Part 4: Fixed Effects [ 1/96]
Econometric Analysis of Panel Data
William Greene
Department of Economics
Stern School of Business
Part 4: Fixed Effects [ 2/96]
Estimation with Fixed Effects

The fixed 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 arbitrarily correlated with xit but E[εit|Xi,ci]=0
Dummy variable representation
N
y it = x it β +  j= 1  j d ijt + ε it , d ijt = 1 (i= j)
Part 4: Fixed Effects [ 3/96]
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
Part 4: Fixed Effects [ 4/96]
Useful Analysis of Variance Notation
D ecom position of T otal variation :
N
Ti
2
N
Ti
2
Σ i= 1 Σ t= 1 (z it  z )  Σ i= 1  Σ t= 1 (z it  z i . ) 


N
 Σ i= 1 T i  z i .  z 


2
Total variation = Within groups variation
+ Between groups variation
Part 4: Fixed Effects [ 5/96]
Baltagi and Griffin’s Gasoline Data
World Gasoline Demand Data, 18 OECD Countries, 19 years
Variables in the file are
COUNTRY = name of country
YEAR = year, 1960-1978
LGASPCAR = log of consumption per car
LINCOMEP = log of per capita income
LRPMG = log of real price of gasoline
LCARPCAP = log of per capita number of cars
See Baltagi (2001, p. 24) for analysis of these data. The article on which the
analysis is based is Baltagi, B. and Griffin, J., "Gasoline Demand in the OECD: An
Application of Pooling and Testing Procedures," European Economic Review, 22,
1983, pp. 117-137. The data were downloaded from the website for Baltagi's
text.
Part 4: Fixed Effects [ 6/96]
Analysis of Variance
Per Capita Gasoline Use for 18 OECD Countries
6 .5 0
6 .0 0
L GA S P C A R
5 .5 0
5 .0 0
4 .5 0
4 .0 0
3 .5 0
3 .0 0
0
2
4
6
8
10
COUNT RY
12
14
16
18
Part 4: Fixed Effects [ 7/96]
Analysis of Variance
SETPANEL ; Group = country ; Pds = T $
REGRESS ; Lhs=lgaspcar ; Rhs=one ; Panel $
Part 4: Fixed Effects [ 8/96]
X
 X1

X2

D   X3


 X N
 X
i1
d1
0
0
0
d2
0
0
0
d3
0
0
0
i2
iN 





0

d N 
(T1 row s)
(T 2 row s)
(T3 row s)
(T N row s)
  i= 1 Ti row s 
N
Part 4: Fixed Effects [ 9/96]
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 
Part 4: Fixed Effects [ 10/96]
Fixed Effects Estimator (cont.)
MD
 M 1D

0




 0
0 

0 
(T h e d u m m y v a ria b le s a re o rth o g o n a l)


N
M D 
0
2
MD
0
1
i
M D  I T  d i ( d i d i ) d i = I T  (1 /T i ) d i d i
i
i
N
i
X M D X =  i=1 X iM D X i ,
N
i
X M D y =  i=1 X iM D y i ,


i
X iM D X i
i
X iM D y i


T
k ,l
  t=i 1 ( x it,k -x i.,k )( x it,l -x i.,l )
T
  t=i 1 ( x it,k -x i.,k )( y it -y i. )
k
0
If a ll g ro u p s h a v e th e sa m e T i , M D  M  I w h e re M
0
X M D X = X [M  I ] X a n d b =

0
X [M  I ] X

1
0
0
 I T  (1 /T ) d d 
X [M  I ] y .
Part 4: Fixed Effects [ 11/96]
The Within Transformation
Removes the Effects
y it  x it β  c i + ε it
y i  x iβ  c i + ε i
y it  y i  ( x it - x i ) β  ( ε it  ε i )
y it  x it β  ε it
Wooldridge notation for data in deviations from group means
Part 4: Fixed Effects [ 12/96]
Least Squares Dummy Variable Estimator


b is obtained by ‘within’ groups least
squares (group mean deviations)
Normal equations for a are D’Xb+D’Da=D’y
a = (D’D)-1D’(y – Xb)
Ti
a i = (1 /T i )Σ t= 1 (y it - x it b )= e i
Notes: This is simple algebra – the estimator is just OLS
Least squares is an estimator, not a model. (Repeat twice.)
Note what ai is when Ti = 1. Follow this with yit-ai-xit’b=0 if Ti=1.
Part 4: Fixed Effects [ 13/96]
Inference About OLS



Assume strict exogeneity: Cov[εit,(xjs,cj)]=0. Every
disturbance in every period for each person is
uncorrelated with variables and effects for every person
and across periods.
Now, it’s just least squares in a classical linear
regression model.
Asy.Var[b] = (  2 /  Ni= 1 Ti )plim[(  2 /  Ni= 1 Ti )  Ni= 1 X iM Di X i ]  1
w h ich is th e u su a l e stim a to r fo r O LS
T
N
2

ˆ 
 i= 1  t=i 1 ( y it -a i -x it b )

N
i= 1
Ti - N - K
2

(N o te th e degrees o f freedo m co rrectio n )
Part 4: Fixed Effects [ 14/96]
Application Cornwell and Rupert
Part 4: Fixed Effects [ 15/96]
LSDV Results
Note huge changes in
the coefficients. SMSA
and MS change signs.
Significance changes
completely!
Pooled OLS
Part 4: Fixed Effects [ 16/96]
The Effect of the Effects
Part 4: Fixed Effects [ 17/96]
On Computation


“Sometimes it is useful to obtain the cˆi even when the
(dummy variable) regression is infeasible.” (W: p.
273) It is never infeasible.
The degrees of freedom correction must include the
constants (N of them).
Part 4: Fixed Effects [ 18/96]
The Estimated Fixed Effects
Fre que nc y
Fixed E ffects fr om C or nw ell and R uper t W age M odel
.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
Part 4: Fixed Effects [ 19/96]
A Kernel Density Estimator
*

x

x
i
m
ˆf ( x * ) 
K


m
i1
n
h 
h
1
n
1

*
 , fo r a se t o r p o in ts x m

h  " b a n d w id th " ch o se n b y th e a n a lyst. A c o m m o n
ch o ice is S ilv e rm a n 's ru le o f th u m b = 1 .0 6 ˆ x /n
1/ 5
K  th e k e rn e l fu n ctio n , su ch a s th e n o rm a l
o r lo g istic d e n sity (o r o n e o f se v e ra l o th e rs)
x*  th e p o in t a t w h ich th e d e n sity is a p p ro xim a te d .
Part 4: Fixed Effects [ 20/96]
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
.0 0 0
0
1
2
3
4
5
6
AI
Ke rn e l d e n s i ty e s ti m a te fo r
AI
Mean = 4.819, standard deviation = 1.054.
7
Part 4: Fixed Effects [ 21/96]
Histogram vs. KDE
Fixed E ffects fr om C or nw ell and R uper t W age M odel
Fixed Effects from Cornwell and Rupert Wage Model
.3 4 5
.2 7 6
D e n s i ty
Fre que nc y
.2 0 7
.1 3 8
.0 6 9
.0 0 0
0
1
2
3
4
5
6
7
AI
.8 5 6
1 .6 8 8
2 .5 2 0
3 .3 5 1
4 .1 8 3
5 .0 1 5
5 .8 4 7
6 .6 7 8
Ke rn e l d e n s i ty e s ti m a te fo r
AI
AI
CREATE
; ID=TRN(7,0)$
SETPANEL ; GROUP=ID ; PDS=TI $
REGRESS ;lhs=lwage;rhs=occ,smsa,ms,exp ; panel ; fixed $
? Creates 595 by 1 matrix named ALPHAFE
HISTOGRAM; rhs=alphafe ;title=Fixed Effects from Cornwell and Rupert Wage Model$
KERNEL;rhs=alphafe ; title=Fixed Effects from Cornwell and Rupert Wage Model$
Part 4: Fixed Effects [ 22/96]
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 =

"W ithin S um of S quares"

=
N
i 1
N
i 1
W hich should appear in the denom inator o f R

Ti

Ti
t 1
t 1
 y it
 y
 y it
 yi 
2
R2 = 0.90542 areg
2
R2 = 0.65142 xtreg fe
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 R reported is obtained by only fitting a
2
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.
Part 4: Fixed Effects [ 23/96]
Robustness of the LSDV Estimator



Under the full Gauss-Markov assumptions, b is
unbiased and consistent (and even efficient).
If Var[εi] = Ωi ≠ε2ITi then b is consistent but
inefficient. (We’ll return to robust estimation
below.)
Under all assumptions, Var[ai] is O(1/Ti).
ai is unbiased but inconsistent.

Inconsistent not because it estimates the wrong
parameter, but because it converges to a random
variable, not a constant. Ti is not increasing.
Part 4: Fixed Effects [ 24/96]
Robust Counterpart to White Estimator?
Assumes Var[εi] = Ωi ≠2ITi
ei = yi – aiiTi - Xib = MDyi – MDXib
(Ti x 1 vector of group residuals)
N
i
E st.A sy.V ar[ b ]=   i= 1 X iM D X i 
H
1
1
  Ni= 1 ( X iM Di e i )( e iM Di X i )    Ni= 1 X iM Di X i 



 N
Ti


 i= 1 t= 1 ( x it  x i )e it



  1
T
 t=i 1 ( x it  x i )e it  H


T
N
H    i= 1  t=i 1 ( x it  x i )( x it  x i )  


Resembles (and is based on) White, but treats a full vector of
disturbances at a time. Robust to heteroscedasticity and
autocorrelation (within the groups).
1
Part 4: Fixed Effects [ 25/96]
Robust Covariance Matrix for LSDV
Cluster Estimator for Within Estimator
Part 4: Fixed Effects [ 26/96]
Asymptotics for ai
1
a i  ( d i d i ) d i ( y i  X ib ) fro m th e L S n o rm a l e q u a tio n s
= y i  x ib
= (y i  x iβ ) - x i ( b - β )
=  i + i - x i ( b - β )
E [a i | X ]   i  0  0 =  i ( b is u n b ia se d )
2
V a r[a i | X ]  0    / T i  x i V a r[( b - β ) | X ] x i
lim N   E [a i | X ]   i
2
lim N   V a r[a i | X ]    / T i + 0 ( b is co n siste n t so V a r[ ( b - β ) | X ]  0 )
2
(S e e slid e 1 3 , lim N  


N
i 1
Ti
= 0)
Part 4: Fixed Effects [ 27/96]
LSDV is an IV Estimator
y it  x it β  c i + ε it
 x it β  ( c i + ε it )
 x it β  w it
C ov [ x it , w it ]  C ov [ x it , ( c i + ε it )]  g ( x it )  0
x it is corre la te d w ith th e F E s e m b e d d e d in w it
Part 4: Fixed Effects [ 28/96]
y it  x it β  c i + ε it
(1 ob se rv a tion )
y i  X iβ  c i d i + ε i (T i ob se rv a tion s)
y i  X iβ  w i
y
(  i T i ob se rv a tion s)
 Xβ  w
p lim ( b )= p lim  X X 
1
X y
 X X 
= β  p lim  N

  i= 1 T i 
 X X 
= β  p lim  N

  i= 1 T i 
 X X 
= β  p lim  N


T
 i= 1 i 
 X X 
= β  p lim  N


T
 i= 1 i 
N ote
Ti
N
 i= 1 T i
=
1
N
1
1
1
 1
p lim  N
  i= 1 T i

N
i= 1


x it c i 
t= 1

Ti
 N
 1
Ti
p lim  
c
i 
N
i= 1

T
 Ti
i=
1
i


 t  1 x it  

Ti
 N

Ti
p lim  
c
x
i i. 
N
i= 1

T
i= 1 i


1
N
p lim   fi c i x i. 
 i= 1

if b a la n ce d p a n e l
N
0 < fi < 1 ,  i= 1 fi  1
Part 4: Fixed Effects [ 29/96]
MD
 M 1D

0
 


 0
0 

0 
(T h e d u m m y variab les are orth og on al)

N 
MD 
0
2
MD
0
1
i
M D  I T  d i ( d id i ) d i = I T  (1 /T i ) d i d i
i
i
N
i
X M D X =  i=1 X iM D X i ,
N
i
X M D y =  i=1 X iM D y i ,

i
X iM D X i
 X M
i
i
yi
D

k,l

k
T
  t=i 1 ( x it,k -x i.,k )( x it,l -x i.,l )
T
  t=i 1 ( x it,k -x i.,k )( y it -y i. )
Part 4: Fixed Effects [ 30/96]
b LS D V   X M D X 
1
X M D y
D e fin e Z = M D X .
b LS D V
  Z X 
1
Z y (L o o k s lik e a n IV e stim a to r.)
 Z X 
(1 ) P lim 
  0?
 Σ i Ti 

i
P lim   X iM D X i


   T i ( x -x )( x -x )
t= 1
it,k
i.,k
it,l
i.,l
k,l 
N o n sin g u la r P D m a trix if th e r e is n o m u ltico llin e a rity a n d if
e v e ry co lu m n o f X h a s w ith in g ro u p v a ria t io n .
 Z w 
(2 ) P lim 
= 0?
 Σ i Ti 
 Σ i X iM Di ( c ii   i ) 
 Z w 
P lim 

  P lim 
Σ
T
Σ
T
i i
 i i 


i
c iM D i = 0 b e ca u se i h a s n o w ith in g ro u p v a ria tio n
1
Σ i Ti
P lim
 Σ X M   
i
i
i
D
i
0 b y th e a ssu m p tio n o f th e m o d e l.
Part 4: Fixed Effects [ 31/96]
LSDV is a Control Function Estimator
y it  x it β  c i + ε it
 x it β  (c i + ε it )
 x it β  w it
C ov [ x it , w it ]  C ov [ x it , (c i + ε it )]  g ( x it )  0
x it is corre la te d w ith th e F E s e m b e d d e d in w it .
L S re g re ssion of y on X is in con siste n t b e ca u se X is
corre la te d w ith w . W e se e k a con trol fu n ction h (.) su ch th a t
X | h (.) is u n corre la te d w ith w . (In th e p re se n ce of h (.), X is
n ot corre la te d w ith w .)
U sin g th e F risch -W a u g h th e ore m
b
= [ X M D X ]
1
 X M D y 
C on sid e r re g re ssion of y on [ X , X ]. I.e ., a d d g rou p
m e a n s to th e re g re ssion .
Part 4: Fixed Effects [ 32/96]
LSDV is a Control Function Estimator
C on sid er reg ression of y on [ X , X ]. I.e., ad d g rou p
m ean s to th e reg ression .
 x 11

x 21

[ X ,X ] 


 x N1
x 12
x 1K
x 1 1 .i1
x 1 2 .i1
x 22
x 2K
x 2 1 .i 2
x 2 2 .i 2
x N2
x NK
x N1 .iN
x N 2 .iN
= [X , (I-M D ) X ]
= [ X , PD X ]
= [ X ,F ]
x 1 1 i1 

x 1 1i2



x N K .iN 
Part 4: Fixed Effects [ 33/96]
LSDV is a Control Function Estimator
U sin g th e F risch -W a u g h th e ore m
b C on trolF u n ction
= [ X M F X ]
1
 X M F y 
1
X M F X  X [ I  F ( F F ) F ] X
 X [ I  PD X ( X PD PD X )
1
X PD ] X
PD is sym m e tric a n d id e m p ote n t. A n d PD = I - M D
 X [ I  ( I - M D ) X ( X ( I - M D ) X )
1
X ( I - M D ) ] X
M u ltip ly th is ou t in fu ll a n d colle ct so m e te rm s
= X IX -
 X ( I - M D ) X   X ( I - M D ) X 
1
X ( I - M D ) X
T h e tw o la rg e m a trice s ca n ce l. O n e m ore ste p
= X X - X ( I - M D ) X = X X - X X + X M D X
= X M D X . Like w ise ,  X M F y    X M D y  . T h e re fore ,
b C on trolF u n ction
= b LS D V
Part 4: Fixed Effects [ 34/96]
Note the usual problem with control function estimators. The standard
errors need to be corrected.
Part 4: Fixed Effects [ 35/96]
The problem here is the estimator of the disturbance variance. The matrix is OK.
Note, for example, .01374007/.01950085 (top panel)
= .16510 /.23432 (bottom panel).
Part 4: Fixed Effects [ 36/96]
Part 4: Fixed Effects [ 37/96]
Generalized Least Squares?
If Var[εi] = Ωi ≠ε2ITi then b is consistent but
inefficient.
ˆ = [ X Ω -1 X ]  1 [ X Ω -1 y ]
G LS : β
N
-1
-1
N
-1
= [  i= 1 X i Ω i X i ] [  i= 1 X i Ω i y i ]
E stim a te Ω ?
N
(1 ) B a la n ce d pa n e l ca se : (1 /N )  i= 1 e i e i fro m fix e d e ffe cts
(2 ) U n ba la n ce d ca se ? P u t ze ro s in e i in a ppro pria te pla ce s?
 E le m e n ts o f Ωˆ a re n o w ba se d o n diffe re n t T . 
i
ˆ is T x T w ith ra n k a t m o st N . If T > N , Ω
ˆ is
N o te Ω
sin gu la r a n d G LS ca n n o t be co m pu te d. N w ill be > > T .
Part 4: Fixed Effects [ 38/96]
Maximum Likelihood Estimation
W ith n orm a lly d istrib u te d d istu rb a n ce s, th e F E m od e l is th e
ord in a ry cla ssica l n orm a l lin e a r re g re ss ion m od e l. O L S is th e
m a x im u m lik e lih ood e stim a tor of β . T h e m a x im u m lik e lih ood
2
e stim a tor of   is
T
N
2

ˆ 
2
 i  1  t i 1 e it
Ti
 t 1 Ti
, th e u su a l m e a n sq u a re d re sid u a l, w ith n o
corre ction for d e g re e s of fre e d om . F rom sta n d a rd re su lts for
th e lin e a r m od e l (e .g ., G re e n e , p . 5 1 ), th e e x a ct e x p e cta tion is
2
2
E[ 
ˆ ]  
 (  Ni  1 T i )  N  K 

1
K 
1 N  K
2
2 


1




1




 
 

N
N

T

T
T
T
 N

i1 i
i1







Part 4: Fixed Effects [ 39/96]
ML Estimation (cont.)
2
2
E[ 
ˆ ]  
 (  Ni  1 T i )  N  K 

1
K 
1  N  K 
2
2 


1




1




 
 


N
N
N

T

T
T
T



i1 i
i1




2
T h is is a 're g u la r' p rob le m , so 
ˆ  con v e rg e s to a
p rob a b ility lim it - it is con siste n t for som e th in g . N ote , a s
2
2
N in cre a se s, 
ˆ  con v e rg e s to   [1 - 1 /T ]. T (or T i ) is
2
fix e d in th is m od e l. S o, 
ˆ  is n o t a con sist e n t e stim a tor
2
of   u n le ss T in cre a se s. S u p p ose T i  2 . T h e n
2
2
p lim 
ˆ 

2
. T h e in con siste n cy d oe s n o t g o a w a y a s N
in cre a se s. T h is is T H E e x a m p le of th e I n c id e n ta l P a ra m e te rs
P ro b le m . (N e y m a n a n d S cott (1 9 4 8 ). It occu rs b e ca u se th e
n u m b e r of p a ra m e te rs b e in g e stim a te d is g row in g a s N g row s.
Part 4: Fixed Effects [ 40/96]
Between Groups Estimator
Inconsistency of the group means estimator
y i  x iβ  c i + ε i
= x iβ  w i
C o v [w i , x i ]  C o v [c i + ε i , x ]
 0
Part 4: Fixed Effects [ 41/96]
Time Invariant Regressors



Time invariant xit is defined as invariant for all i.
E.g., SEX dummy variable. ED (education in the
Cornwell/Rupert data).
If xit,k is invariant for all i, then xit,k = ihidi for
the set of dummy variables and some set of his.
If xit,k is invariant for all i, then the group mean
deviations are all 0.
Part 4: Fixed Effects [ 42/96]
FE With Time Invariant Variables
+----------------------------------------------------+
| There are 2 vars. with no within group variation. |
| FEM
ED
|
+----------------------------------------------------+
+--------+--------------+----------------+--------+--------+----------+
|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).......
+--------------------------------------------------------------------+
|
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 |
+--------------------------------------------------------------------+
Part 4: Fixed Effects [ 43/96]
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
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
Part 4: Fixed Effects [ 45/96]
Introduction: The Pledge
[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.
This article discusses a remedy to the related
problems of estimating time invariant and rarely
changing variables in FE models with unit
effects
Part 4: Fixed Effects [ 46/96]
The Model
y it = α i +

K
k =1
β k x kit +

M
m =1
 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 cts.
Part 4: Fixed Effects [ 47/96]
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
Part 4: Fixed Effects [ 48/96]
Step 2
Regress ai on zi and compute residuals
ai =

M
m =1
 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 .
Part 4: Fixed Effects [ 49/96]
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 kit +

M
m =1
 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.
Part 4: Fixed Effects [ 50/96]
The Turn:
Based on Cornwell and Rupert
namelist ; x = exp,wks,occ,ind,south,smsa,union
; z = fem,ed $
(1) Step 1.
regress ; lhs=lwage;rhs=x,z;panel;fixed;pds=7 $
create
; uhi = alphafe(_stratum) $
(2) Step 2
regress ; lhs = uhi ; rhs = one,z ; res = hi $
(3) Step 3.
regress ; lhs = lwage ; rhs = one,x,z,hi $
Part 4: Fixed Effects [ 51/96]
Step 1 (Based on full sample)
These 2 variables have no within group variation.
FEM
ED
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
--------+---------------------------------------------------------
Part 4: Fixed Effects [ 52/96]
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
--------+---------------------------------------------------------
Part 4: Fixed Effects [ 53/96]
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
HI|
1.00000***
.00670
149.26 .0000 -.103D-13
--------+---------------------------------------------------------
Part 4: Fixed Effects [ 54/96]
Part 4: Fixed Effects [ 55/96]
Part 4: Fixed Effects [ 56/96]
http://davegiles.blogspot.com/2012/06/fixed-effects-vector-decomposition.html
Part 4: Fixed Effects [ 57/96]
Paul Allison, 2005
Part 4: Fixed Effects [ 58/96]
What happened here?
y it = α i +

K
k =1
β k x k it +

M
m =1
 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 x a 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  . It is n o t p a rt o f th e o rig in a l m o d e l.
Part 4: Fixed Effects [ 59/96]
Two Way Fixed Effects

A two way FE model. Individual dummy variables and time
dummy variables.







yit = αi + t + xit’β + εit
Normalization needed as the individual and time dummies both
sum to one. Reformulate model:
yit = μ + αi* + t* + xit’β + εit with
i αi* =0, t t* = 0
y it  y it  y i .  y. t  y
Full estimation:
Practical estimation. Add T-1 dummies
Complication: Unbalanced panels are complicated
Complication in recent applications: Vary large N and very large T
Part 4: Fixed Effects [ 60/96]
Fixed Effects Estimators
Slope estimators, as usual with transformed data
ˆ = y- x b
μ
ˆ
α i *  ( y i .  y )  ( x i .  x ) b
ˆ t *  ( y.t  y )  ( x . t  x ) b
Part 4: Fixed Effects [ 61/96]
Two Way Fixed Effects Application
Spanish Dairy Farms; N=247, T=6
+--------+--------------+----------------+--------+--------+----------+
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
+--------+--------------+----------------+--------+--------+----------+
No Effects
Constant|
11.5774868
.00364586 3175.515
.0000
X1
|
.59517558
.01958331
30.392
.0000
0
X2
|
.02305014
.01122274
2.054
.0400
0
X3
|
.02319244
.01303099
1.780
.0751
0
X4
|
.45175783
.01078465
41.889
.0000
0
Firm Dummies
X1
|
.66200103
.02467845
26.825
.0000
0
X2
|
.03735244
.01613309
2.315
.0206
0
X3
|
.03039947
.02320776
1.310
.1902
0
X4
|
.38251038
.01201690
31.831
.0000
0
Firm and Time Dummies
X1
|
.63796531
.02379854
26.807
.0000
0
X2
|
.04127557
.01544463
2.672
.0075
0
X3
|
.02819226
.02217322
1.271
.2036
0
X4
|
.30816028
.01322571
23.300
.0000
0
REGRESS ; Lhs = yit ; Rhs = one,x1,x2,x3,x4 ; pds=6 ; period=t $
Marginal changes in the estimates.
Why?
Part 4: Fixed Effects [ 62/96]
Analysis of Variance (FIT)
+--------------------------------------------------------------------+
|
Test Statistics for the Classical Model
|
+--------------------------------------------------------------------+
|
Model
Log-Likelihood Sum of Squares
R-squared |
|(1) Constant term only
-1448.90832 .6131518321D+03
.0000000 |
|(2) Group effects only
412.25944 .4974526192D+02
.9188696 |
|(3) X - variables only
809.67611 .2909570093D+02
.9525473 |
|(4) X and group effects
1751.64437 .8161093811D+01
.9866899 |
|(5) X ind.&time effects
1826.23878 .7379537558D+01
.9879646 |
+--------------------------------------------------------------------+
|
Hypothesis Tests
|
|
Likelihood Ratio Test
F Tests
|
|
Chi-squared
d.f. Prob.
F
num. denom.
P value |
|(2) vs (1) 3722.336
246 .00000
56.859
246
1235
.00000 |
|(3) vs (1) 4517.169
4 .00000 7412.185
4
1477
.00000 |
|(4) vs (1) 6401.105
250 .00000 365.021
250
1231
.00000 |
|(4) vs (2) 2678.770
4 .00000 1568.114
4
1231
.00000 |
|(4) vs (3) 1883.937
246 .00000
12.836
246
1231
.00000 |
|(5) vs (4)
149.189
5 .00000
25.969
5
1226
.00000 |
|(5) vs (3) 2033.125
252 .00000
14.317
252
1226
.00000 |
+--------------------------------------------------------------------+
Part 4: Fixed Effects [ 63/96]
Unbalanced Panel Data
(First 10 households in healthcare data)
Ti
zi .   t  1 z it
Nt
z.t   i 1 z it
Part 4: Fixed Effects [ 64/96]
Two Way FE with Unbalanced Data
T h is co m p u ta tio n is n o t a p p ro p ria te in t w o w a y F E
m o d e ls w ith u n b a la n ce d p a n e ls:
N
b=   
 i1
 N
  i1 
ˆ
μ
Ti
t 1
Ti
t 1
( x it - x i . - x .t  x )( x it - x i . - x .t  x )  

1
( x it - x i . - x .t  x )( y it - y i . - y. t  y ) 

= y - x b
ˆ
α i *  ( y i .  y )  ( x i .  x ) b
ˆ t *  ( y.t  y )  ( x . t  x ) b
T h e m o d e l m u st b e fit a s a o n e w a y F E M w ith tim e
d u m m y v a ria b le s

Part 4: Fixed Effects [ 65/96]
y it  y it  y i .  y.t  y an d likew ise for x it .
D oes n ot w ork correctly for u n balan ced p an els.
Fit tw o w ay m odels as on e w ay w ith tim e du m m ies.
Part 4: Fixed Effects [ 66/96]
Textbook formula application. This is incorrect.
Two way fixed effects as one way with time dummies
Part 4: Fixed Effects [ 67/96]
Different Normalizations



Separate constants: using D
Overall constant and N-1 constrasts
Overall constant, N constants, i i = 0
y = X β + D α+ ε
= X β+ C α * + ε
1
= y , s o C α * = D α = (D P )(P α )
Part 4: Fixed Effects [ 68/96]
Renormalizing Fixed Effects
N Dummy Variables vs. a Constant and N-1 Dummy
Variables
U se 4 g ro u p s fo r e x a m p le
i

0
D  
0

0
i

i
C  
i

i
0
0
i
0
0
i
0
0
0
0
i
0
0
i
0
0
0

0

0

i
0
1


0
1
 D
1
0


i
1
P
0
0
1
0
0
1
0
0
0

0
  DP
0

1
1
1

1


1

1
0
0
1
0
0
1
0
0
0

0

0

1
-1
P a = α 1 , α 2  α 1 , α 3  α 1 , ..., α N  α 1
Implication: No change in other coefficients, no change in sum of squares or R2
Part 4: Fixed Effects [ 69/96]
A “Hierarchical” Model
L o w e r le v e l stru ctu ra l m o d e l
y it  x it β  c i + ε it
U p p e r le v e l m o d e l fo r e ffe cts
c i  z i δ + w i
H o w d o e s th is a ffe ct th e fix e d e ffe cts m o d e l?
y it  x it β  α i + ε it
N o ch a n g e in th e m o d e l, b u t it in v ite s a se co n d ste p .
Part 4: Fixed Effects [ 70/96]
Estimating a Hierarchical Model

Classical assumptions at both levels
2
y it  x it β  c i + ε it , E [ε it | X i , c i ]  0 , V a r[ε it | X i , c i ]=   , e tc.
2
c i  z i δ + w i , E [w i| z i ]  0 , V a r[w i| z i ]=  w


Two step estimation
Fixed effects, dummy variables at top level

Regress ai on zi to estimate δ at the 2nd level. The regression
is heteroscedastic. Use OLS/White or Weighted LS with
ˆ i  c i )  c i  v i  z i δ  (w i  v i )
a i  c i  (c
2
i
i
A sy.V a r[a i | X i , z i ]  [  / T i ]  x it ( I - M D ) A sy.V a r[ b ]( I  M D ) x it
2
= [  / T i ]  x i A sy.V a r[ b ] x i
Part 4: Fixed Effects [ 71/96]
A Two Step Regression
Sample ; all$
Create ; person=trn(7,0) ; year=trn(-7,0)$
Namelist; varyingX=occ,smsa,ms,exp$
Namelist; fixedX=one,fem,ed$
? FE regression to compute dummy variable coefficients
Regress ; lhs=lwage ; rhs=varyingX ; panel ; fixed ; pds=7$
Create ; ai=alphafe(person)$
Create ; occb= GroupMean(occ,pds=7)$
Create ; msb = GroupMean(ms,pds=7)$
Create ; smsab=GroupMean(smsa,pds=7)$
Create ; expb= GroupMean(exp,pds=7)$
? Standard errors for dummy variable coefficient estimates
Namelist; means=occb,smsab,msb,expb$
Create ; varai=ssqrd/_Groupti + qfr(means,varb) ; wt=1/varai$
? Weighted least squares regression of dummy variable coefficients
? on time invariant variables.
Regress ; if[year = 7] ; lhs=ai;rhs=FixedX;wts=wt$
Regress ; if[year = 7] ; lhs=ai;rhs=FixedX;Het $
Part 4: Fixed Effects [ 72/96]
First Stage Fixed Effects Model
Part 4: Fixed Effects [ 73/96]
Second Stage Regressions
Weighted Least Squares
OLS with White Estimator
Part 4: Fixed Effects [ 74/96]
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 4: Fixed Effects [ 75/96]
Hierarchical Linear 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 + w i ,
2
E [w i| z i ]  0 , V a r[w i| z i ]   w
y it  x it β  [   z i δ  w i ]  ε it
Part 4: Fixed Effects [ 76/96]
HLM (Simulation Estimator) vs. REM
---------+ Nonrandom parameters
OCC
|
-.02461285
.00566374
-4.346
.0000
.51116447
SMSA
|
-.06076787
.00490494
-12.389
.0000
.65378151
MS
|
-.04446541
.00850068
-5.231
.0000
.81440576
EXP
|
.08508257
.00046901
181.409
.0000
19.8537815
---------+ Means for random parameters
Constant|
2.89358963
.02426391
119.255
.0000
---------+ Scale parameters for dists. of random parameters
Constant|
.86092728
.00448368
192.014
.0000
---------+ Heterogeneity in the means of random parameters
cONE_FEM|
-.54972521
.01030773
-53.331
.0000
cONE_ED |
.16915125
.00122320
138.286
.0000
========================================================================
---------+Variance parameter given is sigma
Std.Dev.|
.15681703
.00074231
211.256
.0000
(REM Estimated by two step FGLS) Sigma(u) = 0.3303
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 4: Fixed Effects [ 77/96]
Mundlak’s Approach
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  + 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   w i  ε it
= ra n do m e ff e cts m o de l in clu din g gro u p m e a n s.
Part 4: Fixed Effects [ 78/96]
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
Estimates: Var[e]
=
.0235632
Var[u]
=
.0773825
Part 4: Fixed Effects [ 79/96]
Application
Passmore,W. et al., “The Effect of Housing
Government Sponsored Enterprises on
Mortgage Rates,”
Federal Reserve Board, Division of Research &
Statistics and Monetary Affairs, 2004, rev.
1/2005
Part 4: Fixed Effects [ 80/96]
First Stage – Rate Difference
M o rtga ge R a te i,t   0 + " lo a n to va lu e ra tio te rm s "
+ "n e w h o m e " (du m m y va ria ble )
+ "sm a ll lo a n " (du m m y va ria ble )
+ "u p fro n t fe e s pa id" (du m m y va ria ble )
+ "m o rtga ge ba n k" vs. de po sito ry in st. (du m m y va ria ble )
+ α 1 ,i J u m b o L o a n i,t( d u m m y v a ria b le fo r lo a n > $ 31 7 , 0 0 0 )
+  i,t
" i " = sta te ,ye a r gro u pin g
"t"= in dividu a l lo a n in spe cifie d sta te ,ye a r
N e a rly a ll "co n fo rm in g" lo a n s (u n de r $ 3 1 7 ,0 0 0 ) a re h e ld by Fa n n ie M a e .
E x pe ct  1 ,i to be > 0 a s Fa n n ie M a e is a ble to fin a n ce a t lo w e r co st th a n
o th e r in stitu tio n s, a n d Fa n n ie M a e do e s n o t fin a n ce Ju m bo lo a n s. In te re st
is in "pa ss th ro u gh " o f th e co st a dva n ta ge .
Part 4: Fixed Effects [ 81/96]
An Algebraic Aspect
Ji is not quite a group dummy variable. For the group, Ji
is one for some members of the group – those with a
“jumbo” mortgage.
i
M J  I  J i ( J i J i ) J i
i
M J y i  y i  J i * m e a n o f th o se w ith ju m bo lo a n s
y it
 y it  y i, ju m bo if ju m bo lo a n 
 

if n o t a ju m bo lo a n 
 y it
O th e rw ise , th is co u ld be tre a te d like a fixe d e ffe cts m o de l.
Part 4: Fixed Effects [ 82/96]
Second Stage – Pass Through
a 1 ,i =  0 +  1 "E stim a te d C a pita l C o st A dva n ta ge "
+ "m a rke t ch a ra cte ristics"
+ "sta te " a n d "qu a rte r" du m m y va ria ble s
+ wi
P rim a ry in te re st is in  1 w h ich is th e a m o u n t o f th e
ca pita l co st a dva n ta ge th a t is pa sse d th ro u gh to
m o rtga ge e s.
R e su lt: Le ss th a n h a lf o f co st a dva n ta g e w a s pa sse d
th ro u gh to bo rro w e rs.
Part 4: Fixed Effects [ 83/96]
Time Varying Fixed Effects
911
Rescue
Part 4: Fixed Effects [ 84/96]
Need for Clarification
Part 4: Fixed Effects [ 85/96]
Time Varying Fixed Effects
Part 4: Fixed Effects [ 86/96]
Munnell State Production Model
Part 4: Fixed Effects [ 87/96]
No Effects
Part 4: Fixed Effects [ 88/96]
Quadratic Fixed Effects
Correct DF: 816-6-3(48)=666
Multiply standard errors by sqr(810/666) = 1.103
Part 4: Fixed Effects [ 89/96]
Appendix II. Fixed Effects Algebra
Part 4: Fixed Effects [ 90/96]
Panel Data Algebra
MD
 M 1D

0




 0
0 

0 
(T h e du m m y va ria ble s a re o rth o go n a l)


N
M D 
0
2
MD
0
i
1
M D  I T  d i ( d i d i ) d  = I T  (1 /T i ) d i d 
i
i
N
i
X M D X =  i=1 X iM D X i ,
N
i
X M D y =  i=1 X iM D y i ,
b LS D V
1
  X M D X 


i
X iM D X i
i
X iM D y i
X M D y


T
k,l
  t=i 1 ( x it,k -x i.,k )( x it,l -x i.,l )
T
k
  t=i 1 ( x it,k -x i.,k )( y it -y i. )
Part 4: Fixed Effects [ 91/96]
Balanced Panel Data Algebra
MD
M D , T

 0
 

 0

0 

0 
 (E a c h m a trix is T  T )

M D , T 
0
M D,T
0
1
M D , T  I T  d ( d  d ) d  = I T  (1 /T ) d d 
MD
1  M D,T

0  M D,T
 

0  M D,T

N o te : d d 
0  M D,T
1  M D,T
0  M D,T
0  M D,T 

0  M D,T 
  IN  M D,T

1  M D , T 
= a m a trix o f o n e s is B a lta g i's J T ;
(1 /T ) d d  = a m a trix o f 1 /T is h is J T
Part 4: Fixed Effects [ 92/96]
Balanced Panel
1
M D , T  I T  d ( d  d ) d  = I T  (1 /T ) d d  = I T  J T
1  J T

0  JT

P is


 0  J T
1  J T

0  JT
PX = 


 0  J T
0  JT
1  JT
0  JT
0  JT
1  JT
0  JT
0  JT 

0  JT 
; P X cre a te s g ro u p m e a n s


1  J T 
0  JT 

0  JT 


1  J T 
X1  X1 


 
X2

  X2 


 


 
X
 N   X N 
T h is is T ro w s e a ch
w ith m e a n s re p e a te d .
Part 4: Fixed Effects [ 93/96]
Balanced Panel
I T

0

INT =


0
INT
0
IT
0
0 
1  I T


0
0  IT
 = 




IT 
0  I T
 1  I T - 1  JT

0  I T  0  JT

-P =


 0  I T  0  J T
0  IT
1  IT
0  IT
0  I T  0  JT
1  I T - 1  JT
0  I T  0  JT
IN  I T  IN  JT  IN  I T - JT   Q


0  IT 

0  IT
 =I  I
N
T


1  IT 
0  I T  0  JT 

0  I T  0  JT 


1  I T - 1  J T 
Part 4: Fixed Effects [ 94/96]
Balanced Panel
1  J T

0  JT

PX =


 0  J T
0  JT
1  JT
0  JT
0  JT 

0  JT 


1  J T 
X1  X1 


 
X2
X

   2


 


 
X
 N   X N 
T h is is T ro w s e a ch
w ith m e a n s re pe a te d.
P X = I  J  X


X1  X1 


 
X2
  X2 
Q X = X - I  J  X = 




 


 
 X N   X N 
T h is is T ro w s e a ch
w ith m e a n de via tio n s.
Part 4: Fixed Effects [ 95/96]
Balanced Panel
Q X = X - IN  JT  X


= INT X - IN  JT  X


=  I N  I T  X -  I N  J T  X


X1  X1 


 
X2
X
   2
= 


 


 
 X N   X N 
1
b L S D V   X Q X 
T h is is T ro w s e a ch
w ith m e a n d e v ia tio n s
X Q y
 QX
Part 4: Fixed Effects [ 96/96]
Balanced Panel
 JT

0
P is 


0
0 

0 
; P X cre a te s g rou p m e a n s


JT 
0
JT
0
I - JT

0
Q is 


 0
0
I - JT
0


0 
1


 IN  I d d 

T



I - JT 
0
H om e w ork:
(1) V e rify th a t b oth P a n d Q a re id e m p ote n t
(2 ) S h ow th a t P Q = 0
(3 ) W h a t is th e tra ce o f Q ?