Open Journal of Statistics, 2011, 1, 185-198
doi:10.4236/ojs.2011.13022 Published Online October 2011 (http://www.SciRP.org/journal/ojs)
185
Double Autocorrelation in Two Way Error Component
Models
Jean Marcelin Bosson Brou1, Eugene Kouassi2, Kern O. Kymn3
1
Department of Economics, University of Cocody, Abidjan, Cote-d’Ivoire
2
Resource Economics, West Virginia University, Morgantown, USA
3
Division of Finance and Economics, West Virginia University, Morgantown, USA E-mail: [email protected]
Received September 15, 2011; revised October 16, 2011; revised October 30,2011
Abstract
In this paper, we extend the works by [1-5] accounting for autocorrelation both in the time specific effect as well as the remainder error term. Several transformations are proposed to circumvent the double autocorrela-tion problem in some specific cases. Estimaautocorrela-tion procedures are then derived.
Keywords:Two Way Random Effect Model, Double Autocorrelation, GLS, FGLS
1. Introduction
Following the works of [6], the regression model with error components or variance components has become a popular method for dealing with panel data. A summary of the main features of the model, together with a discus-sion of some applications, is available in [7-10] among others.
However, relatively little is known about the two way error component models in the presence of double auto-correlation, i.e, autocorrelation in the time specific effect
and in the remainder error term as well.
This paper extends the works by [2-5] on the one-way random effect model in the presence of serial autocorre-lation, and by [1] on the single autocorrelation two-way approach. It investigates some potential transformations to circumvent the double autocorrelation issue, along with some estimation procedures. In particular, we derive several transformations when the two disturbances fol-low various structures: from autoregressive and mov-ing-average processes of order 1 to a general case of double serial correlation. We deduce several GLS esti-mators as well as their asymptotic properties and provide a FGLS version.
The remainder of this paper is organized as follows: Section 2 considers simple transformations on the pres-ence of relatively manageable double autocorrelation structure. In Section 3, general transformations are con-sidered when the double autocorrelation is more complex. GLS estimators are derived in Section 4. Asymptotic
properties of the GLS estimators are considered in Sec-tion 5. SecSec-tion 6 provides a FGLS counterpart approach. Finally, some concluding remarks appear in Section 7.
2. Simple Transformations
To circumvent the double autocorrelation issue, we first need to transform the model based on the variance-co- variance matrix. The general regression model consid-ered is yit 0xituit, ;
where 0
1, ,
i N t1, , T
is the intercept and is a vector of slope coefficients, it
1
k
x is a row vector of
ex-planatory variables which are uncorrelated with the usual two-way error components disturbances
1k
it i
u t
it
(see [7]). In matrix form, we writeyXu.
2.1. When the Errors Follow AR(1) Structures
If the time specific term follows an AR(1) structure, 1
t t t
, 1, with
0, 2 t IID
, and the remainder error term also follow an AR(1) structure
, 1 it i t eit
, 1, with eit IID
0, e
2 , we
can define two transformation matrices of dimensions
T 2
T1
and
T 1
T respectively,1 0 0
0 1
and 0
0 0 1
C
1
1 0 0
0 1
0
0 0
C
(1)
and since we have
3 2
, 1
e e
e e
i i
i
iT i T
C C
and
3 2
*
1
T T
C C
(2)
the transformed errors * i
and * follow two different MA(1) processes, of parameters and respective- ly. Thus, by applying the appropriate transformation ma-trices, the autoregressive error structure can be changed into a moving-average one. The only cost is the loss of the initial and first pseudo-differences, which has no se-rious consequence for a long time dimension. As a result, we focus on the MA(1) error structure.
2.2. When the Errors Follow MA(1) Structures
Here, the time specific term t follows an MA(1)
struc-ture, t t t1 , 1 with
2 0,t IID
while the remainder error term, it also follows an MA
(1) structure, i t, eitei t, 1 , 1 with
it
2 e
e IID 0, . For convergence purpose and assuming normality, the initial values are defined
2 2
0 0, 1 e
i N
2
and
2 2
0 N 0, 1 2
The variance-covariance matrix of the three components error term is given by,
2 2 ' 2 '
e IN IN i iT T i iN N
(3)
where
and
are positive defi-nite matrices of order T and where
.
x x,0, ,0is defined by . The exact inverse of such matrices suggested by [11] and [1] does not involve the parameters
x Toeplitz 1 2,
and . Following [11], let be the Pesaran orthogonal matrix whose t-th row is given by,
P
2 π 2π
sin ,sin , ,sin
1 1 1
t
t t Tt
L
T T T T
π
1
where
'
PP , diag
1, , T
,2
12cos π 1
t
t T
, and
' PP D
1, , t
g d d
Ddia with 2 cos π 1
t T
1 2
t
d
.
Pre-multiplying the model by
INP
of *
u
yields the fol-lowing variance-covariance matrix
INP u
,
'
* 2 2
I D I i i
2
'
e N N T T i iN N
(4)
where .
eral T
sformations
eral case of double
T T
i Pi
. Gen
ran
3
e are now in the context of a gen W
autocorrelation issue and lead to a suitable error covari-ance matrix similar to Equation (4) and its inverse.
.1. First Transformation 3
et P
L
tri
denote the matrix such that ' . Such
T
P P I
a ma x does exist for and is efinite matrix. Transformation of the initial model
a positive-d yXu by
INP
yields
P
y X u* N
y I * * (5)
and the variance-covariance of the transformed errors is
* * *' 2 '
N
2 ' ' 2 '
E u u I P P
N T T N N T
I P i i P i i I
(6)
This transformation has removed the autocorrelation in the time-specific effect t. Unfortunately, by doing
so it has infected the its a worsened the initial cor-
relation in the remaind disturbances. An additional “treatment” is therefore needed.
.2. Second Transformation
nd er
3
e now consider an orthogona
W l matrix P and a
di-P
agonal matrix D such that P P
P'
'D (di-agonalization of 'P P ). Thu cond
transformation
s, applying a se
IN P
yields,
*y** INP y X**u** (7) The underlying variance-covariance matrix of the errors is,
** **'
2
'
'
N
J. M. B. BROU ET AL. 187
'
2 ' ' ' 2 '
N T T N N
I P P i i P P i i P
P (8a)
or,
** ** **' 2
1 2
2 ' 2 '
2 3
1
N
N T T N N
E u u I D
I i i i i
(8b)
where iT PP iT ,
, PP', P P
P P'
' D 22 2
1
,
2 2
2 2
, and
2
2 2
3
2 2 1 2 1
,
if 2 2 2 2
.
Here, because of the choice of matrices P and end up with
P,
we IT
elemen
since is an or tri
ng the procedure de-ra, one gets
P
to
thogo eed
nal ma-x. Generally speaking, and D just n to have zero off-diagonal ts, i.e., be diagonal matrices.
The double autocorrelatio structure is thus absorbed, and one can easily accommodate with the non-spherical form of ** by means of an accurate inversion process.
3.3. Computing the Inverse
n
The inverse of ** is obtained usi eloped by [1]. After a bit of algeb v
** 1
2 2
1
1 1 1
N T N T
E K E L J S
d N
N T
(9) where
' 1
2 22 1
T T
d i D i , JN i iN N' ,
N
N N
J
E I
N
, 1
T T
K D L ,
1 ' 1T D i iT T
,
' 1 1
T T
L
i D
D
i
2 2
4 2
1N 1 i
' 2
1
1
T T T
S S Si i S
and
s with
2 ' 2 T SiT
1
g s, , T
dia S
2 1
2 2
1 3
T
t t
N s
d N
.
Proof: (see the A
4.
ition of the estimator followed ghted average property.
The GLS estimator is,
ppendix)
GLS Estimation
We begin with the defin y its interpretation and wei b
4.1. The GLS Estimator
Proposition 1:
**' ** 1 **
1 **'
** 1 **GLS X X X y
(10)
Proof: (Straightforward)
4.
ression models, [12,13] provide n interpretation of the GLS estimator, which is
appeal-2. Interpretation
In classical two-way reg a
ing in view of the sources of variation in sample data. In the straight line of their work, the GLS estimator may be viewed as obtained by pooling three uncorrelated esti-mators: the covariance estimator (or within estimator), the between-individual estimator and the within-indi- vidual estimator. They are the same as those suggested by [1] except for the last one which was labeled be-tween-time estimator. We have,
1) The covariance estimator,
**' **
1 C
'
** ** 1
X A X
X A y1 ,
where A1
EN KT
;2) The between-indivi ual estimator, d
**' **
1 **' *2 2
B
*
X A X X A y
,
2 2
1
N T
A J S
N
where and,
e within-individual or,
3) Th estimat
**' **
1 **'3 T
** 3
X A X X A
y ,
where A3
ENLT
.It is im
from so on
portant to note that these estimators are obtained me transformati s of the regression Equation (7),
i.e.,
** ' * ** **
y INP y X u .
The covariance estimator, C
y
is obtained when Equa- tion (7) is pre-multiplied b M1
ENKT
A1; the traones in the matrix o nsformation annihilates the individual- and time-ef- fects as well as the column of f ex-planatory variables. It is equivalent to the within estima-tor in the classical two-way error component model (see [1-7]).
The between-individual estimator B comes from the
'
2
1
N T
M i I
N
. This is equivalent to averaging
indi-r each time peindi-riod. ividual estimator T
vidual equations fo
The within-ind is derived when Equation (7) is transformed byM3
ENLT
A3. The presence of the idempotent matrix EN indicates thatthis transformation wipes out th ell as the time specific error term t
e constant term as w
. Howe , the individual effect i
ver
remains.
4.3. GLS as a Weighted Average Estimator
ge of e three estimators defined above.
As in [14], the GLS estimator is a weighted avera th
Proposition 2:
C C B B
F F F
GLS T T
(11) with,
**' ** 1 **
1 **' **1 2
1 1
C
F X X X A
,
**' ** 1 **
1 **' **2 B
X
F X X X A X
,
and
**' ** 1 **
1 **' **3
1
T C B
F X X X A X I F F
d
Proof:
From Equation (10), it comes that
X**' ** 1y** X**' ** 1X** GLS
with
1**' ** ** **' ** **' **
1 2
2 1
**' ** 3
1
1
X y X A y X A
X A y d
y
By definition, the estimators C, B and T are
re-spectively such that
**' ** **' *
X A y1 X A1X * C,
**' ** **' **
2 2 B
X A y X A X ,
and
**' ** **' **
3 3 T
X A y X A X .
Therefore,
1
**' ** ** **' **
1 2 1
**' ** **' **
2 3
1
1
C
B T
X y X A X
X A X X A X
d
Or,
1
**' ** ** **' **
1 2 1
**' ** **' **
2 3
1
1
GLS C
B T
X X X A X
X A X X A X
d
Thus,
1 1
**' ** ** **' **
1 2
1
1 1
**' ** ** **' **
2 1 1
**' ** ** **' **
3 1
1
GLS C
B
T
C C B B T T
X X X A X
X X X A X
X X X A X
d
F F F
with FC, FB and FT
no
defined according to Equation 11). ld also te that the three estimators
(
We shou C, B
and T are uncorrelated. In fact,
2
** 1
1 2 N N T T 0
A A E J K DS
N
and
** 2
1 3 1 N T T 0
A A E K DL
because K iT T 0 EN N 0
i , while A2**A3 0 since
N N
J E . As a result,
cov C, B cov C, T cov B, T 0 (12)
Moreover, following [1], the fact that
1
rank M r M
2
ank rank
1 1 1
N T T N N
3 M
T (13)
le information gives evidence on the use of all availab
from the sample. The estimators C, B and T
to-gether use up the entire set of to GLS estimator
information build the
GLS
with no loss at
rs o tor, say
all.
5. Asymptotic Properties
Under regular assumptions, the GLS and the three pseudo estimato f the coefficient vec GLS,
C
, B and T are all consistent and asympto
the one obtain lassical two-way error component model (see [15]).
tically ed in the equivalent. It is a result similar to
c
5.1. Assumptions
We assume that the x sit are weakly non-stochastic, i.e.
189 J. M. B. BROU ET AL.
' '
** X A X
** ** **
1
plim plim X EN KT X
a
1
NT NT
for the first transformation;
'
' ** **
** ** 2
2 1
1
plim plim
N T
X J S X
X A X N
b
T T
for the second transformation; and
'
' ** **
** **
3
1 plim plim
N T
X E L X
X A X c
NT NT
more, in the straight for the third transformation. Further
line of [1], we also assume that,
'
' ** **
** ** 1
2 plim plim 0
N T
X E K u
X A u a
NT NT
for the first transformation;
''
** ** 2 2
** **
2
plim X
N
plim
1
0
N T
X A u b
T
J S u
T
for the second transformation;
'
' ** **
** ** 3
2 plim plim 0
N T
X E L u
X A u c
NT NT
addition,
the variance-components for the third transformation. In
' 1
lim i D iT T , so that T
quantity ' 1
T T
i D i
2 22 1
. The lim and N
symptotic
infinite as T taken as T
garding the error v sures the a
denotes by remains its and pro lities a
, ss tions
re-ptotic erty of
The covariance estimator
d
babi
. All along this section ump
re
following [1], we consider the “usual” a ector **
u , as stated in [16] and [17],
which en normality.
5.2. Asym Prop the Covariance
Estimator
Proposition 3:
C
is consistent.
T Proof:
Since,
**' **
1 **'
**C X EN KT X X EN K u
Hence,
1
**' K X**
**' **
plim
plim 0
N T
C
N T
X E
NT
X E K u
NT
plim
Making use of assumptions (a1) and (a2), we establish the consistency of the covariance estimator, plim
C .Proposition 4:
The covariance estimator C has an asymptotic nor-mal distribution given by,
1**' **
2 1
1
, plim N T
a
C N
NT NT
X E K X
1 ation, we have
(14)
Proof:
Under the M -transform
**
** 0N T N T
E E K u E K E u .
Moreover its variance is given by
** 2
1
N T N
V E K u E K
eq
T and its inverse is
ual to 12
END
while assumption (a2) states t 1he
absence of correlation between regressors and istur-. We have
d bances under the M1 transformation
**' **
* 2
N T d
N T
X E K u
NT
X
*' **
1
0, plim E K X
N
NT
(15)
and,
1
**' ** **' **
C
N T N T
NT
X E K X X E
NT NT
hich we deduce that
K u
from w
1**' **
2 1
0, plim
d C
N T
NT
X E K X
N
NT
.
Thus, the asymptotic normality of the covariance esti-mator immediately follows,
1**' **
2 1 1
, plim N T
a C
X E K X
N
NT NT
5.3. Asymptotic Property of the Between-Time Estimator
Proposition 5:
The between time estimator B is consistent. Proof:
Since,
**' **
1 **' **2 2
B X A X X A u
Hence, according to assumptions (b1) and (b2),
1
**' **
plim
N
J
X S X
2
**' **
2
plim
plim 0
T
B
N T
N T
J
X S u
N T
the consistency of the between time estimator,
Making use of assumptions (b1) and (b2), we establish
plim B .
Proposition 6:
The between-time estimator B has an asymptotic
normal distribution given by,
1
**' **
2 1 1
, lim
N T
a B
X J S X
N
N p
T T
Proof:
nsformation, we get
(16)
Under the M2-tra
' '
**
N N
T
i i
E I u
N N
** 0
T
I E u
The variance of this error term is written as
' 2
** 2 1 2 '
1 N
T T
i
V I u S i
N N
iT
Its inverse is Again, ption (b2) states the ab-se rrelation between regressors and disturbances un
T
S . assum nce of co
der the M2 transformation. We get
**' **
2 N
T
d
J
X S u
N T
**' **
2 0, plim
N T
J
X S X
N N
T
In addition, we have
1
**' **
2
**' **
2
N T
B
N T
J
X S X
N T
T
J
X S u
N
T
that from which we deduce
1
**' **
2 0, plim
d B
N T
T
J
X S X
N N
T
Thus, the asymptotic normality of the between-time es-timator immediately follows,
1
**' **
2 1 1
, plim
N T
a B
X
J S X
N N
T T
.
5.4. Asymptotic Property of the Within-Individual Estimator
ator
Proposition 7:
The within individual estim T is a consistent
estimator.
Proof:
Since,
**' **
1 **' **3 3
T X A X X A u
Hence,
1 **'
plim plim N
T
X E
NT
**
**' **
T
L X
X E L u
T
plim N T 0 N
J. M. B. BROU ET AL. 191
f assumptions (c1) and (c2), we establish the consistency of the covariance estimator,
Making use o
plim T
Proposition 8:
thin individual estimator
The wi T has an
asymp-totic normal distribution given by,
1**' **
, plim N T
a T
X E L X
d N
NT NT
(17)
Th is obtained as
Proof:
Under the M3-transformation, we obtain
**
** 0N T N T
E E L u E L E u
e variance of
**N T
E L u
**
N T
V E L u d ENLT
The inverse of this matrix is 1
EN LT
d . Assumption (c2) states the absence of correlation between regressors and disturbances under the M3 transformation. We have
**' **
**' **
0, plim
N T d
N T
X E L u
NT
X E L X
N d
and,
NT
1
**' **
N T
X E L X
NT
from which we deduce that
**' **
T
N T
NT
X E L u
NT
1**' **
0, plim
d T
N T
NT
X E L X
N d
NT
Thus, the asymptotic normality of the within individual estimator immediately follows,
1**' **
, plim N T
a T
X E L X
d N
NT NT
of the GLS Estimator
Proposition 9:
The GLS estimator
5.5. Asymptotic Property
GLS
ator
is asymptotically equivalent to the covariance estim C and therefore,
1**' **
2
1
, lim X EN KT X
N p
NT
(18)
1 a GLS
NT
Proof:
From Equation (10), we get
1
1 1
**' ** ** **' ** **
X X X u
GLS
NT
NT NT
On the one hand, we have
1
**' ** ** **' **
2 1
**' ** **' **
2 1
1 1
N T
N T N T
X X X E K X
NT NT
X E L X X J S X
d NT N N T
here
' 1
2 22 1
T T
d i D i
w
om assumption (a1), we find
, as .
There-fore, fr that
T
** 1
wise
' **
0
N T
X E L X
d NT
, when N T, . Like ,
assumption (a2) leads us to
**'X J **
1
0
X
N ,
when . Hence,
2 N ST
N T
,
N T
1
**' ** ** **' **
2 1 1
plim X X plim X EN KT X
NT NT
On the other hand, we can write
1
**' ** ** **' **
** **
1 N T
X u X E K u
NT
2 1
**' **'
2
1 1
N T N T
NT
X E L u X J S u
N NT
the M1 and mations, we get
d NT
.
Under M2 transfor
**' **
**' **
1 plim
1
2
plim 0
N T
N T
X E L u
d NT
X J S u
T
N N
leading to
1
**' ** ** **' **
2 1
1
plim X u plim X EN KT u
NT NT
.
As a result,
1
**' **
**' **
plim
plim
plim
GLS
N T
N T
NT
X E K X
NT
X E K u
NT
i.e.,
plim NT GLS plim NT C
Finally, NT
GLS
has the same limiting distribu-tion as NT
C
of the two
. This shows the asymptotic equivalence estimators GLS and C. We then deduce that,
1**' **
2 1 1
, plim N T
a GLS
X E K X
N
NT NT
.
Thus, the GLS estimator suggested under the double au- ation error structure has the desired asymptotic ies.
6. FGLS Estimation
the
variance-n it i
approach is required. The method used consists in removing the time specific effect to ob-tain a one-way error component model where only tocorrel
propert
In practice, covariance matrix is unknown, as well as all the parameters involved i s determ nation. Therefore, a FGLS
it
his carries the serial correlation (see [18] and [3]). T method has been directly applied to AR(1) and MA(1) processes in separate subsections.
6.1. Feasible Double AR(1) Model
We assume that it i, 1t eit, t t1t,
1
, 1,
2
e e IID 0
it
, , 0, 2
t IID
.
The within error term is,
N T N T N T T
u E I u E i E I i (19) The associated variance-covariance matrix is,
' 2
'
2
N T T N
E uu E i i E
(20)
Since it follows an AR(1) process of parameter ,
we define the matrix C as the famili mation matrix with parameter
ar [19] transfor-
. This matrix is su ch that,
2
'
2
2 2 e1 T T
C C I I
r is given by The resulting GLS estimato
*' *
1 *' *W X X X y
(21)
where *
Ny I C y and X*
IN C X
. us The covariance matrix of *
N
u I C u, ing [20] trick, is
* 2 2 2 2
e e
1 d EN JT EN ET
(22) where
'
'
1
1 1
1
1 1 1 1
T T
i Ci
i
1
T
' 2
1
T T T
J i i
d
,
T
T T
E I J
and
2
2 ' '
2 2 2 2
1
1 1 1
T T T T
d i i i i
T d
Following [21], another GLS estimator can be derived. We label this estimator the within-type estimator and is given by
**' **
1 **' ** WGLS X X X y (23)
with 1 *
** 2 *
y y
and
1 *
** 2 *
X X
. In order to get the estimates of numerous parameters involved in the we first need an estimate of the correlation coef-model,
ficient . The autocorrelation function of the error term u is given by
2 2
, 1
for 0,1, ,
N
h t
h it i t h
N
h E u u
We deduce from it that
(24)
1 2
0 1
J. M. B. BROU ET AL. 193
to a convergent estimator of (see [7]), i.e.,
1 2
0 1
where
,
1 1
1 N T
it i t h i t h
h u u
N T h
with uitde-fined s the OLS ra esiduals of the within equation
yXu. Hence, we get
1
1
2 e BQU estimate of
(25a)
(25b)
Furthermore, th and
1
1
d T
2
2
2 e
is also avail-able as
*' *
2 e
1 1
N T
u E E u
N T
(26)
u being the OLS estimate of u*. As a consequence, we get
*
2e
1
2
2
(27a)
and
1
2 2
0
N N
W
(27b)
e now need to find 2 and . Th autocovariance function of the initial error term u is gi n
e ve
2 2 2
21
h h N h
h h
N
for
0,1, , h t. It comes that,
2
0
0
2 20 1
N N
(28)
We immediately deduce a c erge second correlation coefficient, ,
onv
i.e.
nt estimator of the
1 2 1 2
1
0 1 0 1
1 N N
N
N
where
(29)
1
N T
u
it i t h, withN Th i1t h1
h u
de-
noting the OLS residuals of yXu. The varian s
it
u
ce 2
e
is estimated by,
2
2 21
e
ators mention ch as the within
(30)
addition to the GLS estim ed in Se r GLS estimators su estimator In
othe
ction 4,
W
and the within-type estimator WGLS can all be
per-formed as well. Actually, the knowledge of the AR(1) parameters and
termin
entitles us to bu
n of
** 1ild the ma lved in e de atio , say matrices trices invo
C
th
, C, C, P, , D, KT, LT, S
, 1
e ei t
and
6.2. Feasible Double MA(1) Model
state that
T.
S
We now it it , with 1 and
2
e
eit IID 0, . Again, deviations from individual means lead to the model
yXu with it i
u it
trix of u . The
Equ
variance-covariance ma ation (20), with now
is still given by
2
Toeplitz 1, ,0, ,0 1
r
(31)
Here, we set CCT, CT denoting the correlation
cor-rection matr s defi by [8] in their orthogonalizing algorithm. We then transform th hin model by
i
e wit x a ned
IN C
riance
. The ew ern ha the fo win atrix,
ror term s llo g
m
* u cova
* 2 2 2 2
T
N EN ET
d E J
(32)
Because of the
moving average nature of the process, linear estimation of the correlation parameter is not
easily obtainable. Instead, 2 1
r
proves useful.
The autocorrelation function of the within error term uit
is given by,
2
h, 1
it i t h
N
h E u u
N
(33)
for h0,1, , t
with
h denoting the autocovariance function of it
. As a consequence, 2
1N j N
for some
and 2
j
2
20 N 0
1
N
(34)
where
, 11 N T
it i t h i t h
h u u
N T h
1 is the pirical emautocovariance function and u sit are the OLS residuals
of the within equation. We also get, for some j2,
2 1
1
N r
N
j
(35)
We then apply the [8] matrix to the data (for in-stance to the within transformed dependent vector T
C
y).
r,
to get estim of the
Moreove CT will be applied to the vector of constants
t
ates s. W , in the straight e
of
Ste pute
e have lin
[8], the following steps:
p 1: Com
* 1
1 ,1 i i
y y
g
and
*
, 1
, 1 *
, i t it
t it
t
r y y
g y
g
for t2, ,T
where 2 ,t 1
r
g
for t 2, ,T
, 1t
g
.
Step 2: Compute 1 *
** 2 *
y y
knowing that
'1 2
T T T T
i C i . The estimates of the tsare
obtained as 1 1 and
, 1 ,
1 r
g
t
t
g
fo ,T.
We then obtain the estimate of
t
r t2,
2
d as 2 2
1 T
t t
d
. of the initial pirical counterpart The autocovariance function
2
h E u u
it i t h,
h
composite error term u and its em
h
,it i
h u u
1 1
T
h
1 N
t h i t
N Th
, (it b
del)
u
mo
eing the OLS
residuals of the in -way mation of
itial two permit the esti-2
and r,
2
2 20
and
(36a)
2 2 2 1 r
r
(
The within estimator
36b)
W
and the within-type one
WGLS
GLS
are now obtainable. However, the GLS estimator
can be estimated, provided the MA(1) parameters
and 2
are known, especially under the conditions 1 4r 0
and 1 4r2 0. In other words, the estimates r and r should b h lie e open interval
ot inside th 1 1,
2 2
as a
GLS
pre-requisite to a direct
estimation of , C, Band T.
7. Final Remarks
au dealt with some
par-simonious models, especially the AR(1) and MA(1) ones, as well as the general framework. T
a of the variance-covariance matrix of the errors, rived the GLS estimator and related asymptotic properties. An investigation of the FGLS
ered in the paper.
8.
[1]
la
[2] B. H. Baltagi . Li, “A Transform e Problem
Component Model,” Journal of Econometric, Vol. 48, No.
-393.
This paper has considered a complex but realistic corre-lation structure in the two-way error component model: the double tocorrelation case. It
hrough a precise formul
we de
is also
References
N. S. Revankar, “Error Component Models with Serial Corre ted Time Effects,” Journal of the Indian Statisti-cal Association, Vol. 17, 1979, pp. 137-160.
and Q ation that will Cir-cumvent th of Autocorrelation in an Error 3, 1991, pp. 385 doi:10.1016/0304-4076(91)90070-T
[3] B. H. Balt d Q. L Predic in
ror Component Model with Seragi an i, “ ial Correlation,” tion the One-Way Er-Journal of Forecasting, Vol. 11, No. 6, 1992, pp. 561-567.
0110605 doi:10.1002/for.398
] B. H. Baltagi and Q. Li, “Estimating Error Component [4
Models with General MA(q) Disturbances,” Econometric Theory, Vol. 10, No. 2, 1994, pp. 396-408.
doi:10.1017/S026646660000846X
[5] J. W. Galbraith and V. Zinde-Walsh, “Transforming the Error Component Model for Estimation with general ARMA Disturbances,” Journal of Econometrics, Vol. 66, No. 1-2, 1995, pp. 349-355.
doi:10.1016/0304-4076(94)01621-6
[6] P. Balestra and M. Nerlove, “Pooling Cross-Section and Time-Series Data in the Estimation of a Dynamic Model: for Natural Gas,” Econometrica, Vol. 34,
. 585-612.doi:10.2307/1909771
The Demand No. 3, 1966, pp
[7] B. H. Baltagi, “Econometric Analysis of Panel Data,” 3rd Edition, John Wiley and Sons, New York, 2008.
[8] C. Hsiao, “Analysis of Panel Data,” Cambridge Univer-sity Press, Cambridge, 2003.
