Spatial Econometric Analysis Using GAUSS
8
Kuan-Pin LinPortland State University
Panel Data AnalysisA Review
Model Representation N-first or T-first representation
Pooled Model Fixed Effects Model Random Effects Model
Asymptotic Theory N→∞, or T→∞ N→∞, T→∞ Panel-Robust Inference
Panel Data AnalysisA Review
The Model
One-Way (Individual) Effects: Unobserved Heterogeneity Cross Section and Time Series Correlation
''it it it
it it i t itit i t it
yy u v e
u v e
xx
'it it i ity u e x
( , ) 0, ( , ) 0,
( , ) 0,i j it jt
it i
Cov u u Cov e e i j
Cov e e t
Panel Data AnalysisA Review
N-first Representation
Dummy Variables Representation
T-first Representation'
1,2,..., ; 1, 2,...,
( )
it it i it
i i i T i
N T
y u ei N t T
u
x β
y X β i e
y Xβ I i u e
'
1,2,..., ; 1,2,...,
( )
ti ti i ti
t t t
T N
y u et T i N
x β
y X β u e
y Xβ i I u e
N T T Nor
y Xβ Du eD I i D i I
Panel Data AnalysisA Review
Notations'
1, 1 2, 1 , 11 1 11'
1, 2 2, 2 , 22 2 22
'1, 2, ,
1
2
, , ,
,
i i K ii ii
i i K ii iii i i
iT iT K iTiT iT KiT
t t
tt t
tN
x x xy ex x xy e
x x xy e
yy
y
xx
y X e β
x
x
y X
'1, 1 2, 1 , 1 1 11
'1, 2 2, 2 , 2 2 22
'1, 2, ,
, ,
t t K t t
t t K t ttt
tN tN K tN tN NtN
x x x e ux x x e u
x x x e u
xe u
x
Pooled (Constant Effects) Model
'
'
'
2
( 1,2,..., ; 1, 2,..., )
assuming
1 ,
( | ) , ( | )
it it i it
i
it it it
it it
it it it
e
y u e i N t T
u u i
y u e
uy e
E Var
x β
x ββ
w x δ
w δ y Wδ e
e X 0 e X I
Fixed Effects Model
ui is fixed, independent of eit, and may be correlated with xit.
' ( 1, 2,..., ; 1, 2,..., )it it i ity u e i N t T x β
( , ) 0, ( , ) 0i it i itCov u e Cov u x
,
, 1, 2,...,1,2,...,
i i i T i
t t t
u i i Nt T
y X ey X u e
Fixed Effects Model Fixed Effects Model
Classical Assumptions Strict Exogeneity: Homoschedasticity: No cross section and time series correlation:
Extensions: Panel Robust Variance-Covariance Matrix
( | , ) 0itE e u X2( | , )it eVar e u X
2( | , ) e NTVar e u X I
( | , )Var e u X
Random Effects Model Error Components
ui is random, independent of eit and xit.
Define the error components as it = ui + eit
'
( 1, 2,..., ; 1,2,..., )it it it
it i it
yu e i N t T
x β
( , ) 0, ( , ) 0, ( , ) 0i it i it it itCov u e Cov u Cov e x x
( ), 1, 2,...,( ), 1, 2,...,
i i i T i
t t t
u i i Nt T
y X ey X u e
Random Effects Model
Random Effects Model Classical Assumptions
Strict Exogeneity
X includes a constant term, otherwise E(ui|X)=u.Homoschedasticity
Constant Auto-covariance (within panels)
( | ) 0, ( | ) 0 ( | ) 0it i itE e E u E X X X
2 2 '( | )i e T u T TVar ε X I i i
2 2
2 2 2
( | ) , ( | ) , ( , ) 0
( | )it e i u i it
it e u
Var e Var u Cov u e
Var
X X
X
Random Effects Model
Random Effects Model Classical Assumptions (Continued)
Cross Section Independence
Extensions:Panel Robust Variance-Covariance Matrix
2 2 '( | )( | )
i e T u T T
N
VarVar
ε X I i iε X Ω I
Fixed Effects Model Estimation
Within Model Representation'
'
' '
'
( ) ( )
it it i it
i i i i
it i it i it i
it it it
y u e
y u e
y y e e
y e
x β
x β
x x β
x β
'1 , ( 0, ' )
i i i
i i i
T T T T
orQ Q Q
where Q Q Q Q QT
y X β ey X β e
I i i i
Fixed Effects Model Estimation
Model Assumptions
2
2
2 2 '
( | ) 0
( | ) (1 1/ )
( , | , ) ( 1/ ) 0,
1( | ) ( )
( | )
it it
it it e
it is it is e
i i e e T T T
N
E e
Var e T
Cov e e T t s
Var QT
Var
x
x
x x
e X I i i
e X Ω I
Fixed Effects Model Estimation: OLS
Within Estimator: OLS
1' 1 ' ' '
1 1
' 1 ' ' 1
1 12 ' ' '
1 1 1
12 '
1
2
ˆ ( )
ˆˆ ( ) ( ) ( )
ˆ
ˆ
ˆˆ '
i i i
N NOLS i i i ii i
OLS
N N Ne i i i i i ii i i
Ne i ii
e
Var
Q
y X β e y Xβ e
β XX Xy X X X y
β XX XΩX XX
X X X X X X
X X
e
ˆ ˆ/ ( ),NT N K e e y Xβ
Fixed Effects Model Estimation: ML
Normality Assumption'
2
'
2 2
( 1, 2,..., )( 1,2,..., )
~ ( , )
, , ,1
~ (0, ), '
i
it it i it
i i i T i
i e T
i i i i i i i i i
T T T
i e e
y u e t Tu i N
normal iid
with Q Q Q
QT
normal where QQ Q
x βy X β i e
e 0 I
y X β e y y X X e e
I i i
e
Fixed Effects Model Estimation: ML
Log-Likelihood Function
Since Q is singular and |Q|=0, we maximize
2 ' 1
2 ' 12
1 1( , | , ) ln 2 ln2 2 2
1 1ln 2 ln( ) ln2 2 2 2
i e i i i i
e i ie
Tll
T T Q Q
β y X e e
e e
2 2 '2
1( , | , ) ln 2 ln( )2 2 2i e i i e i i
e
T Tll
β y X e e
Fixed Effects Model Estimation: ML
ML Estimator2 2
1
'2 21
2 2
ˆ( , ) argmax ( , | , )
ˆ ˆ 1 ˆ ˆˆ ˆ1 ,
ˆ ˆ'ˆ ˆ1 ( 1)
Ne ML i e i ii
Ni ii
e e i i i
e e
ll
NT T
TT N T
β β y X
e ee y X β
e e
Fixed Effects ModelHypothesis Testing
Pool or Not Pool F-Test based on dummy
variable model: constant or zero coefficients for D w.r.t F(N-1,NT-N-K)
F-test based on fixed effects (unrestricted) model vs. pooled (restricted) model
'
'
. ( , )it it i it
i
it it it
y u evs u u i
y u e
x β
x β
' '
( ) / 1 ~ ( 1, )/ ( )
ˆ ˆ ˆ ˆ,
R UR
UR
UR FE FE R PO PO
RSS RSS NF F N NT N KRSS NT N K
RSS RSS
e e e e
Fixed Effects ModelHypothesis Testing
Based on estimated residuals of the fixed effects model: Heteroscedasticity
Breusch and Pagan (1980) Autocorrelation: AR(1)
Breusch and Godfrey (1981)
' ˆ , 1,...,i i i i N e y x β
2221' ~ (1)
1 'NTLMT
e ee e
Random Effects Model Estimation: GLS
The Model
2 2 '
2 22
2
' '
,( | )
( | )
1 1,
i i i i i T i
i i
i i e T u T T
e ue T
e
T T T T T T
uE
Var
TQ Q
where Q QT T
y X β ε ε i eε X 0
ε X I i i
I
I i i I i i
Random Effects Model Estimation: GLS
GLS
11 1 1 1 1
1 1
11 1 1
1
2 21 '
2 2 2 2 2 2
1 22
2 2
ˆ ( )
ˆ( ) ( )
1 1
1
N NGLS i i i ii i
NGLS i ii
u eT T T T
e e u e e u
eT
e e u
Var
where Q QT T
and Q QT
β XΩ X XΩ y X X X y
β XΩ X X X
I i i I
I
Random Effects Model Estimation: GLS
Feasible GLS Based on estimated residuals of fixed effects
model
1 1 1
1 1
1 2 2 212 2
1
ˆ ˆ ˆ( )ˆ ˆ( ) ( )
1 1ˆ ˆ ˆ ˆ,ˆ ˆ
GLS
GLS
T e ue
Var
where Q Q T
β XΩ X XΩ y
β XΩ X
I
2
2 2 21 1
ˆ ˆˆ ' / ( 1)1ˆ ˆ ˆ ˆˆ ˆ ˆ ' / ,
e
Tu e i itt
N T
T T N where e eT
e e
e e
Random Effects Model Estimation: ML
Log-Likelihood Function
' '
2 2 1
( ) ( 1, 2,..., )( 1, 2,..., )
~ ( , )
1 1( , , | , ) ln 2 ln2 2 2
it it i it it it
i i i
i
i e u i i i i
y u e t Ti N
normal iid
Tll
x β x βy X β εε 0
β y X ε ε
Random Effects Model Estimation: ML
where2 2
2 2 '2
2 21 '
2 2 2 2 2 2
2 22 ' 2
2 2
( )
1 1 ( )
| | ( ) ( ) 1
e ue T u T T T
e
u eT T T T
e u e e u e
T Tu ue T T T e
e e
TQ Q
Q QT T
T
I i i I
I i i I
I i i
Random Effects Model Estimation: ML
ML Estimator
2 2 2 21
2 2 1
2 22
2
2 2' 2 '
2 2 21 1
ˆ ˆ ˆ( , , ) argmax ( , , | , )
1 1( , , | , ) ln 2 ln2 2 2
1ln 2 ln2 2
1 ( ) ( )2
Ne u ML i e u i ii
i e u i i i i
e ue
e
T Tuit it it itt t
e e u
ll
whereTll
TT
y yT
β β y X
β y X ε ε
x β x β
Random Effects ModelHypothesis Testing
Pool or Not Pool Test for Var(ui) = 0, that is
For balanced panel data, the Lagrange-multiplier test statistic (Breusch-Pagan, 1980) is:
, , ,( ) ( ) ( )it is i it i is it isCov Cov u e u e Cov e e
Random Effects ModelHypothesis Testing
Pool or Not Pool (Cont.)
2
22
1 1
21 1
'
ˆ ˆ'( ) 1 ~ (1)ˆ ˆ2 1 '
ˆ1
2 1 ˆ
ˆˆ 1
ˆ
T N
N Titi t
N Titi t
it it it
Pooled
NTLMT
eNTT e
where e yu
e J I ee e
βx
Random Effects ModelHypothesis Testing
Fixed Effects vs. Random Effects '
0
'1
: ( , ) 0 ( )
: ( , ) 0 ( )i it
i it
H Cov u random effects
H Cov u fixed effects
x
x
Estimator Random EffectsE(ui|Xi) = 0
Fixed EffectsE(ui|Xi) =/= 0
GLS or RE-OLS(Random Effects)
Consistent and Efficient
Inconsistent
LSDV or FE-OLS(Fixed Effects)
ConsistentInefficient
ConsistentPossibly Efficient
Random Effects ModelHypothesis Testing
Fixed effects estimator is consistent under H0 and H1; Random effects estimator is efficient under H0, but it is inconsistent under H1.
Hausman Test Statistic ' 1
2
ˆ ˆ ˆ ˆ ˆ ˆ( ) ( )
ˆ ˆ ˆ~ (# ), # # ( )
RE FE RE FE RE FE
FE FE RE
H Var Var
provided no intercept
β β β β β β
β β β
Random Effects ModelHypothesis Testing
Alternative Hausman Test Estimate the random effects model
F Test that = 0
' ' ' '( ) ( ) ( )it i it i it i ity y e x x β x x γ
0 0: 0 : ( , ) 0i itH H Cov u γ x
Random Effects ModelHypothesis Testing
Heteroscedasticity H0: θ2=0 | θ1=0 H0: θ1=0 | θ2=0 H0: θ2=0, θ1=0
'
2
2 2 '1
2 2 '1
2
2 2 '2
~ (0, )
( ), 1,..., , 1,...,
( ), 1,..., ,
~ (0, )
( ), 1,...,
it
it
it
i
i
it it it
it i it
it e
e e it
e e i
i u
u u i
yu e
e
h i N t T
or h i N t
u
h i N
x β
z
h
f
Random Effects ModelHypothesis Testing
Heteroscedasticity (Cont.) Based on random effects model with
homoscedasticity:2 2 2 2 2
1ˆˆ ˆ ˆ ˆ ˆ ˆ, 1,..., ; , ,i i i u e u ei N T e y X β
2 1
1 20| 4
1
' '
' '
1 ' ( ' ) ' ~ (# )ˆ2
[ , 1,... ], ( / )ˆ ˆ[ , 1,..., ], ( / )
i N N N
i i i T T i
LM S F F F F S F
F i N F N F
S S i N S T
f I i i
e i i e
Random Effects ModelHypothesis Testing
Heteroscedasticity (Cont.)
1 2
1 20|
' '
4 414 4 4 41 1
' '
' '
1 ' ( ' ) ' ~ (# )2
[ , 1,... ], ( / )
ˆ ˆ ( 1) 1 1,ˆ ˆ ˆ ˆ
ˆ ˆ[ , 1,..., ], ( / )ˆ ˆ[ , 1,..., ], ( / )
i N N N
e
e e
i i i T T i
i i i T T T i
LM S H H H H S Ha
H i N H N H
Ta S S S
S S i N S T
S S i N S T
h I i i
e i i e
e I i i e
Random Effects ModelHypothesis Testing
Heteroscedasticity (Cont.)
Baltagi, B., Bresson, G., Pirotte, A. (2006) Joint LM test for homoscedasticity in a one-way error component model. Journal of Econometrics, 134, 401-417.
1 2 2 1
20, 0 0 0 ~ (# # )LM LM LM F H
Random Effects ModelHypothesis Testing
Autocorrelation: AR(1) Based on random effects model with no
autocorrelation:
LM test statistic is tedious, see Baltagi, B., Li, Q. (1995) Testing AR(1) against
MA(1) disturbances in an error component model. Journal of Econometrics, 68, 133-151.
2 2 2 2 21
ˆˆ , 1,...,
ˆ ˆ ˆ ˆ ˆ, ,i i i
u e u e
i N
T
e y X β
Random Effects ModelHypothesis Testing
Joint Test for AR(1) and Random Effects Based on OLS residuals:
Marginal Test for AR(1) & Random Effects
2
22 2 2
0, 0
'1
4 2 ~ (2)2( 1)( 2)
ˆ ˆ ˆ ˆ'( ) '1,ˆ ˆ ˆ ˆ' '
u
N T T
NTLM A AB TBT T
A B
ε I i i ε ε ε
ε ε ε ε
ˆˆ ε y - Xβ
2
22 2 2 2
00~ (1); ~ (1)
2( 1) 1u
NT NTLM A LM BT T
Random Effects ModelHypothesis Testing
Robust LM Tests for AR(1) and Random Effects Because
2 2 2* *0 00, 0 0 0u u u
LM LM LM LM LM
2* 2 2
0
2* 2 20
(2 ) ~ (1)2( 1)(1 2 / )
( / ) ~ (1)( 1)(1 2 / )
u
NTLM B AT TNTLM B A T
T T
Panel Data AnalysisAn Example: U. S. Productivity
The Model (Munnell [1988]):
0 1 2
3 4
ln( ) ln( ) ln( )ln( ) ( )
it it it
it it i it
it it it it
gsp public privateemp unemp u e
public hwy water util
Panel Data AnalysisAn Example: U. S. Productivity
Productivity Data 48 Continental U.S. States, 17 Years:1970-1986
STATE = State name, ST_ABB = State abbreviation, YR = Year, 1970, . . . ,1986, PCAP = Public capital, HWY = Highway capital, WATER = Water utility capital, UTIL = Utility capital, PC = Private capital, GSP = Gross state product, EMP = Employment, UNEMP = Unemployment rate
U. S. ProductivityBaltagi (2008) [munnell.1, munnell.2]
Panel Data Model ln(GSP) = + ln(Public) + 2ln(Private) + 3ln(Labor) + 4(Unemp) +
FixedEffects s.e
RandomEffects s.e
-0.026 0.029 0.003 0.024
0.292 0.025 0.310 0.020
3 0.768 0.030 0.731 0.026
4 -0.005 0.001 -0.006 0.001
0 2.144 0.137
F(47,764) =75.82 LM(1) = 4135
Hausman LM(4) = 905.1
Panel Data AnalysisAnother Example: China Provincial Productivity
Cobb-Douglass Production Function ln(GDP) = + ln(L) + ln(K) +
Fixed Effects s.e.
Random Effects s.e
0.30204 0.078 0.4925 0.078
0.04236 0.0178 0.0121 0.0176
2.6714 0.6254
F(29,298) = 158.81 LM(1) = 771.45
Hausman LM(2) = 48.4
References B. H. Baltagi, Econometric Analysis of Panel Data, 4th
ed., John Wiley, New York, 2008. W. H. Greene, Econometric Analysis, 6th ed., Chapter 9:
Models for Panel Data, Prentice Hall, 2008. C. Hsiao, Analysis of Panel Data, 2nd ed., Cambridge
University Press, 2003. J. M. Wooldridge, Econometric Analysis of Cross Section
and Panel Data, The MIT Press, 2002.
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