Semiparametric Estimation of Fixed Eects Panel Data Varying Coecient Models Yiguo Sun Department of Economics, University of Guelph Guelph, ON, Canada N1G2W1 Raymond J. Carroll Department of Statistics, Texas A&M University College Station, TX 77843-3134, USA Dingding Li Department of Economics, University of Windsor Windsor, ON, Canada N9B3P4 April 2, 2009 Abstract We consider the problem of estimating a varying coecient panel data model with xed eects using a local linear regression approach. Unlike rst-dierenced estimator, our proposed estimator removes xed eects using kernel-based weights. This results a one-step estimator without using back-tting technique. The computed estimator is shown to be asymptotically normally distributed. A modied least-squared cross-validatory method is used to select the optimal bandwidth automatically. Moreover, we propose a test statistic for testing the null hypothesis of a random eects against a xed eects varying coecient panel data model. Monte Carlo simulations show that our proposed estimator and test statistic have satisfactory nite sample performance. Key words: Consistent test; Fixed eects; Panel data; Varying coecients model.
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Semiparametric Estimation of Fixed E�ects Panel Data Varying
Coe�cient Models
Yiguo SunDepartment of Economics, University of Guelph
Guelph, ON, Canada N1G2W1
Raymond J. CarrollDepartment of Statistics, Texas A&M University
College Station, TX 77843-3134, USA
Dingding LiDepartment of Economics, University of Windsor
Windsor, ON, Canada N9B3P4
April 2, 2009
Abstract
We consider the problem of estimating a varying coe�cient panel data model with �xede�ects using a local linear regression approach. Unlike �rst-di�erenced estimator, our proposedestimator removes �xed e�ects using kernel-based weights. This results a one-step estimatorwithout using back-�tting technique. The computed estimator is shown to be asymptoticallynormally distributed. A modi�ed least-squared cross-validatory method is used to select theoptimal bandwidth automatically. Moreover, we propose a test statistic for testing the nullhypothesis of a random e�ects against a �xed e�ects varying coe�cient panel data model.Monte Carlo simulations show that our proposed estimator and test statistic have satisfactory�nite sample performance.
lying between Zit and z for each i and t. Both An and Bn contribute to the bias term of the
estimator. Also, ifPni=1 �i = 0 holds true, Bn = 0; if we only assume �i being iid with zero mean
and �nite variance, the bias due to the existence of unknown �xed e�ects can be asymptotically
ignored.
To derive the asymptotic distribution of vecfb�(z)g, we �rst give some regularity conditions.Throughout this paper, we use M > 0 to denote a �nite constant, which may take a di�erent value
at di�erent places.
Assumption 1: The random variables (Xit; Zit) are independently and identically distributed
(i.i.d.) across the i index, and
(a) EkXitk2(1+�) �M <1 and EkZitk2(1+�) �M <1 hold for some � > 0 and for all i and t.
(b) The Zit are continuous random variables with a p.d.f. ft(z). Also, for each z 2 Rq,
f (z) =Pmt=1 ft (z) > 0.
(c) Denote �it = KH(Zit; z) and $it = �it=Pmt=1 �it 2 (0; 1) for all i and t. (z) = jHj
�1Pmt=1
E�(1�$it)�itXitXT
it
�is a nonsingular matrix.
7
(d) Let ft (zjXit) be the conditional pdf of Zit at Zit = z conditional onXit and ft;s (z1; z2jXit; Xjs)
be the joint conditional pdf of (Zit; Zjs) at (Zit; Zjs) = (z1; z2) conditional on (Xit; Xjs) for t 6= s
and any i and j. Also, � (z), ft (z), ft (�jXit), ft;s (�; �jXit; Xjs) are uniformly bounded in the domain
of Z and are all twice continuously di�erentiable at z 2 Rq for all t 6= s, i and j.
Assumption 2: BothX and Z have full column rank; fXit;1; :::; Xit;p; fXit;lZit;j : l = 1; :::; p; j =
1; :::; qgg are linearly independent. If Xit;l � Xi;l for at most one l 2 f1; � � � ; pg, i.e., Xi;l does not
depend on t, we assume E(Xi;l) 6= 0: The unobserved �xed e�ects �i are i.i.d. with zero mean and
a �nite variance �2� > 0. The random errors vit are assumed to be i.i.d. with a zero mean, �nite
variance �2v and independent of Zit and Xit for all i and t. Yit is generated by equation (1).
If Xit contains a time invariant regressor, say the lth component of Xit is Xit;l = Wi. Then
the corresponding coe�cient �l(�) is estimable if MH(z)(W em) 6= 0 for a given z, where W =
(W1; :::;Wn)>. Simple calculations give MH(z)(W em) = (n�1
Pni=1Wi)MH(z) �(en em). The
proof of Lemma A.2 in Appendix 7.1 can be used to show that MH(z)(en em) 6= 0 for any
given z with probability one. Therefore, �l(�) is asymptotically identi�able if n�1Pni=1Xit;l �
n�1Pni=1Wi 9 0 while ��
a:s:! 0. For example, if Xit contains a constant, say, Xit;1 =Wi � 1; then
�1(�) is estimable because n�1Pni=1Wi = 1 6= 0.
Assumption 3: K(v) =Qqs=1 k(vs) is a product kernel, and the univariate kernel function
k(�) is a uniformly bounded, symmetric (around zero) probability density function with a compact
support [�1; 1]. In addition, de�ne jHj = h1 � � �hq and kHk =qPq
j=1 h2j : As n ! 1, kHk ! 0,
njHj ! 1.
The assumptions listed above are regularity assumptions commonly seen in nonparametric es-
timation literature. Assumption 1 apparently excludes the case of either Xit or Zit being I(1);
other than the moment restrictions, we do not impose I(0) structure on Xit across time, since
this paper considers the case that m is a small �nite number. Also, instead of imposing the
smoothness assumption on ft (�jXit) and ft;s (�; �jXit; Xis) as in Assumption 1(d), we can assume
that ft (z)E�XitX
Tit jz�and ft;s (z1; z2)E
�XitX
Tjsjz1; z2
�are uniformly bounded in the domain of
Z and are all twice continuously di�erentiable at z 2 Rq for all t 6= s and i and j. Our version of
the smoothness assumption simpli�es our notation in the proofs.
Assumption 2 indicates that Xit can contain a constant term of ones. The kernel function
8
having a compact support in Assumption 3 is imposed for the sake of brevity of proof and can be
removed at the cost of lengthy proofs. Speci�cally, the Gaussian kernel is allowed.
We use b�(z) to denote the �rst column of b�(z). Then �(z) estimates �(z).THEOREM 3.1 Under Assumptions 1-3, we obtain the following bias and variance for b�(z),given a �nite integer m > 0:
The �rst term of bias(b�(z)) results from the local approximation of � (z) by a linear function of
z, which is of order O�kHk2
�as usual. The second term of bias(b�(z)) results from the unknown
�xed e�ects �i: (a) if we assumedPni=1 �i = 0, this term is zero exactly; (b) the result indicates
that the bias term is dominated by the �rst term and will vanish as n!1.
In Appendix, we show that
jHj�1mXt=1
E��itXitX
Tit
�= �(z) + o(kHk2),
jHj�1mXt=1
Eh�itXitX
Tit rH
�~Zit; z
�i= �2� (z)�H (z) + o(kHk2),
jHj�1mXt=1
E��2itXitX
Tit
�=
�ZK2 (u) du
�� (z) + o(kHk2),
where �2 =Rk (v) v2dv, � (z) =
Pmt=1 ft (z)E
�X1tX
T1tjz�, and �H (z) =
htr�H @2�1(z)
@z@zTH�; � � � ;
tr�H@2�p(z)@z@zT
H�iT
. Since $it 2 [0; 1) for all i and t, the results above imply the existence of (z),
� (z), and � (z). However, given a �nite integer m > 0, we can not obtain explicitly the asymptotic
bias and variance due to the random denominator appearing in $it.
Further, the following Theorem gives the asymptotic normality results for b�(z).
9
THEOREM 3.2 Under Assumptions 1-3, and assuming in addition that Ejvitj2+� <1 for some
� > 0, and thatpnjHjkHk2 = O (1) as !1, we havep
njHjfb�(z)� �(z)�(z)�1 � (z) =2g d! N(0;��(z));
where ��(z) = �2v limn!1(z)
�1 � (z) (z)�1. Moreover, a consistent estimator for ��(z) is given
as follows:
b��(z) = Spb(z;H)�1 bJ(z;H)b(z;H)�1S>p p! ��(z);b(z;H) = n�1jHj�1R(z;H)>SH(z)R(z;H)bJ(z;H) = n�1jHj�1R(z;H)>SH(z)bV bV >SH(z)R(z;H)where bV is the vector of estimated residuals and Sp includes the �rst p rows of the p(q+1) identify
matrix. Finally, a consistent estimator for the leading bias can be easily obtained based on a
nonparametric local quadratic regression result.
4 TESTING RANDOM EFFECTS VERSUS FIXED EFFECTS
In this section we discuss how to test for the presence of random e�ects versus �xed e�ects in
a semiparametric varying coe�cient panel data model. The model remains as (1). The random
e�ects speci�cation assumes that �i is uncorrelated with the regressors Xit and Zit, while for the
�xed e�ects case, �i is allowed to be correlated with Xit and/or Zit in an unknown way.
We are interested in testing the null hypothesis (H0) that �i is a random e�ect versus the
alternative hypothesis (H1) that �i is a �xed e�ect. The null and alternative hypotheses can be
written as
H0 : PrfE(�ijZi1; :::; Zim; Xi1; � � � ; Xim) = 0g = 1 for all i, (14)
H1 : PrfE(�ijZi1; :::; Zim; Xi1; � � � ; Xim) 6= 0g > 0 for some i , (15)
while we keep the same setup given in model (1) under both H0 and H1.
Our test statistic is based on the squared di�erence between the FE and RE estimators, which
is asymptotically zero under H0 and positive under H1: To simplify the proofs and save computing
time, we use local constant estimator instead of local linear estimator for constructing our test.
10
Then following the argument in Section 2 and Appendix 7.2, we have
Z eU(z)>SH(z)XX>SH(z)eU(z)dzsince fX>SH(z)Xgfb�FE(z)�b�RE(z)g = X>SH(z)fY �Xb�RE(z)g � X>SH(z)eU(z): To simplify thestatistic, we make several changes in Tn. Firstly, we simplify the integration calculation by replacingeU(z) by bU , where bU � bU(Z) = Y � BfX; b�RE(Z)g and BfX; b�RE(Z)g stacks up X>
itb�RE(Zit) in
the increasing order of i �rst then of t: Secondly, to overcome the complexity caused by the random
denominator in MH(z), we replace MH(z) by MD = In�m �m�1In (eme>m) such that the �xed
e�ects can be removed due to the fact that MDD0 = 0. With the above modi�cation and also
removing the i = j terms in Tn (since Tn contains two summationsPi
Pj), our further modi�ed
test statistic is given by
eTn def= nXi=1
Xj 6=i
bU>i Qm Z KH(Zi; z)X>i XjKH(Zj ; z)dzQm
bUj ;where Qm = Im �m�1eme>m: If jHj ! 0 as n!1, we obtain
jHj�1ZKH(Zi; z)X
>i XjKH(Zj ; z)dz (16)
=
264�KH(Zi;1; Zj;1)X
>i;1Xj;1 � � � �KH(Zi;1; Zj;m)X
>i;1Xj;m
.... . .
...�KH(Zi;m; Zj;1)X
>i;mXj;1 � � � �KH(Zi;m; Zj;m)X
>i;mXj;m
375 ,where �KH(Zit; Zjs) =
RKfH�1(Zit�Zjs)+!gK(!)d!. We then replace �KH(Zit; Zjs) byKH(Zit; Zjs);
this replacement will not a�ect the essence of the test statistic since the local weight is untouched.
Now, our proposed test statistic is given by
bTn = 1
n2jHj
nXi=1
nXj 6=i
bU>i QmAi;jQm bUj (17)
11
whereAi;j equals to the right-hand side of equation (16) after replacing �KH(Zit; Zjs) byKH(Zit; Zjs).
Finally, to remove the asymptotic bias term of the proposed test statistic, we calculate the leave-
one-unit-out random-e�ects estimator of �(Zit); that is, for a given pair of (i; j) in the double
summation of (17) with i 6= j, b�RE(Zit) is calculated without using the observations on the jth-unit, f(Xjt; Zjt; Yjt)gmt=1 and b�RE(Zjt) is calculated without using the observations on the ith-unit.
We present the asymptotic properties of this test below and delay the proofs to Appendix 7.3.
THEOREM 4.1 Under Assumptions 1-3, and ft (z) has a compact support S for all t, and
npjHj kHk4 ! 0 as n!1, then we have under H0 that
Jn = npjHj bTn=b�0 d! N(0; 1) (18)
where b�20 = 2n2jHj
Pni=1
Pnj 6=i(
bV >i QmAi;jQm bVj)2 is a consistent estimator of�20 = 4(1� 1=m)2�4v
ZK2(u)du
mXt=2
t�1Xs=1
Ehft(Z1s)(X
>1sX2t)
2i,
where bVit = Yit � X>itb�FE(Zit) � b�i and for each pair of (i; j), i 6= j, b�FE(Zit) is a leave-two-
unit-out FE estimator without using the observations from the ith and jth units and b�i = �Yi �
m�1Pmt=1X
>itb�FE(Zit). Under H1, Pr[Jn > Bn] ! 1 as n ! 1, where Bn is any nonstochastic
sequence with Bn = o(npjHj).
Assuming that ft (z) has a compact support S for all t is to simplify the proof of supz2S jjb�RE(z)�� (z) jj = op (1) as n ! 1; otherwise, some trimming procedure has to be placed to show the
uniform convergence result and the consistency of b�20 as an estimator of �20. Theorem 4.1 states
that the test statistic Jn = npjHj bTn=b�0 is a consistent test for testing H0 against H1. It is a
one-sided test. If Jn is greater than the critical values from the standard normal distribution, we
reject the null hypothesis at the corresponding signi�cance levels.
5 MONTE CARLO SIMULATIONS
In this section we report some Monte Carlo simulation results to examine the �nite sample perfor-
mance of the proposed estimator. The following data generating process is used:
Yit = �1(Zit) + �2(Zit)Xit + �i + vit; (19)
12
where �1(z) = 1 + z + z2, �2(z) = sin(z�), Zit = wit + wi;t�1, wit is i.i.d. uniformly distributed in
[0; �=2], Xit = 0:5Xi;t�1 + �it, �it is i.i.d. N(0; 1). In addition, �i = c0 �Zi� + ui for i = 2; � � � ; n with
c0 = 0; 0:5; and 1.0, ui is i.i.d. N(0; 1):When c0 6= 0, �i and Zit are correlated; we use c0 to control
the correlation between �i and �Zi� = m�1Pmt=1 Zit. Moreover, vit is i.i.d. N(0; 1), wit, �it, ui and
vit are independent of each other.
We report estimation results for both the proposed �xed-e�ects (FE) estimator and the random-
e�ects (RE) estimator, see Appendix 7.2 for the asymptotic results of the RE estimator. To learn
how the two estimators perform when we have �xed-e�ects model and when we have random-e�ects
model, we use the integrated squared error as a standard measure of estimation accuracy:
ISE(b�l) = Z fb�l(z)� �l(z)g2f(z)dz; (20)
which can be approximated by the average mean squared error
AMSE(b�l) = (nm)�1 nXi=1
mXt=1
[b�l(Zit)� �l(Zit)]2for l = 1; 2. In Table 1 we present the average value of AMSE(b�l) from 1000 Monte Carlo
experiments. We choose m = 3 and n = 50, 100, and 200.
Since the bias and variance of the proposed FE estimator do not depend on the values of the
�xed e�ects, our estimates are the same for di�erent values of c0; however, it is not true under the
random-e�ects model. Therefore, the results derived from the FE estimator are only reported once
in Table 1 since it is invariant to di�erent values of c0.
It is well-known that the performance of non/semiparametric models depends on the choice of
bandwidth. Therefore, we propose a leave-one-unit-out cross validation method to automatically
select the optimal bandwidth for estimating both the FE and RE models. Speci�cally, when
estimating �(�) at a point Zit; we remove f(Xit; Yit; Zit)gmt=1 from the data and only use the rest
of (n � 1)m observations to calculate b�(�i)(Zit). In computing the RE estimate, the leave-one-unit-out cross validation method is just a trivial extension of the conventional leave-one-out cross
validation method. The conventional leave-one-out method fails to provide satisfying result due to
the existence of unknown �xed e�ects. Therefore, when calculating the FE estimator, we use the
13
following modi�ed leave-one-unit-out cross validation method:
bHopt = argminH[Y �BfX; b�(�1)(Z)g]>M>
DMD[Y �BfX; b�(�1)(Z)g]; (21)
where MD = In�m � m�1In (eme>m) satis�es MDD0 = 0; this is used to remove the unknown
�xed e�ects. In addition, BfX; b�(�1)(Z)g stacks up X>itb�(�i)(Zit) in the increasing order of i �rst
where the last term does not depend on the bandwidth. If vit is independent of the fXjs; Zjsg for all
i, j, s and t, or (Xit; Zit) is strictly exogenous variable, then the second term has zero expectation
because the linear transformation matrix MD removes a cross-time not cross-sectional average
from each variable, e.g. eYit = Yit �m�1Pms=1 Yis for all i and t. Therefore, the �rst term is the
dominant term in large samples and (21) is used to �nd an optimal smoothing matrix minimizing
a weighted mean squared error of fb�(Zit)g: Of course, we could use other weight matrices in (21)instead of MD as long as the weight matrices can remove the �xed e�ects and do not trigger a
non-zero expectation of the second term in (22).
Table 1 shows that the RE estimator performs better than the FE estimator when the true
model is a random e�ects model. However, the FE estimator performs much better than the RE
estimator when the true model is a �xed-e�ects model. This is expected since the RE estimator
is inconsistent when the true model is the �xed e�ects model. Therefore, our simulation results
indicate that a test for random e�ects against �xed e�ects will be always in demand when we
analyze panel data models. In Table 2 we report simulation results of the proposed nonparametric
test of random e�ects against �xed e�ects.
For the selection of the bandwidth h, for univariate case, Theorem 4.1 indicates that h ! 0,
nh!1, and nh9=2 ! 0 as n!1; if we take h � n��; Theorem 4.1 requires � 2 (29 ; 1): To ful�ll
both conditions nh ! 1 and nh9=2 ! 0 as n ! 1, we use � = 2=7. Therefore, in producing
Table 2, we use h = c(nm)�2=7b�z to calculate the RE estimator with c taking a value from :8 , 1:0,
and 1:2. Since the computation is very time consuming, we only report results for n = 50 and 100.
14
With m = 3, the e�ective sample size is 150 and 300, which is small but moderate sample size.
Although the bandwidth chosen this way may not be optimal, the results in Tables 2, 3, and 4 show
that the proposed test statistic is not very sensitive to the choice of h when c changes and that a
moderate size distortion and decent power are consistent with the �ndings in the nonparametric
tests literature. We conjecture that some bootstrap procedures can be used to reduce the size
distortion in �nite samples. We will leave this as a future research topic.
6 CONCLUSION
In this paper we proposed a local linear least squares method to estimate a �xed e�ects varying
coe�cient panel data model when the number of observations across time is �nite; a data-driven
method was introduced to automatically �nd the optimal bandwidth for the proposed FE estimator.
In addition, we introduced a new test statistic to test for a random e�ects model against a �xed
e�ects model. Monte Carlo simulations indicate that the proposed estimator and test statistic have
good �nite sample performance.
7 APPENDIX
7.1 Proof of Theorem 3.1
To make our mathematical formula short, we introduce some simpli�ed notations �rst: for each i
and t; �it = KH (Zit; z) and cH (Zi; z)�1 =
Pmt=1 �it, and for any positive integers i; j; t; s
Applying (A.2), (A.3), (A.6), and (A.7) toAn1, we have n�1 jHj�1An1 �
Pmt=1E
�St;1;1
�XitX
Tit
��+Op
�kHk2
�+Op
�n�
12 jHj�
12
�if kHk ! 0 and n jHj ! 1 as n!1.
Apparently,Pmt=1$it = 1 for all i. In addition, since the kernel function K (�) is zero out-
side the unit circle by Assumption 3, the summations in An2 are taken over units such that H�1 (Zit � z) � 1. By Lemma A.1 and by the LLN given Assumption 1 (a), we obtain 1
n jHjPni=1 cH (Zi; z)
nXi=1
mXt=1
mXs=1
$it$is [�]it;is �XitX
Tis
� = Op �n�1 ln (lnn)�and
1njHj
Pni=1
Pmt=1
Pms 6=t
�it�isPmt=1 �it
[�]it;is �XitX
Tis
� � 12njHj
Pni=1
Pmt=1
Pms 6=tp�it�is
[�]it;is �XitXTis
� = Op (jHj), where we use
Pmt=1 �it � �it + �is � 2
p�it�is for any t 6= s.
Hence, we have n�1 jHj�1An2 = n�1 jHj�1Pni=1
Pmt=1$it�it [�]it;it
�XitX
Tit
�+ Op (jHj). De-
note dit = $it�it [�]it;it �XitX
Tit
�and �n = n
�1 jHj�1Pni=1
Pmt=1 (dit � Edit). It is easy to show
that n�1 jHj�1�n = Op�n�1=2 jHj�1=2
�. Since E (kditk) � E
h�it
[�]it;it �XitXTit
� i � M jHj
holds for all i and t, n�1 jHj�1An2 = jHj�1Pmt=1E
h$it�it [�]it;it
�XitX
Tit
�i+ op (1) exists, but
we can not calculate the exact expectation due to the random denominator.
Consider An3. We have n�1 jHj�1 kAn3k = Op
�jHj2 ln (lnn)
�by Lemma A.1, Assump-
tion 1, and the fact that n�1 jHj�1Pni=1
Pmt=1 I
� H�1 (Zit � z) � 1� = 2f (z) + Op
�kHk2
�+
18
Op
�n�1=2 jHj�1=2
�.
Hence, we obtain
n�1 jHj�1An � n�1 jHj�1An1 � n�1 jHj�1nXi=1
mXt=1
$it�it [�]it;it �XitX
Tit
�= n�1 jHj�1
nXi=1
mXt=1
(1�$it)�it [�]it;it �XitX
Tit
�= jHj�1
mXt=1
Eh(1�$it)�it [�]it;it
�XitX
Tit
�i+ op (1) :
This will complete the proof of this Lemma.
Lemma A.3 Under Assumptions 1-3, we have
n�1 jHj�1R (z;H)T SH (z)� (z;H) � jHj�1mXt=1
Eh(1�$it)�it (Git Xit)XT
it rH
�~Zit; z
�i:
Proof: Simple calculations give
Bn = R (z;H)T SH (z)� (z;H)
=nXi=1
mXt=1
�it (Git Xit)XTit rH
�~Zit; z
��
nXj=1
nXi=1
qij
mXs=1
mXt=1
�js�it (Git Xit)XTjsrH
�~Zjs; z
�=
nXi=1
mXt=1
�it (Git Xit)XTit rH
�~Zit; z
��
nXi=1
qii
mXt=1
�2it (Git Xit)XTit rH
�~Zit; z
��
nXi=1
qii
mXt=1
mXs 6=t
�is�it (Git Xit)XTisrH
�~Zis; z
��
nXj=1
nXi6=j
qij
mXt=1
mXs=1
�js�it (Git Xit)XTjsrH
�~Zjs; z
�= Bn1 �Bn2 �Bn3 �Bn4;
where � (z;H) is de�ned in Section 3. Using the same method in the proof of Lemma A.2, we show
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Table 1: Average mean squared errors (AMSE) of the �xed and random e�ects estimators whenthe data generation process is a random e�ects model and when it is a �xed e�ects model.
Data Process Random E�ects Estimator Fixed E�ects Estimatorn = 50 n = 100 n = 200 n = 50 n = 100 n = 200