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Transcript of Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model...
![Page 1: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/1.jpg)
Panel Data AnalysisIntroduction
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
![Page 2: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/2.jpg)
Panel Data AnalysisIntroduction
The Model
One-Way (Individual) Effects: Unobserved Heterogeneity Cross Section and Time Series Correlation
''it it it
it it i t it
it 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
![Page 3: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/3.jpg)
Panel Data Analysis Introduction
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 e
i 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 e
t T i N
x β
y X β u e
y Xβ i I u e
N T T Nor
y Xβ Du e
D I i D i I
![Page 4: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/4.jpg)
Panel Data Analysis Introduction
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 e
x x xy e
x x xy e
y
y
y
x
xy 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 u
x x x e u
x x x e u
xe u
x
![Page 5: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/5.jpg)
Example: Investment Demand
Grunfeld and Griliches [1960]
i = 10 firms: GM, CH, GE, WE, US, AF, DM, GY, UN, IBM; t = 20 years: 1935-1954
Iit = Gross investment
Fit = Market value
Cit = Value of the stock of plant and equipment
it i it it itI F C
![Page 6: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/6.jpg)
Pooled (Constant Effects) Model
'
'
'
2
( 1, 2,..., ; 1, 2,..., )
assuming
,
1
( | ) , ( | )
it it i it
i
it it it
it it it
e
y u e i N t T
u u i
y u e or
y eu
E Var
x β
x β
βx y Xβ e
e X 0 e X I
![Page 7: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/7.jpg)
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 N
t T
y X e
y X u e
![Page 8: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/8.jpg)
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
![Page 9: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/9.jpg)
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
y
u 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 N
t T
y X e
y X u e
![Page 10: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/10.jpg)
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
![Page 11: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/11.jpg)
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
Var
Var
ε X I i i
ε X Ω I
![Page 12: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/12.jpg)
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
or
Q Q Q
where Q Q Q Q QT
y X β e
y X β e
I i i i
![Page 13: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/13.jpg)
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
![Page 14: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/14.jpg)
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 N
OLS i i i ii i
OLS
N N N
e i i i i i ii i i
N
e 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 ˆ/ ( ),NT N K e e y Xβ
![Page 15: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/15.jpg)
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 T
u i N
normal iid
with Q Q Q
QT
normal where QQ Q
x β
y X β i e
e 0 I
y X β e y y X X e e
I i i
e
![Page 16: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/16.jpg)
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 ln
2 2 21 1
ln 2 ln( ) ln2 2 2 2
i e i i i i
e i ie
Tll
T TQ Q
β y X e e
e e
2 2 '2
1( , | , ) ln 2 ln( )
2 2 2i e i i e i ie
T Tll
β y X e e
![Page 17: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/17.jpg)
Fixed Effects Model Estimation: ML
ML Estimator2 2
1
'2 21
2 2
ˆ( , ) argmax ( , | , )
ˆ ˆ 1 ˆ ˆˆ ˆ1 ,
ˆ ˆ'ˆ ˆ
1 ( 1)
N
e ML i e i ii
N
i iie e i i i
e e
ll
NT T
T
T N T
β β y X
e ee y X β
e e
![Page 18: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/18.jpg)
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 e
vs 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 K
RSS NT N K
RSS RSS
e e e e
![Page 19: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/19.jpg)
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
u
E
Var
TQ Q
where Q QT T
y X β ε ε i e
ε X 0
ε X I i i
I
I i i I i i
![Page 20: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/20.jpg)
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 N
GLS i i i ii i
N
GLS 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
![Page 21: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/21.jpg)
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
T
u e i itt
N T
T T N where e eT
e e
e e
![Page 22: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/22.jpg)
Random Effects Model Estimation: GLS
Feasible GLS Within Model Representation
2 2
2 2 21
' '
'
2
1 1
( ) ( )
(1 ) ( )
( ) 0, ( )
( , ) ( , ) 0
e e
u e
it i it i it i
it it it
it it i i it i
it it e
it i it jt
T
y y
y
u e e
E Var
Cov Cov
x x
x
![Page 23: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/23.jpg)
Random Effects Model Estimation: ML
Log-Likelihood Function
' '
2 2 1
( ) ( 1, 2,..., )
( 1, 2,..., )
~ ( , )
1 1( , , | , ) ln 2 ln
2 2 2
it it i it it it
i i i
i
i e u i i i i
y u e t T
i N
normal iid
Tll
x β x β
y X β ε
ε 0
β y X ε ε
![Page 24: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/24.jpg)
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
![Page 25: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/25.jpg)
Random Effects Model Estimation: ML
ML Estimator
2 2 2 2
1
2 2 1
2 22
2
2 2' 2 '
2 2 21 1
ˆ ˆ ˆ( , , ) argmax ( , , | , )
1 1( , , | , ) ln 2 ln
2 2 2
1ln 2 ln2 2
1( ) ( )
2
N
e 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
where
Tll
TT
y yT
β β y X
β y X ε ε
x β x β
![Page 26: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/26.jpg)
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
![Page 27: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/27.jpg)
Random Effects ModelHypothesis Testing
Pool or Not Pool (Cont.)
2
22
1 1
2
1 1
'
ˆ ˆ'( )1 ~ (1)
ˆ ˆ2 1 '
ˆ1
2 1 ˆ
ˆˆ 1
ˆ
T N
N T
iti t
N T
iti t
it it it
Pooled
NTLM
T
eNT
T e
where e yu
e J I e
e e
βx
![Page 28: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/28.jpg)
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 Effects
E(ui|Xi) = 0
Fixed Effects
E(ui|Xi) =/= 0
GLS or RE-OLS
(Random Effects)
Consistent and Efficient
Inconsistent
LSDV or FE-OLS
(Fixed Effects)
Consistent
Inefficient
Consistent
Possibly Efficient
![Page 29: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/29.jpg)
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
![Page 30: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/30.jpg)
Random Effects ModelHypothesis Testing
Alternative Hausman Test Estimate the random effects model
F Test that = 0
' ' ( )it it i i ity u e x β x γ
0 0: 0 : ( , ) 0i itH H Cov u γ x
![Page 31: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/31.jpg)
Extensions
Random Coefficients Model
Mixed Effects Model
Two-Way EffectsNested Random Effects
'' '( )it it i it
it it it i it
i i
y ey e
x βx β x u
β β u
' '( )it it it i ity e x β z γ
'it it i t ity u v e x β
'ijt ijt i ij ijty u v e x β
![Page 32: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/32.jpg)
Example: U. S. Productivity
Munnell [1988] Productivity Data48 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,
![Page 33: Panel Data Analysis Introduction Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic.](https://reader035.fdocuments.net/reader035/viewer/2022062720/56649f165503460f94c2ba8b/html5/thumbnails/33.jpg)
References
B. H. Baltagi, Econometric Analysis of Panel Data, 4th ed., John Wiley, New York, 2008.
W. H. Greene, Econometric Analysis, 7th ed., Chapter 11: Models for Panel Data, Prentice Hall, 2011.
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.