Adaptive Estimation of the Dynamic Linear Model
with Fixed E¤ects¤
By Tiemen Woutersen and Marcel Voiay
May 2002
1
Introduction
The analysis of the dynamic linear Model with …xed e¤ects has been subject of some
attention in econometrics for almost two decades. The popularity of this linear model might
be due to the fact that it is the simplest model in which Heckman’s (1981a and 1981b)
spurious correlation and state dependence can be studied. Also, in practice growth models
such as the dynamic version of the Solow’s (1956) are …tting very well into the class of
dynamic linear models with …xed e¤ects. Sollow’s model became important once the paper
of Mankiw, Romer and Weil (1992) used it to …t data on GDP; savings and labor force for
121 countries during 1960 ¡ 1985: They found then that aproximatively 80 percent of the
variation in the above mentioned variables con…rm model’s predictions.
Refering to the model estimation, Nickell (1981) showed that the maximum likelihood
su¤ers from the incidental parameter problem of Neyman and Scott (1948). In particular, Nickell shows that the inconsistency of the maximum likelihood estimator is O(T ¡1):
Econometricians have subsequently developed moment estimators. Examples include Anderson and Hsiao (1982), Holtz-Eakin, Newey and Rosen (1988), Arrelano and Bond (1991), and
Ahn and Schmidt (1995). Combining moment restrictions can be di¢cult, especially, when
some moments become uninformative for particulr regions of the parameter space. Neway
¤
We gratefully acknowlegde stimulating suggestions from Jan Kieviet. The CIBC Human Capital and
Productivity Program provided …nancial support. All errors are ours.
y
Correspondence addresses: Brown University, Box B, RI, 02912, USA, and University of Western Ontario, Department of Economics, Social Science Centre, London, Ontario, N6A 5C2, Canada. Email:
Tony_Lancaster@brown.edu, mvoia@uwo.ca, and twouters@uwo.ca.
1
and Smith (2001) also give a general discussion on how combining moments usually causes
a higher order bias. The small sample bias of these moment estimators motivates the bias
corrected least squares estimator of Kieviet (1995). Alvarez and Arellano (1998) developed
an alternative asymptotic where the number of individuals, N; increases as well as the number of observations per individual, T: Using this alternative asymptotics, Hahn, Hausman
and Kuersteiner (2001) develop an expression for the bias for the case in which N / T then,
Hahn et al. developed an estimator for the dynamic linear model without regressors. Hahn
and Kuersteiner (2002) developed a bias corrected MLE which is asymptotically unbiased
and e¢cient.
Lancaster (1997 and 2000) proposes to approximately separate the parameters of interest from the nuisance parameters. In particular, Lancaster uses a parametrization of the
likelihood that has a block diagonal information matrix. That is, the cross derivatives of
the log likelihood of the nuisance parameters and parameters of interest is zero in expectation. Lancaster (1997 and 2000) then integrates out all …xed e¤ects. Woutersen (2001)
shows that information-orthogonality reduces the bias of this integrated likelihood estimator
to O(T ¡2 ) and that the integrated likelihood estimator is asymptotically unbiased and adaptive if T _ N ® where ® > 13 : That is, the integrated likelihood estimator is as e¢cient as the
infeasible maximum likelihood estimator that assumes the values of the nuisance parameters
to be known.
This paper presents a new adaptiveness result for the integrated likelihood estimator that
integrates out individual speci…c e¤ects. We show that the integrated likelihood estimator is
adaptive for any asymptotic in which T increases as long as the integrated likelihood estimator
is consistent for N increasing and T being …xed. This theorem implies adaptiveness of
Lancaster’s (1997) estimator of the dynamic linear model without regressors. We also derive
an information orthogonal parametrization for the dynamic model with regressors. For this
model, the integrated likelihood estimator is asymptotically unbiased and adaptive as long
as T _ N ® where ® > 13 :
{Lancaster (1997) gives a parametrization of the dynamic linear model that approximates
2
an information-orthogonal parametrization if N ! 1: However, information-orthogonality
is a …nite sample property. Simulations show that an implicit parametrization works better;
this was somewhat surprising to me (tw) since N was larger than T in all simulations}
This paper is organized as follows. Section 2 reviews the integrated likelihood estimator
and information orthogonality. Section 3 derives an information orthogonal parametrization
for the dynamic linear model with …xed e¤ects and regressors. Section 4 gives adaptiveness
results for the integrated likelihood estimator (case without regressors). Section 5 gives
simulation results and section 6 concludes.
2
The Integrated Likelihood Estimator and Orthogonality
Suppose we observe N individuals for T periods. Let the log likelihood contribution of the
tth spell of individual i be denoted by Lit : Summing over the contribution of individual i
yields the log likelihood contribution,
Li(¯; ¸i ) =
X
Lit (¯; ¸i );
t
where ¯ is the common parameter and ¸i is the individual speci…c e¤ect. Suppose that the
parameter ¯ is of interest and that the …xed e¤ect ¸i is a nuisance parameter that controls for
heterogeneity. This paper considers elimination of nuisance parameters by integration. This
Bayesian treatment of nuisance parameters is straightforward: formulate a prior on all the
nuisance parameters and then integrate the likelihood with respect to that prior distribution
of the nuisance parameters, (see Gelman et al. (1995) for an overview). For a panel data
model with …xed e¤ects this means that we have to specify priors on the common parameters
and all the …xed e¤ects. Berger et al. (1999) review integrated likelihood methods in which
‡at priors are used for both the parameter of interest and the nuisance parameters. The
individual speci…c nuisance parameters are then eliminated by integration. We denote the
logarithm of the integrated likelihood contribution by Li;I ; i.e.
i;I
L (¯) = ln
3
Z
i
eL d¸i :
Summing over i yields the logarithm of the integrated likelihood,
X
X Z
i
I
i;I
L (¯) =
L (¯) =
ln eL d¸i:
i
(1)
i
After integrating out the …xed e¤ects, the mode of the integrated likelihood can be used as
an estimator1 We thus de…ne the integrated likelihood estimator ¯^ to be the mode of LI (¯):
¯^ = arg max LI (¯):
¯
A parametrization of the likelihood is information-orthogonal if the information matrix is
block diagonal. That is
EL¯¸(¯ 0; ¸0) = 0
Z
i.e.
tmax
tmin
L¯¸ (¯ 0; ¸0 )eL(¯0 ;¸0 ) dt = 0;
where t denotes the dependent variable, t 2 [t min ; t max] and ¯ 0 ; ¸0 denote the true value of
the parameters. Cox and Reid (1987) and Je¤reys (1961) use this concept and refer to it
as ‘orthogonality’. We prefer the term information-orthogonality to distinguish it from the
other orthogonality concepts and to stress that it is de…ned in terms of the properties of
the information matrix. See Tisbshirani and Wasserman (1994) and Woutersen (2000) for
an overview of orthogonality concepts. Consider the log likelihood L(¯; f (¯; ¸)) where the
nuisance parameter f is written as a function of ¯ and the orthogonal nuisance parameter
¸: Di¤erentiating L(¯; f (¯; ¸)) with respect to ¯ and ¸ yields
@L(¯; f (¯; ¸))
@f
= L¯ + Lf
@¯
@¯
@ 2L(¯; f (¯; ¸))
@f
@f @f
@ 2f
= Lf ¯
+ Lf f
+ Lf
@¸@¯
@¸
@¸ @¯
@¸@¯
where Lf is a score and therefore ELf = 0: Information orthogonality requires that the
cross-derivative
@2 L(¯;f (¯;¸))
@¸@¯
is zero in expectation, i.e.
EL¯¸ = ELf ¯
@f
@f @f
+ ELf f
= 0:
@¸
@¸ @¯
1
As N ! 1; using the marginal posteriors is asymptotically equivalent. Considering the mode of the
posterior, however, simpli…es the algebra.
4
This condition implies the following di¤erential equation
ELf ¯ + ELff
@f
= 0:
@¯
(2)
If equation (2) has an analytical solution then L(¯; f (¯; ¸)) is an explicit function of f¯; ¸g
and we refer to such a parametrization as an explicit parametrization. In most cases, however,
equation (2) has an implicit solution and we have to recover the Jacobian
@¸
@f
from this
implicit solution. In this case, L(¯; ¸) has an implicit parametrization. The informationorthogonal parametrization of the dynamic linear model without regressors is explicit and
given by Lancaster (1997). We need an implicit parametrization to deal with regressors
and will derive that parametrization in the next section. For both implicit and explicit
parametrization, Woutersen (2001) shows that the integrated likelihood estimator is adaptive
if T _ N ® ® >
3
1
3
and ¯^ ¡ ¯ is Op(T ¡2 ) if ® · 13 .
Orthogonality in the Dynamic Linear Model
Consider the dynamic linear model with …xed e¤ects without regressors,
yis = yi;s¡1½ + fi + "is where E"is = 0; E"2is = ¾2; E"is "it = 0 for s 6= t and s = 1; :::; T:
Lancaster (2000) conditions on yi0 and suggests the following parametrization2
fi = yi0(1 ¡ ½) + ¸ie¡b(½) where b(½) =
T
1 X T ¡s s
½:
T s=1 s
Analogue to quasi-maximum likelihood estimators, normality of the error terms is assumed
in order to derive the integrated likelihood estimator. The estimator depends only on the
…rst two moments of yis and is given by Lancaster (1997). In particular, the score of the
integrated likelihood has the following form (see apendix 1 for derivation),
1 X
1
(ys ¡ ys¡1 ½)ys¡1 ¡ 2 T (y s ¡ ys¡1 ½)ys¡1
¾2 s
¾
1 T ¡1
1 X
T
=
f
¡
(ys ¡ ys¡1½)2 + (ys ¡ ys¡1½)2 g
¾2
2
2¾2 s
2
Li;I
= b 0(½) +
½
Li;I
¾2
where b(½)0 =
1
T
PT
s=1(T
¡ s)½s¡1 :
2
Appendix 5 of Woutersen (2001) gives information-orthogonal parametrizations for linear models with
more then one autoregressive term.
5
3.1
Dynamic linear model with regressors
Now consider the dynamic linear model with …xed e¤ects and regressors,
yis = yi;s¡1½+xis¯ +fi +"is where E"is = 0; E"2is = ¾2; E"is "it = 0 for s 6= t and s = 1; :::; T:
We assume normality in order to derive the following likelihood contribution for individual
i:
Li = ¡T ln ¾ ¡
1 X
(y ¡ yi;s¡1½ ¡ xis ¯ ¡ f i)2.
2¾2 s is
It follows from Woutersen (2001, appendix 5) that the information-orthogonal nuisance parameter has the following expression,
Z yi;s¡1½+xis¯+fi
1X
¸=
E
d¹:
T s
¡1
Thus, ¸ is expressed as an unbounded integral. Fortunately, its derivatives are …nite,
@¸
@f
@¸
@½
@¸
@¾2
= 1
=
1X
Ey i;s¡1
T s
= 0:
Normalizing the regressors to have mean zero,
and
Thus,
P
s xis
= 0, gives
@¸ X
=
xis = 0
@¯
s
@¸
df
1X
@½
= ¡ @¸ = ¡
Eyi;s¡1:
d½
T s
@f
R i
R Li
Li
(L½ + Lif df
d
ln
e
d¸
d½ )e df
i;I
2
R i
L½ (¯; ½; ¾ ) =
=
(3)
d½
eL df
R i Li
R Li
L¯ e df
d
ln
e
d¸
i;I
L¯ (¯; ½; ¾2 ) =
= R Li
(4)
d¯
e df
1 T ¡1
1 X
T
i;I
L¾2 (¯; ½; ¾2 ) =
f
¡ 2
(yis ¡ yi;s¡1½ ¡ xis¯)2 + (yis ¡ yi;s¡1 ½ ¡ xis ¯)2g:
2
¾
2
2¾ s
2
6
Thus, the estimate for the variance is
4
c
¾2 =
X X
1
f (yis ¡ yi;s¡1½ ¡ xis ¯)2 + T (yis ¡ yi;s¡1 ½ ¡ xis ¯)2g:
N(T ¡ 1)
s
i
Adaptiveness
Woutersen (2001) shows that the integrated likelihood estimator is adaptive in the sense that
it is equivalent to the infeasible maximum likelihood estimator that assumes the nuisance
parameters to be known. The conditions for this result are a regularity condition that the
integrated likelihood can be approximated by a Laplace formula and the substantial condition
that T _ N ® where ® > 13 : Suppose that T is rather small so that an asymptotics in which
T _ N ® where ® >
1
3
is not satisfactory. Woutersen (2001) gives su¢cient conditions for
the integrated likelihood estimator to be consistent for T …xed and N ! 1: We now show
that consistency for …xed T implies that the integrated likelihood estimator is adaptive for
an asymptotics with T increasing.
Proposition (case without regressors)
Suppose the asymptotic variance of ½ML = arg max½ L(½; ¸0) equals ª =
£
¤
1
0 g ¡1
EfL
L
½
½
NT
and that T / N ® where ® > 0: Then the integrated likelihood estimator b
½ is an adaptive
estimator and
p
NT (b½I ¡ ½0) !d N(0; ª):
Proof: See appendix 2 .
5
Simulation Results
We use the same simulation designs as Hahn, Hausman and Kuersteiner (2001), (HH K),
Hahn and Kuersteiner (2002), (H K) ; and Kiviet (1995). When we compare the performances of the integrated likelihood estimator with the performances of HHK estimator
we considered the case without regressors, looking only at the performances of the autoregressive parameter estimator (½). We designed the Monte Carlo experiment by looking at
7
values of the autoregressive parameter ½ = f0:1; 0:3; 0:5; 0:8; 0:9; 0:95; 0:99g for samples of
size n = f100; 500g and T = f5; 10g and assuming unknown ¾2: Thus, we extended the
analysis of HHK by looking at values of ½ closed to the unit root. The number of simulation
replications was set at 5000. The simulations results are presented in Table.1.
Looking at the RMSE of the estimates, we observe that the integrated likelihood estimator is performing better than the other estimators in all situations. Also, we observe
that for small values of ½ = f0:3; 0:5g ; for n = f500g and T = f10g ; the bias for integrated
likelihood estimator is slighty higher than the bias of HHK estimator , and that in any other
cases the bias is lower. We can observe very good performances of the integrated likelihood
estimator in the vicinity of unit root. Thus, in Table 3, we presented the performances of the
integrated likelihood estimator for high values of ½ = f0:75; 0:8; 0:85; 0:9; 0:95g for n = 100
and T = 5 : The results show that our estimator is performing better in terms of mean and
median bias, exception at ½ = 0:9 for the mean bias, and much better in terms of MSE
in comparison with the other available estimators: Thus, our results con…rm the prediction
about the MSE in comparison to the other available estimators.
Also, we did Monte Carlo simulations using the design of HK (2002) ; where ½ =
f0:0; 0:3; 0:6; 0:9g ; n = f100; 200g and T = f5; 10; 20g : Our simulations results are presented
in Table 2. In all situations our results are presenting lower bias and lower RMSE: Thus,
even for the cases where we have the same RMSE (½ = f0:0; 0:3g for n = 200 and T = 20) ;
our results are better due to lower bias.
When we compare the performances of the integrated likelihood estimator with the performances of the Kieviet estimator we are considering the case with covariates. We also compare
the results of the integrated likelihood estimator using Lancaster’s (1997) approximation, to
the Kieviet results, see Table 3.
We are considering 10 out of 14 cases Kieviet considered, this is due to the fact that we
are not assuming any distribution for the …xed e¤ect as Kieviet assumed. Thus, we present
results for 10 di¤erent parameter combinations which are matching Kieviet’s combinations
for ¹ = 1. The results for 5000 replications are presented in Table 4.
8
Comparing the results obtained using Lancaster’s (1997) approximation to the Kieviet’
results, we observe that for values of ½ = 0; Kiviet’s estimator is performing better than
integrated likelihood estimator and as ½ increases the reverse is true, exception is at ½ = 0:4
where the bias of the integrated likelihood ½ estimator is higher than Kiviet’s at ½ = 0:8;
where RMSE of the integrated likelihood ¯ estimator is larger than Kieviet’s.
Looking at the integrated likelihood estimator results obtained by using the implicit
parametrization of Woutersen (2001) ; see Table 5; we observe improvements, thus we are not
…nding any cases for which Kiviet.s estimator is doing better than the integrated likelihood
estimator.
Overall, the simulations show that the integrated likelihood estimator is very good in
terms of MSE and bias, see section 9 for the tables.
6
Conclusion
This paper derived new adaptiveness result for the integrated likelihood estimator. The
integrated likelihood estimator is adaptive for an asymptotic with T increasing. Simulations
show the relevance of the adaptiveness results since the MSE of the integrated likelihood
estimator was smaller then the MSE of competing estimators for several versions of the
dynamic linear model with …xed e¤ects. Also, our simulation results showed very good
performances of the estimator for slower adjustment processes, ½ close to unit root.
Then, we derived a new estimator for the dynamic linear model with …xed e¤ects and
regressors. The new estimator performed very well, showing in simulations very good results
for both MSE and bias.
9
7
Appendices
Appendix 1.
¡
¡
¢¢
We assume normality of the error term "is s N 0; ¾2 to derive moment coditions.
i;I
De…ning the integrated likelihood by eL :
e
Li;I
=
=
=
_
Z
P
1 b(½)
¡ 1 2 s(ys¡ys¡1½¡f )2
2¾
e
e
df
¾T
Z
1 P
2
1 b(½)
¡ 2¾
2
s (ys ¡ys¡1½¡f ) df
e
e
T
¾
Z
T
2
1 b(½)¡ 12 Ps(ys¡ys¡1 ½) 2
2¾
e
e¡ 2¾2 ff ¡2f(ys ¡ys¡1 ½)gdf
T
¾
1 b(½)¡ 12 Ps(ys ¡ys¡1 ½)2 + T2 (ys¡ys¡1½)2
2¾
e
¾T ¡1
This implies that the log-of the integrated likelihood for individual i and the moments we
are using to estimate parameters are:
T ¡1
1 X
T
ln(¾2 ) + b(½) ¡ 2
(ys ¡ y s¡1½)2 + 2 (ys ¡ ys¡1 ½)2
2
2¾ s
2¾
X
1
= b 0(½) + 2 f (ys ¡ ys¡1 ½)ys¡1 ¡ T(ys ¡ ys¡1½)ys¡1g
¾
s
X
1
T
2
= b 00 (½) ¡ 2
ys¡1
+ 2 ys¡12
¾ s
¾
1 T¡1
1 X
T
=
f
¡ 2
(ys ¡ ys¡1 ½)2 + (ys ¡ ys¡1½)2g
2
¾
2
2¾ s
2
Li;I =
Li;I
½
Li;I
½½
Li;I
¾2
i;I
The functions Li;I
½ and L¾ 2 can be used as a moment function.We can derive the same result
when we integrate the score function:
R i L
R i L
L½e d¸
L½e df
@¸
i;I
L½ = R L
= R L
since
= eb(½) does not depend on ¸:
e d¸
e df
@f
1
1 P
2
T
2
constant eL = p
e¡ 2 f s (ys¡ys¡1½) g¡ 2 ff ¡2f(ys ¡ys¡1 ½)g
2¼T
1
1 P
2
T
2 T
2
= p
e¡ 2 f s (ys¡ys¡1½) g+ 2 (ys¡ys¡1½) ¡ 2 (f ¡ys ¡ys¡1 ½)
2¼T
10
R L
f e df
Note that R L
e df
Li;I
½
R
(f ¡ (~
ys ¡ y~s¡1½))2eLdf
1
R L
= y~s ¡ y~s¡1½ and
=
T
e df
X
0
= b (½) +
(ys ¡ ys¡1½)ys¡1 ¡ T(ys ¡ ys¡1½)y s¡1
s
Li;I
½½
00
= b (½) ¡
X
2
ys¡1
+ T ys¡12
s
Appendix 2. Proposition
To be shown:
p
NT (b½I ¡ ½0) !d N(0; ª);
where ª is the asymptotic variance of the ML estimator.
Proof:
We are considering the case without regressors.
a) Determining the asymptotic variance of the ML estimator for orthogonal
…xed e¤ect
¡
¡
¢¢
We assume normality of the error term "is s N 0; ¾2 to derive moment coditions.
After deriving the moment conditions we show that under E"is = 0; E"2is = ¾2 and E"is "ij =
0 for j 6= s our Proposition holds.
De…ne the known orthogonal …xed e¤ects by ¸i ; with initial …xed e¤ects de…ned by
fi = (1 ¡ ½) yi0 ¡ ¸ie¡b(½) :
If we de…ne the loglikelihood with known orthogonal …xed e¤ects for individual i by Li ;
we have:
1
1 X
Li = ¡ ln(¾2 ) ¡ 2
(~
ys ¡ y~s¡1½ ¡ ¸e¡b(½) )2 where y~s = ys ¡ y0 :
2
2¾ s
Now, we can derive the …rst and second moments necessary to determine the asymptotic
11
variance for ML estimator:
1 X
(~
y ¡ y~s¡1½ ¡ ¸e¡b(½) )(~
ys¡1 ¡ ¸b0 (½)e¡b(½) )
¾2 s s
1 X
=
(~
ys ¡ y~s¡1½ ¡ ¸e¡b(½) )(~
ys¡1 ¡ ¸b0 (½)e¡b(½) )
¾2 s
¡
¢
1 X
¸ X
= ¡ 2
(~
ys¡1 ¡ ¸b0 (½)e¡b(½) )2 + 2
(~
ys ¡ y~s¡1½ ¡ ¸e¡b(½) ) b02(½) ¡ b 00(½) e¡b(½)
¾ s
¾ s
Li½ =
Li½½
1
T
where b(½) =
s) (s ¡ 1) ½s¡2:
De…ning, by
@f
@½
PT
s=1
T ¡s s
s ½ ;
b 0(½) =
1
T
PT
s=1(T
¡ s)½s¡1 and b 00 (½) =
1
T
PT
s=1(T
¡
= ¡y0 + ¸b 0(½)e¡b(½) we can rewrite:
¢
1 X
@f 2
¸ X ¡ 02
(y
+
)
+
"s b (½) ¡ b 00(½) e¡b(½)
s¡1
2
2
¾ s
@½
¾ s
Ã
µ ¶2!
¢
1 X 2
@f
@f
¸ X ¡ 02
= ¡ 2
ys¡1 + 2T y s¡1 + T
+ 2
"s b (½) ¡ b00 (½) e¡b(½)
¾
@½
@½
¾ s
s
µ
¶
¢
1 X 2
2T @f
T @f 2
¸ X ¡ 02
= ¡ 2
ys¡1 ¡ 2
y s¡1 ¡ 2
+ 2
"s b (½) ¡ b00 (½) e¡b(½) :
¾ s
¾ @½
¾
@½
¾ s
Li½½ = ¡
Li½½
Taking expectations we have:
ELi½½ = ¡ ¾12 E
ELi½½ = ¡ ¾12 E
P
P
2
2T
s ys¡1 ¡ ¾2
@f
T
@½ Ey s¡1 ¡ ¾ 2
2T
2
s ys¡1 ¡ ¾2
@f
T
@½ Ey s¡1 ¡ ¾ 2
³
³
@f
@½
@f
@½
´2
´2
+ ¾¸2
:
P
s E ("s )
¡ 02
¢
b (½) ¡ b00 (½) e¡b(½)
Given that E"is = 0; E"2is = ¾2 and E"is"ij = 0 for j 6= s and
1 X
@f
(yis ¡ yi;s¡1½ ¡ fi)(yi;s¡1 + i )
¾2 s
@½
1 X
@fi
=
"is (yi;s¡1 +
) and
2
¾ s
@½
Ã
!2
1 X
@fi
=
"is (yi;s¡1 +
)
¾4
@½
s
µ
¶µ
¶
1 X 2
@fi 2
2 X
@fi
@fi
=
" (yi;s¡1 +
) + 4
("is ) ("ij ) yi;s¡1 +
yi;j ¡1 +
;
¾4 s is
@½
¾
@½
@½
Li½ =
¡
Li½
¢ ¡ i ¢0
L½
j6=s
12
¡
¢
we have E (L½) (L½)0 that is
à µ
¶2 !
1 XX
@f
i
E "2is yi;s¡1 +
¾4
@½
s
i
µ
µ
¶µ
¶¶
2 X
@fi
@fi
+ 4
E ("is ) ("ij ) yi;s¡1 +
yi;j¡1 +
¾
@½
@½
j6=s
à µ
!
¶
µ
¶
1 XX
@fi 2
¾2 X X
@fi 2
2
=
E "is yi;s¡1 +
= 4
E yi;s¡1 +
¾4 i s
@½
¾ i s
@½
µ
¶
1 XX 2
2T @fi X
T X @fi 2
=
Ey
+
Ey
+
:
i;s¡1
i¡
¾2 i s
¾2 @½ i
¾2 i
@½
¡
¢
E (L½) (L½)0 =
Thus,
¡
¢
E (L½ ) (L½ )0 = ¡EL½½;
therefore, the asymptotic variance of ML estimator for orthogonal …xed e¤ects will be de…ned
as:
·
¸¡1 ·
¸·
¸¡1
¡
1
1
1
0¢
ª = Asy:var (b
½ML) =
EL½½
E (L½) (L½)
EL½½
NT
NT
NT
·
¸¡1 ·
¸¡1
¡
1
1
0¢
=
E (L½) (L½)
= ¡
EL½½
NT
NT
·
¸¡1
1
ª = Asy:var (b
½ML) = ¡
EL½½
NT
where EL½½ is
µ
¶
1 XX 2
2T @fi X
T X @fi 2
EL½½ = ¡ 2
Eyi;s¡1 ¡ 2
Ey i¡ ¡ 2
:
¾ i s
¾ @½ i
¾ i
@½
b) Determining the asymptotic variance of the integrated likelihood estimator
¡
¢
Assuming again "is s N 0; ¾2 we derived the moment conditions for the integrated
i;I
likelihood estimator. If we de…ne the integrated likelihood for individual i by eL ; we have
eL
i;I
_
1
¾
1
eb(½)¡ 2¾2
T ¡1
P
T
2
2
s (ys ¡ys¡1½) + 2¾ 2 (ys ¡ys¡1 ½)
:
This implies that the log of integrated likelihood to be:
Li;I =
T ¡1
1 X
T
ln(¾2) + b(½) ¡ 2
(ys ¡ ys¡1 ½)2 + 2 (ys ¡ ys¡1½)2
2
2¾ s
2¾
13
and the corresponding …rst and second moments for the integrated likelihood are:
1 X
f (ys ¡ ys¡1½)ys¡1 ¡ T(ys ¡ ys¡1½)y s¡1g
¾2 s
1 X 2
T
= b 00(½) ¡ 2
y
+
ys¡12:
¾ s s¡1 ¾2
Li;I
= b 0(½) +
½
Li;I
½½
Taking expectations we have:
00
ELi;I
½½ = b (½) ¡
1 X 2
T
Eys¡1 + 2 Eys¡12:
2
¾ s
¾
Relying on the work of Newey, W.K. and D. McFadden (1994) on the asymptotic variance
of moment estimators, we can express the asymptotic variance of the integrated likelihood
estimator as:
·
1
Asy:var (b
½I ) =
ELI½½
NT
¸¡1 ·
where ELI½½ is
¸¡1
³¡ ¢ ¡ ¢ ´¸ · 1
1
I
I 0
I
E L½ L½
EL½½
NT
NT
Ã
!
Ã
!
XX
X
1
T
2
ELI½½ = N b00 (½) ¡ 2 E
yi;s¡1
+ 2 E
yi¡2
¾
¾
s
i
i
and
ELI½
Ã
!
X X
1
= Nb (½) + 2 E
f (y is ¡ yi;s¡1½)yi;s¡1 ¡ T(yis ¡ yi;s¡1½)yi;s¡1 g :
¾
s
i
0
Given the fact that
compute …rst:
1
NT E
³¡
LI½
¢¡
LI½
¢0 ´
=
P
1
i
NT E
14
³
Li;I
½
´³
´0
Li;I
when ELI½ = 0; we will
½
¡ i;I ¢ ¡ i;I ¢0
2b0 (½) X
2b0 (½)
L½
L½
= b0 (½)2 +
(y
¡
y
½)y
¡
T(ys ¡ ys¡1½)y s¡1
s
s¡1
s¡1
¾2
¾2
s
Ã
!2
1 X
+ 4
(ys ¡ ys¡1½)y s¡1 ¡ T(ys ¡ ys¡1½)ys¡1
¾
s
2b0 (½) X
2b0 (½)
= b0 (½)2 +
(f
+
"
)
y
¡
T (f + "s ) ys¡1
s s¡1
¾2
¾2
s
Ã
!2
1 X
+ 4
(f + "s ) ys¡1 ¡ T (f + "s ) ys¡1
¾
s
Ã
!
0 (½) X
2b
= b0 (½)2 +
("s ) (ys¡1) ¡ T ("s ) (ys¡1)
¾2
s
Ã
!2
1 X
+ 4
("s ) ys¡1 ¡ T ("s ) ys¡1 :
¾
s
¡ ¢ ¡ ¢0
Thus, E LI½ LI½ is determined by:
Ã
!
¡ I ¢ ¡ I ¢0
2b0 (½) X X
0
2
E L½ L½
= Nb (½) +
E
("is ) (yi;s¡1 ) ¡ T ("is ) (yi;s¡1)
¾2
s
i
Ã
!2
1 X X
+ 4E
("is ) yi;s¡1 ¡ T ("is ) yi;s¡1 ;
¾
s
i
0
where b (½) =
0
lim b (½)
T !1
2
lim b0 (½)2
T !1
=
1
T
PT
s=1(T
lim
T !1
Ã
Ã
s¡1
¡ s)½
0
; b (½) =
T
1X
(T ¡ s)½s¡1
T
s=1
2
!2
³ P
T
1
T
s=1 (T
¡ s)½
s¡1
´2
;
!2
T
1 X s¡1
= lim
½
¡
s½
T !1
T
s=1
s=1
à T
!2
à T
!2
T
T
X
2 X s¡1 X s¡1
1 X s¡1
s¡1
= lim
½
¡ lim
½
s½
+ lim
s½
T !1
T !1 T
T !1 T
s=1
s=1
s=1
s=1
¡ ¡1 ¢
1
=
+
O
T
+ O(T ¡2 ):
(1 ¡ ½)2
T
X
s¡1
15
and
00
lim b (½) =
T !1
=
=
=
=
lim
T !1
lim
T !1
lim
T !1
Ã
Ã
T
1X
(T ¡ s) (s ¡ 1) ½s¡2
T
s=1
T
X
1
T
T
X
s=1
s=1
T !1
1
+ lim
½ T !1
lim
T !1
T
X
s=2
T
X
¡
¢
T s ¡ T ¡ s2 + s ½s¡2
s½s¡2 ¡ lim
T
X
s=2
!
T
X
s=1
T
T
1 X 2 s¡2
1 X s¡2
s½
+ lim
s½
T !1 T
T !1 T
½s¡2 ¡ lim
1
¡ lim
½ T !1
(s) ½s¡2 ¡
!
(s ¡ 1 + 1) ½s¡2 ¡
T
X
s=2
s=1
s=1
¡
¢
½s¡2 + O T ¡1
¡
¢
1
+ O T ¡1
1¡ ½
T
X
¡
¢
1
+ O T ¡1
T !1
T !1
1¡ ½
s=2
s=2
¡
¢
¡
¢
1
1
1
1
=
+
¡
+ O T ¡1 =
+ O T ¡1 :
2
2
1 ¡½ 1¡ ½
(1 ¡ ½)
(1 ¡ ½)
=
lim
(s ¡ 1) ½s¡2 + lim
½s¡2 ¡
The second moment of the integrated likelihood estimator is de…ned as:
LI½½ = Nb 00 (½) ¡
1 XX 2
T X
y
+
yi;s¡12
i;s¡1
2
¾2
¾
s
i
i
and after taking expectations we have
ELI½½ = N b00 (½) ¡
1 XX 2
T X
E
y
+
E
yi;s¡1 2:
i;s¡1
2
¾2
¾
s
i
i
Adding up the two moments we have:
¡ I ¢ ¡ I ¢0
1
1 ELI = b0 (½)2 + 2b0 (½) E P (P (" ) (y
L½ + NT
i;s¡1) ¡ T ("is ) (yi;s¡1 ))
½½
NT E L½
T
i
s is
NT ¾ 2
P P
00
+ NT1¾4 E i ( s ("is ) yi;s¡1 ¡ T ("is ) yi;s¡1 )2 + b T(½)
P P
1 P
2
2
¡ NT1¾2 i s Eyi;s¡1
+ N¾
2
i E ys¡1
b0(½) 2
b00 (½)
2b0(½) P P
= T + T + NT ¾2 i ( s E ("is ) (yi;s¡1 ¡ y i;s¡1))
P P
P P
+ NT1¾4 E i ( s ("is ) (yi;s¡1 ¡ yi;s¡1))2 ¡ NT1¾2 i s Ey2i;s¡1
1 P
2
+ N¾
2
i E ys¡1
¡
¢
¡
¢
P P
= O T ¡1 +O T ¡1 + NT1¾4 i s E ("is )2 (yi;s¡1 ¡ yi;s¡1 )2
P P P
+ NT2¾4 i s j6=s E ("is ) ("ij) (yi;s¡1 ¡ yi;s¡1 ) (y i;j¡1 ¡ yi;j¡1)
16
P P
2
1 P E y
2
¡ NT1¾2 i s Eyi;s¡1
+ N¾
2
i;s¡1
i
¡
¢
¡
¢
2
¾2 P P
= O T ¡1 + O T ¡1 + NT
i
s E (yi;s¡1 ¡ yi;s¡1)
¾4
P P
1 P
2
2
¡ NT1¾2 i s Eyi;s¡1
+ N¾
2
i E yi;s¡1
¡ ¢ ¡ I ¢0
¡
¢
¡
¢
1 E LI
1 ELI = O T ¡1 + O T ¡1 +
1 P P Ey2
L½ + NT
½
½½
i
s
i;s¡1
NT
N T ¾2
1 P
1 P P
1 P
2
2
2
¡ N¾2 i E yi;s¡1 ¡ NT ¾2 i s Eyi;s¡1 + N¾
2
i E yi;s¡1 :
¡ ¢ ¡ ¢0 1
¡
¢
1
Thus, we get NT
E LI½ LI½ + NT
ELI½½ that is O T ¡1 : This means that, when T ! 1;
we can write:
³¡ ¢ ¡ ¢0´
1
1
ELI½½ = ¡
E LI½ LI½ ;
NT
NT
therefore, the asymptotic variance of the integrated likelihood estimator is given by
·
¸¡1
³¡ ¢ ¡ ¢0 ´¸¡1 · 1
1
I
I
I
Asy:var (b
½I ) =
E L½ L½
= ¡
EL½½
:
NT
NT
Comparing the two expectations
1
I
NT EL½½
and
1
N T EL½½
we can establish the rate at which
the asymptotic variances are equal.
i
Taking the di¤erence between ELi;I
½½ ¡ EL½½ we have:
³ ´2
00
P
P @fi
1
1 P @fi
I ¡ 1 EL = b (½) + 1
2+ 2
EL
Ey
Ey
+
½½
i;s¡1
2
2
2
½½
i
i
i;s¡1
i
NT
NT
T
@½
@½
N¾
N¾
N¾
´2
P ³
P
¡
¢
b00(½)
@fi
1
1
= T + N¾2 i Eyi;s¡1 + @½ + N¾2 i V ar yi;s¡1
³ P
´2
³P
´
00
1 P E s yi;s¡1 + @f i
1 P V ar
s yi;s¡1
= b T(½) + N¾
+
2
2
i
T
@½
i
T
N¾
µ P
¶2
P T ¡1 j ¡s
s¡1
s¡1
00
P
yi0 +(T ¡s+1)½
f i +"i;s¡1 j =1 ½
) @fi
b (½)
1
s (½
= T + N¾2 i E
+ @½
T
µ P s¡1
¶
P ¡1 j¡s
yi0+(T ¡s+1)½s¡1 fi +" i;s¡1 T
)
1 P
s (½
j=1 ½
+ N¾
V
ar
2
i
T
Now, considering T is increasing, when T ! 1; we have
PT
P T ¡1 j¡s
s¡1y +(T ¡s+1)½s¡1 f +"
)
i0
i
i;s¡1
s=1(½
j=1 ½
lim T !1 Ey s¡1 = lim T !1 E
T
PT
PT ¡1 j ¡s
PT
s¡1f
½s¡1 yi0
i
s=1 E"i;s¡1
j =1 ½
s=1(T ¡s+1)½
+lim
+lim
T !1
T !1
T
T
T
PT
1 y
i0
(s¡1)½s¡1f i
1
limT !1 1¡½T + 1¡½
fi ¡ limT !1 s=1 T
PT
1
¡ ¡1 ¢
¡ ¡1¢
s¡1f
¡
¢
1
1¡½ yi0
i
s=1(s¡1)½
f
+O
T
;
because
is
O
T
and
is O T ¡1 :
i
1¡½
T
T
= limT !1
=
=
PT
s
17
Thus, Eys¡1 !
f
1¡½
@f
@½
and
0
lim b (½) =
T !1
=
= ¡f b0 (½) ; but
lim
Ã
lim
T
X
T !1
T !1
T
1X
(T ¡ s)½s¡1
T
s=1
s=1
!
T
1 X s¡1
s½
T !1 T
½s¡1 ¡ lim
s=1
1
1
1
=
¡ lim
1 ¡ ½ T !1 T (1 ¡ ½)2
¡
¢
1
@f
f
lim b0 (½) =
+ O T ¡1 =)
!¡
as T ! 1
T !1
1 ¡½
@½
1¡½
³
´
³
´
³
´2
¡ ¡1¢
@f
@f
Then, Ey s¡1 + @f
is
O
T
and
y
+
!
0
as
T
!
1
and
)
Ey
+
s¡1
s¡1
@½
@½
@½
¡ ¡2 ¢
¡
¢
is O T
. Also, if we are looking at V ar yi;s¡1 when T ! 1 we have:
0P ³
PT ¡1 j¡s ´ 1
s¡1y + (T ¡ s + 1) ½s¡1f + "
½
i0
i
i;s¡1
¡
¢
s
j =1 ½
A
V ar y i;s¡1 = V ar @
T
0
0
11
0 0
12 1
T
T
¡1
T
T
¡1
X
X
X
X
1
1
@"i;s¡1
@¾2 @
=
V ar @
½j¡s AA = 2
½j¡s A A
T2
T
s=1
j=1
s=1
j=1
and
1
N¾ 2
P
i
¡
¢
V ar y i;s¡1 =
1
NT
P
i
µ
1
T
Using the results derived, we have
PT
s=1
³P
1
I
NT EL½½
´
T ¡1 j¡s 2
j=1 ½
1 EL
¡ NT
½½
¶
¡
¢
is O T ¡1 :
¡
¢
is O T ¡1 :
Given the fact that the di¤erence between the asymptotic variances:
Asy:var (b½I ) ¡ Asy:var (b½ML ) =
Asy:var (b½I ) ¡ Asy:var (b½ML ) =
1
1
¡ 1
1
I
NT EL½½
NT EL½½
1 ELI ¡ 1 EL
½½
½½
¡NT
¢ ¡ NT
¢
1
1
I
NT EL½½
NT EL½½
¡
¢
is of order O T ¡1 or o (1) as N ! 1 and T _ N ® ; where the only condition in ® is ® > 0;
we can conclude that the two asymptotic variances are equal:
Asy:var (b
½I ) = Asy:var (b
½ML ) = ª:
18
8
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23
9
Tables
Table 1 Performances of integrated likelihood estimator for
the case without exogenous regressors comparative
to estimators found in Hahn, Hausman and Kuersteiner (2001)
(¾2 is unknown; 5000 data sets)
T
5
10
5
10
5
10
5
10
5
10
5
10
5
10
5
10
5
10
5
10
5
10
5
10
5
10
5
10
n
100
100
500
500
100
100
500
500
100
100
500
500
100
100
500
500
100
100
500
500
100
100
500
500
100
100
500
500
½
0:10
0:10
0:10
0:10
0:30
0:30
0:30
0:30
0:50
0:50
0:50
0:50
0:80
0:80
0:80
0:80
0:90
0:90
0:90
0:90
0:95
0:95
0:95
0:95
0:99
0:99
0:99
0:99
RMSE ½^GMM
0:08
0:05
0:04
0:02
0:10
0:05
0:04
0:02
0:13
0:06
0:06
0:03
0:32
0:14
0:13
0:05
0:55
0:25
0:28
0:10
RMSE ^½BC2
0:08
0:05
0:04
0:02
0:10
0:05
0:04
0:02
0:13
0:06
0:06
0:03
0:34
0:11
0:13
0:04
0:78
0:23
0:30
0:08
24
RMSE ^½LIML
0:082
0:045
0:036
0:020
0:099
0:050
0:044
0:023
0:130
0:058
0:057
0:026
0:327
0:109
0:127
0:044
0:604
0:229
0:277
0:080
RMSE ^½I
0:056
0:035
0:025
0:016
0:061
0:036
0:027
0:016
0:067
0:037
0:031
0:016
0:089
0:045
0:036
0:018
0:099
0:057
0:038
0:023
0:111
0:059
0:042
0:026
0:118
0:058
0:044
0:028
T
5
10
5
10
5
10
5
10
5
10
5
10
5
10
5
10
5
10
5
10
5
10
5
10
5
10
5
10
n
100
100
500
500
100
100
500
500
100
100
500
500
100
100
500
500
100
100
500
500
100
100
500
500
100
100
500
500
½
0:10
0:10
0:10
0:10
0:30
0:30
0:30
0:30
0:50
0:50
0:50
0:50
0:80
0:80
0:80
0:80
0:90
0:90
0:90
0:90
0:95
0:95
0:95
0:95
0:99
0:99
0:99
0:99
%bias ^½GMM
¡14:96
¡14:06
¡3:68
¡3:15
¡8:86
¡7:06
¡2:03
¡1:58
¡10:05
¡6:76
¡2:25
¡1:53
¡27:65
¡13:45
¡6:98
¡3:48
¡50:22
¡24:27
¡20:50
¡8:74
%bias ^½BC2
0:25
¡0:77
¡0:77
¡0:16
¡0:47
¡0:66
¡0:16
¡0:10
¡1:14
¡0:93
¡0:15
¡0:11
¡11:33
¡4:55
¡0:72
¡0:37
¡42:10
¡15:82
¡6:23
¡2:02
%bias ^½LIML
¡3
¡1
¡1
¡1
¡3
¡1
¡1
0
¡3
¡1
¡1
0
¡15
¡5
¡3
¡1
¡41
¡15
¡10
¡2
%bias ^½I
2:66
0:45
¡0:20
¡0:74
1:04
¡0:25
¡0:37
¡0:14
0:38
¡0:15
¡0:29
¡0:19
0:80
0:25
¡0:12
¡0:06
1:36
0:83
¡0:07
¡0:07
1:45
0:78
¡0:05
0:02
1:69
0:44
0:15
0:22
The …xed e¤ects ®i and the innovations "it are assumed to have independent standard
normal distributions. Initial observations yi0 are assumed to be generated by the stationary
³
´
®i
1
distribution N 1¡½
;
:
2
0 1¡½
0
25
Table 2 Performances of Integrated likelihood estimator for the case without
exogenous regressors comparative to GMM and BCML estimator
of Hahn and Kuersteiner (2002)
(¾2 is unknown; 5000 data sets)
T
5
5
5
5
5
5
5
5
10
10
10
10
10
10
10
10
20
20
20
20
20
20
20
20
n
100
100
100
100
200
200
200
200
100
100
100
100
200
200
200
200
100
100
100
100
200
200
200
200
½
0:0
0:3
0:6
0:9
0:0
0:3
0:6
0:9
0:0
0:3
0:6
0:9
0:0
0:3
0:6
0:9
0:0
0:3
0:6
0:9
0:0
0:3
0:6
0:9
bias ^½GMM
¡0:011
¡0:027
¡0:074
¡0:452
¡0:006
¡0:014
¡0:038
¡0:337
¡0:011
¡0:021
¡0:045
¡0:218
¡0:006
¡0:011
¡0:025
¡0:152
¡0:011
¡0:017
¡0:029
¡0:100
¡0:006
¡0:009
¡0:016
¡0:065
bias b^½
-0:039
¡0:069
¡0:115
¡0:178
¡0:041
¡0:071
¡0:116
¡0:178
¡0:010
¡0:019
¡0:038
¡0:079
¡0:011
¡0:019
¡0:037
¡0:079
¡0:003
¡0:005
¡0:011
¡0:032
¡0:003
¡0:005
¡0:010
¡0:031
bias ^½I
¡0:0004
0:003
0:002
0:012
¡0:001
0:001
0:003
0:006
¡0:001
¡0:0007
0:001
0:007
¡0:0007
¡0:001
¡0:0001
0:004
¡0:000
0:000
0:000
0:0005
¡0:0002
¡0:0002
¡0:0004
0:0008
RMSE ^½GMM
0:074
0:099
0:160
0:552
0:053
0:070
0:111
0:443
0:044
0:053
0:075
0:248
0:031
0:038
0:051
0:181
0:029
0:033
0:042
0:109
0:020
0:022
0:027
0:074
RMSE b
^½ RMSE ^½I
0:065
0:054
0:089
0:061
0:129
0:070
0:187
0:099
0:055
0:038
0:081
0:042
0:124
0:048
0:183
0:067
0:036
0:035
0:040
0:036
0:051
0:037
0:085
0:058
0:027
0:025
0:032
0:026
0:045
0:027
0:082
0:041
0:024
0:023
0:024
0:023
0:024
0:022
0:037
0:026
0:017
0:017
0:017
0:017
0:018
0:015
0:034
0:018
The …xed e¤ects ®i and the innovations "it are assumed to have independent standard
normal distributions. Initial observations yi0 are assumed to be generated by the stationary
³
´
®i
1
distribution N 1¡½
;
:
2
0 1¡½
0
26
Table 3. Performances of Integrated likelihood estimator for high values of ½ and T = 5
N = 100
½ = 0:75
½ = 0:75
½ = 0:75
½ = 0:80
½ = 0:80
½ = 0:80
½ = 0:85
½ = 0:85
½ = 0:85
½ = 0:90
½ = 0:90
½ = 0:90
½ = 0:95
½ = 0:95
½ = 0:95
Actual mean % Bias
Actual median % Bias
RMSE
Actual mean % Bias
Actual median % Bias
RMSE
Actual mean % Bias
Actual median % Bias
RMSE
Actual mean % Bias
Actual median % Bias
RMSE
Actual mean % Bias
Actual median % Bias
RMSE
bLIML;1
½
1:297
¡3:087
0:181
¡0:112
¡5:725
0:213
¡3:899
¡10:117
0:233
¡9:757
¡15:389
0:246
¡15:203
¡19:637
0:252
bI2SLS;LD
½
5:533
1:381
0:176
4:304
1:457
0:173
1:966
0:065
0:160
¡0:771
¡2:346
0:153
¡3:367
¡4:776
0:149
b½CU E;LD
11:553
7:470
0:213
10:413
8:651
0:205
7:983
7:558
0:194
6:138
6:114
0:180
3:124
3:136
0:165
b½I
1:158
0:475
0:084
0:802
¡0:112
0:089
0:995
¡0:022
0:093
1:363
0:288
0:099
1:455
¡0:045
0:111
Table 4 Integrated likelihood estimator for the exogenous regressors case
using Lancaster’s approximation (1997)
(¾2 is unknown; 5000 data sets, ¾2» is the variance of the exogenous variable generation
BIAS ½
I
0:065
II
0:034
III 0:006
IV 0:050
V
0:027
V I 0:005
BIAS ½
V II
0:028
V III 0:027
XI
0:020
XII
0:023
BIAS ¯
¡0:044
¡0:018
¡0:001
¡0:042
¡0:019
¡0:002
BIAS ¯
¡0:010
¡0:007
¡0:008
¡0:010
process)
RMSE ½ RMSE ¯
0:079
0:156
0:053
0:135
0:033
0:174
0:064
0:063
0:046
0:045
0:032
0:033
RMSE ½ RMSE ¯
0:072
0:186
0:071
0:136
0:062
0:051
0:063
0:052
27
T
6
6
6
6
6
6
T
3
3
3
3
½
0:0
0:4
0:8
0:0
0:4
0:8
½
0:4
0:4
0:4
0:4
¯
1:0
0:6
0:2
1:0
0:6
0:2
¯
0:6
0:6
0:6
0:6
®
0:80
0:80
0:80
0:99
0:99
0:99
®
0:80
0:80
0:99
0:99
¾»
0:85
0:88
0:40
0:20
0:19
0:07
¾»
0:88
1:84
0:19
0:40
Kieviet Results for LSDVc estimator in the exogenous regressors case
BIAS ½
I
¡0:019
II
¡0:038
III ¡0:125
IV ¡0:017
V
¡0:042
V I ¡0:127
BIAS ½
V II
¡0:205
V III ¡0:033
XI
¡0:249
XII
¡0:248
BIAS ¯
¡0:018
¡0:002
¡0:011
0:005
0:019
0:050
BIAS ¯
0:004
0:061
0:055
0:046
RMSE ½
0:043
0:059
0:135
0:050
0:067
0:136
RMSE ½
0:220
0:077
0:263
0:263
RMSE ¯
0:057
0:052
0:114
0:217
0:231
0:643
RMSE ¯
0:093
0:077
0:433
0:212
T
6
6
6
6
6
6
T
3
3
3
3
½
0:0
0:4
0:8
0:0
0:4
0:8
½
0:4
0:4
0:4
0:4
¯
1:0
0:6
0:2
1:0
0:6
0:2
¯
0:6
0:6
0:6
0:6
®
0:80
0:80
0:80
0:99
0:99
0:99
®
0:80
0:80
0:99
0:99
¾»
0:85
0:88
0:40
0:20
0:19
0:07
¾»
0:88
1:84
0:19
0:40
Table 5. Integrated likelihood estimator for the exogenous regressors case
BIAS ½
I
¡0:0003
II
0:0002
III 0:0002
IV 0:0004
V
¡0:0001
V I ¡0:0004
BIAS ½
V II
¡0:0009
V III ¡0:0001
XI
¡0:0002
X II
¡0:0020
(¾2
BIAS ¯
0:0004
0:0006
0:0005
¡0:0004
0:0004
¡0:0007
BIAS ¯
0:0010
¡0:0010
¡0:0008
0:0003
is unknown; 5000 data sets)
RMSE ½ RMSE ¯ T
0:0288
0:0288
6
0:0289
0:0283
6
0:0290
0:0288
6
0:0289
0:0290
6
0:0287
0:0289
6
0:0290
0:0289
6
RMSE ½ RMSE ¯ T
0:0287
0:0287
3
0:0290
0:0288
3
0:0287
0:0289
3
0:0283
0:0289
3
28
½
0:0
0:4
0:8
0:0
0:4
0:8
½
0:4
0:4
0:4
0:4
¯
1:0
0:6
0:2
1:0
0:6
0:2
¯
0:6
0:6
0:6
0:6
®
0:80
0:80
0:80
0:99
0:99
0:99
®
0:80
0:80
0:99
0:99
¾»
0:85
0:88
0:40
0:20
0:19
0:07
¾»
0:88
1:84
0:19
0:40