Part 5: Random Effects [ 1/54]
Econometric Analysis of Panel Data
William Greene
Department of Economics
Stern School of Business
Part 5: Random Effects [ 2/54]
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
Part 5: Random Effects [ 3/54]
Random vs. Fixed Effects

Random Effects




Fixed Effects





Small number of parameters
Efficient estimation
Objectionable orthogonality assumption (ci  Xi)
Robust – generally consistent
Large number of parameters
More reasonable assumption
Precludes time invariant regressors 
Which is the more reasonable model?
Part 5: Random Effects [ 4/54]
Error Components Model
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
2
E[u i | X i ]  σ u
  2   u2

2

u
V ar[ ε i + u ii ]  


2

u

y i = X iβ + ε i + u ii fo r T i o b se rv a tio n s
2
2
u
2
2
  u
2
u


2
u


2
2
  u 
u
Part 5: Random Effects [ 5/54]
Notation
y1  X1 
 ε 1   u 1 i1 

 

  

y2
X2
ε2
u 2 i2

  
β     


 

  


 

  

y
X
ε
u
i
 N  N
 N  N N
= X β+ ε+ u
T1 o b se rv a tio n s
T 2 o b se rv a tio n s
TN o b se rv a tio n s
N
 i=1 T i o b se rv a tio n s
= X β+ w
In a ll th a t fo llo w s , e x c e p t w h e re e x p lic itly n o te d , X , X i
a n d x it c o n ta in a c o n s ta n t te rm a s th e fir s t e le m e n t.
T o a v o id n o ta tio n a l c lu tte r, in th o s e c a s e s , x it e tc . w ill
s im p ly d e n o te th e c o u n te rp a rt w ith o u t th e c o n s ta n t te rm .
U s e o f th e s y m b o l K fo r th e n u m b e r o f v a ria b le s w ill th u s
b e c o n te x t s p e c ific b u t w ill u s u a lly in c lu d e th e c o n s ta n t te rm .
Part 5: Random Effects [ 6/54]
Notation
  2   u2

2

u
V a r[ ε i + u ii ]  


2
 u
2
2
2
u
2


2
u


2
2 
  u 
u
2
  u
2
u
2
=   I T   u ii  T i  T i
i
2
2
=   I T   u ii 
i
= Ωi
Ω1

0
V a r[ w | X ]  


 0
0
Ω2
0
0 

0 (N o te th e se d iffe r o n ly


in th e d im e n sio n T i )

ΩN
2
2
    I T   u ii   I N
Part 5: Random Effects [ 7/54]
Regression Model-Orthogonality
p lim
p lim
1
# o b se rv a tio n s
1
N
 i 1 Ti
X 'w  0
1
N
 i=1 X i w i  p lim
 N
X i ε i
p lim N
  i=1 T i
 i 1 Ti 
Ti
1
N
 i 1 Ti
N
 i=1 X i ( ε i + u ii)  0
X ii i 
+  i=1 T iu i

Ti 
N
 N
X i ε i
p lim   i=1 fi
Ti

+  i=1 f i
 N
X i ε i
p li m   i=1 fi
Ti


+  i=1 fi x iu i   0

N
N

Ti
u i  , 0 < fi  N
< 1
Ti
 i 1 Ti

X ii i
=
1
N
if T i  T  i
Part 5: Random Effects [ 8/54]
Convergence of Moments
X X
N
 i 1 T
X Ω X
N
 i 1 T
N
  i  1 fi
N
  i  1 fi
2
X i X i
Ti
 a w e ig h te d su m o f in d iv id u a l m o m e n t m a t rice s
X i Ω i X i
Ti
N
=    i  1 fi
X i X i
Ti
 a w e ig h te d su m o f in d iv id u a l m o m e n t m a t rice s
  u  i  1 f i x i x i
2
N
N o te a sy m p to ti cs a re w ith re sp e ct to N . E a ch m a trix
X i X i
Ti
is th e
m o m e n ts fo r th e T i o b se rv a tio n s. S h o u ld b e 'w e ll b e h a v e d ' in m icro
le v e l d a ta . T h e a v e ra g e o f N su ch m a tric e s sh o u ld b e lik e w ise .
T o r T i is a ssu m e d to b e fix e d (a n d sm a ll).
Part 5: Random Effects [ 9/54]
Ordinary Least Squares

Standard results for OLS in a GR model




Consistent
Unbiased
Inefficient
True Variance
V a r[b | X ] 
 X X 
 N

N
 i1 Ti   i1 Ti 
1
 0   Q
1
-1
X Ω X  X X 
 N

N
 i1 Ti   i1 Ti 
 Q * Q
1
-1
 0 a s N   w ith o u r co n ve rg e n ce a ssu m p tion s
Part 5: Random Effects [ 10/54]
Estimating the Variance for OLS
 X X 
V a r[ b | X ]  N
 N

 i 1 Ti   i 1 Ti 
1
X Ω X
N
 i 1 T
N
  i  1 fi
X i Ω i X i
Ti
1
 X Ω X   X X 
 N
 N


T

T
 i 1 i   i 1 i 
1
, w h e re = Ω i = E [ w i w i | X i ]
In th e sp irit o f th e W h ite e stim a to r, u s e
X Ω X
N
 i 1 T
N
  i  1 fi
ˆ iw
ˆ i X i
X i w
Ti
ˆ i = y i - X ib
, w
H y p o th e sis te sts a re th e n b a se d o n W a ld sta tistics.
T H IS IS T H E 'C L U S T E R ' E S T IM A T O R
Part 5: Random Effects [ 11/54]
Mechanics
1
E st.V a r[ b | X ]   X X 

N
i 1

ˆ iw
ˆ i X i  X X 
X i w
1
ˆ i = se t o f T i O LS re sidu a ls fo r in dividu a l i.
w
X i = T i xK da ta o n e xo ge n o u s va ria ble fo r in dividu a l i.
ˆ i = K x 1 ve cto r o f pro du cts
X i w
ˆ i )( w
ˆ i X i )  K xK m a trix (ra n k 1 , o u te r pro du ct)
( X i w
   X wˆ   wˆ X  
N
i 1
i
i
i
i
= su m o f N ra n k 1 m a trice s. R a n k  K .
Part 5: Random Effects [ 12/54]
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 5: Random Effects [ 13/54]
Alternative OLS 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
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
Part 5: Random Effects [ 14/54]
Generalized Least Squares
ˆ = [ X Ω -1 X ]  1 [ X Ω -1 y ]
β
N
-1
1
N
-1
= [  i  1 X i Ω i X i ] [  i 1 X i Ω i y i ]
-1
Ωi
2



1
u
 2 I T  2
ii 
2
  i
   Ti  u 
(n o te , d e p e n d s o n i o n ly th ro u g h T i )
Part 5: Random Effects [ 15/54]
Panel Data Algebra (1)
2
2
Ω i = σ ε I + σ u ii , d e p e n d s o n 'i' b e ca u se it is T i  T i
2
2
2
2
Ω i = σ ε [ I +  ii], 
2
2
2
= σu / σε
2
Ω i = σ ε [ I +  ii] = σ ε [ A  b b ], A = I , b =  i.
U sin g (A -6 6 ) in G re e n e (p . 9 6 4 )
-1
Ωi

1  -1
1
-1
-1 

A
A
b
b
A
2
-1

σ ε 
1 + b A b

2



σ
1 
1
1
2
u
=
I ii  =
I- 2
ii 
2 
2
2 
2
σε 
1 + Ti
σε 
σ ε + Tiσ u 

Part 5: Random Effects [ 16/54]
Panel Data Algebra (2)
(B a se d o n W o o ldridge p. 2 8 6 )
2
2
2
2
-1
2
2
i
Ω i  σ ε I + σ u ii  σ ε I + T i σ u i(i i) i  σ ε I + T i σ u PD
2
2
i
 σ ε I + Ti σ u ( I  M D )
2
2
i
2
2
i
2
2
i
2
2
2
 (σ ε + T i σ u )[ PD   ( I  PD )],  i = σ ε /(σ ε + T i σ u )
i
 (σ ε + T i σ u )[ PD   M D ]
 (σ ε + Ti σ u ) S i
1
Si
1 / 2
Si
i
i
i
i
 PD  (1 /  i )M D (Pro ve by m u ltiplyin g. PD M D  0 .)
i
 PD  (1 /
a
(N o te S i
i
i
 i )M D 
a
i
 PD   i M D )
1
 I   iPDi  ,  i = 1 -  i

1  i 
Part 5: Random Effects [ 17/54]
Panel Data Algebra (3)
1 / 2
1

Ωi
i
2
i
(PD  (1 /
2
   Ti  u
1
 i )M D )
1
 I   iPDi  ,


2
2
   Ti  u 1   i

2
i

= 1-
1 / 2

Ωi
1 / 2
V a r[ Ω i
1 / 2
Ωi
2
2
   Ti  u
1

1

2
2
   Ti  u
i
[ I   iPD ]
2
( ε i  u ii)]    I
y i  (1 /   )( y i   i y i .i) fo r th e G LS tra n sf o rm a tio n .
Part 5: Random Effects [ 18/54]
GLS (cont.)
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

Part 5: Random Effects [ 19/54]
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


Part 5: Random Effects [ 20/54]
Feasible GLS
2
2
Fe a sible G LS re qu ire s (o n ly) co n siste n t e stim a to rs o f   a n d  u .
C a n dida te s:
T
N
 i 1  t i 1 ( y it  a i  x it b LS D V )
2
Fro m th e ro bu st LS D V e stim a to r: 
ˆ 
2
N
 i 1 Ti  K  N
T
N
2
2
Fro m th e po o le d O LS e stim a to r: E st(     u ) 
 i 1  t i 1 ( y it  a O LS  x it b O LS )
N
 i 1 Ti  K  1
N
2

2
u
Fro m th e gro u p m e a n s re gre ssio n : E st(  / T   ) 
 i 1 ( y it  a  x ib M E A N S )
N K 1
N
( W o o ldridge ) B a se d o n E [w it w is | X i ]  
2
u
2
2
u
if t  s, 
ˆ 
T h e re a re m a n y o th e rs.
x´ does not contain a constant term in the preceding.
T 1
T
ˆ it w
ˆ is
 i 1  t i 1  s i t  1 w
N
 i 1 Ti  K  N
2
Part 5: Random Effects [ 21/54]
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
Part 5: Random Effects [ 22/54]
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.
Part 5: Random Effects [ 23/54]
Computing Variance Estimators
U sin g fu ll list o f va ria b le s (FE M a n d E D a re tim e in va ria n t)
O LS su m o f sq u a re s = 5 2 2 .2 0 0 8 .
2
2
E st(   +  u ) = 5 2 2 .2 0 0 8 / (4 1 6 5 - 9 ) = 0 .1 2 5 6 5 .
U sin g fu ll list o f va ria b le s a n d a g e n e r a lize d in ve rse (sa m e
a s d ro p p in g FE M a n d E D ), LS D V su m o f sq u a re s = 8 2 .3 4 9 1 2 .
2

ˆ  = 8 2 .3 4 9 1 2 / (4 1 6 5 - 8 -5 9 5 ) = 0 .0 2 3 1 1 9 .
2

ˆ u  0 .1 2 5 6 5 - 0 .0 2 3 1 1 9 = 0 .1 0 2 5 3
2
B o th e stim a to rs a re p o sitive . W e ca n sto p h e re . If 
ˆ u w e re
n e g a tive , w e w o u ld u se e stim a to rs w ith o u t D F co r re ctio n s.
Part 5: Random Effects [ 24/54]
Application
+--------------------------------------------------+
| Random Effects Model: v(i,t) = e(i,t) + u(i)
|
| Estimates: Var[e]
=
.231188D-01 |
|
Var[u]
=
.102531D+00 |
|
Corr[v(i,t),v(i,s)] =
.816006
|
| (High (low) values of H favor FEM (REM).)
|
|
Sum of Squares
.141124D+04 |
|
R-squared
-.591198D+00 |
+--------------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|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
Part 5: Random Effects [ 25/54]
Testing for Effects: An LM Test
B reu sch a n d P a g a n La g ra n g e M u ltip lier st a tistic
y it    x it  u i   it , u i a n d  it
  0    u2
~ N orm a l    , 
  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 )
=
2
N
2
 T  i1 ei


1
 N T 2

2( T  1)   i  1  t  1 e it

NT
2
if T is con sta n t
0 
2 
   
Part 5: Random Effects [ 26/54]
LM Tests
+--------------------------------------------------+
| Random Effects Model: v(i,t) = e(i,t) + u(i)
|
| Estimates: Var[e]
=
.216794D+02 |
|
Var[u]
=
.958560D+01 |
|
Corr[v(i,t),v(i,s)] =
.306592
|
| Lagrange Multiplier Test vs. Model (3) = 4419.33 |
| ( 1 df, prob value = .000000)
|
| (High values of LM favor FEM/REM over CR model.) |
| Baltagi-Li form of LM Statistic =
1618.75 |
+--------------------------------------------------+
+--------------------------------------------------+
| Random Effects Model: v(i,t) = e(i,t) + u(i)
|
| Estimates: Var[e]
=
.210257D+02 |
|
Var[u]
=
.860646D+01 |
|
Corr[v(i,t),v(i,s)] =
.290444
|
| Lagrange Multiplier Test vs. Model (3) = 1561.57 |
| ( 1 df, prob value = .000000)
|
| (High values of LM favor FEM/REM over CR model.) |
| Baltagi-Li form of LM Statistic =
1561.57 |
+--------------------------------------------------+
Unbalanced Panel
#(T=1) = 1525
#(T=2) = 1079
#(T=3) = 825
#(T=4) = 926
#(T=5) = 1051
#(T=6) = 1200
#(T=7) = 887
Balanced Panel
T = 7
REGRESS ; Lhs=docvis ; Rhs=one,hhninc,age,female,educ ; panel $
Part 5: Random Effects [ 27/54]
Testing for Effects: Moments
W o o ld rid g e (p a g e 2 6 5 ) su g g e sts b a se d o n th e o ff d ia g o n a l e le m e n ts
N
Z=
T -1
T
 i= 1  t= 1  s= t+ 1 e it e is
N

T -1
T
 i= 1  t= 1  s= t+ 1 e it e is
d

2
  N[0 , 1]
w h ich co n v e rg e s to sta n d a rd n o rm a l. ("W e a re n o t a ssu m in g a n y
p a rticu la r d istrib u tio n fo r th e  it . In ste a d , w e d e riv e a sim ila r te st th a t
h a s th e a d v a n ta g e o f b e in g v a lid fo r a n y d istrib u tio n ...") It's co n v e n ie n t
to e x a m in e Z
2
w h ich , b y th e S lu tsk y th e o r e m co n v e rg e s (a lso ) to ch i-
sq u a re d w ith o n e d e g re e o f fre e d o m .
Part 5: Random Effects [ 28/54]
Testing (2)
T -1
T
 t= 1  s= t+ 1 e it e is = 1 /2 o f th e su m o f a ll o ff d ia g o n a l e le m e n ts o f
o f e i e i o r 1 /2 th e su m o f a ll th e e le m e n ts m in u s th e d ia g o n a l e le m e n ts.
T -1
T
T -1
T
2

 t= 1  s= t+ 1 e it e is = 1 /2 [ i ( e i e i ) i  e ie i ]. B u t, i  e i = T e i . S o ,
2
 t= 1  s= t+ 1 e it e is = (1 /2 )[(T e i )  e ie i ]
Z

N
2
N
2
2

 i  1 [(T e i )  e ie i ]
 i  1 [(T e i )  e ie i ]
2
 LM 
N
[  i  1 e ie i ]
NT
 i  1 [(T e i )  e ie i ]
N
N o te , a lso
N
Z=
 i  1 ri
N
 i  1 ri
2

N r
sr
2
2( T  1)
2
, w h e re ri  (T e i )  e ie i .
2
2
Part 5: Random Effects [ 29/54]
Testing for Effects
? Obtain OLS residuals
Regress; lhs=lwage;rhs=fixedx,varyingx;res=e$
? Vector of group sums of residuals
Calc
; T = 7 ; Groups = 595
Matrix ; tebar=T*gxbr(e,person)$
? Direct computation of LM statistic
Calc
; list;lm=Groups*T/(2*(T-1))*
(tebar'tebar/sumsqdev - 1)^2$
? Wooldridge chi squared (N(0,1) squared)
Create ; e2=e*e$
Matrix ; e2i=T*gxbr(e2,person)$
Matrix ; ri=dirp(tebar,tebar)-e2i ; sumri=ri'1$
Calc
; list;z2=(sumri)^2/ri'ri$
LM
Z2
= .37970675705025540D+04
= .16533465085356830D+03
Part 5: Random Effects [ 30/54]
Two Way Random Effects Model
y it  x it   u i  v t   it
H o w to e stim a te th e v a ria n ce co m p o n e n ts?
2
(1 ) T w o w a y F E M re sid u a l v a ria n ce e stim a te s  
2
2
2
(2) S im p le O L S re sid u a l v a ria n ce e stim a t e s     u   v
(3) T h e re a re n u m e ro u s w a y s to g e t a th i rd e q u a tio n .
E .g ., th e o n e w a y F E M re sid u a l v a ria n ce in e ith e r d im e n sio n
2
2
O n e w a y F E M b a se d o n g ro u p s e stim a t e s (     v ) / (1  1 / T )
E .g ., th e g ro u p m e a n re g re ssio n s in e ith e r d im e n sio n .
2
2
2
B a se d o n g ro u p m e a n s e stim a te s  u  (     v )/T
(P e rio d m e a n s re g re ssio n m a y h a v e a tin y n u m b e r o f o b se rv a tio n s.)
(A n d a w h o le lib ra ry o f o th e rs - se e B a lta g i, se c. 3 .3 .)
N e g a tiv e e stim a to rs o f c o m m o n v a ria n ce s a re co m m o n .
S o lu tio n s a re co m p lica te d .
Part 5: Random Effects [ 31/54]
One Way REM
Part 5: Random Effects [ 32/54]
Two Way REM
Note sum
= .102705
Part 5: Random Effects [ 33/54]
Hausman Test for FE vs. RE
Estimator
FGLS
(Random Effects)
LSDV
(Fixed Effects)
Random Effects
E[ci|Xi] = 0
Consistent and
Efficient
Fixed Effects
E[ci|Xi] ≠ 0
Inconsistent
Consistent
Inefficient
Consistent
Possibly Efficient
Part 5: Random Effects [ 34/54]
Hausman Test for Effects
ˆ -β
ˆ
B a sis fo r th e te st, β
FE
RE
ˆ -β
ˆ ; W = q
ˆ = β
ˆ [V a r( q
ˆ )] -1 q
ˆ
W a ld C rite rio n : q
FE
RE
A le m m a (H a u sm a n (1 9 7 8 )): U n d e r th e n u ll h y p o th e sis (R E )
d
ˆ - β ]  
n T [β
N [ 0 , V R E ] (e fficie n t)
RE
d
ˆ - β ]  
n T [β
N [ 0 , V F E ] (in e fficie n t)
FE
ˆ - β )-( β
ˆ  β ). T h e le m m a sta te s th a t in th e
ˆ = (β
N o te : q
FE
RE
jo in t lim itin g d istrib u tio n o f
ˆ - β] and
nT [β
RE
ˆ , th e
nT q
lim itin g co v a ria n ce , C Q ,R E is 0 . B u t, C Q ,R E = C F E ,R E - V R E . T h e n ,
V a r[ q ] = V F E + V R E - C F E ,R E - C F E ,R E . U sin g th e le m m a , C F E ,R E = VR E .
It fo llo w s th a t V a r[ q ]= V F E - V R E . B a se d o n th e p re ce d in g
ˆ -β
ˆ )  [E st.V a r( β
ˆ ) - E st.V a r( β
ˆ )] -1 ( β
ˆ -β
ˆ )
H = (β
FE
RE
FE
RE
FE
RE
β does not contain the constant term in the preceding.
Part 5: Random Effects [ 35/54]
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.
Part 5: Random Effects [ 36/54]
Part 5: Random Effects [ 37/54]
Part 5: Random Effects [ 38/54]
Hausman Test?
What went wrong?
The matrix is not positive definite.
It has a negative characteristic root.
The matrix is indefinite.
(Software such as Stata and
NLOGIT find this problem and
refuse to proceed.)
Properly, the statistic cannot be
computed.
The naïve calculation came out
positive by the luck of the draw.
Part 5: Random Effects [ 39/54]
A Variable Addition Test



Asymptotically 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 standard F or Wald test to test for coefficients
on means equal to 0. Large F or chi-squared weighs
against random effects specification.
Part 5: Random Effects [ 40/54]
Variable Addition
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 .
Part 5: Random Effects [ 41/54]
Application: Wu Test
NAMELIST
create ;
create ;
create ;
create ;
create ;
create ;
create ;
create ;
create ;
namelist
; XV = exp,expsq,wks,occ,ind,south,smsa,ms,union,ed,fem$
expb=groupmean(exp,pds=7)$
expsqb=groupmean(expsq,pds=7)$
wksb=groupmean(wks,pds=7)$
occb=groupmean(occ,pds=7)$
indb=groupmean(ind,pds=7)$
southb=groupmean(south,pds=7)$
smsab=groupmean(smsa,pds=7)$
unionb=groupmean(union,pds=7)$
msb = groupmean(ms,pds=7) $
; xmeans = expb,expsqb,wksb,occb,indb,southb,smsab,msb,
unionb $
REGRESS ; Lhs = lwage ; Rhs = xmeans,Xv,one ; panel ; random $
MATRIX ; bmean = b(1:9) ; vmean = varb(1:9,1:9) $
MATRIX ; List ; Wu = bmean'<vmean>bmean $
Part 5: Random Effects [ 42/54]
Means Added
Part 5: Random Effects [ 43/54]
Wu (Variable Addition) Test
Part 5: Random Effects [ 44/54]
Basing Wu Test on a Robust VC
? Robust Covariance matrix for REM
Namelist ; XWU = wks,occ,ind,south,smsa,union,exp,expsq,ed,blk,fem,
wksb,occb,indb,southb,smsab,unionb,expb,expsqb,one $
Create ; ewu = lwage - xwu'b $
Matrix ; Robustvc = <Xwu'Xwu>*Gmmw(xwu,ewu,_stratum)*<XwU'xWU>
; Stat(b,RobustVc,Xwu) $
Matrix ; Means = b(12:19);Vmeans=RobustVC(12:19,12:19)
; List ; RobustW=Means'<Vmeans>Means $
Part 5: Random Effects [ 45/54]
Robust Standard Errors
Part 5: Random Effects [ 46/54]
Fixed vs. Random Effects
ˆ
β
M odel
 N


ˆ i,M o d e l 
   i 1 X i  I 
ii  X i 
Ti




-1
 N

ˆ i,M o d e l  
ii  y i 
  i 1 X i  I 
Ti


 
ˆ M o d e l  1 fo r fix e d e ffe cts.
2
ˆ i,M o d e l 
Ti 
ˆu
2


ˆ  Ti 
ˆ
2
u
fo r ra n d o m e ffe cts.
A s T i   , ˆ i,R E  1, ra n d o m e ffe cts b e co m e s fix e d e ffe cts
2
As 
ˆ u  0 , ˆ i,R E  0 , ra n d o m e ffe cts b e co m e s O LS (o f c o u rse )
2
As 
ˆ u   , ˆ i,R E  1, ra n d o m e ffe cts b e co m e s fix e d e ffe cts
2
2
Fo r th e C & R a p p lica tio n , 
ˆ u = .1 3 3 1 5 6 , 
ˆ  = .0 2 3 5 2 3 1 , ˆ  .9 7 5 3 8 4 .
Lo o ks like a fix e d e ffe cts m o d e l. N o te th e H a u sm a n sta tistic a g re e s.
β does not contain the constant term in the preceding.
Part 5: Random Effects [ 47/54]
Another Comparison (Baltagi p. 20)
ˆ
ˆ
ˆ

 W
 ( I  W )
G LS
LS D V ( W ith in )
B etw een
Lim itin g an d S p ecial C ases
2
(1 ) If  u  0, G LS = O LS
(2) A s T    , G LS
ˆ
  
LS D V ( W ith in )
ˆ
ˆ
(3 ) A s b etw een variation   0, 
  
G LS
LS D V ( W ith in )
ˆ
ˆ
(4 ) A s w ith in variation   0, 
  
G LS
B etw een
Part 5: Random Effects [ 48/54]
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
Part 5: Random Effects [ 49/54]
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
Part 5: Random Effects [ 50/54]
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
Part 5: Random Effects [ 51/54]
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.
Part 5: Random Effects [ 52/54]
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
Part 5: Random Effects [ 53/54]
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.
Part 5: Random Effects [ 54/54]
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