Econometric Analysis of Panel Data
description
Transcript of Econometric Analysis of Panel Data
Econometric Analysis of Panel Data
• Panel Data Analysis– Fixed Effects
• Dummy Variable Estimator• Between and Within Estimator• First-Difference Estimator• Panel-Robust Variance-Covariance Matrix
– Heteroscedasticity and Autocorrelation– Cross Section Correlation
– Hypothesis Testing• To pool or Not to pool
Panel Data Analysis
• Fixed Effects Model
– ui is fixed, independent of eit, and may be correlated with xit.
' ( 1, 2,..., )
( 1,2,..., )i
it it i it i
i i i T i
y u e t T
u i N
x β
y X β i e
( , ) 0, ( , ) 0i it i itCov u e Cov u x
Fixed Effects Model
• Classical Assumptions– Strict Exogeneity
– Homoschedasticity
– No cross section and time series correlation
( | ) 0itE e X
2( | ) e NTVar e X I
2( | )it eVar e X
Fixed Effects Model
• Extensions– Weak Exogeneity
1 2
1 2
( | , ,..., ) ( | ) 0
( | , ,..., ) 0( | ) 0
iit i i iT it i
it i i it
it it
E e E e
E eE e
x x x X
x x xx
Fixed Effects Model
• Extensions– Heteroschedasticity
1
2
0 00 0
( | )
0 0 N
Var
e X Ω
21,1
21,22
21,
0 00 0
( | ) ,
0 0i
it i it i
T
Var e
X
Fixed Effects Model
• Extensions– Time Series Correlation (with cross section
independence for short panels)
2
( , | , ) ,( , | , ) 0,
( | ) ( | ) ( | )
it is it is ts
it js it js
it it tt t i i i
Cov e e t sCov e e i j
Var e Var Var
x xx x
x e X e X Ω
11 12 1
21 22 2
1 2
i
i
i i i i
T
Ti
T T TT
1
2
0 00 0
0 0 N
Ω
Fixed Effects Model
• Extensions– Cross Section Correlation (with time series
independence for long panels)
2
( , | , ) ,
( , | , ) 0,
( | ) ( | )
it jt it jt ij
it js it js
it it i T
Cov e e i j
Cov e e t s
Var e Var
x x
x x
x e X I Ω
2 21 12 1 1 12 1
2 221 2 2 21 2 2
2 21 2 1 2
,
N N
N N
N N N N N N
I I II I I
Ω
I I I
Dummy Variable Model
• Dummy Variable Representation
– Note: X does not include constant term, otherwise one less number of dummy variables should be used.
1
2
1 1 1 1
2 2 2 2
0 0
0 0
0 0N
T
T
N N N NT
uu
u
iy X eiy X e
β
y X ei
βy Xβ Du e X D e
uy Wδ e
Dummy Variable Model
• Dummy Variable Estimator (LSDV)
• Heteroscedasticity and Autocorrelation
1' 1 ' ' '
1 1
12 ' 1 2 '
1
2
ˆ ( )
ˆˆ ˆ ˆ( ) ( )
ˆ ˆˆ ' / ( )ˆˆ
N NOLS i i i ii i
NOLS e e i ii
e
Var
NT N K
δ WW Wy WW Wy
δ WW WW
e e
e y Wδ
' 1 ' ' 1
1 1' ' ' '
1 1 1
ˆ ˆ ˆ( ) [( )( ) '] ( ) ( ') ( )
( )N N Ni i i i i i i ii i i
Var E E
E
δ δ δ δ δ WW W ee W WW
WW W e e W WW
Dummy Variable Model
• Panel-Robust Variance-Covariance Matrix
1 1' ' ' '
1 1 1
1 1' ' '
1 1 1 1 1 1 1
ˆ ˆ ˆˆ ( ) [( )( ) ']
ˆ ˆ
ˆ ˆ
ˆˆˆˆ
i i i i
N N Ni i i i i i i ii i i
N T N T T N Tit it it is it is it iti t i t s i t
i i i
it it it
Var E
e e
e y
δ δ δ δ δ
WW We e W WW
w w w w w w
e y Wδ
w δ
Within Model
• Within Model Representation'
' '
'
( ) ( )i i i i
it i it i it i
it i it
y u e
y y e e
y e
x β
x x β
x β
' '1 , ( 0, )i i i i
i i i
i i i i i i
i T T T i T i i ii
orQ Q Q
where Q Q QQ QT
y X β ey X β e
I i i i
Within Model
• Model Assumptions
2
2
2 2 '
1
2
( | ) 0
( | ) (1 1 / )
( , | , ) ( 1/ ) 0,
1( | ) ( )
0 00 0
( | )
0 0
i i i
it it
it it i e
it is it is i e
i i i e i e T T Ti
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
2
1 1/ 1/ 1/1/ 1 1/ 1/
1/ 1/ 1 1/
i i i
i i ii e
i i i
T T TT T T
T T T
Within Model
• Within Estimator: FE-OLS
1' 1 ' ' '
1 1
2 ' 1 ' ' 1
1 12 ' ' '
1 1 1
12 '
1
ˆ ( )
ˆˆ ˆ( ) ( ) ( )
ˆ
ˆ
ˆ
i i i
N NOLS i i i ii i
OLS e
N N Ne i i i i i i ii i i
Ne i ii
Var
y X β e y Xβ e
β XX Xy X X X y
β XX XQX XX
X X XQ X X X
X X
2 ˆ ˆ ˆ ˆ' / ( ),e NT N K e e e y Xβ
Within Model
• Within Estimator: GLS
• GLS = FE-OLS– Note:
1' 1 1 ' 1 ' 1 ' 1
1 1
1' '
1 1
12 ' 1 1 2 ' 1
1
12 '
1
ˆ ( )
ˆˆ ˆ ˆ( ) ( )
ˆ
N NGLS i i i i i ii i
N Ni i i ii i
NGLS e e i i ii
Ne i ii
Var
β XQ X XQ y XQ X XQ y
X X X y
β XQ X XQ X
X X
' 1 ' ' 1 ' ' ' ' '
' 1 ' ' 1 ' ' ' ' '
i i i i i i i i i i i i i i i i i
i i i i i i i i i i i i i i i i i
Q QQ Q Q QQ
Q QQ Q Q QQ
X X X X X X X X X X
X y X y X y X y X y
Within Model
• Normality Assumption'
2
'
2 ' 2
( 1,2,..., )( 1, 2,..., )
~ ( , )
, , ,1
~ (0, ),
i
i
i i i
it it i it i
i i i T i
i e T
i i i i i i i i i i i i
i T T Ti
i i i e i i e i
y u e t Tu i N
iidn
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
Within Model
• Log-Likelihood Function
• ML Estimator
2 ' 1
2 '2
1 1( , | , ) ln 2 ln2 2 2
1ln 2 ln( )2 2 2
ii e i i i i i i
i ie i i
e
Tll
T T
β y X e e
e e
2 21
' '2 1 1
1 1
ˆ ˆ( , ) argmax ( , | , )
ˆ ˆˆ ,
Ne ML i e i ii
N Ni i i i ii i
eML ML FE OLSN Ni ii i
ll
Q
T T
β β y X
e e e eβ β
Within Model
• ML Estimator of e2 is downward biased
even for large N:
• For balanced panel (T=Ti: ), e2 should be
estimated as:
' '2 2 21 1 1
1 1 1
ˆ ˆ ˆ( 1) ( 1)
N N Ni i i i ii i i
eFE eML eMLN N Ni i ii i i
T
T T T
e e e e
'2 21ˆ ˆ
( 1) ( 1)
Ni ii
e eMLT
N T T
e e
Within Model
• Estimated Fixed Effects
– For , is consistent but is inconsistentunless .
' ˆˆi i iu y x β
2 ' ˆˆ ˆˆ ˆ( ) / ( )i i i i iVar u T Var x β x
N β̂
ˆiuiT
Within Model
• Panel-Robust Variance-Covariance Matrix– Consistent statistical inference for general
heteroscedasticity, time series and cross section correlation.
1 1' ' ' '
1 1 1
1 1' ' '
1 1 1 1 1 1 1
ˆ ˆ ˆˆ ( ) [( )( ) ']
ˆ ˆ
ˆ ˆ
ˆˆ ˆ,
i i i i
N N Ni i i i i i i ii i i
N T N T T N Tit it it is it is it iti t i t s i t
i i i it it
Var E
e e
e y
β β β β β
X X X e e X X X
x x x x x x
e y X β
' ˆitx β
First-Difference Model
• First-Difference Representation
• Model Assumptions
' ' '1 1 1( ) ( )it it it it it it it it ity y e e y e x x β x β
2
2
( | ) 0
( | ) 2
| | 1( , | , )
0
it it
it it e
eit is it is
E e
Var e
if t sCov e e
otherwise
x
x
x x
1
22 2 2
2 1 0 0 00 01 2 1 0 0
0 00 1 2 1 0( | ) , ( | )
0 00 0 1 2 10 0 0 1 2
( )
i i e i e e
N
Var Var
Toepliz form
e X e X Ω
First-Difference Model• First-Difference Estimator: FD-OLS
• Consistent statistical inference for general heteroscedasticity, time series and cross section correlation should be based on panel-robust variance-covariance matrix.
1' 1 ' ' '
1 1
2 ' 1 ' ' 1
1 12 ' ' '
1 1 1
22 2
ˆ ( )
ˆˆ ˆ( ) ( ) ( )
ˆ
ˆ ˆ ˆˆ ˆ, ' / (2
i i i
N NOLS i i i ii i
OLS e
N N Ne i i i i i i ii i i
ee e
Var
N
y X β e y Xβ e
β X X X y X X X y
β X X XΩ X X X
X X X X X X
e e ˆˆ),T N K e y Xβ
First-Difference Model
• First-Difference Estimator: GLS' 1 1 ' 1
1' 1 ' 1
1 1
2 ' 1 1
12 ' 1
1
22 2
ˆ ( )
ˆˆ ˆ( ) ( )
ˆ
ˆ ˆˆ ˆ ˆˆ ˆ, ' / ( ),2
GLS
N Ni i i i i ii i
GLS e
Ne i i ii
ee e
Var
NT N K
β XΩ X XΩ y
X X X y
β XΩ X
X X
e e e y Xβ
Hypothesis Testing
• To Pool or Not to 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
R
UR FE FE R PO PO
RSS RSS NF F N NT N K
RSS NT N K
RSS RSS
e e e e
Hypothesis Testing
• Heteroscedasticity
• Serial Correlation
• Spatial Correlation
2'
2, ( ) itit it i it it
i
y u e Var e
x β
1'
1
, it itit it i it it
i it it
v ey u v v
v e
x β
' ,it it i it it ij jt itj
y u v v w v e x β
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