i c i  ?


p lim b = β if th e c o v a ria n c e o f x i a n d c i c o n v e rg e s to ze ro .
Estimating the Sampling Variance of b
•
s2(X ́X)-1? Inappropriate because
•
•
•
Correlation across observations
(Possibly) Heteroscedasticity
A ‘robust’ covariance matrix
•
•
•
Robust estimation (in general)
The White estimator
A Robust estimator for OLS.
Cluster Estimator
R o bu st va ria n ce e stim a to r fo r V a r[ b ]
E st.V a r[ b ]
= ( X 'X )
1
Ti
Ti
1
  Ni= 1 (  t=


x it ˆ
v it )(  t= 1 x it ˆ
v it ) ( X 'X )
1


= ( X 'X )
1
1
  N  Ti  Ti ˆ


ˆ
v v x it x is ( X 'X )
i= 1
t= 1 s= 1 it is




ˆ
v it  a le a st squ a re s re sidu a l =  it  c i
(If T i = 1 , th is is th e W h ite e stim a to r.)
Application: Cornwell and Rupert
Bootstrap variance for a
panel data estimator
•
Panel Bootstrap =
Block Bootstrap
•
Data set is N groups
of size Ti
•
Bootstrap sample is N
groups of size Ti
drawn with
replacement.
Difference in Differences
With two periods,
 y it = y i2 -y i1 =  0 + ( x i2 - x i1 ) β + u i
C o n sid e r a "tre a tm e n t, D i ," th a t ta k e s p la c e b e tw e e n
tim e 1 a n d tim e 2 fo r so m e o f th e in d iv id u a ls
 y i =  0 + (  x i )  β +  1D i + u i
D i = th e "tre a tm e n t d u m m y "
This is a linear regression model. If there are no regressors,
ˆ
 1   y | tre a tm e n t -  y | co n tro l
= "diffe re n ce in diffe re n ce s" e stim a to r.
ˆ
 0  A ve ra ge ch a n ge in y i fo r th e "tre a te d"
Difference-in-Differences Model
With two periods and strict exogeneity of D and T,
y it =  0   1D it   2 T t   3 T t D it   it
D it = d u m m y v a ria b le fo r a tre a tm e n t th a t ta k e s p la c e
b e tw e e n tim e 1 a n d tim e 2 fo r so m e o f th e in d iv id u a ls,
T t = a tim e p e rio d d u m m y v a ria b le , 0 in p e rio d 1 ,
1 in p e rio d 2 .
This is a linear regression model. If there are no regressors,
U sing least squares,
b 3  ( y 2  y 1 ) D 1  ( y 2  y 1 ) D  0
Difference in Differences
y it =  0   1D it   2 Tt   3 D it Tt  β x it   it , t  1, 2
 y it =  2   3 D i 2   ( β x it )    it
=  2   3 D i 2  β (  x it )  u i
  y it | D
 1     y it | D  0 
  3  β   (  x it | D  1)  (  x it | D  0 ) 
If th e sa m e in d iv id u a l is o b se rv e d in b o th sta te s ,
th e se co n d te rm is ze ro . If th e e ffe ct is e stim a te d b y
a v e ra g in g in d iv id u a ls w ith D = 1 a n d d iffe re n t in d iv id u a ls
w ith D = 0 , th e n p a rt o f th e 'e ffe ct' is e x p la in e d b y ch a n g e
in th e co v a ria te s, n o t th e tre a tm e n t.
The Fixed Effects Model
yi = Xi + diαi + εi, for each individual
 y1

y
 2


 yN
  X1
 
X
   2
 
 
  X N
d1
0
0
0
d2
0
0
0
0
0 

0 β 

 ε
α

d N 
β 
= [X , D ]    ε
α
= Zδ  ε
E[ci | Xi ] = g(Xi); Effects are correlated with included variables.
Cov[xit,ci] ≠0
Estimating the Fixed Effects Model
•
•
The FEM is a plain vanilla regression model but
with many independent variables
Least squares is unbiased, consistent, efficient,
but inconvenient if N is large.
 b   X X
   
 a   D X
X D 

D D 
1
 X y 



D
y


U sin g th e Frisc h -W a u g h th e o re m
b
= [ X M D X ]
1
 X M D y 
The Within Groups Transformation
Removes the Effects
y it  x it β  c i + ε it
y i  x iβ  c i + ε i
y it  y i  ( x it - x i )  β  (ε it  ε i )
U se le a st squ a re s to e stim a te β .
Least Squares Dummy Variable Estimator
•
b is obtained by ‘within’ groups least
squares (group mean deviations)
•
a is estimated using the normal equations:
D’Xb+D’Da=D’y
a = (D’D)-1D’(y – Xb)
T
a i = (1 /T i )Σ t=i 1 (y it - x it b )= e i
Application Cornwell and Rupert
LSDV Results
Note huge changes in
the coefficients. SMSA
and MS change signs.
Significance changes
completely!
Pooled OLS
The Effect of the Effects
A Caution About Stata and R2
R squared = 1 -
R esidual S um of S quares
T otal S um of S quares
For the FE model above,
O r is it? W hat is the total sum of squa res?
C onventional: T otal S um of S quares =

i 1
"W ithin S um of S quares"

i 1
=
N
N
W hich should appear in the denom inator o f R

Ti

Ti
t 1
t 1
 y it
 y
2
R2 = 0.90542
 y it
 yi 
R2 = 0.65142
2
2
The coefficient estimates and standard errors are the same. The calculation of the R 2 is different. In the
areg procedure, you are estimating coefficients for each of your covariates plus each dummy variable
for your groups. In the xtreg, fe procedure the R2 reported is obtained by only fitting a mean deviated
model where the effects of the groups (all of the dummy variables) are assumed to be fixed quantities.
So, all of the effects for the groups are simply subtracted out of the model and no attempt is made to
quantify their overall effect on the fit of the model.
Since the SSE is the same, the R2=1−SSE/SST is very different. The difference is real in that we are
making different assumptions with the two approaches. In the xtreg, fe approach, the effects of the
groups are fixed and unestimated quantities are subtracted out of the model before the fit is
performed. In the areg approach, the group effects are estimated and affect the total sum of squares of
the model under consideration.
Examining the Effects with a KDE
Fixed Effects from Cornwell and Rupert Wage Model
.3 4 5
.2 7 6
D e n s i ty
.2 0 7
.1 3 8
.0 6 9
Fixed E ffects fr om C or nw ell and R uper t W age M odel
.0 0 0
0
1
2
3
4
5
6
7
AI
AI
Fre que nc y
Ke rn e l d e n s i ty e s ti m a te fo r
Mean = 4.819,
Standard deviation = 1.054.
.8 5 6
1 .6 8 8
2 .5 2 0
3 .3 5 1
4 .1 8 3
AI
5 .0 1 5
5 .8 4 7
6 .6 7 8
Robust Covariance Matrix for LSDV
Cluster Estimator for Within Estimator
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
|OCC
|
-.02021
.01374007
-1.471
.1412
.5111645|
|SMSA
|
-.04251**
.01950085
-2.180
.0293
.6537815|
|MS
|
-.02946
.01913652
-1.540
.1236
.8144058|
|EXP
|
.09666***
.00119162
81.114
.0000
19.853782|
+--------+------------------------------------------------------------+
+---------------------------------------------------------------------+
| Covariance matrix for the model is adjusted for data clustering.
|
| Sample of
4165 observations contained
595 clusters defined by |
|
7 observations (fixed number) in each cluster.
|
+---------------------------------------------------------------------+
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
|DOCC
|
-.02021
.01982162
-1.020
.3078
.00000|
|DSMSA
|
-.04251
.03091685
-1.375
.1692
.00000|
|DMS
|
-.02946
.02635035
-1.118
.2635
.00000|
|DEXP
|
.09666***
.00176599
54.732
.0000
.00000|
+--------+------------------------------------------------------------+
Time Invariant Regressors
•
Time invariant xit is defined as
invariant for all i. E.g., sex dummy
variable, FEM and ED (education in
the Cornwell/Rupert data).
•
If xit,k is invariant for all t, then the
group mean deviations are all 0.
FE With Time Invariant Variables
+----------------------------------------------------+
| There are 3 vars. with no within group variation. |
| FEM
ED
BLK
|
+----------------------------------------------------+
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
EXP
|
.09671227
.00119137
81.177
.0000
19.8537815
WKS
|
.00118483
.00060357
1.963
.0496
46.8115246
OCC
|
-.02145609
.01375327
-1.560
.1187
.51116447
SMSA
|
-.04454343
.01946544
-2.288
.0221
.65378151
FEM
|
.000000
......(Fixed Parameter).......
ED
|
.000000
......(Fixed Parameter).......
BLK
|
.000000
......(Fixed Parameter).......
+--------------------------------------------------------------------+
|
Test Statistics for the Classical Model
|
+--------------------------------------------------------------------+
|
Model
Log-Likelihood
Sum of Squares R-squared |
|(1) Constant term only
-2688.80597
886.90494
.00000 |
|(2) Group effects only
27.58464
240.65119
.72866 |
|(3) X - variables only
-1688.12010
548.51596
.38154 |
|(4) X and group effects
2223.20087
83.85013
.90546 |
+--------------------------------------------------------------------+
Drop The Time Invariant Variables
Same Results
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
EXP
|
.09671227
.00119087
81.211
.0000
19.8537815
WKS
|
.00118483
.00060332
1.964
.0495
46.8115246
OCC
|
-.02145609
.01374749
-1.561
.1186
.51116447
SMSA
|
-.04454343
.01945725
-2.289
.0221
.65378151
+--------------------------------------------------------------------+
|
Test Statistics for the Classical Model
|
+--------------------------------------------------------------------+
|
Model
Log-Likelihood
Sum of Squares R-squared |
|(1) Constant term only
-2688.80597
886.90494
.00000 |
|(2) Group effects only
27.58464
240.65119
.72866 |
|(3) X - variables only
-1688.12010
548.51596
.38154 |
|(4) X and group effects
2223.20087
83.85013
.90546 |
+--------------------------------------------------------------------+
No change in the sum of squared residuals
Fixed Effects Vector Decomposition
Efficient Estimation of Time
Invariant and Rarely Changing
Variables in Finite Sample Panel
Analyses with Unit Fixed Effects
Thomas Plümper and Vera Troeger
Political Analysis, 2007
Introduction
[T]he FE model … does not allow the estimation of
time invariant variables. A second drawback of
the FE model … results from its inefficiency in
estimating the effect of variables that have very
little within variance.
problems of estimating time invariant and rarely
changing variables in FE models with unit
effects
The Model
y it = α i +

K
β
x
+
k
k
it
k =1

M
γ
z
+
ε
m
m
i
it
m =1
w h e re α i d e n o te th e N u n it e ffe cts.
Fixed Effects Vector Decomposition
Step 1: Compute the fixed effects regression to
get the “estimated unit effects.” “We run this
FE model with the sole intention to obtain
estimates of the unit effects, αi.”
FE
αˆ i = y i -  k =1 b k x k i
K
Step 2
Regress ai on zi and compute residuals
ai =

M
m =1
γ m z im + h i
h i is o rth o g o n a l to z i (sin ce it is a re sid u a l)
V e cto r h i is e xp a n d e d so e a ch e le m e n t
h i is re p lica te d Ti tim e s - h is th e le n g th o f
th e fu ll sa m p le .
Step 3
Regress yit on a constant, X, Z and h using
ordinary least squares to estimate α, β, γ, δ.
y it = α +

K
k =1
β k x k it +

M
m =1
γ m z m i + δ h i + ε it
N otice that  i in the original m odel has
becom e  +  h i in the revised m odel.
Step 1 (Based on full sample)
These 3 variables have no within group variation.
FEM
ED
BLK
F.E. estimates are based on a generalized inverse.
--------+--------------------------------------------------------|
Standard
Prob.
Mean
LWAGE| Coefficient
Error
z
z>|Z|
of X
--------+--------------------------------------------------------EXP|
.09663***
.00119
81.13 .0000
19.8538
WKS|
.00114*
.00060
1.88 .0600
46.8115
OCC|
-.02496*
.01390
-1.80 .0724
.51116
IND|
.02042
.01558
1.31 .1899
.39544
SOUTH|
-.00091
.03457
-.03 .9791
.29028
SMSA|
-.04581**
.01955
-2.34 .0191
.65378
UNION|
.03411**
.01505
2.27 .0234
.36399
FEM|
.000
.....(Fixed Parameter).....
.11261
ED|
.000
.....(Fixed Parameter).....
12.8454
BLK|
.000
.....(Fixed Parameter).....
.07227
--------+---------------------------------------------------------
Step 2 (Based on 595 observations)
--------+--------------------------------------------------------|
Standard
Prob.
Mean
UHI| Coefficient
Error
z
z>|Z|
of X
--------+--------------------------------------------------------Constant|
2.88090***
.07172
40.17 .0000
FEM|
-.09963**
.04842
-2.06 .0396
.11261
ED|
.14616***
.00541
27.02 .0000
12.8454
BLK|
-.27615***
.05954
-4.64 .0000
.07227
--------+---------------------------------------------------------
Step 3!
--------+--------------------------------------------------------|
Standard
Prob.
Mean
LWAGE| Coefficient
Error
z
z>|Z|
of X
--------+--------------------------------------------------------Constant|
2.88090***
.03282
87.78 .0000
EXP|
.09663***
.00061
157.53 .0000
19.8538
WKS|
.00114***
.00044
2.58 .0098
46.8115
OCC|
-.02496***
.00601
-4.16 .0000
.51116
IND|
.02042***
.00479
4.26 .0000
.39544
SOUTH|
-.00091
.00510
-.18 .8590
.29028
SMSA|
-.04581***
.00506
-9.06 .0000
.65378
UNION|
.03411***
.00521
6.55 .0000
.36399
FEM|
-.09963***
.00767
-13.00 .0000
.11261
ED|
.14616***
.00122
120.19 .0000
12.8454
BLK|
-.27615***
.00894
-30.90 .0000
.07227
HI|
1.00000***
.00670
149.26 .0000 -.103D-13
--------+---------------------------------------------------------
The Magic
Step 1
Step 2
Step 3
What happened here?
y it = α i +

K
k =1
β k x k it +

M
m =1
γ m z m i + ε it
w h e re α i d e n o te th e N u n it e ffe c ts .
A n a s s u m p tio n is a d d e d a lo n g th e w a y
C o v (α i , Z i ) = 0 . T h is is e xa c tly th e n u m b e r o f
o rth o g o n a lity a s s u m p tio n s n e e d e d to
id e n tify γ . It is n o t p a rt o f th e o rig in a l m o d e l.
The Random Effects Model
•
The random effects model
y it = x it β + c i + ε it , o bse rva tio n fo r pe rso n i a t tim e t
y i = X iβ + c ii+ ε i , T i o bse rva tio n s in gro u p i
= X iβ + c i + ε i , n o te c i  (c i , c i ,...,c i ) 
N
y = X β + c + ε ,  i= 1 T i o bse rva tio n s in th e sa m ple
N
c= ( c 1 , c 2 , ...c N ) ,  i= 1 Ti by 1 ve cto r
•
ci is uncorrelated with xit for all t;
E[ci |Xi] = 0
E[εit|Xi,ci]=0
Error Components Model
A Generalized Regression Model
y it  x it b + ε it + u i
E[ε it | X i ]  0
2
2
E[ε it | X i ]  σ 
E[u i | X i ]  0
2
 σ 2ε + σ u2

2
σ
u
V a r[ ε i + u ii ]  
 ...

2
 σ u
2
E[u i | X i ]  σ u
y i = X iβ + ε i + u ii fo r T i o b se rv a tio n s
2
σu
2
...
2
σε + σu
...
...
2
σu
…
2


2
σu
  Ω
i
... 

2
2
σ ε + σ u 
σu
Random vs. Fixed Effects
•
Random Effects
•
•
•
•
Small number of parameters
Efficient estimation
Objectionable orthogonality assumption (ci  Xi)
Fixed Effects
•
•
Robust – generally consistent
Large number of parameters
Ordinary Least Squares
•
Standard results for OLS in a GR model
•
•
•
•
Consistent
Unbiased
Inefficient
True variance of the least squares estimator
 X X 
V a r[b | X ]  N
 N

 i 1 Ti   i 1 Ti 
1
1
-1
X Ω X  X X 
 N

N
 i 1 Ti   i 1 Ti 
 0   Q   Q* Q
 0 as N  
1
-1
Estimating the Variance for OLS
 X X 
V a r[ b | X ]  N
 N

 i 1 Ti   i 1 Ti 
1
1
 X Ω X   X X 
 N
 N


T

T
 i 1 i   i 1 i 
1
In th e sp irit o f th e W h ite e stim a to r, u s e
ˆX
X Ω
N
 i 1 Ti
N
  i  1 fi
ˆ iw
ˆ i X i
X i w
Ti
ˆ i = y i - X ib , fi 
, w
Ti
N
 i 1 Ti
H y p o th e sis te sts a re th e n b a se d o n W a ld sta tistic s.
T H IS IS T H E 'C L U S T E R ' E S T IM A T O R
OLS Results for Cornwell and Rupert
+----------------------------------------------------+
| Residuals
Sum of squares
=
522.2008
|
|
Standard error of e =
.3544712
|
| Fit
R-squared
=
.4112099
|
|
=
.4100766
|
+----------------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Constant
5.40159723
.04838934
111.628
.0000
EXP
.04084968
.00218534
18.693
.0000
19.8537815
EXPSQ
-.00068788
.480428D-04
-14.318
.0000
514.405042
OCC
-.13830480
.01480107
-9.344
.0000
.51116447
SMSA
.14856267
.01206772
12.311
.0000
.65378151
MS
.06798358
.02074599
3.277
.0010
.81440576
FEM
-.40020215
.02526118
-15.843
.0000
.11260504
UNION
.09409925
.01253203
7.509
.0000
.36398559
ED
.05812166
.00260039
22.351
.0000
12.8453782
Alternative Variance Estimators
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
Constant
5.40159723
.04838934
111.628
.0000
EXP
.04084968
.00218534
18.693
.0000
EXPSQ
-.00068788
.480428D-04
-14.318
.0000
OCC
-.13830480
.01480107
-9.344
.0000
SMSA
.14856267
.01206772
12.311
.0000
MS
.06798358
.02074599
3.277
.0010
FEM
-.40020215
.02526118
-15.843
.0000
UNION
.09409925
.01253203
7.509
.0000
ED
.05812166
.00260039
22.351
.0000
Robust – Cluster___________________________________________
Constant
5.40159723
.10156038
53.186
.0000
EXP
.04084968
.00432272
9.450
.0000
EXPSQ
-.00068788
.983981D-04
-6.991
.0000
OCC
-.13830480
.02772631
-4.988
.0000
SMSA
.14856267
.02423668
6.130
.0000
MS
.06798358
.04382220
1.551
.1208
FEM
-.40020215
.04961926
-8.065
.0000
UNION
.09409925
.02422669
3.884
.0001
ED
.05812166
.00555697
10.459
.0000
Generalized Least Squares
G LS is e qu iva le n t to O LS re gre ssio n o f
y it *  y it   i y i . o n x it *  x it   i x i .,
w h e re  i  1 

2
2
   Ti  u
ˆ ]  [ X Ω -1 X ] -1   2 [ X  * X * ] -1
A sy.V a r[β

Estimators for the Variances
y it  x it β   it  u i
U sin g th e O L S e stim a to r o f β , b O LS ,
N
T
 i 1  t i 1 ( y it - a - x it b )

N
i 1
2
2
2
e stim a te s     U

Ti -1 -K
W ith th e L S D V e stim a te s, a i a n d b LS D V ,
N
T
 i 1  t i 1 ( y it - a i - x it b )

N
i 1

Ti -N -K
2
2
e stim a te s  
U sin g th e d iffe re n ce o f th e tw o ,
  N  Ti ( y - a - x  b ) 2
it
 i 1 t 1 it
N

 i 1 T i -1 -K



   N  Ti ( y - a - x  b ) 2
i
it
   i 1 t 1 it
N
 
 i 1 T i -N -K
 



 e stim a te s  U2


Practical Problems with FGLS
2
 T h e p re c e d in g re g u la rly p ro d u c e n e g a tiv e e stim a te s o f  u .
 E stim a tio n is m a d e v e ry c o m p lic a te d in u n b a la n c e d p a n e ls.
A b u lle tp ro o f so lu tio n (o rig in a lly u se d in T S P , n o w N L O G IT a n d o th e rs).
T
N
 i  1  t i 1 ( y it  a i  x it b L S D V )
2
Fro m th e r o b u st L S D V e stim a to r: 
ˆ 
N
 i1 Ti
N
2
2
Fro m th e p o o le d O L S e stim a to r: E st(     u ) 
N
2

ˆu 
T
2
N
T
T
 i  1  t i 1 ( y it  a O L S  x it b O L S )
N
 i1 Ti
 i  1  t i 1 ( y it  a O L S  x it b O L S )   i 1  t i 1 ( y it  a i  x it b L S D V )
N
2
 i1 Ti
2
 0
2
2
 
ˆ
Stata Variance Estimators
N
2


ˆ 
T
 i 1  t i 1 ( y it  a i  x it b LS D V )
N
 i 1 Ti  K  N
2
> 0 ba se d o n FE e stim a te s
2


(N

K
)

S
S
E
(gro
u
p
m
e
a
n
s)
ˆ
2



ˆ u  M ax 0 ,

N

A
(N

A
)
T


 0
2
w h e re A = K o r if 
ˆ u is n e ga tive ,
A = tra ce o f a m a trix th a t so m e w h a t re se m ble s I K .
M a n y o th e r a dju stm e n ts e x ist. N o n e gu a ra n te e d to be
po sitive . N o o ptim a lity pro pe rtie s o r e v e n gu a ra n te e d co n siste n cy.
Application
+--------------------------------------------------+
| Random Effects Model: v(i,t) = e(i,t) + u(i)
|
| Estimates: Var[e]
=
.231188D-01 |
No problems arise
|
Var[u]
=
.102531D+00 |
|
Corr[v(i,t),v(i,s)] =
.816006
| in this sample.
| Variance estimators are based on OLS residuals. |
+--------------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
EXP
.08819204
.00224823
39.227
.0000
19.8537815
EXPSQ
-.00076604
.496074D-04
-15.442
.0000
514.405042
OCC
-.04243576
.01298466
-3.268
.0011
.51116447
SMSA
-.03404260
.01620508
-2.101
.0357
.65378151
MS
-.06708159
.01794516
-3.738
.0002
.81440576
FEM
-.34346104
.04536453
-7.571
.0000
.11260504
UNION
.05752770
.01350031
4.261
.0000
.36398559
ED
.11028379
.00510008
21.624
.0000
12.8453782
Constant
4.01913257
.07724830
52.029
.0000
Testing for Effects: An LM Test
B re u sc h a n d P a g a n La g ra n g e M u ltip lie r sta tistic
y it    x it  u i   it , u i a n d  it
 0    2
~ N o rm a l    ,  u
  0   0
2
H0 : u  0
N
2
N
2
2
  i 1 ( Ti e i )

2
LM =

1



[1]
 N T 2

N
2  i  1 T ( T i  1)   i 1  t  1 e it

i
(  i1 Ti )
0 

2 
   
Application: Cornwell-Rupert
Hausman Test for FE vs. RE
Estimator
Random Effects
E[ci|Xi] = 0
Fixed Effects
E[ci|Xi] ≠ 0
FGLS
(Random Effects)
LSDV
(Fixed Effects)
Consistent and
Efficient
Inconsistent
Consistent
Inefficient
Consistent
Possibly Efficient
Computing the Hausman Statistic
 N



1
2
ˆ
E st.V a r[ β F E ]  
ii  X i 
ˆ    i 1 X i  I 
T



i

 N



ˆ i
2
ˆ


E st.V a r[ β R E ]  
ii  X i 
ˆ    i 1 X i  I 
T



i

1
-1
2
, 0  ˆ i =
Ti 
ˆu
2
2

ˆ   Ti 
ˆu
 1
2
2
ˆ ]  E st.V a r[ β
ˆ ]
A s lo n g a s 
ˆ  and 
ˆ u a re c o n siste n t, a s N   , E st.V a r[ β
FE
RE
w ill b e n o n n e g a tiv e d e fin ite . In a fin ite sa m p le , to e n su re th is, b o th m u st
2
b e c o m p u te d u sin g th e sa m e e stim a te o f 
ˆ  . T h e o n e b a se d o n L S D V w ill
g e n e ra lly b e th e b e tte r c h o ic e .
ˆ ] if th e re a re tim e
N o te th a t c o lu m n s o f ze ro s w ill a p p e a r in E st.V a r[ β
FE
in v a ria n t v a ria b le s in X .
β does not contain the constant term in the preceding.
Hausman Test
+--------------------------------------------------+
| Random Effects Model: v(i,t) = e(i,t) + u(i)
|
| Estimates: Var[e]
=
.235368D-01 |
|
Var[u]
=
.110254D+00 |
|
Corr[v(i,t),v(i,s)] =
.824078
|
| Lagrange Multiplier Test vs. Model (3) = 3797.07 |
| ( 1 df, prob value = .000000)
|
| (High values of LM favor FEM/REM over CR model.) |
| Fixed vs. Random Effects (Hausman)
= 2632.34 |
| ( 4 df, prob value = .000000)
|
| (High (low) values of H favor FEM (REM).)
|
+--------------------------------------------------+
A Fixed E ffects M odel
y it   i    x it   it
LS D V estim ator - D eviations from group m eans:
T o estim ate  , regress (y it  y i ) on ( x it  x i )
A lgebraic equivalent: O LS regress y it on ( x it , x i )
M undlak interpretation:  i     x i  u i
M odel becom es y it     x i  u i    x it   it
=    x i    x it   it  u i
 a random effects m odel w ith the group m eans.
E stim ate by FG LS .
•
•
•
Asymptotic equivalent to Hausman
Also equivalent to Mundlak formulation
In the random effects model, using FGLS
•
•
•
Only applies to time varying variables
Add expanded group means to the regression (i.e.,
observation i,t gets same group means for all t.
Use Wald test to test for coefficients on means
equal to 0. Large chi-squared weighs against
random effects specification.
Fixed Effects
+----------------------------------------------------+
| Panel:Groups
Empty
0,
Valid data
595 |
|
Smallest
7,
Largest
7 |
|
Average group size
7.00 |
| There are 3 vars. with no within group variation. |
| ED
BLK
FEM
|
| Look for huge standard errors and fixed parameters.|
| F.E. results are based on a generalized inverse.
|
| They will be highly erratic. (Problematic model.) |
| Unable to compute std.errors for dummy var. coeffs.|
+----------------------------------------------------+
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
|WKS
|
.00083
.00060003
1.381
.1672
46.811525|
|OCC
|
-.02157
.01379216
-1.564
.1178
.5111645|
|IND
|
.01888
.01545450
1.221
.2219
.3954382|
|SOUTH
|
.00039
.03429053
.011
.9909
.2902761|
|SMSA
|
-.04451**
.01939659
-2.295
.0217
.6537815|
|UNION
|
.03274**
.01493217
2.192
.0283
.3639856|
|EXP
|
.11327***
.00247221
45.819
.0000
19.853782|
|EXPSQ
|
-.00042***
.546283D-04
-7.664
.0000
514.40504|
|ED
|
.000
......(Fixed Parameter).......
|
|BLK
|
.000
......(Fixed Parameter).......
|
|FEM
|
.000
......(Fixed Parameter).......
|
+--------+------------------------------------------------------------+
Random Effects
+--------------------------------------------------+
| Random Effects Model: v(i,t) = e(i,t) + u(i)
|
| Estimates: Var[e]
=
.235368D-01 |
|
Var[u]
=
.110254D+00 |
|
Corr[v(i,t),v(i,s)] =
.824078
|
| Lagrange Multiplier Test vs. Model (3) = 3797.07 |
| ( 1 df, prob value = .000000)
|
| (High values of LM favor FEM/REM over CR model.) |
+--------------------------------------------------+
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
|WKS
|
.00094
.00059308
1.586
.1128
46.811525|
|OCC
|
-.04367***
.01299206
-3.361
.0008
.5111645|
|IND
|
.00271
.01373256
.197
.8434
.3954382|
|SOUTH
|
-.00664
.02246416
-.295
.7677
.2902761|
|SMSA
|
-.03117*
.01615455
-1.930
.0536
.6537815|
|UNION
|
.05802***
.01349982
4.298
.0000
.3639856|
|EXP
|
.08744***
.00224705
38.913
.0000
19.853782|
|EXPSQ
|
-.00076***
.495876D-04
-15.411
.0000
514.40504|
|ED
|
.10724***
.00511463
20.967
.0000
12.845378|
|BLK
|
-.21178***
.05252013
-4.032
.0001
.0722689|
|FEM
|
-.24786***
.04283536
-5.786
.0000
.1126050|
|Constant|
3.97756***
.08178139
48.637
.0000
|
+--------+------------------------------------------------------------+
The Hausman Test, by Hand
--> matrix; br=b(1:8) ; vr=varb(1:8,1:8)\$
--> matrix ; db = bf - br ; dv = vf - vr \$
--> matrix ; list ; h =db'<dv>db\$
Matrix H
has 1 rows and
1
+-------------1| 2523.64910
1 columns.
--> calc;list;ctb(.95,8)\$
+------------------------------------+
| Listed Calculator Results
|
+------------------------------------+
Result =
15.507313
Means Added to REM - Mundlak
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
|WKS
|
.00083
.00060070
1.380
.1677
46.811525|
|OCC
|
-.02157
.01380769
-1.562
.1182
.5111645|
|IND
|
.01888
.01547189
1.220
.2224
.3954382|
|SOUTH
|
.00039
.03432914
.011
.9909
.2902761|
|SMSA
|
-.04451**
.01941842
-2.292
.0219
.6537815|
|UNION
|
.03274**
.01494898
2.190
.0285
.3639856|
|EXP
|
.11327***
.00247500
45.768
.0000
19.853782|
|EXPSQ
|
-.00042***
.546898D-04
-7.655
.0000
514.40504|
|ED
|
.05199***
.00552893
9.404
.0000
12.845378|
|BLK
|
-.16983***
.04456572
-3.811
.0001
.0722689|
|FEM
|
-.41306***
.03732204
-11.067
.0000
.1126050|
|WKSB
|
.00863**
.00363907
2.371
.0177
46.811525|
|OCCB
|
-.14656***
.03640885
-4.025
.0001
.5111645|
|INDB
|
.04142
.02976363
1.392
.1640
.3954382|
|SOUTHB |
-.05551
.04297816
-1.292
.1965
.2902761|
|SMSAB
|
.21607***
.03213205
6.724
.0000
.6537815|
|UNIONB |
.08152**
.03266438
2.496
.0126
.3639856|
|EXPB
|
-.08005***
.00533603
-15.002
.0000
19.853782|
|EXPSQB |
-.00017
.00011763
-1.416
.1567
514.40504|
|Constant|
5.19036***
.20147201
25.762
.0000
|
+--------+------------------------------------------------------------+
--> matrix ; bm=b(12:19);vm=varb(12:19,12:19)\$
--> matrix ; list ; wu = bm'<vm>bm \$
Matrix WU
has 1 rows and
1
+-------------1| 3004.38076
1 columns.
A Hierarchical Linear Model
Interpretation of the FE Model
y it  x it β  c i + ε it , ( x d o e s n o t co n ta in a co n sta n t)
2
E [ε it | X i , c i ]  0 , V a r[ε it | X i , c i ]=  
c i   + z i δ + u i ,
2
E [u i| z i ]  0 , V a r[u i| z i ]   u
y it  x it β  [   z i δ  u i ]  ε it
Hierarchical Linear Model as REM
+--------------------------------------------------+
| Random Effects Model: v(i,t) = e(i,t) + u(i)
|
| Estimates: Var[e]
=
.235368D-01 |
|
Var[u]
=
.110254D+00 |
|
Corr[v(i,t),v(i,s)] =
.824078
|
|
Sigma(u)
= 0.3303
|
+--------------------------------------------------+
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
OCC
|
-.03908144
.01298962
-3.009
.0026
.51116447
SMSA
|
-.03881553
.01645862
-2.358
.0184
.65378151
MS
|
-.06557030
.01815465
-3.612
.0003
.81440576
EXP
|
.05737298
.00088467
64.852
.0000
19.8537815
FEM
|
-.34715010
.04681514
-7.415
.0000
.11260504
ED
|
.11120152
.00525209
21.173
.0000
12.8453782
Constant|
4.24669585
.07763394
54.702
.0000
Evolution: Correlated Random Effects
U nknow n param eters
y it   i    x it   it ,   [  1 ,  2 , ...,  N ,  ,   ]
2
S tandard estim ation based on L S (dum m y v ariables)
A m biguous definition of the distribution of y it
E ffects m odel, nonorthogonality, heterog eneit y
y it   i    x it   it , E[  i | X i ]  g ( X i )  0
C ontrast to random effects E[  i | X i ]  
S tandard estim ation (still) based on L S (dum m y variables)
C orrelated random effects, m ore detailed m odel
y it   i    x it   it , P[  i | X i ]  g ( X i )  0
L inear projection?  i    x i  u i C or(u i , x i )  0
Mundlak’s Estimator
Mundlak, Y., “On the Pooling of Time Series and Cross Section
Data, Econometrica, 46, 1978, pp. 69-85.
W rite c i = x i δ  u i , E [c i | x i1 , x i1 , ... x iT ] = x i δ
i
A ssu m e c i c o n ta in s a ll tim e in v a ria n t in fo rm a tio n
y i = X iβ + c ii+ ε i , T i o b se rv a tio n s in g ro u p i
= X iβ + ix i δ + ε i + u ii
L o o k s lik e ra n d o m e ffe c ts.
2
V a r[ ε i + u ii]= Ω i + σ u ii
T h is is th e m o d e l w e u se d fo r th e W u te s t.
Correlated Random Effects
M u n d la k
c i = x i δ  u i , E [c i | x i1 , x i1 , ... x iT ] = x i δ
i
A ssu m e c i co n ta in s a ll tim e in v a ria n t in fo rm a tio n
y i = X iβ + c ii+ ε i , T i o b se rv a tio n s in g ro u p i
= X iβ + ix i δ + ε i + u ii
C h a m b e rla in / W o o ld ri d g e
c i = x i1 δ 1  x i2 δ 2  ...  x iT δ T  u i
y i = X iβ  ix i1 δ 1  ix i1 δ 2  ...  ix iT δ T  i u i + ε i
TxK  TxK
 TxK

TxK
e tc.
P ro b le m s: R e q u ire s b a la n ce d p a n e ls
M o d e rn p a n e ls h a v e la rg e T ; m o d e ls h a v e la rg e K
Mundlak’s Approach for an FE Model
with Time Invariant Variables
y it  x it β + z i δ  c i + ε it , ( x do e s n o t co n ta in a co n sta n t)
2
E [ε it | X i , c i ]  0 , V a r[ε it | X i , c i ]=  
c i   + x i θ + w i ,
2
E [w i| X i , z i ]  0 , V a r[w i| X i , z i ]   w
y it  x it β  z i δ    x i θ  w i  ε it
 ra n do m e ffe cts m o de l in clu din g gro u p m e a n s o f
tim e va ryin g va ria ble s.
Mundlak Form of FE Model
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
x(i,t)=================================================================
OCC
|
-.02021384
.01375165
-1.470
.1416
.51116447
SMSA
|
-.04250645
.01951727
-2.178
.0294
.65378151
MS
|
-.02946444
.01915264
-1.538
.1240
.81440576
EXP
|
.09665711
.00119262
81.046
.0000
19.8537815
z(i)===================================================================
FEM
|
-.34322129
.05725632
-5.994
.0000
.11260504
ED
|
.05099781
.00575551
8.861
.0000
12.8453782
Means of x(i,t) and constant===========================================
Constant|
5.72655261
.10300460
55.595
.0000
OCCB
|
-.10850252
.03635921
-2.984
.0028
.51116447
SMSAB
|
.22934020
.03282197
6.987
.0000
.65378151
MSB
|
.20453332
.05329948
3.837
.0001
.81440576
EXPB
|
-.08988632
.00165025
-54.468
.0000
19.8537815
Variance Estimates=====================================================
Var[e]|
.0235632
Var[u]|
.0773825
Panel Data Extensions
•
•
•
•
•
Dynamic models: lagged effects of the
dependent variable
Endogenous RHS variables
Cross country comparisons– large T
More general parameter heterogeneity – not
only the constant term
Nonlinear models such as binary choice
The Hausman and Taylor Model
y it  x 1 it β 1  x 2 it β 2  z 1 i α 1  z 2 i α 2   it  u i
M o d e l: x 2 a n d z 2 a re c o rre la te d w ith u .
D e v ia tio n s fro m g ro u p m e a n s re m o v e s a ll tim e in v a ria n t v a ria b le s
y it  y i  ( x 1 it - x 1 i ) 'β 1  ( x 2 it - x 2 i ) 'β 2   it
Im p lic a tio n : β 1 , β 2 a re c o n siste n tly e stim a te d b y L S D V .
( x 1 it - x 1 i ) = K 1 in stru m e n ta l v a ria b le s
( x 2 it - x 2 i ) = K 2 in stru m e n ta l v a ria b le s
z1i
?
= L 1 in stru m e n ta l v a ria b le s (u n c o rre la te d w ith u )
= L 2 in stru m e n ta l v a ria b le s (w h e re d o w e g e t th e m ?)
H & T : x 1 i = K 1 a d d itio n a l in stru m e n ta l v a ria b le s. N e e d s K 1  L 2 .
H&T’s 4 Step FGLS Estimator
2
(1) LS D V e stim a te s o f β 1 , β 2 ,  
(2) ( e * ) ' = ( e 1 , e 1 , ..., e 1 ), ( e 2 , e 2 , ..., e 2 ), ..., ( e N , e N , . .., e N )
IV re g re ssio n o f e * o n Z * w ith in stru m e n ts
W i co n siste n tly e stim a te s α 1 a n d α 2 .
2
2
(3) W ith fix e d T , re sid u a l va ria n ce in (2 ) e stim a te s  u    / T
2
2
W ith u n b a la n ce d p a n e l, it e stim a te s  u    (1 /T ) o r so m e th in g
2
re se m b lin g th is. (1 ) p ro vid e d a n e stim a te o f   so u se th e tw o
2
2
to o b ta in e stim a te s o f  u a n d   . Fo r e a ch g ro u p , co m p u te
ˆ
i  1 
2
2
2

ˆ  / (
ˆ   Ti
ˆu)
( 4 ) T ra n sfo rm
[ x it1 , x it2 , z i1 , z i2 ] to
W i * = [ x it1 , x it2 , z i1 , z i2 ] - ˆ
 i [ x i1 , x i2 , z i1 , z i2 ]
and
y it to y it * = y it - ˆ
 i y i.
H&T’s 4 STEP IV Estimator
In stru m e n ta l V a ria b le s V i 
( x 1 it - x 1 i ) = K 1 in stru m e n ta l va ria b le s
( x 2 it - x 2 i ) = K 2 in stru m e n ta l va ria b le s
z1i
= L 1 in stru m e n ta l va ria b le s ( u n co rre la te d w ith u )
x1i
= K 1 a d d itio n a l in stru m e n ta l va ria b le s.
N o w d o 2 S LS o f y * o n W * w ith in stru m e n ts V to e stim a te
a ll p a ra m e te rs. I.e .,
ˆ * W
ˆ * ) -1 W
ˆ * y * .
[ β 1 , β 2 , α 1 , α 2 ]= ( W
Arellano/Bond/Bover’s Formulation
Builds on Hausman and Taylor
y it  x 1 it β 1  x 2 it β 2  z 1 i α 1  z 2 i α 2   it  u i
In stru m e n ta l v a ria b le s fo r p e rio d t
( x 1 it - x 1 i ) = K 1 in stru m e n ta l v a ria b le s
( x 2 it - x 2 i ) = K 2 in stru m e n ta l v a ria b le s
z1i
= L 1 in stru m e n ta l v a ria b le s ( u n c o rre la te d w ith u )
x1i
= K 1 a d d itio n a l in stru m e n ta l v a ria b le s. K 1  L 2 .
L e t v it   it  u i
L e t z it  [( x 1 it - x 1 i ) ' ,( x 2 it - x 2 i ) ' , z 1 i , x 1']
T h e n E [ z it v it ]  0
W e fo rm u la te th is fo r th e T i o b se rv a tio n s in g ro u p i.
Arellano/Bond/Bover’s Formulation
Adds a Lagged DV to H&T
y it   y i,t 1 + x 1 it β 1  x 2 it β 2  z 1 i α 1  z 2 i α 2   it  u i
P a ra m e te rs : θ = [  , β 1 , β 2 , α 1 , α 2 ]'
T h e d a ta
 y i,2 
 y i,1 x 1 i2



y
 i,3 
 y i,2 x 1 i3
yi 
, Xi 






 y i, T 
 y i,T -1 x 1 iT

i 

i
1
K1
x 2 i2
z 1 i
x 2 i3
z 1 i
x 2 iT
z 1 i
K2
i
L1
z 2 i 

z 2 i 
, T i -1 ro w s


z 2 i 

L2
c o lu m n s
This formulation is the same as
H&T with yi,t-1 contained in x2it .
Dynamic (Linear) Panel
Data (DPD) Models
•
•
•
•
Application
Bias in Conventional Estimation
Development of Consistent Estimators
Efficient GMM Estimators
Dynamic Linear Model
B a le stra -N e rlo ve (1 9 6 6 ), 3 6 S ta te s, 1 1 Y e a rs
D e m a n d fo r N a tu ra l G a s
S tru ctu re
*
N e w D e m a n d: G i,t  G i,t  (1   )G i,t 1
*
D e m a n d Fu n ctio n G i,t   1   2 Pi,t   3  N i,t   4 N i,t   5  Yi,t   6 Yi,t   i,t
G = ga s de m a n d
N = po pu la tio n
P = price
Y = pe r ca pita in co m e
R e du ce d Fo rm
G i,t   1   2 Pi,t   3  N i,t   4 N i,t   5  Yi,t   6 Yi,t   7 G i,t 1   i   i,t
A General DPD model
y i,t  x i, t β   y i,t 1  c i   i,t
E[  i,t | X i ,c i ]  0
2
2
E[  i,t | X i , c i ]    , E[  i,t  i,s | X i , c i ]  0 if t  s.
E [c i | X i ]  g( X i )
N o co rre la tio n a cro ss in dividu a ls
O LS a n d G LS a re bo th in co n siste n t.
Arellano and Bond Estimator
B a se o n first d iffe re n ce s
y i,t  y i,t 1  ( x i,t  x i,t 1 ) 'β +  (y i,t 1  y i,t  2 )  (  i,t   i,t 1 )
In stru m e n ta l v a ria b le s
y i,3  y i,2  ( x i,3  x i,2 ) 'β +  (y i,2  y i,1 )  (  i,3   i,2 )
C a n u se y i1
y i,4  y i,3  ( x i,4  x i,3 ) 'β +  (y i,3  y i ,2 )  (  i,4   i,3 )
C a n u se y i,1 a n d y i2
y i,5  y i,4  ( x i,5  x i,4 ) 'β +  (y i,4  y i,3 )  (  i,5   i,4 )
C a n u se y i,1 a n d y i2 a n d y i,3
Arellano and Bond Estimator
M o re in stru m e n ta l v a ria b le s - P re d e te rm in e d X
y i,3  y i,2  ( x i,3  x i,2 ) 'β +  (y i,2  y i,1 )  (  i,3   i,2 )
C a n u se y i1 a n d x i,1 , x i,2
y i,4  y i,3  ( x i,4  x i,3 ) 'β +  (y i,3  y i,2 )  (  i,4   i,3 )
C a n u se y i,1 , y i2 , x i,1 , x i,2 , x i,3
y i,5  y i,4  ( x i,5  x i,4 ) 'β +  (y i,4  y i,3 )  (  i,5   i,4 )
C a n u se y i,1 , y i2 , y i,3 , x i,1 , x i,2 , x i,3 , x i,4
Arellano and Bond Estimator
E v e n m o re in stru m e n ta l v a ria b le s - S tric tly e x o g e n o u s X
y i,3  y i,2  ( x i,3  x i,2 ) 'β +  (y i,2  y i,1 )  (  i,3   i,2 )
C a n u se y i1 a n d x i,1 , x i,2 , ..., x i, T (a ll p e rio d s)
y i,4  y i,3  ( x i,4  x i,3 ) 'β +  (y i,3  y i,2 )  (  i,4   i, 3 )
C a n u se y i,1 , y i2 , x i,1 , x i,2 , ..., x i, T
y i,5  y i,4  ( x i,5  x i,4 ) 'β +  (y i,4  y i,3 )  (  i,5   i,4 )
C a n u se y i,1 , y i2 , y i,3 , x i,1 , x i,2 , ..., x i, T
T h e n u m b e r o f p o te n tia l in stru m e n ts is h u g e .
T h e se d e fin e th e ro w s o f Z i . T h e se c a n b e u se d fo r
sim p le in stru m e n ta l v a ria b le e stim a tio n .