Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Chapter 1. Linear Panel Models and Heterogeneity
Master of Science in Economics - University of Geneva
Christophe Hurlin, Université d’Orléans
Université d’Orléans
January 2010
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Introduction
The outline of the chapter:
1
Speci…cation tests and analysis of covariance
2
Linear unobserved e¤ects panel data models
3
Fixed or random methods?
4
Fixed-E¤ects methods: Least Squares dummy variable
approach
5
Random e¤ects methods
6
Heterogeneous panels: random coe¢ cients models
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
Section 1.
Speci…cation tests and analysis of
covariance
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
A linear model commonly used to assess the e¤ects of both
quantitative and qualitative factors is postulated as
yi ,t = αi ,t + βi0 ,t xi ,t + εi ,t
where 8 i = 1, .., N, 8 t = 1, .., T
αit and β = ( β1it , β2it , ..., βKit ) are 1 1 and 1 K vectors of
parameters that vary across i and t,
xit = (x1it , ..., xKit ) is a 1 K vector of exogenous variables,
uit is the error term.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
Three aspects of the estimated regression coe¢ cients can be
tested:
1
the homogeneity of regression slope coe¢ cients
2
the homogeneity of regression intercept coe¢ cients.
3
the time stability of parameters (slopes and constants). We
will not consider this issue (not speci…c to panel data models)
here.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
We assume that parameters are constant over time, but can vary
across individuals.
yi ,t = αi + βi0 xi ,t + εi ,t
Three types of restrictions can be imposed on this model.
Regression slope coe¢ cients are identical, and intercepts are not
(model with individual / unobserved e¤ects).
yi ,t = αi + β0 xi ,t + εi ,t
Regression intercepts are the same, and slope coe¢ cients are not
(unusual).
yi ,t = α + βi0 xi ,t + εi ,t
Both slope and intercept coe¢ cients are the same (homogeneous
/ pooled panel).
yi ,t = α + β0 xi ,t + εi ,t
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
De…nition
An heterogeneous panel data model is a model in which all
parameters (constant and slope coe¢ cients) vary accross
individuals.
De…nition
An homogeneous panel data model (or pooled model) is a
model in which all parameters (constant and slope coe¢ cients) are
common
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
Speci…cation Tests
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
Figure: Hsiao (2003)
TestH01 : αi =α βi =β ∀i∈[1, N]
H01 rejetée
H01 vraie
TestH0 : βi =β ∀i ∈[1, N]
2
2
H0 rejetée
yi,t =αi +βi ' xi,t +εi,t
yi,t =α+β' xi,t +εi,t
2
H0 vraie
TestH03 : αi =α ∀i ∈[1, N]
H03 vraie
yi,t =α+β' xi,t +εi,t
C. Hurlin
H03 rejetée
yi,t =αi +β' xi,t +εi,t
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
Lemma
Under the assumption that the εit are independently normally
distributed over i and t with mean zero and variance σ2ε , F tests
can be used to test the restrictions postulated
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
First step (homogeneous/ pooled assumption)
Let us consider the general model
yi ,t = αi + β0 xi ,t + εi ,t
The hypothesis of common intercept and slope can be viewed as a
set of (K + 1)(N 1) linear restrictions:
H01 : βi = β αi = α 8 i 2 [1, N ]
Ha1 : 9 (i, j ) 2 [1, N ] / βi 6= βj ou αi 6= αj
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
Under the alternative H1 , there are NK estimated slope
coe¢ cients for the N vectors βi (K 1) and estimated N
constants.
Under H1 , the unrestricted residual sum of squares S1 divided
by σ2ε has a chi-square distribution with NT N (K + 1)
degrees of freedom.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
De…nition
Under the homogeneous assumption H01 ,
H01 : βi = β
(K ,1 )
(K ,1 )
αi = α 8 i 2 [1, N ]
the F statistic, denoted F1 ,and de…ned by:
F1 =
(RSS1,c RSS1 ) / [(N 1) (K + 1)]
RSS1 / [NT N (K + 1)]
has a Fischer distribution with (N 1) (K + 1) and
NT N (K + 1) degrees of freedom. RSS1 denotes the residual
sum of squares of the model and RSS1,c the residual sum of
squares of the constrained model:
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
Under H1 , the residual sum of squares is equal to the sum of the N
residual sum of squares associated to the N individual regressions:
N
N
RSS1 =
∑ RSS1,i = ∑
Syy ,i
y i )2 with x i =
1
T
0
1
Sxy
,i Sxx ,i Sxy ,i
i =1
i =1
with
T
Syy ,i =
∑ (yi ,t
t =1
T
∑ xi ,t
t =1
T
Sxx ,i =
∑ (xi ,t
and y i =
x i ) (xi ,t
x i )0
x i ) (yi ,t
y i )0
t =1
T
Sxy ,i =
∑ (xi ,t
t =1
C. Hurlin
Panel Data Econometrics
1
T
T
∑ yi ,t
t =1
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
Under H01 , the model is :
yi ,t = α + β0 xi ,t + εi ,t
the least-squares regression of the pooled model yields parameter
estimates
b
β = Sxx1 Sxy
N
Sxx =
T
∑ ∑ (xi ,t
x ) (xi ,t
x )0
with x =
i =1 t =1
N
Sxy =
∑
T
∑ (xi ,t
x ) (yi ,t
y )0
with y =
i =1 t =1
C. Hurlin
Panel Data Econometrics
1
NT
1
NT
N
T
∑ ∑ xi ,t
i =1 t =1
N
T
∑ ∑ yi ,t
i =1 t =1
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
Under H01 , the overall RSS is de…ned by
SCR1,c = Syy
with
N
Syy =
T
0
Sxy
Sxx1 Sxy
∑ ∑ (yi ,t
y i )2
i =1 t =1
N
Sxx ,i =
T
∑ ∑ (xi ,t
x i ) (xi ,t
x i )0
x i ) (yi ,t
y i )0
i =1 t =1
N
Sxy ,i =
T
∑ ∑ (xi ,t
i =1 t =1
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
Second step (individual/unobserved e¤ects)
Let us consider the general model
yi ,t = αi + βi0 xi ,t + εi ,t
The hypothesis of heterogeneous intercepts but homogeneous
slopes can be reformulated as subject to (N 1)K linear
restrictions (non restrictions on αi ).
H02 : βi = β 8 i = 1, ..N
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
De…nition
Under the assumption H02 ,
H02 : βi = β 8 i = 1, ..N
the F statistic, denoted F2 , and de…ned by:
F2 =
(RSS1,c 0 RSS1 ) / [(N 1) K ]
RSS1 / [NT N (K + 1)]
has a Fischer distribution with (N 1) K et NT N (K + 1)
degrees of freedom under H02 . RSS1 denotes the residual sum of
squares of the model and RSS1,c 0 the residual sum of squares of
the constrained model (model with individual e¤ects):
yi ,t =
αi + β0Panel
xi ,t Data
+ εEconometrics
C. Hurlin
i ,t
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
Under H02 , the residual sum of squares is:
N
RSS1,c 0 =
T
Syy ,i =
∑ Syy ,i
i =1
∑ (yi ,t
N
∑ Sxy ,i
i =1
!0
y i )2 with x i =
t =1
N
∑ Sxx ,i
i =1
1
T
∑ (xi ,t
∑ xi ,t
t =1
x i ) (xi ,t
x i )0
x i ) (yi ,t
y i )0
t =1
T
Sxy ,i =
∑ (xi ,t
t =1
C. Hurlin
1
N
∑ Sxy ,i
i =1
T
T
Sxx ,i =
!
Panel Data Econometrics
yi =
1
T
T
!
∑ yi ,t
t =1
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
Last step
If H02 is not rejected, one can also apply a conditional test for
homogeneous intercepts (N 1 linear restrictions).
H03 : αi = α 8 i = 1, .., N
given
βi = β
Under the null, the model is homogeneous (pooled) and the
restricted residual sum of squares is SCR1,c . .
Under the alternative, the model is yi ,t = αi + β0 xi ,t + εi ,t ,
and there is NT K N degrees of freedom
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
De…nition
Under the assumption H03 ,
H03 : αi = α 8 i = 1, .., N
given
βi = β
the F statistic, denoted F3 , and de…ned by:
F3 =
(RSS1,c RSS1,c 0 ) / (N 1)
RSS1,c 0 / [N (T 1) K ]
(1)
has a Fischer distribution with N 1 and N (T 1) K degrees
of freedom under H02 . RSS1,c 0 denotes the residual sum of squares
of the model with individual e¤ects and SCR1,c the residual sum of
squares of the pooled model previously de…ned.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
Application: Strikes in OECD
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
Example
Let us consider a simple panel regression model for the total
number of strikes days in OECD countries. We have a balanced
panel data set for 17 countries (N = 17) and annual data form
1951 to 1985 (T = 35). General idea: evaluate the link between
strikes and some macroeconomic factors (in‡atiion, unemployment
etc..)
s,t = αi + βi u,t + γi pit + εit 8 i = 1, .., 17
si ,t the number of strike days for 1000 workers for the country
i at time t.
uitt the unemployement rate
pi ,t , the in‡ation rate
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Figure: Spéci…cation tests with TSP 4.3A
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
C. Hurlin
Speci…cation tests
Application: strikes in OECD
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
C. Hurlin
Speci…cation tests
Application: strikes in OECD
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Speci…cation tests
Application: strikes in OECD
Speci…cation tests and analysis of covariance
Conclusion of section 1.
Fact
It is possible to test the heterogeneity / homogeneity of the
parameters under some speci…c assumptions (normality of
residuals). More generaly, the assumption of heterogenity /
homogeneity of the parameters (slope coe¢ cients and constants)
has to be evaluated with an economic interpretation.
Example
Example: It is reasonnable to assume that the slope parameters of
the production function are the same accros countries? what does
it imply? Should I impose a common mean for the TFP for France
and Germany?
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Section 2.
Linear unobserved/individual e¤ects
panel data models
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
2.1. De…nitions
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
We now consider a model with individual e¤ects and common
slope parameters.
De…nition
A linear unobserved / individual e¤ects panel data model is de…ned
as follows:
yit = αi + β0 xit + εit
where 8 i = 1, .., N, 8 t = 1, .., T
αi and β = ( β1 , β2 , ..., βK ) are 1 1 and 1 K vectors of
parameters,
xit = (x1it , ..., xKit ) is a 1 K vector of exogenous variables,
εit is the error term, assumed to be i.i.d., with 8 i = 1, .., N,
8 t = 1, .., T
E (εit ) = 0 E ε2it = σ2ε
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
De…nition
There are many names for αi : (1) unobserved e¤ects, (2)
individual e¤ects, (3) unobserved component, (4) latent variable
(for random e¤ects models), (5) individual heterogeneity.
De…nition
The εit are called the idiosyncratic errors or idiosyncratic
disturbances because these change accross t as well as accross i.
C. Hurlin
Panel Data Econometrics
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Linear unobserved e¤ects panel data models
Writting in vector form. Let us denote
0
1
yi ,1
B yi ,2 C
C
yi = B
@ ... A
(T ,1 )
yi ,T
0
x1,i ,1
B x1,i ,2
Xi = B
@ ...
(T ,K )
x1,i ,T
x2,i ,1
x2,i ,2
...
x2,i ,T
Let us denote e a unit vector and εi the vector of
0
1
0
1
εi ,1
B 1 C
B εi ,2
C
e =B
εi = B
@ ... A
@ ...
(T ,1 )
(T ,1 )
1
εi ,T
C. Hurlin
Panel Data Econometrics
...
...
...
...
1
xK ,i ,1
xK ,i ,2 C
C
A
...
xK ,i ,T
errors:
1
C
C
A
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Linear unobserved e¤ects panel data models
De…nition
For a given individual i, the linear unobserved / individual
e¤ects panel data model is de…ned as follows:
yi = eαi + Xi β + εi
8 i = 1, .., N
E ( εi ) = 0
E εi εi0 = σ2ε IT
E εi εj0 =
C. Hurlin
0
(T ,T )
if i 6= j
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Example
Let us consider the case of a Cobb Douglas production function in
log, as de…ned previously, for the case T = 3. We have:
yi ,t = αi + βk ki ,t + βn ni ,t + εi ,t 8 i, 8 t 2 [1, 3]
or in a vectorial form for a country
0
1 0 1
0
yi ,1
1
ki ,1
@ yi ,2 A = @ 1 A αi + @ ki ,2
yi ,3
1
ki ,3
C. Hurlin
i:
1
ni ,1
ni ,2 A
ni ,3
βk
βn
Panel Data Econometrics
0
1
εi ,1
+ @ εi ,2 A
εi ,3
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Linear unobserved e¤ects panel data models
It is also possible to stackle all this vectors/matrix as follows
1
y1
B y2 C
C
Y =B
@ ... A
(TN ,1 )
yN
0
Y =e
ee
α+Xβ+ε
1
0
X1
B X2 C
C
X =B
@ ... A
(TN ,K )
XN
where 0T is the null vector
0
e
0T
B 0T e
e
e =B
@ ... ...
(TN ,N )
0T 0T
(T , 1) .
...
...
...
...
C. Hurlin
1
0T
0T C
C
0T A
e
1
ε1
B ε2 C
C
ε =B
@ ... A
(TN ,1 )
εN
0
0
1
α1
B α2 C
C
e
α =B
@ ... A
(N ,1 )
αN
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Example
Consider
N=2
0
y1,1
B y1,2
B
B y
B 1,3
B
B y2,1
B
@ y2,2
y2,3
the case of the production function with T = 3 and
1
0
C B
C B
C B
C B
C=B
C B
C @
A
1
1
1
0
0
0
0
0
0
1
1
1
1
C
C
C
C
C
C
A
0
α1
α2
k11
k12
k13
B
B
B
B
+B
B k21
B
@ k22
k,3
() Y = e
ee
α+Xβ+ε
C. Hurlin
n11
n12
n13
n21
n22
n23
Panel Data Econometrics
1
C
C
C
C
C
C
C
A
0
βk
βn
B
B
B
B
+B
B
B
@
ε1
ε1
ε1,
ε2,
ε2,
ε2,
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
2.2. Fixed and random e¤ects
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Especially in methodological papers, but also in applications, one
often sees a discussion about whether αi will be treated as a
random e¤ect or a …xed e¤ect.
De…nition
In the traditional approach to panel data models, αi is called a
“random e¤ect” when it is treated as a random variable and a
“…xed e¤ect” when it is treated as a parameter to be estimated
for each cross section observation i.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Fact
For Wooldridge (2003), these discussions about whether the αi
should be treated as random variables or as parameters to be
estimated are wrongheaded for microeconometric panel data
applications. With a large number of random draws from the cross
section, it almost always makes sense to treat the unobserved
e¤ects, αi , as random draws from the population, along with
yit and xit . This approach is certainly appropriate from an omitted
variables or neglected heterogeneity perspective.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Fact
As suggested by Mundlak (1978), the key issue involving αi is
whether or not it is uncorrelated with the observed explanatory
variables xit , for t = 1, .., T .
Mundlak Y. (1978), ”On the Pooling of Time Series and Cross
Section Data”, Econometrica, 46, 69-85
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
De…nition
In modern econometric parlance, “random e¤ect” is synonymous
with zero correlation between the observed explanatory variables
and the unobserved e¤ect:
cov (xit , αi ) = 0,
C. Hurlin
t = 1, .., T
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Actually, a stronger conditional mean independence assumption,
E ( αi j xi 1 , .., xiT ) = 0
will be needed to fully justify statistical inference. In applied
papers, when αi is referred to as, say, an “individual random
e¤ect,” then αi is probably being assumed to be uncorrelated with
the xit .
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
De…nition
In microeconometric applications, the term “…xed e¤ect” does not
usually mean that αi is being treated as nonrandom; rather, it
means that one is allowing for arbitrary correlation between the
unobserved e¤ect αi and the observed explanatory variables xit .
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Wooldridge (2003) avoids referring to αi as a random e¤ect or a
…xed e¤ect. Instead, we will refer to αi as unobserved e¤ect,
unobserved heterogeneity, and so on. Nevertheless, later we will
label two di¤erent estimation methods random e¤ects
estimation and …xed e¤ects estimation. This terminology is so
ingrained that it is pointless to try to change it now.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Example (Production function)
Let us consider the simple example of the Cobb Douglass
production function.
yi ,t = βi ki ,t + γi ni ,t + αi + vi ,t
In this case, αi corresponds to the unobserved e¤ect on TFP due to
scountry speci…c omitted factor (climate, institutions, organiszation
etc..). In this case, we might expect that the the level of factors
are positivily correlated with this component of TFP: the more a
country is productive, the more it invests in capital for instance.
cov (αi , ki ,t ) > 0
C. Hurlin
cov (αi , ni ,t ) > 0
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
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De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Example (Patents and R&D)
Hausman, Hall, and Griliches (1984) estimate (nonlinear)
distributed lag models to study the relationship between patents
awarded to a …rm and current and past levels of R&D spending. A
linear version of their model is:
patentsit = θ t + zit γ + δ0 RDit + δ1 RDi ,t
1
+ .. + δ5 RDi ,t
5
+ αi + vi ,t
where RDit is spending on R&D for …rm i at time t and zit
contains other explanatory variables. αi is a …rm heterogeneity
term that may in‡uence patentsit and that may be correlated with
current, past, and future R&D expenditures.
cov (αi , RDi ,t
C. Hurlin
k)
6= 0
8k
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
De…nition
The Hausman’s test (1978), is a test of the null hypothesis
cov (xit , αi ) = 0,
t = 1, .., T
and is generally presented as test of speci…cation (…xed or
random) of the unobserved e¤ects (see later, section 5.).
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
2.3. Fixed e¤ects methods
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Let us consider the following model:
yi = eαi + Xi β + εi
0
8 i = 1, .., N
where αi is a constant term, β = ( β1 β2 ....βK ) 2 RK and:
yi =
εi ,1 εi ,2 ... εi ,T
0
εi ,1 εi ,2 ... εi ,T
0
(T ,1 )
εi =
(T ,1 )
e
(T ,1 )
=
1 1 ... 1
0
x1,i ,1
B x1,i ,2
Xi = B
@ ...
(T ,K )
x1,i ,T
C. Hurlin
x2,i ,1
x2,i ,2
...
x2,i ,T
...
...
...
...
0
1
xK ,i ,1
xK ,i ,2 C
C
A
...
xK ,i ,T
Panel Data Econometrics
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Linear unobserved e¤ects panel data models
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Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Assumtions (H1) Let us assume that errors terms εi ,t are
i.i.d. 8 i 2 [1, N ] , 8 t 2 [1, T ] with:
E (εi ,t ) = 0
σ2ε
t=s
E (εi ,t εi ,s ) =
, or E (εi εi0 ) = σ2ε IT
0
8t 6 = s
where It denotes the identity matrix (T , T ) .
E (εi ,t εj ,s ) = 0, 8j 6= i, 8 (t, s ) , or E εi εj0
where 0 denotes the null matrix (T , T ) .
C. Hurlin
Panel Data Econometrics
=0
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Theorem
Under assumption H1 , the ordinary-least-squares (OLS) estimator
of β is the best linear unbiased estimator (BLUE).
De…nition
In this context, the OLS estimator b
β is called the least-squares
dummy-variable (LSDV) of Fixed E¤ect (FE) estimator,
because the observed values of the variable for the coe¢ cient αi
takes the form of dummy variables.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
OLS estimators of αi and β and are obtained by minimizing
n
b
αi , b
βLSDV
o
N
arg min S
=
fαi ,βgN
i =1
∑ εi0 εi
i =1
N
=
∑ (yi
eαi
Xi β)0 (yi
i =1
C. Hurlin
Panel Data Econometrics
eαi
Xi β)
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
FOC1 (with respect to αi ):
b
αi = y i
with
xi =
1
T
T
∑ xi ,t
t =1
C. Hurlin
0
b
βLSDV x i
yi =
1
T
T
∑ yi ,t
t =1
Panel Data Econometrics
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Linear unobserved e¤ects panel data models
Given the second FOC (with respect to β) and the previous result,
we have:
De…nition
Under assumption H1 , the …xed e¤ect estimator or LSDV estimator
of parameter β is de…ned by:
b
βLSDV
=
"
N
T
∑ ∑ (xi ,t
x i ) (xi ,t
xi )
i =1 t =1
"
N
#
1
T
∑ ∑ (xi ,t
x i ) (yi ,t
i =1 t =1
C. Hurlin
0
Panel Data Econometrics
yi)
#
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
1
The computational procedure for estimating the slope
parameters in this model does not require that the dummy
variables for the individual (and/or time) e¤ects actually be
included in the matrix of explanatory variables.
2
We need only …nd the means of time-series observations
separately for each cross-sectional unit, transform the
observed variables by subtracting out the appropriate
time-series means, and then apply the leastsquares method to
the transformed data.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
The foregoing procedure is equivalent to premultiplying the i th
equation
yi = eαi + Xi β + εi
by a T T idempotent (covariance) transformation matrix
(within operator)
1 0
Q = IT
ee
T
to “sweep out” the individual e¤ect αi so that individual
observations are measured as deviations from individual means
(over time).
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Qyi and QXi correspond to the observations are measured as
deviations from individual means :
Qyi
=
IT
= yi
0
1 0
ee yi
T
1 0
e
e yi
T
1
yi ,1
B yi ,2 C
C
= B
@ ... A
yi ,T
C. Hurlin
1
T
T
∑ yi ,t
t =1
!
0
1
1
B 1 C
B
C
@ ... A
1
Panel Data Econometrics
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Linear unobserved e¤ects panel data models
QXi
1 0
ee Xi
T
1
x1,i ,1 x2,i ,1 ... xK ,i ,1
B x1,i ,2 x2,i ,2 ... xK ,i ,2 C
C
= B
@ ...
A
...
... ...
x1,i ,T x2,i ,T ... xK ,i ,T
0
1
1
C
1 B
B 1 C ∑T x1,i ,t ∑T x2,i ,t
t =1
t =1
T @ ... A
1
= Xi
0
C. Hurlin
Panel Data Econometrics
... ∑Tt=1 xK ,i ,t
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Linear unobserved e¤ects panel data models
Finally, when the transformation Q is applied to a vector of
constant (or a time invariant variable), it lead to a null vector.
Qe =
IT
1 0
ee e
T
e=0
= e
= e
since
e 0e =
1
..
C. Hurlin
1 0
ee e
T
1
0
1
1
@ .. A = T
1
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
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Fixed and random e¤ects
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Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
So, we have:
yi = eαi + Xi β + εi
() Qyi = Qeαi + QXi β + Qεi
() Qyi = QXi β + Qεi
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
De…nition
Under assumption H1 , the …xed e¤ect estimator or LSDV estimator
of parameter β is de…ned by:
where
b
βLSDV =
"
N
∑
Xi0 QXi
i =1
Q = IT
C. Hurlin
#
1
"
N
∑
Xi0 Qyi
i =1
1 0
ee
T
Panel Data Econometrics
#
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Example
Let us consider a simple panel regression model for the total
number of strikes days in OECD countries. We have a balanced
panel data set for 17 countries (N = 17) and annual data form
1951 to 1985 (T = 35). General idea: evaluate the link between
strikes and some macroeconomic factors (in‡atiion, unemployment
etc..)
s,t = αi + βi u,t + γi pit + εit 8 i = 1, .., 17
si ,t the number of strike days for 1000 workers for the country
i at time t.
uitt the unemployement rate
pi ,t , the in‡ation rate
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Linear unobserved e¤ects panel data models
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C. Hurlin
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
C. Hurlin
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Panel Data Econometrics
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Linear unobserved e¤ects panel data models
Theorem
The LSDV estimator b
β is unbiased and consistent when either N or
T or both tend to in…nity. Its variance–covariance matrix is:
V b
βLSDV
=
b
βLSDV
C. Hurlin
σ2ε
"
N
∑
Xi0 QXi
i =1
#
1
p
! β
NT !∞
Panel Data Econometrics
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Linear unobserved e¤ects panel data models
Theorem
However, the estimator for the intercept, b
αi , although unbiased, is
consistent only when T ! ∞.
b
αi
p
! αi
T !∞
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
2.4. Random e¤ects methods
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
De…nition
The random speci…cation of unobserved e¤ects corresponds to a
particular case of variance-component or error-component
model, in which the residual is assumed to consist of three
components
yi ,t = β0 xit + εi ,t 8 i 8t
εi ,t = αi + λt + vi ,t
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Assumptions (H2) Let us assume that error terms εi ,t =
αi + λt + vi ,t are i.i.d. with
8 i = 1, .., N, 8 t = 1, ., T
E (αi ) = E (λt ) = E (vi ,t ) = 0
E (αi λt ) = E (λt vi ,t ) = E (αi vi ,t ) = 0
σ2α
i =j
E ( αi αj ) =
0
8i 6 = j
σ2λ
t=s
E ( λt λs ) =
0
8t 6 = s
σ2v
t = s, i = j
E (vi ,t vj ,s ) =
0
8t 6= s, 8i 6= j
E αi xi0,t = E λt xi0,t = E vi ,t xi0,t = 0
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Panel Data Econometrics
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Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
De…nition
As suggested by Wooldridge (2001), the "…xed e¤ect"
speci…cation can be viewed as a case in which αi is a random
parameter with cov αi , xi0,t 6= 0, whereas the "random e¤ect
model" correspond to the situation in which cov αi , xi0,t = 0.
De…nition
The variance of yit conditional on xit is the sum of three
components:
σ2y = σ2α + σ2λ + σ2v
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
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De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
De…nition
Sometimes, the individual e¤ects αi are supposed to have a non
zero mean, with E (αi ) = µ, then we can de…ned a individual
e¤ects αi with zero mean. A linear unobserved e¤ects panel data
models is then de…ned as follows:
yi = eµ + Xi β + εi
εi
(T ,1 )
= e
αi + vi
(T ,1 )(T ,1 )
(T ,1 )
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
De…nition
The vectorial expression of the individual e¤ects model is then:
yi
(T ,1 )
=
εi
(T ,1 )
ei
X
γ + εi
(T ,K +1 )(K +1,1 )
(T ,1 )
= e
αi + vi
(T ,1 )(T ,1 )
(T ,1 )
with
ei = (e : Xi ) and γ0 = µ : β0
X
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
De…nition
Under assumptions H2 , the variance-covariance matrix of εi is
equal to:
V = E εi εi0 = E (αi e + vi ) (αi e + vi )0 = σ2α ee 0 + σ2v IT
Its inverse is:
V
1
=
1
IT
σ2v
C. Hurlin
σ2v
σ2α
+ T σ2α
ee 0
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
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De…nitions
Fixed and random e¤ects
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Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
The presence of αi produces a correlation among residuals of the
same crosssectional unit, though the residuals from di¤erent
cross-sectional units are independent
V = E εi εi0 = σ2α ee 0 + σ2v IT
0
B
V =B
@
σ2α + σ2v
σ2α
2
σα + σ2v
C. Hurlin
...
...
...
1
σ2α
C
σ2α
C
2
A
σα
2
2
σ α + σv
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
However, regardless of whether the αi are treated as …xed or as
random,the individual-speci…c e¤ects for a given sample can be
swept out by the idempotent (covariance) transformation matrix Q
Qyi = Qeµ + QXi β + Qeαi + Qvi
Since Qe = IT
T
1 ee 0
e = 0, we have
Qyi = Qeµ + QXi β + Qvi
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Theorem
Under assumptions H2 , when αi are treated as random, the LSDV
estimator is unbiased and consistent either N or T or both tend to
in…nity. However, whereas the LSDV is the BLUE under the
assumption that αi are …xed constants, it is not the BLUE in …nite
samples when αi are assumed random. The BLUE in the latter
case is the generalized-least-squares (GLS) estimator.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Fact
Moreover, if the explanatory variables contain some time-invariant
variables zi , their coe¢ cients cannot be estimated by LSDV,
because the covariance transformation eliminates zi .
C. Hurlin
Panel Data Econometrics
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Linear unobserved e¤ects panel data models
Let us consider the model
ei γ + εi
yi = X
8 i = 1, .., N
e
where εi = αi e + vi , Xi = (e, Xi ) and γ0 = µ, β0 . The variance
covariance matrix V = E (εi εi0 ) is known.
De…nition
If the variance covariance matrix V is known, the GLS estimator of
b GLS , is de…ned by:
the γ vector, denoted γ
b GLS =
γ
"
N
∑ Xei0 V
i =1
1e
Xi
#
1
"
N
∑ Xei0 V
i =1
Under assumptions H2 , this estimaor is BLUE.
C. Hurlin
Panel Data Econometrics
1
yi
#
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
De…nition
Following Maddala (1971), we write V as:
V
where Q = (IT
1
=
1
1
Q + ψ ee 0
2
σv
T
ee 0 /T ) and where parameter ψ is de…ned by:
ψ=
σ2v
σ2v + T σ2α
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
1,
Given this de…nition of V
b GLS =
γ
b GLS
γ
"
"
N
∑
i =1
N
ei0
X
we have:
1
ei
Q + ψ ee 0 X
T
ei0 Q X
ei + ψ 1
= ∑X
T
i =1
N
#
1
∑ Xei0 ee 0 Xei
i =1
ei = (e Xi ) and γ0 = µ β0
with X
C. Hurlin
"
#
N
∑
i =1
1
ei0
X
"
1
Q + ψ ee 0 yi
T
N
1
∑ Xei0 Qyi + ψ T
i =1
Panel Data Econometrics
N
#
∑ Xei0 ee 0 yi
i =1
#
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
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Linear unobserved e¤ects panel data models
bGLS
µ
b
βGLS
2
N
0
3
ψT ∑ x i
7
6 ψNT
i =1
7
6
= 4
N
N
N
0 5
0
ψT ∑ x i ∑ Xi QXi + ψT ∑ x i x i
i =1
i =1
i =1
2
3
ψNT y
N
4 N 0
5
∑ Xi Qyi + ψT ∑ x i y i
i =1
1
i =1
Using the formula of the partitioned inverse, we can derive b
βGLS .
C. Hurlin
Panel Data Econometrics
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Linear unobserved e¤ects panel data models
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Linear unobserved e¤ects panel data models
De…nition
If the variance covariance matrix V is known, the GLS estimator of
vector β is de…ned by:
b
βGLS
=
"
"
1
T
1
T
N
∑
Xi0 QXi
i =1
N
+ ψ ∑ (x i
x ) (x i
x)
i =1
N
N
i =1
i =1
∑ Xi0 Qyi + ψ ∑ (x i
x ) (y i
where ψ is a constant de…ned by ψ = σ2v σ2v + T σ2α
C. Hurlin
Panel Data Econometrics
y)
1
0
#
#
1
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
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Linear unobserved e¤ects panel data models
This estimator can be expressed as a weigthed average of the
LSDV (OLS) estimator and the between estimator.
De…nition
The between-group estimator or between estimator,denoted b
βBE ,
corresponds to the OLS estimator obtained in the model:
y i = c + β 0 x i + εi
b
βBE =
"
N
∑ (x i
x ) (x i
x )0
i =1
#
8 i = 1, .., N
1
"
N
∑ (x i
x ) (y i
i =1
y)
#
The estimator b
βBE is called the between-group estimator because
it ignores variation within the group
C. Hurlin
Panel Data Econometrics
De…nitions
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Linear unobserved e¤ects panel data models
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Linear unobserved e¤ects panel data models
De…nition
The pooled estimator,denoted b
βpooled , corresponds to the OLS
estimator obtained in the model:
yit = α + β0 xit + εit
b
βpooled
=
"
T
8 i = 1, .., N 8 t = 1, .., T
N
∑ ∑ (xi ,t
x ) (x i
t =1 i =1
"
T
x)
0
#
N
∑ ∑ (xit
x ) (yit
t =1 i =1
C. Hurlin
1
Panel Data Econometrics
y)
#
De…nitions
Fixed and random e¤ects
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Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
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Linear unobserved e¤ects panel data models
Theorem
Under assumptions H2 , the GLS estimator b
βGLS is a weighted
b
average of the between-group βBE and the within-group (LSDV)
estimators b
βLSDV .
b
βGLS = ∆ b
βBE + (IK
βLSDV
∆) b
where ∆ denotes a weigth matrix de…ned by:
∆ = ψT
"
"
N
∑
Xi0 QXi
N
∑ (x i
+ ψT
i =1
N
∑ (x i
x ) (x i
i =1
x ) (x i
i =1
C. Hurlin
x)
0
#
Panel Data Econometrics
x)
0
#
1
De…nitions
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1
2
If ψ ! 1, then GLS converges to the OLS pooled estimator.
p
b
βGLS
ψ !1
b
βGLS
ψ !0
!b
βpooled
If ψ ! 0, the GLS estimator converges to LSDV estimator.
C. Hurlin
p
!b
βLSDV
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Fact
The constant ψ measures the weight given to the between-group
variation.
In the LSDV (or …xed-e¤ects model) procedure, this source of
variation is completely ignored.
The OLS procedure corresponds to ψ = 1. The between-group and
within-group variations are just added up.
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Panel Data Econometrics
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Fact
The procedure of treating αi as random provides a solution
intermediate between treating them all as di¤erent and
treating them all as equal.
C. Hurlin
Panel Data Econometrics
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Linear unobserved e¤ects panel data models
Given the de…nition of ψ, we have:
ψ=
σ2v
σ2v + T σ2α
lim ψ = 0
T !∞
Fact
When T tends to in…nity, the GLS estimator converges to the
LSDV estimator:
b
βLSDV
βGLS ! b
T !∞
This is because when T ! ∞, we have an in…nite number of
observations for each i.Therefore, we can consider each αi as a
random variable which has been drawn once and forever, so that
for each i we can pretend that they are just like …xed parameters.
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Panel Data Econometrics
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Fact
Computation of the GLS estimator can be simpli…ed by introducing
a transformation matrix P such that
i
h
P = IT
1 ψ1/2 (1/T ) ee 0
We have
V
1
=
1 0
PP
σ2v
Premultiplying the model by the transformation matrix P, we
obtain the GLS estimator by applying the least-squares method to
the transformed model (Theil (1971, Chapter 6)).
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Panel Data Econometrics
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This is equivalent to …rst transforming the data by subtracting a
fraction 1 ψ1/2 of individual means y i and x i from their
corresponding yit and xit , then regressing yit
1 ψ1/2 y i on
1/2
aconstant and xit
1 ψ
xi .
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Panel Data Econometrics
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We can show that:
var b
βGLS
=
σ2v
"
N
∑
Xi0 QXi
N
+ ψT
∑ (x i
x ) (x i
i =1
i =1
x)
0
#
1
Because ψ > 0, we see immediately that the di¤erence between
the covariance matrices of b
βLSDV and b
βGLS is a positive
semide…nite matrix. For K = 1, we have:
var b
βGLS
C. Hurlin
var b
βLSDV
Panel Data Econometrics
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De…nition
If the variance components σ2ε and σ2α are unknown, we can use
two-stepGLS estimation.
1
In the …rst step, we estimate the variance components using
some consistent estimators.
2
In the second step, we substitute their estimated values into
b GLS =
γ
"
N
∑ Xei0 Vb
i =1
or its equivalent form.
C. Hurlin
1e
Xi
#
1
"
N
∑ Xei0 Vb
i =1
Panel Data Econometrics
1
yi
#
Speci…cation tests and analysis of covariance
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Lemma
When the sample size is large (in the sense of either N ! ∞,or
T ! ∞), the two-step GLS estimator will have the same
asymptotic e¢ ciency as the GLS procedure with known variance
components.
Lemma
Even for moderate sample size [for T
3, N (K + 1) 9; for
T
2, N (K + 1) 10], the two-step procedure is still more
e¢ cient than the covariance (or within-group) estimator in the
sense that the di¤erence between the covariance matrices of the
covariance estimator and the two-step estimator is nonnegative
de…nite
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Panel Data Econometrics
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Noting that y i = αi + β0 x i + εi and
(yit y i ) = (xit x i ) + (vit v i ), we can use the within- and
between-group residuals to estimate σ2ε and σ2α by
b2v =
σ
N T h
∑ ∑ (yi ,t
yi)
i =1 t =1
N (T
N
b2α =
σ
∑
i =1
yi
N
C. Hurlin
0
b
βLSDV (xi ,t
1)
0
b
βLSDV x i
K
1
K
2
b2v
σ
Panel Data Econometrics
xi )
i2
Speci…cation tests and analysis of covariance
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Then, we have an estimate of ψ and V
b=
ψ
b
V
1
=
b2v
σ
1
b2v + T σ
b2α
σ
!
1
b 1 ee 0
Q+ψ
2
T
bv
σ
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Panel Data Econometrics
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Speci…cation tests and analysis of covariance
Example
Let us consider a simple panel regression model for the total
number of strikes days in OECD countries. We have a balanced
panel data set for 17 countries (N = 17) and annual data form
1951 to 1985 (T = 35).
s,t = αi + βu,t + γpit + εit 8 i = 1, .., 17
si ,t the number of strike days for 1000 workers for the country
i at time t.
uitt the unemployement rate
pi ,t , the in‡ation rate
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Figure: Random e¤ects method
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Panel Data Econometrics
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Linear unobserved e¤ects panel data models
2.5. Fixed or random methods?
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Panel Data Econometrics
De…nitions
Fixed and random e¤ects
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Random e¤ects methods
Fixed or random methods?
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
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Linear unobserved e¤ects panel data models
Fact
Whether to treat the e¤ects as …xed or random makes no
di¤erence when T is large, because both the LSDV estimator and
the generalized least-squares estimator become the same estimator:
b
βGLS
! b
βLSDV
T !∞
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Panel Data Econometrics
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Linear unobserved e¤ects panel data models
When T is …nite and N is large, whether to treat the e¤ects as
…xed or random is not an easy question to answer. It can make a
surprising amount of di¤erence in the estimates of the parameters.
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Panel Data Econometrics
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Example
For example, Hausman (1978) found that using a …xed-e¤ects
speci…cation produced signi…cantly di¤erent results from a
random-e¤ects speci…cation when estimating a wage equation
using a sample of 629 high school graduates followed over six years
by the Michigan income dynamics study. The explanatory variables
in the Hausman wage equation include a piecewise-linear
representation of age, the presence of unemployment or poor
health in the previous year, and dummy variables for
self-employment, living in the South, or living in a rural area.
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Panel Data Econometrics
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Figure: Hausman (1978)
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
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Linear unobserved e¤ects panel data models
Fact
In the random-e¤ects framework, there are two fundamental
assumptions. One is that the unobserved individual e¤ects αi are
random draws from a common population. The other is that the
explanatory variables are strictly exogenous. That is, the error
terms are uncorrelated with (or orthogonal to) the past, current,
and future values of the regressors:
E ( εit j xi 1 , .., xiK ) = E ( αi j xi 1 , .., xiK ) = E ( vit j xi 1 , .., xiK ) = 0
C. Hurlin
Panel Data Econometrics
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What happens when this condition is violated?
E ( αi j xi 1 , .., xiK ) 6= 0 or E αi xi0,t 6= 0
1
The Mundlak’s speci…cation (1978)
2
The Hausman test
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Linear unobserved e¤ects panel data models
a. The Mundlak’s speci…cation
C. Hurlin
Panel Data Econometrics
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Linear unobserved e¤ects panel data models
Mundlak (1978) criticized the random-e¤ects formulation on the
grounds that it neglects the correlation that may exist between the
e¤ects αi and the explanatory variables xit . There are reasons to
believe that in many circumstances αi and xit are indeed correlated.
Mundlak Y. (1978), ”On the Pooling of Time Series and Cross
Section Data”, Econometrica, 46, 69-85.
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Linear unobserved e¤ects panel data models
The properties of various estimators we have discussed thus far
depend on the existence and extent of the relations between the
X’s and the e¤ects. Therefore, we have to consider the joint
distribution of these variables. However, αi are unobservable.
=) Mundlak (1978a) suggested that we approximate
E (αi xi ,t ) by a linear function.
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Linear unobserved e¤ects panel data models
De…nition
Let us assume that the individual e¤ects satisfy:
αi = x i0 a + αi
with a 2 RK and E (αi xit0 ) = 0. Under this assumption, the model
is:
yi ,t = µ + β0 xi ,t + x i0 a + εi ,t
εi ,t = αi + vi ,t
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Assumptions H4 The error term εi ,t = αi + vi ,t satis…es,
8 i 2 [1, N ] , 8 t 2 [1, T ]
E (αi ) = E (vi ,t ) = 0
E (αi vi ,t ) = 0
σ2α
i =j
E αi αj =
0
8i 6 = j
σ2v
t = s, i = j
E (vi ,t vj ,s ) =
0
8t 6= s, 8i 6= j
E vi ,t xi0,t = E αi xi0,t = 0
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The model can be rewritten as follows:
yi
(T ,1 )
=
ei
X
γ + εi
(T ,1 )
(T ,K +1 )(K +1,1 )
8 i = 1, .., N
e = (ex 0 , e, Xi ) and γ0 = a, µ, β0 .
with εi = αi e + vi , X
i
i
h
i
0
E εi εj0
= E (αi e + vi ) αj e + vj
=
σ2α ee 0 + σ2v IT = V
0
C. Hurlin
Panel Data Econometrics
i =j
i 6= j
Speci…cation tests and analysis of covariance
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Linear unobserved e¤ects panel data models
Utilizing the expression for the inverse of a partitioned matrix, we
obtain the GLS of (µ, β, a ) as:
bGLS = y
µ
b
aGLS = b
βBE
x 0b
βBE
b
βGLS = ∆ b
βBE + (IK
C. Hurlin
b
βLSDV
∆) b
βLSDV
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Then, the GLS estimator b
βGLS is unbiased and we have:
b
βGLS
βLSDV
! b
T !∞
b
βGLS
C. Hurlin
! β
NT !∞
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Linear unobserved e¤ects panel data models
Now let us assume that the DGP corresponds to the Mundlak’s
model
αi = x i0 a + αi
and we apply GLS to the initial model:
yi ,t = µ + β0 xi ,t + εi ,t
εi ,t = αi + vi ,t
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Linear unobserved e¤ects panel data models
In general, we have:
b
βGLS = ∆ b
βBE + (IK
∆) b
βLSDV
In this case, we can show that:
E b
βBE
= β+a
E b
βLSDV
C. Hurlin
b
βBE
p
! β+a
N !∞
=β
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Linear unobserved e¤ects panel data models
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Linear unobserved e¤ects panel data models
Theorem
So, if αi = x i0 a + αi , the GLS is biaised when T is …xed. More
precisely:
E b
βGLS = β + ∆a
b
βGLS
p
! β + ∆a
N !∞
As usual, the GLS is asymptotically unbiased:
b
βGLS
C. Hurlin
p
! β
T !∞
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Linear unobserved e¤ects panel data models
When T is …xed and N tends to in…nity, the GLS is biased if there
is a correlation between the individual e¤ects and the expanatory
variables:
plim b
βGLS
N !∞
e = plim ∆.
with ∆
N !∞
e plim b
= ∆
βBE + IK
N !∞
e ( β + a) + IK
= ∆
e
= β + ∆a
C. Hurlin
e plim b
∆
βLSDV
N !∞
e β
∆
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Linear unobserved e¤ects panel data models
Summary
E ( αi j xi 1 , .., xiK ) = 0
T Fixed, N ! ∞
T ! ∞ and N ! ∞
E ( αi j xi 1 , .., xiK ) 6= 0
LSDV
GLS
LSDV
GLS
Unbiased
—
Unbiased
Biased
Unbiased
BLUE
Unbiased
Unbiased
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Linear unobserved e¤ects panel data models
b. The Hausman’s speci…cation test
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Linear unobserved e¤ects panel data models
Hausman (1978) proposes a general test of speci…cation, that can
be applied in the speci…c context of linear panel models to the
issue of speci…cation of individual e¤ects (…xed or random).
Hausman J.A., (1978) ”Speci…cation Tests in Econometrics”,
Econometrica, 46, 1251-1271
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
The general idea of the an Hausman’s test is the following.Let us
consider a particular model y = f (x; β) + ε and particular
hypothesis H0 on this model (parameter, error term etc.).Let us
consider two estimators of the K -vector β, denoted b
β1 and b
β2 ,
both consistent under H0 and asymptotically normally distributed.
1
2
Under H0 , the estimator b
β1 attains the asymptotic
Cramer–Rao bound.
Under H1 , the estimator b
β is biased.
2
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Panel Data Econometrics
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Linear unobserved e¤ects panel data models
By examing the "distance" between b
β1 and b
β2 , it is possible to
conclude about H0 :
1
If the "distance" is small, H0 can not be rejected.
2
If the "distance" is large, H0 can be rejected.
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
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Fixed and random e¤ects
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Fixed or random methods?
Linear unobserved e¤ects panel data models
This distance is naturally de…ned as follows:
H= b
β2
b
β1
0
h
Var b
β2
b
β1
i
1
b
β2
b
β1
However, the issue is to compute the variance-covariance
matrix Var b
β2 b
β1 of the di¤erence between both
estimators. Generaly we know V b
β2
Var b
β2
b
β1 .
C. Hurlin
and V b
β1 , but not
Panel Data Econometrics
Speci…cation tests and analysis of covariance
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Linear unobserved e¤ects panel data models
Lemma (Hausman, 1978)
Based on a sample of N observations, consider two estimates b
β1
b
and β2 that are both consistent and asymptotically normally
distributed, with b
β1 attaining the asymptotic Cramer–Rao bound
p
β1 β is asymptotically normally distributed with
so that N b
p
variance–covariance matrix V1 . Suppose N b
β2 β is
asymptotically normally distributed, with mean zero and
b
variance–covariance matrix V2 . Let q
b=b
β
β1 . Then the
p2
p
limiting distribution [under the null] of N b
β
β and Nb
q
1
has zero covariance:
E (b
β1 q
b0 ) = 0K
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Panel Data Econometrics
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Linear unobserved e¤ects panel data models
Theorem
From this lemma, it follows that
b
β2
β1 = Var b
β2
Var b
Thus, Hausman suggests using the statistic
H= b
β2
or equivalently
b
β1
0
h
Var b
β2
Var b
β1
H=q
b0 [Var (q
b)]
C. Hurlin
β1
Var b
1
i
1
q
b
Panel Data Econometrics
b
β2
b
β1
Speci…cation tests and analysis of covariance
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Under the null hypothesis, this statistic is distributed
asymptotically as central chisquare, with K degrees of freedom.
H
HO
! χ2 (K )
N !∞
Under the alternative, it has a noncentral chi-square distribution
with noncentrality parameter q
e0 [Var (q
b)] 1 q
e, where q
e is de…ned
as follows:
q
e = p lim b
β2 b
β1
H 1 /N !∞
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Problem
Now, apply the Hausman’s test to discriminate between …xed
e¤ects methods and random e¤ects methods. We assume that αi
are random variable and the key assumption tested is here de…ned
as:
H0 : E ( αi j Xi ) = 0
H1 : E ( αi j Xi ) 6= 0
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
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Linear unobserved e¤ects panel data models
This test can be interpreted as a speci…cation test between "…xed
e¤ect methods" and "random e¤ect methods".
1
If the null is rejected, the correlation between individual
e¤ects and the explicative variables induces a bias in the GLS
estimates. So, a standard LSDV approach (…xed e¤ect
method) has to be privilegiated.
2
If the null is not rejected, we can use a GLS estilmator
(random e¤ect method) and specify the individual e¤ects as
random variables (random e¤ects model).
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
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De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
How to implement this test? Let us consider the standard model
with random e¤ects (µ = 0):
yi = Xi β + eαi + vi
1
2
3
Under H0 (and assumptions H2 ) we know that b
βLSDV and
b
βGLS are consistent and asymptotically normally distributed.
Under H0 , b
βGLS is BLUE and attains asymptotic Cramer–Rao
bound.
Under H1 , b
β
is biased.
GLS
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
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Linear unobserved e¤ects panel data models
According to the Hausman’s lemma, we have:
cov b
βGLS , b
βLSDV
b
βGLS
βLSDV
Var b
βLSDV + Var b
βGLS
= Var b
Since,
We have:
b
βGLS
Var b
βLSDV
b
βGLS
= 0 () cov b
βLSDV , b
βGLS
= Var b
βLSDV
C. Hurlin
βLSDV , b
βGLS
2cov b
Var b
βGLS
Panel Data Econometrics
= Var b
βGLS
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
De…nition
The Hausman’s speci…cation test statistic of individual e¤ect can
be de…ned as follows:
i 1
0h
b
H= b
βLSDV b
βGLS
Var b
βLSDV
Var b
βGLS
βLSDV
Under H0 , we have:
H
HO
! χ2 (K )
NT !∞
C. Hurlin
Panel Data Econometrics
b
βGLS
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
1
2
When N is …xed and T tends to in…nity, b
βGLS and b
βMCG
become identical. However, it was shown by Ahn and Moon
(2001) that the numerator and denominator of H approach
zero at the same speed. Therefore the ratio remains
chi-square distributed. However, in this situation the
…xed-e¤ects and random-e¤ects models become
indistinguishable for all practical purposes.
The more typical case in practice is that N is large relative to
T , so that di¤erences between the two estimators or two
approaches are important problems.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
Example
Let us consider a simple panel regression model for the total
number of strikes days in OECD countries. We have a balanced
panel data set for 17 countries (N = 17) and annual data form
1951 to 1985 (T = 35).
s,t = αi + βi u,t + γi pit + εit 8 i = 1, .., 17
si ,t the number of strike days for 1000 workers for the country
i at time t.
uitt the unemployement rate
pi ,t , the in‡ation rate
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
De…nitions
Fixed and random e¤ects
Fixed e¤ects methods
Random e¤ects methods
Fixed or random methods?
Linear unobserved e¤ects panel data models
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cients models
Section 3.
Random coe¢ cients models
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cients models
There are cases in which there are changing economic structures or
di¤erent socioeconomic and demographic background factors that
imply that the response parameters may be varying over time
and/or may be di¤erent for di¤erent crosssectional units.
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cients models
3.1. Variable coe¢ cient model
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
When data do not support the hypothesis of coe¢ cients being the
same, a single-equation model in its most general form can be
written as:
K
yit =
∑ βkit xkit + vit
i = 1, .., N and t = 1, .., T
k =1
where, in contrast to previous sections, we no longer treat the
intercept di¤erently than other explanatory variables and let
x1it = 1.
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
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Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
Fact
We assume that parameters do not vary with time (the time
stability issue is not speci…c to panel data econometrics)
βkit = βki
K
yit =
∑ βki xkit + vit
i = 1, .., N and t = 1, .., T
k =1
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
In the case in which only cross-sectional di¤erences are
present, this approach is equivalent to postulating a separate
regression for each cross-sectional unit
yi = Xi βi + vi
i = 1, .., N
or
yit = βi0 xit + vit
i = 1, .., N and t = 1, .., T
where βi = ( β1i , β2i , ..., βKi ) is a 1 K vector of parameters, and
xit = (x1it , ..., xKit ) is a 1 K vector of exogenous variables.
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
De…nition
Alternatively, each regression coe¢ cient can be viewed as a
random variable with a probability distribution (e.g., Hurwicz
(1950); Klein (1953); Theil and Mennes (1959); Zellner (1966)).
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
1
The random-coe¢ cient speci…cation reduces the number of
parameters to be estimated substantially, while still allowing
the coe¢ cients to di¤er from unit to unit and/or from time to
time.
2
Depending on the type of assumption about the parameter
variation, it can be further classi…ed into one of two
categories: stationary and nonstationary random-coe¢ cient
models.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
De…nition
Stationary random-coe¢ cient models regard the coe¢ cients as
having constant means and variance–covariances.Namely, the
K
1 vector of parameters βi is speci…ed as:
βi = β + ζ i
for i = 1, .., N, where β is a K
1 vector of constants, and ζ i is a
K
1 vector of stationary random variables with zero means and
constant variance–covariances.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
For this type of model we are interested in
1
Estimating the mean coe¢ cient vector β,
2
Predicting each individual component βi ,
3
Estimating the dispersion of the individual-parameter vector
βi (variance covariance matrix ∆)
4
Testing the hypothesis that the variances of ξ i (and so βi ) are
zero.
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
Fact
Because of the computational complexities, variable-coe¢ cient
models have not gained as wide acceptance in empirical work as
has the variable-intercept model. However, that does not mean
that there is less need for taking account of parameter
heterogeneity in pooling the data.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
De…nition
When regression coe¢ cients are viewed as invariant over time, but
varying from one unit to another, we can write the model as
K
yit =
∑ ( βk + ξ ki ) xkit + vit
i = 1, .., N and t = 1, .., T
k =1
where β = ( β1 , β2 , ..., βK ) can be viewed as the common mean
coe¢ cient 1 K vector, and ξ i = (ξ 1i , ξ 2i , ..., ξ Ki ) as the
individual deviation from the common mean.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
1
If individual observations are heterogeneous or the
performance of individual units from the data base is of
interest, then ξ i are treated as …xed constants.
2
If conditional on xkit , individual units can be viewed as
random draws from a common population or the population
characteristics are of interest, then ξ i are generally treated
as random variables having zero means and constant
variances and covariances.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
De…nition (Fixed-coe¢ cient model)
When βi are treated as …xed and di¤erent constants, we can stack
the NT observations in the form of the Zellner (1962) seemingly
unrelated regression model
0
1 0
10
1 0
1
y1
β1
X1 0 0
v1
@ . A = @ 0 .. .. A @ . A + @ . A
yN
0 .. XN
βN
vN
where yi and vi are T 1 vectors (yit , ..., yiT ) and (vit , ..., viT ),
and Xi is the T K matrix of the time-series observations of the
i th individual’s explanatory variables with the t th row equal to xit .
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
1
2
If the covariances between di¤erent cross-sectional units are
0
not zero, e.g. E (vi vj 0 ) 6= 0, the GLS estimator of β10 , ..., βN
is more e¢ cient than the single-equation estimator of i for
each cross-sectional unit.
If Xi are identical for all i or E (vi vi 0 ) = σ2i IT and
0
is
E (vi vj 0 ) = 0 for i 6= j, the GLS estimator for β10 , ..., βN
the same as applying least squares separately to the
time-series observations of each cross-sectional unit.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
3.2. Random coe¢ cient model
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
De…nition
We now assume that all the coe¢ cients βi are random, with
common mean β.
K
yit =
∑ ( βk + ξ ki ) xkit + vit
i = 1, .., N and t = 1, .., T
k =1
where β = ( β1 , β2 , ..., βK ) can be viewed as the common mean
coe¢ cient 1 K vector, and ξ i = (ξ 1i , ξ 2i , ..., ξ Ki ) as the
individual deviation from the common mean. Let us denote
βki = βk + ξ ki
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
1
The vector xi = (x1i ..xKi ) includes a constant term.The
parameter βki associated to this constant term correspond to
an individual e¤ect
2
An alternative notation is:
K
yit = α +
∑ ( βk + ξ ki ) xkit + αi + vit
k =2
E ( αi ) = 0
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
We will consider a set of assumptions used in the seminal paper of
Swamy (1970)
Swamy P.A. (1970), ”E¢ cient Inference in a Random
Coe¢ cient Regression Model”, Econometrica, 38, 311-323
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
Assumptions (Swamy, 1970) Let us assume that
E (ξ i ) = 0, E (vi ) = 0
∆
i =j
E ξ i ξ j0 =
0 8i 6 = j
E xit ξ j0 = 0, E ξ i vj0 = 0, 8 (i, j )
E (vi vj 0 ) =
σ2i IT
0
C. Hurlin
t = s, i = j
8t 6= s, 8i 6= j
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
Remark : we assume that the error term vi is heteroskedastic:
E vi vi0 = σ2i IT
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Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
De…nition
The two …rst moments of the vector of random parameters
βi = β + ξ i are de…ned by, 8 i = 1, ..N :
E ( βi ) = β
(K ,1 )
(K ,1 )
∆ =E
(K ,K )
βi βi0
=E
ξ i ξ i0
0
B
B
=B
@
C. Hurlin
σ2ξ
1
σξ 2 ,ξ 1
...
σξ K ,ξ 1
σξ 1 ,ξ 2
σ2ξ
2
...
σξ K ,ξ 2
Panel Data Econometrics
...
...
...
...
σξ 1 ,ξ K
σξ 2 ,ξ K
...
σ2ξ
K
1
C
C
C
A
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
Fact
The matrix ∆ corresponds to variance covariance matrix of the
random parameters βi = ( β1i , β2i , .., βKi )0 , which is assumed to be
common to all cross section units:
0 2
1
σβ
σ β1 ,β2 ... σ β1 ,βK
1
B σ
2
... σ β2 ,βK C
B β ,β σ β2
C
∆ =B 2 1
C
@ ...
A
...
... ...
(K ,K )
σ βK ,β1 σ βK ,β2 ... σ2β
K
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
De…nition
For each cross section unit, we have:
yi = Xi β + Xi ξ i + vi
with
βi = β + ξ i
where the vector Xi include a constant term (i.e. the average of
random individual e¤ects, α).
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
De…nition
This model can be rewriten as follows :
yi = Xi β + εi
with
εi = Xi ξ i + vi = Xi ( βi
C. Hurlin
β) + vi
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
For a given cross unit, the covariance matrix for the composite
disturbance term εi = Xi ξ i + vi is de…ned by:
Φi
= E εi εi0
= E (Xi ξ i + vi ) (Xi ξ i + vi )0
= Xi E ξ i ξ i0 Xi0 + E vi vi0
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
De…nition
For a given cross unit, the covariance matrix for the composite
disturbance term εi = Xi ξ i + vi is de…ned by:
Φi = Xi ∆Xi0 + σ2i IT
Stacking all NT observations, the covariance matrix for the
composite disturbance term is either block-diagonal and
heteroskedastic or diagonal and heteroskedastic.
C. Hurlin
Panel Data Econometrics
Speci…cation tests and analysis of covariance
Linear unobserved e¤ects panel data models
Random coe¢ cients models
Variable coe¢ cient model
Random coe¢ cient model
Mixed …xed and random coe¢ cient model
Random coe¢ cient model
1
Under Swamy’s assumption, the simple regression of y on X
will yield an unbiased and consistent estimator of β if
(1/NT )X 0 X converges to a nonzero constant matrix.
2
But the estimator is ine¢ cient, and the usual least-squares
formula for computing the variance–covariance matrix of the
estimator is incorrect, often leading to misleading statistical
inferences.
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De…nition
The best linear unbiased estimator of β is the GLS estimator
b
βGLS =
N
∑ Xi0 Φi
1
i =1
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Xi
!
1
N
∑ Xi0 Φi
i =1
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1
yi
!
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De…nition
The GLS estimator b
βGLS is a matrix-weighted average of the
least-squares estimator b
βi for each cross-sectional unit, with the
weights inversely proportional to their covariance matrices:
Wi =
(
N
∑
i =1
h
b
βGLS =
∆
+ σ2i
Xi0 Xi
1
N
∑ Wi bβi
i =1
i
b
βi = Xi0 Xi
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1
)
1
1
h
∆ + σ2i Xi0 Xi
Xi0 yi
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i
1
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The covariance matrix for the GLS estimator is:
! 1
N
V b
β
=
∑ Xi0 Φ 1 Xi
i
GLS
i =1
=
(
N
∑
i =1
h
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∆
+ σ2i
Xi0 Xi
1
i
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1
)
1
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Swamy proposed using the least-squares estimators
1
b
βi = (Xi0 Xi ) Xi0 yi and their residuals vbi = yi Xi b
βi to obtain
2
unbiased estimators of σv and ∆
b2i =
σ
=
1
T
K
T
1
T
0
with yi ,t = b
βi xi ,t + vbi ,t .
h
yi0 I
K
∑ vbi ,t
Xi Xi0 Xi
1
i
Xi0 yi
t =1
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For the ∆ matrix, we have:
b
∆
(K ,K )
=
1
N
1
N
N
1 i∑
=1
N
"
∑ σb2
i
b
βi
Xi0 Xi
1
N
1
N
∑ bβi
i =1
!
b
βi
i =1
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1
N
N
∑ bβi
i =1
!0 #
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De…nition
b is not necessarily nonnegative
However, the previous estimator ∆
de…nite. In this situation, Swamy (1970) has suggested replacing
this estimator by:
"
!
!0 #
N
N
N
1
1
1
b
b
b
b
b =
βi
βi
βi
βi
∆
N 1 i∑
N i∑
N i∑
(K ,K )
=1
=1
=1
This estimator, although not unbiased, is nonnegative de…nite and
is consistent when T tends to in…nity.
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b for σ2 and ∆ yields an
b2i and ∆
Swamy proved that substituting σ
i
asymptotically normal and e¢ cient estimator of β. The speed of
convergence of the GLS estimator is N 1/2 .
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Summary: how to estimate a random coe¢ cient model?
1
2
0
Run the N individual regressions yi ,t = b
βi xi ,t + vbi ,t .
b as follows
b2 and the Swamy’s estimator ∆
Compute σ
i
b =
∆
(K ,K )
1
N
N
1 i∑
=1
b2i =
σ
"
1
T
K
b
βi
h
yi0 I
C. Hurlin
1
N
N
∑ bβi
i =1
!
Xi Xi0 Xi
b
βi
1
1
N
N
∑ bβi
i =1
i
Xi0 yi
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3. Compute the GLS estimate of the expectation of individual
parameters βi
b
βGLS =
N
∑
i =1
b 1 Xi
Xi0 Φ
i
!
1
N
∑
i =1
b i0 + σ
b i = Xi ∆X
b2i IT
Φ
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b 1 yi
Xi0 Φ
i
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!
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Example
Swamy (1970) used this model to reestimate the Grunfeld
investment function with the annual data of 11 U.S. corporations.
His GLS estimates of the common-mean coe¢ cients of the …rms’
beginning-of-year value of outstanding shares and capital stock are
0.0843 and 0.1961, with asymptotic standard errors 0.014 and
0.0412, respectively. The estimated dispersion measure of these
coe¢ cients is
0.0011
0.0002
b=
∆
0.0187
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Predicting Individual Coe¢ cients
1
Sometimes one may wish to predict the individual component
βi , because it provides information on the behavior of each
individual and also because it provides a basis for predicting
future values of the dependent variable for a given individual.
2
Swamy (1970, 1971) has shown that the best linear unbiased
predictor, conditional on given βi , is the least-squares
estimator b
βi .
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De…nition
Lee and Gri¢ ths (1979) suggest predicting βi by
βGLS + ∆Xi0 Xi ∆Xi0 + σ2ι IT
βi = b
1
yi
Xi b
βGLS
This predictor is the best linear unbiased estimator in the sense
that E ( βi
βi ) = 0, where the expectation is an unconditional
one.
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3.3. Mixed …xed and random coe¢ cient
model
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Many of the previously discussed models can be treated as special
cases of a general mixed …xed- and random-coe¢ cients model.
C. Hurlin
Panel Data Econometrics
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De…nition
We assume that each cross section unit is di¤erent
K
yit =
∑
k =1
m
βki xki + ∑ γli wli + vit
i = 1, .., N and t = 1, .., T
l =1
where xit and wit are each a K 1 and an m 1 vector of
explanatory variables that are independent of the error of the
equation vit .
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The parameters
β
= ( β10 , β20 , ..., βK0 )0 are assumed to be
(NK ,1 )
random and parameters
γ
(Nm,1 )
0 )0 are …xed.
= (γ10 , γ20 , ..., γm
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In a vectorial form, we have
y
(NT ,1 )
0
=
X
β
(NT ,NK )(NK ,1 )
X1 0
B 0 X2
X =B
@
0
1
0
+
W
γ
(NT ,Nm )(Nm,1 )
+
0
C
C
A
W1 0
B 0 W2
W =B
@
0
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XN
v
(NT ,1 )
0
WN
1
C
C
A
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The motivation of a mixed …xed and random coe¢ cients model is
that while there may be fundamental di¤erences among
cross-sectional units, by conditioning on these individual speci…c
e¤ects one may still be able to draw inferences on certain
population characteristics through the imposition of a priori
constraints on the coe¢ cients of xit and wit.
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We assume that there exist two kinds of restrictions, stochastic
and …xed, in the following form:
1
The coe¢ cients of xit are assumed to be subject to stochastic
restrictions of the form
β = A1 β + ζ
where A1 is an NK L matrix with known elements, β is an L 1
vector of constants, and ζ is assumed to be (normally distributed)
random variables with mean 0 and nonsingular constant covariance
matrix C and is independent of xi .
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2. The coe¢ cients of wit are assumed to be subject to
γ = A2 γ
where A2 is an Nm n matrix with known elements, and γ is
an n 1 vector of constants. Since A2 is known, we can
rewrite the model as
y
=
(NT ,1 )
X
β
(NT ,NK )(NK ,1 )
f = WA2 .
where W
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f
+ W
γ +
(NT ,n )(n,1 )
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v
(NT ,1 )
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Many of the linear panel data models with unobserved individual
speci…c but time-invariant heterogeneity can be treated as special
cases of this model
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Example
A common model for all cross-sectional units. If there is no
interindividual di¤erence in behavioral patterns, we may let X = 0,
A2 = eN Im , so model becomes
yit = wit γ + vit
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Example
When each individual is considered di¤erent, then X = 0,
A2 = IN Im , and the model becomes
yit = wit γi + vit
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Example
When the e¤ects of the individual speci…c, time-invariant omitted
variables are treated as random variables just as in the assumption
on the e¤ects of other omitted variables, we can let Xi = eT ,
ζ 0 = (ζ 1 , ..., ζ N ), A1 = eN , C = IN σ2α , β be an unknown constant,
and wit not contain an intercept term. Then the model becomes:
yit = β + γ0 wit + ζ i + vit
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Panel Data Econometrics