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
ci is uncorrelated with xit for all t; E[ci |Xi] = 0
E[εit|Xi,ci]=0
it it i it
i i i i i
i i i i i i i
Ni=1 i
1 2 N
y = +c+ε , observation for person i at time t
= +c + , T observations in group i
= + + , note (c ,c ,...,c )
= + + , T observations in the sample
c=( , ,... ) ,
x β
y Xβ i ε
Xβ c ε c
y Xβ c ε
c c c Ni=1 iT by 1 vector
Part 5: Random Effects [ 3/54]
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 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
it it i
it i
2 2it i
i i
2 2i i u
i i i i
y +ε +u
E[ε | ] 0
E[ε | ] σ
E[u | ] 0
E[u | ] σ
= + +u for T observations
it
i
x b
X
X
X
X
y Xβ ε i
2 2 2 2u u u
2 2 2 2u u u
i i
2 2 2 2u u u
Var[ +u ]ε i
Part 5: Random Effects [ 5/54]
Notation
1 1 1
2 2 2
N N N
i
u T observations
u T observations
u T observations
= + + T observations
= +
1 1 1
2 2 2
N N N
Ni=1
y X ε i
y X ε iβ
y X ε i
Xβ ε u
Xβ w
In all that fo
i
it
it
llows, except where explicitly noted, X, X
and x contain a constant term as the first element.
To avoid notational clutter, in those cases, x etc. will
simply denote the counterpart without the constant term.
Use of the symbol K for the number of variables will thus
be context specific but will usually include the constant term.
Part 5: Random Effects [ 6/54]
Notation
2 2 2 2u u u
2 2 2 2u u u
i i
2 2 2 2u u u
2 2u i i
2 2u
i
1
2
N
Var[ +u ]
= T T
=
=
Var[ | ]
i
i
T
T
ε i
I ii
I ii
Ω
Ω 0 0
0 Ω 0w X
0 0 Ω
2 2u N
i
(Note these differ only
in the dimension T ) TI ii I
Part 5: Random Effects [ 7/54]
Regression Model-Orthogonality
iN Ni 1 i i 1 i
i i iNi 1 i i i
ii i i i N
i i i 1 i
1plim
#observations1 1
plim plim ( +u )T T
1plim T + Tu
T T T
Tplim f + f u , 0 < f < 1
T T T
pli
N Ni=1 i i i=1 i i
N Ni i i ii=1 i=1
N Ni i i ii=1 i=1
X'w 0
X w X ε i 0
Xε Xi
Xε Xi
i i i i i
i
1m f + f u = if T T i
T NN Ni ii=1 i=1
Xεx 0
Part 5: Random Effects [ 8/54]
Convergence of Moments
Ni 1 iN
i 1 i
Ni 1 iN
i 1 i
N Ni 1 i u i 1 i
i
f a weighted sum of individual moment matricesT T
f a weighted sum of individual moment matricesT T
= ffT
Note asymptoti
i i
i i i
2 2ii ii i
X XXX
XΩXXΩX
X Xx x
i
i
i
cs are with respect to N. Each matrix is the T
moments for the T observations. Should be 'well behaved' in micro
level data. The average of N such matrices should be likewise.
T or T is assum
ii iX X
ed to be fixed (and small).
Part 5: Random Effects [ 9/54]
Ordinary Least Squares Standard results for OLS in a GR model
Consistent Unbiased Inefficient
True Variance
1 1
N N N Ni 1 i i 1 i i 1 i i 1 i
Var[ | ]T T T T
as N with our convergence assumptions
-1 -1
1 XX XΩX XXb X
0 Q Q* Q
0
Part 5: Random Effects [ 10/54]
Estimating the Variance for OLS
1 1
N N N Ni 1 i i 1 i i 1 i i 1 i
Ni 1 iN
i 1 i
Ni 1 iN
i 1 i
Var[ | ]T T T T
f , where = =E[ | ]T T
In the spirit of the White estimator, use
ˆ ˆˆf ,
T T
i i ii i i i
i i i ii
1 XX XΩX XXb X
XΩXXΩXΩ w w X
X w w XXΩXw =
Hypothesis tests are then based on Wald statistics.
i iy - Xb
THIS IS THE 'CLUSTER' ESTIMATOR
Part 5: Random Effects [ 11/54]
Mechanics
1 1Ni 1
i
i
ˆ ˆEst.Var[ | ]
ˆ = set of T OLS residuals for individual i.
= TxK data on exogenous variable for individual i.
ˆ = K x 1 vector of products
ˆ ˆ( )( )
i i i i
i
i
i i
i i i i
b X XX X w w X XX
w
X
X w
X w w X
Ni 1
KxK matrix (rank 1, outer product)
ˆ ˆ = sum of N rank 1 matrices. Rank K.i i i iX w w X
Part 5: Random Effects [ 12/54]
Cornwell and Rupert Data
Cornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 YearsVariables in the file are
EXP = work experience, EXPSQ = EXP2
WKS = weeks workedOCC = occupation, 1 if blue collar, IND = 1 if manufacturing industrySOUTH = 1 if resides in southSMSA = 1 if resides in a city (SMSA)MS = 1 if marriedFEM = 1 if femaleUNION = 1 if wage set by unioin contractED = years of educationLWAGE = 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 .0000Robust 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
i
1
N 1 Ni 1 i 1
2u
T2 2 2i u
i
ˆ=[ ] [ ]
=[ ] [ ]
1I
T
(note, depends on i only through T)
-1 -1
-1 -1i i i i i i
-1i
β XΩ X XΩ y
XΩ X XΩ y
Ω ii
Part 5: Random Effects [ 15/54]
Panel Data Algebra (1)
2 2ε u i i
2 2 2 2ε u ε
2 2ε ε
2ε
= σ +σ , depends on 'i' because it is T T
= σ [ ], = σ / σ
= σ [ ] = σ [ ], = , = .
Using (A-66) in Greene (p. 964)
1 1σ 1+
=
i
2i
2i
-1 -1 -1 -1i -1
Ω I ii
Ω I + ii
Ω I + ii A bb A I b i
Ω A - A bbAbA b
22 u
2 2 2 2 2ε i ε ε i u
σ1 1 1=
σ 1+T σ σ +TσI - ii I - ii
Part 5: Random Effects [ 16/54]
Panel Data Algebra (2)
2 2 2 2 2 2ε u ε i u ε i u
2 2ε i u
2 2 2 2 2ε i u i ε ε i u
2 2ε i u
2ε
(Based on Wooldridge p. 286)
σ +σ σ +Tσ σ +Tσ
σ +Tσ ( )
(σ +Tσ )[ ( )], = σ /(σ +Tσ )
(σ +Tσ )[ ]
(σ
-1 ii D
iD
i iD D
i iD D
Ω I ii I i(i i) i I P
I I M
P I P
P M2
i u
1i
1/ 2i i i i
i
a ai
+Tσ )
(1/ ) (Prove by multiplying. .)
1 (1/ ) , =1-
1
(Note )
i
i i i ii D D D D
i i ii D D D
i ii D D
S
S P M P M 0
S P M I P
S P M
Part 5: Random Effects [ 17/54]
Panel Data Algebra (3)
1/ 2i2 2
i u
i2 2ii u
2
i 2 2 2 2i u i u
1/ 2i
1/ 2 2i i
1/ 2i i i i
1 (1/ ) )
T
1 1 ,
1T
=1- 1T T
1 [ ]
Var[ ( u )]
(1/ )( y. ) for the GLS transf
i ii D D
iD
ii D
i
i
Ω (P M
I P
Ω I P
Ω ε i I
Ω y y i ormation.
Part 5: Random Effects [ 18/54]
GLS (cont.)
it it i i it it i i
i 2 2i u
2
GLS is equivalent to OLS regression of
y * y y. on * .,
where 1T
ˆAsy.Var[ ] [ ] [ ]
-1 -1 -1
x x x
β XΩ X X * X*
Part 5: Random Effects [ 19/54]
Estimators for the Variances
i
it it i
OLS
TN 22 2i 1 t 1 it
UNi 1 i
i LSDV
Ni 1
y u
Using the OLS estimator of , ,
(y a ) estimates
T -1-K
With the LSDV estimates, a and ,
it
it
x β
β b
- - x b
b
i
i i
T 22t 1 it
Ni 1 i
T TN 2 N 22i 1 t 1 it i 1 t 1 itUN N
i 1 i i 1 i
(y a )estimates
T -N-K
Using the difference of the two,
(y a ) (y a ) estimates
T -1-K T -N-K
i it
it i it
- - x b
- - x b - - x b
Part 5: Random Effects [ 20/54]
Feasible GLS
i
2 2u
TN 22 i 1 t 1 it i it LSDV
Ni 1 i
2 2u
Feasible GLS requires (only) consistent estimators of and .
Candidates:
(y a )From the robust LSDV estimator: ˆ
T K N
From the pooled OLS estimator: Est( )
x b
i
i i
TN 2i 1 t 1 it OLS it OLS
Ni 1 i
N 22 2 i 1 it i MEANS
u
T 1 TN2 2 i 1 t 1 s t 1 it is
it is i u u
(y a )T K 1
(y a )From the group means regression: Est( / T )
N K 1ˆ ˆw w
(Wooldridge) Based on E[w w | ] if t s, ˆ
x b
xb
X Ni 1 iT K N
There are many others.
x´ does not contain a constant term in the preceding.
Part 5: Random Effects [ 21/54]
Practical Problems with FGLS
2u The preceding regularly produce negative estimates of .
Estimation is made very complicated in unbalanced panels.
A bulletproof solution (originally used in TSP, now NLOGIT and others).
From the r
i
i
i
TN 22 i 1 t 1 it i it LSDV
Ni 1 i
TN 22 2 2i 1 t 1 it OLS it OLS
u Ni 1 i
TN 2 N2 i 1 t 1 it OLS it OLS i 1u
(y a )obust LSDV estimator: ˆ
T
(y a )From the pooled OLS estimator: Est( ) ˆ
T
(y a ) ˆ
x b
x b
x b
iT 2t 1 it i it LSDV
Ni 1 i
(y a )0
T
x b
Part 5: Random Effects [ 22/54]
Stata Variance Estimators
iTN 22 i 1 t 1 it i it LSDV
Ni 1 i
22u
2u
(y a ) > 0 based on FE estimatesˆ
T K N
(N K)SSE(group means) ˆMax 0, 0ˆ
N A (N A)T
where A = K or if is negative,ˆ
A=trace of a matrix that somewhat rese
x b
Kmbles .
Many other adjustments exist. None guaranteed to be
positive. No optimality properties or even guaranteed consistency.
I
Part 5: Random Effects [ 23/54]
Computing Variance Estimators
2 2u
Using full list of variables (FEM and ED are time invariant)
OLS sum of squares = 522.2008.
Est( + ) = 522.2008 / (4165 - 9) = 0.12565.
Using full list of variables and a generalized inverse (same
as
2
2u
2u
dropping FEM and ED), LSDV sum of squares = 82.34912.
= 82.34912 / (4165 - 8-595) = 0.023119.ˆ
0.12565 - 0.023119 = 0.10253ˆ
Both estimators are positive. We can stop here. If were ˆ
negative, we would use estimators without DF corrections.
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
i
2u
it it i it i it 2
20 u
2N 2 N 22i 1 i i 1 i i
N N T 2i 1 i i 1 t 1 it
Breusch and Pagan Lagrange Multiplier statistic
0 0y x u , u and ~ Normal ,
0 0
H : 0
( T ) (T e )LM = 1 [1]
2 T (T 1) e
22 N 2i 1 i
N T 2i 1 t 1 it
T eNT = 1 if T is constant
2(T 1) e
Part 5: Random Effects [ 26/54]
LM Tests+--------------------------------------------------+| Random Effects Model: v(i,t) = e(i,t) + u(i) | Unbalanced Panel| Estimates: Var[e] = .216794D+02 | #(T=1) = 1525| Var[u] = .958560D+01 | #(T=2) = 1079| Corr[v(i,t),v(i,s)] = .306592 | #(T=3) = 825| Lagrange Multiplier Test vs. Model (3) = 4419.33 | #(T=4) = 926| ( 1 df, prob value = .000000) | #(T=5) = 1051| (High values of LM favor FEM/REM over CR model.) | #(T=6) = 1200| Baltagi-Li form of LM Statistic = 1618.75 | #(T=7) = 887+--------------------------------------------------++--------------------------------------------------+| Random Effects Model: v(i,t) = e(i,t) + u(i) || Estimates: Var[e] = .210257D+02 | Balanced Panel| Var[u] = .860646D+01 | T = 7| 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 |+--------------------------------------------------+
REGRESS ; Lhs=docvis ; Rhs=one,hhninc,age,female,educ ; panel $
Part 5: Random Effects [ 27/54]
Testing for Effects: Moments
N T-1 Tdi=1 t=1 s=t+1 it is
2N T-1 Ti=1 t=1 s=t+1 it is
Wooldridge (page 265) suggests based on the off diagonal elements
e eZ= N[0,1]
e e
which converges to standard normal. ("We are not assuming any
particular it
2
distribution for the . Instead, we derive a similar test that
has the advantage of being valid for any distribution...") It's convenient
to examine Z which, by the Slutsky theorem converges (also) to chi-
squared with one degree of freedom.
Part 5: Random Effects [ 28/54]
Testing (2)
T-1 Tt=1 s=t+1 it is
T-1 Tt=1 s=t+1 it is
e e = 1/2 of the sum of all off diagonal elements of
of or 1/2 the sum of all the elements minus the diagonal elements.
e e =1/2[ ( ) ]. But, i i
i i i i
ee
i ee i ee i e
i
T-1 T 2t=1 s=t+1 it is i
2N 2 N 2i 1 i2 i 1N 2 2 N 2 2i 1 i i 1 i
N2i 1 i
i iN 2ri 1 i
= T e. So,
e e = (1/2)[(T e ) ]
[(T e ) ] [ ]2(T 1)Z LM
NT[(T e ) ] [(T e ) ]
Note, also
r N rZ= ,where r (T e )
sr
i
i i
i i i i
i i i i
ee
ee eeee ee
e .i ie
Part 5: Random Effects [ 29/54]
Testing for Effects
? Obtain OLS residualsRegress; lhs=lwage;rhs=fixedx,varyingx;res=e$? Vector of group sums of residualsCalc ; T = 7 ; Groups = 595Matrix ; tebar=T*gxbr(e,person)$? Direct computation of LM statisticCalc ; 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 = .37970675705025540D+04 Z2 = .16533465085356830D+03
Part 5: Random Effects [ 30/54]
Two Way Random Effects Model
it it i t it
2
2 2 2u v
y u v
How to estimate the variance components?
(1) Two way FEM residual variance estimates
(2) Simple OLS residual variance estimates
(3) There are numerous ways to get a thi
x
2 2v
rd equation.
E.g., the one way FEM residual variance in either dimension
One way FEM based on groups estimates ( ) / (1 1 / T)
E.g., the group mean regressions in either dimension.
B 2 2 2u vased on group means estimates ( )/T
(Period means regression may have a tiny number of observations.)
(And a whole library of others - see Baltagi, sec. 3.3.)
Negative estimators of common variances are common.
Solutions are complicated.
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 Random EffectsE[ci|Xi] = 0
Fixed EffectsE[ci|Xi] ≠ 0
FGLS (Random Effects)
Consistent and Efficient
Inconsistent
LSDV(Fixed Effects)
ConsistentInefficient
ConsistentPossibly Efficient
Part 5: Random Effects [ 34/54]
Hausman Test for Effects
-1
d
ˆ ˆBasis for the test,
ˆ ˆˆ ˆ ˆ ˆWald Criterion: = ; W = [Var( )]
A lemma (Hausman (1978)): Under the null hypothesis (RE)
ˆ nT[ ] N[ , ] (efficient)
ˆ nT[
FE RE
FE RE
RE RE
FE
β - β
q β - β q q q
β - β 0 V
β d] N[ , ] (inefficient)
ˆ ˆˆNote: = ( )-( ). The lemma states that in the
ˆ ˆjoint limiting distribution of nT[ ] and nT , the
limiting covariance, is . But, =
FE
FE RE
RE
Q,RE Q,RE FE,R
- β 0 V
q β - β β β
β - β q
C 0 C C - . Then,
Var[ ] = + - - . Using the lemma, = .
It follows that Var[ ]= - . Based on the preceding
ˆ ˆ ˆ ˆ ˆ ˆH=( ) [Est.Var( ) - Est.Var( )] (
E RE
FE RE FE,RE FE,RE FE,RE RE
FE RE
-1FE RE FE RE FE RE
V
q V V C C C V
q V V
β - β β β β - β )
β does not contain the constant term in the preceding.
Part 5: Random Effects [ 35/54]
Computing the Hausman Statistic1
2 Ni 1 i
i
-12
2 N i uii 1 i i 2 2
i i u
2 2u
1ˆEst.Var[ ] IˆT
Tˆ ˆˆEst.Var[ ] I , 0 = 1ˆˆT Tˆ ˆ
ˆAs long as and are consistent, as N , Est.Var[ˆ ˆ
FE i
RE i
F
β X ii X
β X ii X
β
2
ˆ] Est.Var[ ]
will be nonnegative definite. In a finite sample, to ensure this, both must
be computed using the same estimate of . The one based on LSDV willˆ
generally be the better choice.
Note
E REβ
ˆthat columns of zeros will appear in Est.Var[ ] if there are time
invariant variables in .FEβ
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
it i it it
it i it i
it it i
A Fixed Effects Model
y
LSDV estimator - Deviations from group means:
To estimate , regress (y y ) on ( )
Algebraic equivalent: OLS regress y on ( )
Mundlak interpretation:
x
x x
x , x
i i i
it i i it it
i it it i
u
Model becomes y u
= u
a random effects model with the group means.
Estimate by FGLS.
x
x x
x x
Part 5: Random Effects [ 41/54]
Application: Wu Test
NAMELIST ; XV = exp,expsq,wks,occ,ind,south,smsa,ms,union,ed,fem$create ; expb=groupmean(exp,pds=7)$create ; expsqb=groupmean(expsq,pds=7)$create ; wksb=groupmean(wks,pds=7)$create ; occb=groupmean(occ,pds=7)$create ; indb=groupmean(ind,pds=7)$create ; southb=groupmean(south,pds=7)$create ; smsab=groupmean(smsa,pds=7)$create ; unionb=groupmean(union,pds=7)$create ; msb = groupmean(ms,pds=7) $namelist ; 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 REMNamelist ; 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-1
i,Model i,ModelN NModel i 1 i i 1 i
i i
Model
2i u
i,Model 2 2i u
i i,RE
ˆ ˆˆ I IT T
1 for fixed effects.ˆ
Tˆ for random effects.ˆ
Tˆ ˆ
As T , 1, random effects beˆ
i iβ X ii X X ii y
2u i,RE
2u i,RE
2 2u
comes fixed effects
As 0, 0, random effects becomes OLS (of course)ˆˆ
As , 1, random effects becomes fixed effectsˆˆ
For the C&R application, =.133156, =.0235231, .975384.ˆˆ ˆ
Lo
oks like a fixed effects model. Note the Hausman statistic agrees.
β does not contain the constant term in the preceding.
Part 5: Random Effects [ 47/54]
Another Comparison (Baltagi p. 20)
GLS LSDV(Within) Between
2u
LSDV(Within)
GLS LSDV(Within)
ˆ ˆ ˆ( )
Limiting and Special Cases
(1) If 0, GLS = OLS
ˆ(2) As T , GLS
ˆ ˆ(3) As between variation 0,
(4) As within variation
W I W
GLS Betweenˆ ˆ0,
Part 5: Random Effects [ 48/54]
A Hierarchical Linear ModelInterpretation of the FE Model
it it i it
2it i i it i i
i i
2i i i i u
it it i it
y c+ε , ( does not contain a constant)
E[ε | ,c ] 0,Var[ε | ,c ]=
c + + u ,
E[u| ] 0, Var[u| ]
y [ u] ε
i
i i
i
x β x
X X
zδ
z z
x β zδ
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
2it i it it 1 2 N
it
Unknown parameters
y , [ , ,..., , , ]
Standard estimation based on LS (dummy variables)
Ambiguous definition of the distribution of y
Effects model, nonorthogonality, heterogeneit
x
it i it it i i i
i i
it i it it i
y
y , E[ | ] g( ) 0
Contrast to random effects E[ | X ]
Standard estimation (still) based on LS (dummy variables)
Correlated random effects, more detailed model
y , P[ |
x X X
x i i
i i i i i
] g( ) 0
Linear projection? u Cor(u , ) 0
X X
x x
Part 5: Random Effects [ 51/54]
Mundlak’s Estimator
ii i i i i1 i1 iT i
i
i i i i i
i i i i
i i
Write c = u , E[c | , ,... ]
Assume c contains all time invariant information
= +c + , T observations in group i
= + + + u
Looks like random effects.
Var[ + u ]=
xδ x x x = xδ
y Xβ i ε
Xβ ixδ ε i
ε i Ω +
This is the model we used for the Wu test.
2i uσ ii
Mundlak, Y., “On the Pooling of Time Series and Cross Section Data, Econometrica, 46, 1978, pp. 69-85.
Part 5: Random Effects [ 52/54]
Correlated Random Effects
ii i i i i1 i1 iT i
i
i i i i i
i i i i
i i1 1 i2 2
c = u , E[c | , ,... ]
Assume c contains all time invariant information
= +c + , T observations in group i
= + + + u
c =
xδ x x x = xδ
y X
Mundlak
Chamberlain/ Wooldri
β i ε
Xβ ixδ ε i
x δ x
dg
δ
e
iT T i
i i i1 1 i1 2 iT T i i
... u
= ... u+
TxK TxK TxK TxK etc.
Problems: Requires balanced panels
Modern panels have large T; models have large K
x δ
y Xβ ix δ ix δ ix δ i ε
Part 5: Random Effects [ 53/54]
Mundlak’s Approach for an FE Model with Time Invariant Variables
it it i it
2it i i it i i
i i
2i i i i w
it it i it
y + c+ε , ( does not contain a constant)
E[ε | ,c ] 0,Var[ε | ,c ]=
c + + w ,
E[w| , ] 0, Var[w| , ]
y w ε
random effe
i
i
i i
i i
x β zδ x
X X
x θ
X z X z
x β zδ x θ
cts model including group means of
time varying variables.
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 19.8537815z(i)=================================================================== FEM | -.34322129 .05725632 -5.994 .0000 .11260504 ED | .05099781 .00575551 8.861 .0000 12.8453782Means 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 19.8537815Variance Estimates===================================================== Var[e]| .0235632 Var[u]| .0773825