Econometrica, Vol. 72, No. 5 (September, 2004), 1519–1563
WAVELET-BASED TESTING FOR SERIAL CORRELATION OF
UNKNOWN FORM IN PANEL MODELS
BY YONGMIAO HONG AND CHIHWA KAO1
Wavelet analysis is a new mathematical method developed as a unified field of science over the last decade or so. As a spatially adaptive analytic tool, wavelets are useful
for capturing serial correlation where the spectrum has peaks or kinks, as can arise from
persistent dependence, seasonality, and other kinds of periodicity. This paper proposes
a new class of generally applicable wavelet-based tests for serial correlation of unknown
form in the estimated residuals of a panel regression model, where error components
can be one-way or two-way, individual and time effects can be fixed or random, and
regressors may contain lagged dependent variables or deterministic/stochastic trending
variables. Our tests are applicable to unbalanced heterogenous panel data. They have a
convenient null limit N(0 1) distribution. No formulation of an alternative model is required, and our tests are consistent against serial correlation of unknown form even in
the presence of substantial inhomogeneity in serial correlation across individuals. This
is in contrast to existing serial correlation tests for panel models, which ignore inhomogeneity in serial correlation across individuals by assuming a common alternative,
and thus have no power against the alternatives where the average of serial correlations among individuals is close to zero. We propose and justify a data-driven method
to choose the smoothing parameter—the finest scale in wavelet spectral estimation,
making the tests completely operational in practice. The data-driven finest scale automatically converges to zero under the null hypothesis of no serial correlation and
diverges to infinity as the sample size increases under the alternative, ensuring the consistency of our tests. Simulation shows that our tests perform well in small and finite
samples relative to some existing tests.
KEYWORDS: Error component, hypothesis testing, serial correlation of unknown
form, spectral peak, static and dynamic panel models, unbalanced panel data, wavelet.
1. INTRODUCTION
PANEL DATA HAVE BEEN WIDELY USED in economics and finance. They often
provide insights not available in pure time-series or cross-sectional data (e.g.,
Baltagi (2002), Granger (1996), Hsiao (2003)). This paper proposes a new class
of generally applicable wavelet-based consistent tests for serial correlation of
unknown form in the errors of panel models. It is important to test serial correlation for panel models because existence of serial correlation will invalidate
conventional tests such as t- and F -tests which use standard covariance estimators of parameter estimators, and will indicate model misspecification when
1
We thank the co-editor and three referees for insightful comments that have lead to significant improvement on a previous version. We also thank Badi Baltagi, Pierre Duchesne,
Jiti Gao, Jerry Hausman, Cheng Hsiao, Heshem Pesaran, Jim Stock, and seminar participants
at MIT-Harvard Econometrics Workshop, 2001 North American Summer Meeting of Econometric Society in Washington, DC, 2001 Far Eastern Meeting of Econometric Society in Kobe,
Japan, Fifth ICSA International Conference in Hong Kong, and 10th International Conference
on Panel Data Models in Berlin for helpful comments. This research is supported by National
Science Foundation via Grant SES-0111769.
1519
1520
Y. HONG AND C. KAO
regressors include lagged dependent variables. Moreover, the choice of estimation methods may depend on whether there exists serial correlation in the errors of panel models. When the errors are serially correlated, for example, the
computation of MLE (e.g., Anderson and Hsiao (1982), Hsiao (2003), Binder,
Hsiao, and Pesaran (1999)) and GMM (e.g., Blundell and Bond (1998)) could
be complicated, and the feasible GLS estimator will be invalid or have to be
modified substantially (e.g., Baltagi and Li (1991)). Some procedures, such as
Breusch and Pagan’s (1980) tests for random effects, also assume serial uncorrelatedness in the errors of panel models.
There have been some tests for serial correlation in panel models. Bhargava,
Franzini, Narendranathan (1982) extend Durbin and Watson’s (1951) test to
static panel models. Breusch and Pagan (1980) propose an LM test for firstorder serial correlation, assuming no random effects. Baltagi and Li (1995)
propose a class of LM tests for first-order serial correlation, allowing random
or fixed individual effects. Bera, Sosa-Escudero, and Yoon (2001) also propose a convenient OLS-based test for first-order serial dependence. Li and
Hsiao (1998) propose tests for first-order and higher-order serial correlation
in a semiparametric partially linear panel model.
All existing tests for serial correlation in panel models assume a known form
of a common alternative, e.g., an AR(1) or MA(1) model. These tests have
optimal power when the assumed model is true, but they are not consistent
against serial correlation of unknown form. It is useful to test serial correlation
of unknown form because prior information about the alternative is usually
not available in practice. This is particularly relevant for panel models because
there may exist significant inhomogeneity in serial correlation across individuals (e.g., Choi (2002)). By assuming a common alternative model, existing
tests ignore inhomogeneity in serial correlation across individuals, and so have
little power against the alternatives where the average of serial correlations
among individuals is close to zero. Moreover, as Granger and Newbold (1977,
p. 92) pointed out, the first few lags of OLS residuals of a misspecified linear dynamic model often appear like a white noise, due to the very nature of
OLS. It is therefore important to check serial correlation at higher order lags.
Little effort has been made on evaluation of dynamic panel models (Granger
(1996)).
Wavelets are a new mathematical tool alternative to the Fourier transform.
They can effectively capture nonsmooth features such as singularities and spatial inhomogeneity (e.g., Donoho and Johnstone (1994, 1995a, 1995b), Donoho
et al. (1996), Gao (1997), Hong and Lee (2001), Jensen (2000), Neumann
(1996), Lee and Hong (2001), Ramsey (1999), Wang (1995)). Many economic
and financial time series have a spectrum with peaks and kinks, as can arise
from persistent dependence, business cycles, seasonality, and other kinds of
periodicity (e.g., Bizer and Durlauf (1992), Granger (1969), Watson (1993)).
In this paper we use wavelets to test serial correlation in estimated residuals of panel models. Unlike existing tests, whose constructions are usually
SERIAL CORRELATION IN PANEL MODELS
1521
model-dependent, our tests are generally applicable. The panel model can be
static or dynamic, and one-way or two-way; both balanced and unbalanced
panel data are covered; individual and time effects can be fixed or random;
regressors can contain lagged dependent variables or deterministic/stochastic
trending variables; and no specific estimation method is required. Our tests
have a convenient limit N(0 1) distribution under the null hypothesis, no
matter whether regressors contain lagged dependent variables or deterministic/stochastic trending variables. In contrast to Durbin and Watson’s (1951)
test and Box and Pierce’s (1970) portmanteau test, parameter estimation uncertainty has no impact on the limit distribution of our test statistics when
applied to dynamic panel models. We do not require an alternative model,
and our tests are consistent against serial correlation of unknown form even in
the presence of substantial inhomogeneity in serial correlation across individuals. No consistent test for serial correlation of unknown form was available
for panel models.
Our asymptotic theory considers a panel model with both large n and T ,
where n is the number of individuals and T is the number of time-series
observations Increasing effort has been devoted to the study of panel models with both large n and T , due to the growing use of cross-country data over
time to study growth convergence, international R&D spillover and purchasing power parity, and to the growing use of firm- or portfolio-level financial
time series. As is well known (e.g., Phillips and Moon (1999), Hahn and
Kuersteiner (2002)), asymptotic analysis in panel models is much more involved than in pure time-series analysis, due to the need to handle double
indices. As a distinct feature, we treat both n → ∞ and T → ∞ simultaneously,
which complements Phillips and Moon (1999) and Hahn and Kuersteiner’s
(2002) joint limit theory for panel models. Our general theory does not require that the ratio n/T go to 0 or a constant. We also show that the use of
the estimated residuals from a possibly nonstationary panel model rather than
the unobservable errors has no impact on the null limit distribution of our test
statistics. In addition, we find several interesting features not available in pure
time-series analysis. Most remarkably, the limit N(0 1) distribution of our test
statistics is obtained without requiring the smoothing parameter—the finest
scale in wavelet estimation to grow with T . This not only leads to reasonable
asymptotic approximation in finite samples, but also makes it possible to use
data-driven methods that deliver a fixed finest scale under the null hypothesis of no serial correlation. This is in sharp contrast to Lee and Hong (2001),
who, in testing serial correlation for observed raw time series data, require the
finest scale to grow as T → ∞ under the null hypothesis. We further develop
a data-driven method to choose the finest scale, making our tests completely
operational in practice. The data-driven finest scale converges to 0 under the
null and grows to ∞ under the alternative, ensuring consistency against serial
correlation of unknown form. We also find that a heteroskedasticity-corrected
test may be less powerful than a heteroskedasticity-consistent test. This differs
1522
Y. HONG AND C. KAO
from the well-known estimation result that heteroskedasticity-corrected estimators (e.g., feasible GLS) are more efficient than heterokedasticity-consistent
estimators (e.g., OLS). Our tests work reasonably well in small and finite samples often encountered in economics.
We describe the panel model and hypotheses in Section 2, introduce wavelets
and test statistics in Section 3, derive the limit distributions of these tests in
Section 4, and establish their consistency in Section 5. Section 6 proposes a
data-driven finest scale. Section 7 is a simulation study. Section 8 concludes.
All proofs are in the Appendix. Throughout, A denotes the Euclidean norm
[tr(A A)]1/2 ; A∗ and Re(A) the complex conjugate and the real part of A;
Z ≡ {0 ±1 } and Z+ ≡ {0 1 } the set of integers and the set of nonnegative integers; and c and C generic bounded constants, with 0 < c < C < ∞.
Unless indicated, all limits are taken as both n T → ∞. A GAUSS program
for implementing our tests is available from the authors upon request. The user
only needs to supply estimated residuals.
2. THE FRAMEWORK
Consider a linear panel process
(2.1)
Yit = α + Xit β + µi + λt + εit
(t = 1 Ti ; i = 1 n; n Ti ∈ Z+ ),
where Yit is a scalar Xit is a p × 1 vector of regressors that may contain lagged
dependent variables, α is an intercept, β is a p × 1 vector of slope parameters,
µi is the individual effect, λt is the time effect, and εit is the error such that
{εit } and {εls } are mutually independent for all i = l and all t s. We allow for
fixed or random effects. We assume Ti = ci T for some integer T and ci ∈ [c C],
thus permitting unbalanced panel data. Moreover, we allow Yit Xit α, and β
to depend on n and T (For notational simplicity, we have suppressed such
dependence.)
The slope parameter β is often of interest. It can be estimated, e.g., by the
within estimator
n T
−1 n T
i
i
X̃it X̃it
X̃it Ỹit β̂ ≡
(2.2)
i=1 t=1
i=1 s=1
Ti
n
where X̃it ≡ Xit − X̄i· − X̄·t + X̄ X̄i· ≡ Ti−1 t=1
Xit X̄·t ≡ n−1 i=1 Xit , and
n
Ti
Xit The variables Ỹit Ȳi· Ȳ·t , and Ȳ are defined simX̄ ≡ n−1 i=1 Ti−1 t=1
ilarly. For interval estimation and hypothesis testing, one often uses the
n Ti
X̃it X̃it )−1 of β̂, where σ̂ε2
standard covariance estimator Ω̂β̂ ≡ σ̂ε2 ( i=1 t=1
n
2
−1
2
is an estimator for σε ≡ limn→∞ n
i=1 E(εit ) This estimator is valid when
SERIAL CORRELATION IN PANEL MODELS
1523
{εit } in (2.1) is serially uncorrelated, among other things. The existence of serial correlation of any form, however, will generally invalidate the covariance
estimator and related inference. In particular, conventional t- and F -tests will
be misleading. Moreover, when the Xit contain lagged dependent variables,
serial correlation will further render inconsistent the within estimator β̂ for β.
We are interested in testing whether the error process {εit } is serially correlated. The hypotheses of interest are H0 : cov(εit εit−|h| ) = 0 for all h = 0 and
all i vs. HA : cov(εit εit−|h| ) = 0 at least for some h = 0 and some i The alternative HA allows some (but not all) individual series to be white noises. Prior
information about the alternative is usually not available in practice, although
there may exist substantial inhomogeneity in serial correlation across i.
To test H0 we will examine serial correlation in the demeaned estimated
residual
(2.3)
v̂it ≡ ûit − ūi· − ū·t + ū
(t = 1 Ti ; i = 1 n)
Ti
n
n
where ûit ≡ Yit −Xit β̂ ūi· ≡ Ti−1 t=1
ûit ū·t ≡ n−1 i=1 ûit ū ≡ n−1 i=1 Ti−1 ×
Ti
t=1 ûit and β̂ is an estimator consistent for β under H0 When β̂ is the within
estimator in (2.2), v̂it is the well known within residual. However, we allow using other estimators that are consistent for β under H0 but not necessarily
under HA .
p
Suppose β̂ → β, as does the within estimator in (2.2) under H0 ; then v̂it will
converge in probability to the true error εit Under HA however, β̂ may not
be consistent for β (as in a dynamic panel model), and v̂it will converge in
probability to the model error
(2.4)
vit ≡ εit + (β − β∗ ) (Xit − E X̄i· − E X̄·t + E X̄)
which contains both the true error εit and the misspecified component, where
β∗ ≡ p lim β̂. This does not invalidate our tests, but it affects the power of the
tests in finite samples, because serial correlation in {vit } may differ from serial
correlation in {εit } However, our tests are still consistent against HA because
H0 holds if and only if {vit } is serially uncorrelated: When H0 holds, {vit } coincides with {εit } and so is serially uncorrelated; on the other hand, if {vit } is
serially uncorrelated in a linear dynamic panel model, we can view {vit } as the
true error for (2.1), and estimation and inference can be implemented in a
standard fashion. (We note that in a linear dynamic panel setup, it is possible
that {εit } is serially correlated but {vit } is serially uncorrelated, due to β∗ = β
This occurs when and only when εit contains the misspecified linear component, (β∗ − β) (Xit − E X̄i· − E X̄·t + E X̄) plus a white noise. In this case,
serial correlation in {εit } is solely caused by the misspecified linear component,
and it is actually more appropriate to view that (2.1) is a correctly specified
linear dynamic panel model, but with vit as the true error and β∗ as the true
1524
Y. HONG AND C. KAO
model parameter. With such an interpretation, H0 holds when {vit } is seriously
uncorrelated.)
Suppose {vit } has autocovariance function Ri h ≡ E(vit vit−|h| ) and power
spectrum
(2.5)
fi (ω) ≡ (2π)−1
∞
Ri (h)e−ihω ω ∈ [−π π]
i≡
−1
h=−∞
Both Ri (h) and fi (ω) contain the same information on serial correlation
of {vit } One can use Ri (h) or fi (ω) to test H0 vs. HA All existing tests for
serial correlation in panel models are based on Ri (h) assuming a common
model with some prespecified lags h for all i (e.g., AR(1) and MA(1)). We
use fi (ω) here, which is a natural tool to test serial correlation of unknown
form, because it contains information on serial correlation at all lags. Under H0 fi (ω) becomes fi0 (ω) ≡ (2π)−1 Ri (0) for all ω ∈ [−π π] Under HA fi (ω) = (2π)−1 Ri (0) at least for some i Thus, a consistent test for H0 vs.
HA can be formed by comparing consistent estimators of fi (ω) and fi0 (ω)
We will use wavelets to estimate fi (ω) which are suitable for time series with
spectral peaks and kinks. Of course, other nonparametric methods (e.g., kernel
smoothing; see Hong (1996) and Section 7 below) could be used.
3. WAVELET METHOD
3.1. Wavelets
The essence of wavelet analysis is to expand a function as a sum of elementary functions called wavelets centered at a sequence of locations. These
wavelets are derived from a single function ψ(·) called the mother wavelet,
by translations and dilations. As a spatially adaptive analytic tool, wavelets are
powerful in capturing singularities of nonsmooth functions, such as spectral
peaks and kinks (e.g., Gao (1997), Neumann (1996)). We first impose a standard condition on ψ(·).
∞ψ : R → R is an∞ orthonormal wavelet such that
∞ASSUMPTION 1:
ψ(x)
dx
=
0
|ψ(x)| dx < ∞ −∞ ψ(x)ψ(x − k) dx = 0 for all k ∈ Z,
−∞
∞ 2 −∞
k = 0 and −∞ ψ (x) dx = 1.
The orthonormality of ψ(·) ensures that the doubly infinite sequence {ψjk(·)}
where
(3.1)
ψjk (x) ≡ 2j/2 ψ(2j x − k)
j k ∈ Z
constitutes an orthonormal basis for L2 (R) the space of square-integrable
functions on R (Daubechies (1992)). The integers j and k are called scale and
1525
SERIAL CORRELATION IN PANEL MODELS
translation parameters. Intuitively, j localizes analysis in frequency and k localizes analysis in time or space.
Assumption 1 ensures that the Fourier transform of ψ(·),
∞
−1/2
(3.2)
ψ(x)e−izx dx z ∈ R
ψ̂(z) ≡ (2π)
−∞
exists and is continuous in z almost everywhere. We impose a condition
on ψ̂(·)
ASSUMPTION 2: (a) |ψ̂(z)| ≤ C(1 + |z|)−τ for some τ > 32 ; (b) ψ̂(z) =
eiz/2 b(z) or ψ̂(z) = −ieiz/2 b(z) where b(·) is real-valued with b(0) = 0.
Many wavelets satisfy this. One example is spline wavelets of positive order
m ∈ Z+ For odd m, this family has ψ̂(z) = eiz/2 b(z), where b(·) is real-valued
and symmetric. For even m, it has ψ̂(z) = −ieiz/2 b(z) where b(·) is real-valued
and odd (e.g., Hernández and Weiss (1996, (2.16), p. 161)). One member in this
family is the first-order spline wavelet, called the Franklin wavelet, whose
(3.3)
ψ̂(z) = e
P3 (z) ≡
iz/2
−1/2
(2π)
1/2
sin4 (z/4) P3 (z/4 + π/4)
(z/4)2 P3 (z/2)P3 (z/4)
where
2 1
+ cos(2z)
3 3
Another member is the second-order spline wavelet, with
(3.4)
ψ̂(z) = −ie
P5 (z) ≡
iω/2
−1/2
(2π)
1/2
sin6 (z/4) P5 (z/4 + π/4)
(z/4)3 P5 (z/2)P5 (z/4)
where
8
1
13
cos2 (2z) +
cos(2z) + 30
30
15
3.2. Wavelet Representation of Spectrum
We now consider wavelet representation of spectral density fi (·) Given an
orthonormal wavelet basis {ψjk (·)} for L2 (R) we define
Ψjk (ω) ≡ (2π)−1/2
∞
m=−∞
ψjk
ω
+m 2π
ω ∈ [−π π]
This constitutes an orthonormal wavelet basis for L2 [−π π] the space of
2π-periodic functions on [−π π] See, e.g., Daubechies (1992, Ch. 9) or
Hernández and Weiss (1996, Ch. 4).
1526
Y. HONG AND C. KAO
compute Ψjk (·) from its Fourier transform Ψ̂jk (h) ≡ (2π)−1/2 ×
πOne can also
−ihω
Ψjk (ω)e
dω via the formula
−π
(3.6)
∞
−1/2
Ψjk (ω) = (2π)
Ψ̂jk (h)eihω h=−∞
Lee and Hong (2001) show that the spectral density fi (·) in (2.5) can be
decomposed as
∞
2
j
(3.7)
−1
fi (ω) = (2π) σ +
2
i
αijk Ψjk (ω)
ω ∈ [−π π]
j=0 k=1
where the wavelet coefficient αijk is the orthogonal projection of fi (·) on the
base Ψjk (·); i.e.,
π
(3.8)
fi (ω)Ψjk (ω) dω
αijk ≡
−π
Unlike the Fourier transforms, αijk depends on the local behavior of fi (·) because Ψjk (·) is effectively 0 outside an interval of width 2−j centered at k/2j Such a spatial adoption feature makes it powerful for capturing nonsmooth
features. We can also express αijk in time domain; i.e.,
(3.9)
αijk = (2π)−1/2
∞
Ri (h)Ψ̂jk∗ (h)
h=−∞
3.3. Wavelet Spectral Density Estimator
Define the sample autocovariance function of {v̂it }:
(3.10)
R̂i (h) ≡ Ti−1
Ti
(h = 0 ±1 ±(Ti − 1))
v̂it v̂it−|h|
t=|h|+1
Then a wavelet estimator of the spectral density fi (·) can be given by
Ji
2
j
(3.11)
fˆi (ω) ≡ (2π)−1 R̂i (0) +
α̂ijk Ψjk (ω)
j=0 k=1
where the empirical wavelet coefficient
(3.12)
−1/2
α̂ijk ≡ (2π)
Ti −1
h=1−Ti
R̂i (h)Ψ̂jk∗ (h)
ω ∈ [−π π]
SERIAL CORRELATION IN PANEL MODELS
1527
and Ji ≡ Ji (Ti ) is the finest scale corresponding to the highest resolution level.
Appropriate conditions on Ji will be given. We allow a different Ji for a different i This is useful because the pattern of serial correlation may vary
substantially across i We will also propose a data-driven method to choose Ji .
Note that (3.11) is a linear wavelet estimator. Nonlinear wavelet estimators
of Donoho et al. (1996) which are popular in curve estimation could be used.
However, under our regularity conditions (see subsequent sections) which are
not stronger than standard assumptions in time series panel econometrics, both
linear and nonlinear wavelet estimators have the same convergence rate. Nonlinear wavelet estimators have no advantage at least in large samples. Masry
(1994, 1997) also finds that the gain from using nonlinear wavelet estimators
rather than linear wavelet estimators is marginal for different models. Moreover, the use of nonlinear wavelet estimators would lead to a much more
complicated analysis in theory.
Put Q(f1 f2 ) ≡
quadratic form
π
−π
3.4. Wavelet-Based Tests
[f1 (ω) − f2 (ω)]2 dω for any f1 (·) and f2 (·) We use the
Ji
2
j
(3.13)
Q(fˆi fˆi0 ) =
α̂2ijk j=0 k=1
where fˆi0 (ω) ≡ (2π)−1 R̂i (0) and the equality follows by Parseval’s identity. Our
first test statistic
n
Ji
2j
2
Vˆ 1/2 (3.14) Ŵ1 ≡
2πTi
α̂ − M̂
ijk
i=1
j=0 k=1
where
M̂ ≡
n
R̂2i (0)Mi0 i=1
V̂ ≡
n
R̂4i (0)Vi0 i=1
Mi0 ≡
Ti −1
(1 − h/Ti )bJi (h h)
h=1
Vi0 ≡ 4
Ti
Ti
(1 − h/Ti )(1 − m/Ti )b2Ji (h m)
h=1 m=1
1528
Y. HONG AND C. KAO
and
bJ (h m) ≡ 2 Re[aJ (h m) + aJ (h −m)]
J
2
j
aJ (h m) ≡
Ψ̂jk (h)Ψ̂jk∗ (m)
j=0 k=1
and Ψ̂jk (·) is as in (3.6)
Ti −1
Put âijk ≡ (2π)−1/2 h=1−T
ρ̂i (h)Ψ̂jk (h) where ρ̂i (h) ≡ R̂i (h)/R̂i (0). Our
i
second test statistic
Ji
n
2j
1 1/2
2
2πTi
(3.15) Ŵ2 ≡ √
âijk − Mi0
Vi0 n i=1
j=0 k=1
Intuitively, Ŵ2 can be viewed as a heteroskedasticity-corrected test while
Ŵ1 is a heteroskedasticity-consistent test, where heteroskedasticity arises from
different variances σi2 and finest scales Ji In Ŵ2 these two forms of heteroskedasticity are corrected first for each i. As is shown below, Ŵ1 and Ŵ2
are asymptotically N(0 1) under H0 but their power properties generally differ. The heteroskedasticity-robust test Ŵ1 may be more powerful than the
heteroskedasticity-consistent test Ŵ2 . This differs from the well-known estimation result that correcting heteroskedasticity leads to more efficient estimation
(e.g., the feasible GLS is more efficient than OLS).
Both Ŵ1 and Ŵ2 apply to one-way or two-way error component models. For
one-way component models, however, one can use v̂it ≡ ûit − ūi· if one knows
λt = 0 and use v̂it ≡ ûit − ū·t if one knows µi = 0 The limit distribution of the
test statistics is unchanged.
4. ASYMPTOTIC DISTRIBUTION
We now impose a set of unified regularity conditions that hold under both
H0 and HA .
ASSUMPTION 3:
√
nT (β̂ − β∗ ) = OP (1) where β∗ = β under H0 .
ASSUMPTION 4: (a) For each i {vit } is covariance-stationary with E(vit ) = 0,
E(vit2 ) = σi2 ∈ [c C] and E(vit8 ) ∈ [c C]; (b) the individual and time effects,
µi and λt can be stochastic (random effects) or deterministic ( fixed effects).
Ti
X̃it ṽit−h if h ≥ 0 and Γ˜ixv (h) ≡
ASSUMPTION 5: Put Γ˜ixv (h) ≡ Ti−1 t=h+1
Γ˜ixv (−h) if h < 0 Γixv (h) ≡ p lim Γ˜ixv (h) where X̃it ≡ Xit − X̄i· − X̄·t + X̄
SERIAL CORRELATION IN PANEL MODELS
1529
Ti
EX̃it 4 ≤ C;
and ṽit ≡ vit − v̄i· − v̄·t + v̄ Then (a) sup1≤i≤n Ti−1 t=1
∞
−1
(b) sup1≤i≤n sup1≤h<Ti EΓ˜ixv (h) − Γixv (h)2 ≤ CTi ; (c) h=−∞ Γixv (h) ≤ C.
We only require that the estimator β̂ be consistent for β under H0 ; it need
not be asymptotically most efficient under H0 and even need not be consistent
for β under HA Thus, we can use the convenient within estimator β̂ in (2.2).
Of course, other estimators such as OLS, feasible GLS, MLE, and IV are also
allowed.
In Assumption 4, no restrictive assumptions on {µi } and {λt } are imposed.
We allow {λt } to be serially correlated if λt is random, and {µi } to be spatially
correlated if µi is random. Given Assumption 3, {vit } coincides with the true
error {εit } under H0 Here, we allow a certain degree of heterogeneity in panel
data under H0 —the process {Yit Xit } need not be stationary for each i, and
{εit } may have different variances across i In particular, we allow some nonstationary processes. One example is the deterministic trend process (e.g., Kao
and Emerson (2004))
(4.1)
Yit = α + γ1 t + γ2 t 2 + · · · + γp t p + µi + λt + εit This is covered by (2.1) with Xit ≡ [t/T (t/T )p ] and β ≡ (T γ1 T pγp ) Another example is the panel cointegration process (e.g., Phillips and Moon
(1999), Kao and Chiang (2000)):
(4.2)
Yit = α + γZit + µi + λt + εit where Zit = Zit−1 + ηit {ηit } is I(0) for each i and {ηit } may or may not be
correlated with {εit } This process is also covered by (2.1) with Xit ≡ T −1 Zit
and β ≡ T γ
However, Assumption 4 and (2.4) generally imply that both {εit } and {Xit }
are covariance-stationary when β∗ = β under HA This more restrictive condition is needed under HA when we investigate the asymptotic power property
of the proposed tests.
THEOREM 1: Suppose Assumptions 1–5 hold and max1≤i≤n (22Ji )/(n2 +
T ) → 0 as n → ∞ and T → ∞ If {εit } in (2.1) is i.i.d. for each i then
d
d
Ŵ1 → N(0 1) and Ŵ2 → N(0 1).
The large sample properties of the proposed tests are due to the n inde√
pendent copies of serially uncorrelated series {εit } and the existence of a nT
consistent estimator for β under H0 . In fact, the assumption of a linear panel
model is not necessary to obtain the large sample properties. One could extend Assumptions 4 and 5 to cover some nonlinear panel models, with more
tedious
algebra in the proof. It is also possible to relax the condition on the
√
nT -convergence rate of β̂ for β at a cost of strengthening conditions on the
finest scales {Ji }
1530
Y. HONG AND C. KAO
Most remarkably, we permit but do not require Ji → ∞ for any i; all Ji can
be fixed as n T → ∞ under H0 This is in sharp contrast to Lee and Hong
(2001), who require J → ∞ as T → ∞ to achieve asymptotic normality in testing serial correlation for observed raw time-series data. The reason that all Ji
can be fixed is that the additional smoothing provided by n ensures asymptotic
normality of Ŵ1 and Ŵ2 Intuitively, Ŵ1 and Ŵ2 are sums of approximately independent random variables {2πTi Q(fˆi fˆi0 )}ni=1 By the central limit theorem,
they will converge to a normal distribution as n → ∞. This occurs no matter whether Ji → ∞. In the time-series or cross-sectional literature it is often
found that the normal approximation is inadequate for the finite sample distributions of quadratic forms involving smoothed nonparametric estimation.
This is because the asymptotic normality of these quadratic forms requires
the smoothing parameter to grow or vanish at a suitable rate as the sample
size grows and the convergence rate of test statistics delicately depends on the
smoothing parameter. The fact that the asymptotic normality of Ŵ1 and Ŵ2
does not depend on whether Ji → ∞ suggests that asymptotic approximation may work well in the panel context. Indeed, our simulation shows that
Ŵ1 and Ŵ2 perform well in finite samples even when Ji = 0 for all i Most
importantly, the fact that Ji may be fixed for all i allows use of data-driven
methods that may deliver a fixed finest scale under H0 Sensible data-driven
methods have this feature because the optimal finest scale J0 = 0 under H0 We will propose a plug-in method to select a finest scale, which automatically
converges to 0 under H0 and grows to ∞ under HA thus ensuring consistency
against serial correlation of unknown form. Such a data-driven method could
not be used for Lee and Hong’s (2001) test.
Although we require that both n and T grow to ∞, we do not impose
a restrictive relative speed limit between them. Also, no specific estimation
method is required. From the proof of Theorem 1, we find that parameter estimation uncertainty for β has no impact on the null limit distribution
of Ŵ1 and Ŵ2 , no matter whether Xit contains lagged dependent variables
or deterministic/stochastic trending variables. This is in contrast to Durbin
and Watson (1951) and Box and Pierce (1970), whose test statistics or limit
distributions have to be modified when applied to estimated residuals of a
stationary dynamic model. If regressors contain deterministic or stochastic
trending variables, the limit distributions of these tests will become nonstandard (e.g., Kao and Chiang (2000), Kao and Emerson (2004)). Intuitively,
parameter estimation uncertainty for β induces an adjustment of a finite number of degrees of freedom for Ŵ1 and Ŵ2 , but this becomes negligible as
n → ∞
The tests Ŵ1 and Ŵ2 are applicable for both small and large Ji When (and
only when) Ji → ∞ for all i = 1 n we can use the following simplified
1531
SERIAL CORRELATION IN PANEL MODELS
versions of test statistics:
n Ji 2j 2
2
Ji +1
− 1)
i=1 2πTi
j=0
k=1 α̂ijk − R̂i (0)(2
W̃1 =
(4.3)
n
1/2
2 i=1 R̂4i (0)(2Ji +1 − 1)
Ji 2j 2
n J +1
1 2πTi j=0 k=1 âijk − (2 i − 1)
W̃2 = √
(4.4)
2(2Ji +1 − 1)1/2
n i=1
These are the generalizations of Lee and Hong’s (2001) test to estimated residuals of panel models. Theorem 2 shows that they are asymptotically N(0 1)
under H0 if Ji → ∞ for all i.
THEOREM 2: Suppose Assumptions 1–5 hold, 2Ji +1 = ai Tiν for ai ∈ [c C] and
ν ∈ (0 12 ) n/T ν log22 T → 0 n/T 2(2τ−1)−2(2τ−1/2)ν → 0 as n T → ∞ where τ ≥ 32
p
is as in Assumption 2. If {εit } in (2.1) is i.i.d. for each i then W̃1 − Ŵ1 → 0 W̃2 −
p
d
d
W2 → 0, W̃1 → N(0 1), and W̃2 → N(0 1).
Thus, for large (and only large) Ji W̃1 and W̃2 are asymptotically equivalent
to Ŵ1 and Ŵ2 respectively. Note that here, n cannot grow faster than T ν where
ν < 12 .
5. CONSISTENCY
We now show that Ŵ1 and Ŵ2 are consistent against HA . We assume the
following condition.
ASSUMPTION
∞
∞zero-mean stationary
∞6: For2 each i, {vit } isa∞ fourth-order
process with
R
(h)
≤
C
and
h=−∞ i
j=−∞
k=−∞
l=−∞ |κi (j k l)| ≤ C
where κi (j k l) is the fourth-order cumulant of the joint distribution of {vit vit+j vit+k vit+l }
Assumption 6 characterizes temporal dependence of {vit } When {vit } is
Gaussian, the cumulant condition holds trivially because κi (j k l) = 0 for all
j k l ∈ Z If for each i {vit } is a fourth-order stationary linear process with
absolutely summable coefficients and i.i.d. innovations whose fourth order moment exists, the cumulant condition also holds (e.g., Hannan (1970, p. 211)).
More primitive conditions (e.g., strong mixing) could be imposed, but such
primitive conditions would rule out long memory processes. Assumption 6 allows long memory processes I(d) with d < 14 for {vit }.
THEOREM 3: Put nA ≡ #(NA ) and ci ≡ Ti /T where N
A ≡ {i : 0 ≤ i ≤ n
n
Q(fi fi0 ) > 0}. Suppose Assumptions 1–6 hold, (nA T )−1 i=1 2Ji → 0, and
1532
Y. HONG AND C. KAO
Ji → ∞ for all i = 1 n as n T → ∞. Then (a) (nA T )−1 V̂ 1/2 Ŵ1 − n−1
A ×
p
Ji +1
ν
= ai Ti for all i where ai ∈
i∈NA 2πci Q(fi fi0 ) → 0; (b) if in addition 2
p
−1
1−ν/2 −1
[c C] and ν ∈ (0 1) then (nA T
) Ŵ2 − nA i∈NA π(ci /ai )1/2 Q(fi fi0 ) → 0.
As discussed earlier, although serial correlation in {vit } may differ from serial correlation in {εit } H0 holds if and only if {vit } is serially uncorrelated.
Consequently, the index set NA is nonempty under HA , at least for large n It
follows that n−1
A
i∈NA ci Q(fi fi0 ) ≥ c for large n. Then P[Ŵ1 > C(n T )] → 1
and P[Ŵ2 > C(n T )] → 1 under HA for any sequence of constants {C(n T ) =
n
o[nA T/( i=1 2Ji )1/2 ]} Thus, Ŵ1 and Ŵ2 are consistent against HA provided
n
(nA T )−1 i=1 2Ji → 0 and Ji → ∞ for all i. For simplicity, we let Ji → ∞ for
all i to ensure consistency against HA This differs from the case under H0 where Ji can be fixed for all i Our data-driven method below will deliver a
finest scale that automatically converges to 0 under H0 but grows to ∞ with T
under HA n
Under HA Ŵ1 and Ŵ2 diverge to ∞ at the rate of nA T/( i=1 2Ji )1/2 Thus,
the larger the set NA is, the more powerful Ŵ1 and Ŵ2 are. In fact, the power
depends on nA /n the proportion of individuals with serial correlation. For
n
2Ji +1 = ai Tiν , nA T ( i=1 2Ji )1/2 ∝ (nA /n)n1/2 T 1−ν/2 This implies that Ŵ1 and Ŵ2
have asymptotic power 1 against HA even if the proportion nA /n → 0 at a
rate slightly slower than n1/2 T 1−ν/2 For Ŵ1 and Ŵ2 serial correlations from
different individuals never cancel each other out when some individuals have
positive autocorrelations and some have negative autocorrelations. In contrast,
cancellation may occur at least in part for existing tests, leading to low or little
power; see Section 7 for more discussion. We emphasize that the ability of our
tests to detect serial correlation in the presence of substantial inhomogeneity
in serial correlation across i is not due to the use of wavelets, but to the use of
the quadratic form in (3.13). On the other hand, we may extend the adaptive
procedures of Fan (1996) and Spokoiny (1996) to further improve the power
of our tests, as one referee pointed out. We leave this for future research.
Theorems 1 and 3 imply that for large n and T , the negative values of
Ŵ1 and Ŵ2 can occur only under H0 Thus, upper-tailed N(0 1) critical values
should be used.
As noted earlier, Ŵ1 and Ŵ2 are heteroskedasticity-consistent and heteroskedasticity-corrected tests respectively. An interesting question is, which test,
Ŵ1 or Ŵ2 is more powerful? For convenience, we assume 2Ji +1 = ai Tiν for all i
where ai ∈ [c C] and ν ∈ (0 1) and assume a larger ai for processes with
stronger serial correlation in terms of a larger Q(fi fi0 ). With this rule, we
have the following theorem.
THEOREM 4: Suppose Assumptions 1–6 hold, n = γT ς for γ ∈ (0 ∞) and
ς ∈ (0 ∞) and 2Ji +1 = ai Tiν for ai ∈ [c C] and ν ∈ (0 1) If ai is monotonically
SERIAL CORRELATION IN PANEL MODELS
1533
increasing in Q(fi fi0 ) and Ti = T for all i then Ŵ1 is more efficient than Ŵ2 in
terms of Bahadur’s asymptotic efficiency criterion.
Bahadur’s (1960) asymptotic slope criterion is pertinent for power comparison of large sample tests under fixed alternatives. The basic idea is to compare
the logarithms of the asymptotic significance levels (i.e., p-values) of the tests
under a fixed alternative. Bahadur’s asymptotic efficiency is defined as the limit
ratio of the sample sizes required by the two tests under comparison to achieve
the same asymptotic significance level (p-value) under a fixed alternative.
Theorem 4 implies that for hypothesis testing, correcting heteroskedasticity
may give poorer power. This is in contrast to the well-known result that correcting heteroskedasticity leads to more efficient estimation.
Intuitively, for Ŵ2 √
a larger Q(fi fi0 ) is more heavily discounted by Vi0 ∼ 2(2Ji +1 − 1)1/2 when
Ji is larger. Thus, it is less powerful than Ŵ1 which puts uniform weighting on
each Q(fi fi0 ) Of course it is possible that Ŵ1 is asymptotically less powerful
than Ŵ2 as will occur when ai is monotonically decreasing in Q(fi fi0 ) However, sensible data-driven methods usually provide a rule that ai is increasing
in Q(fi fi0 ) When Ji = J for all i Ŵ1 and Ŵ2 may still not be asymptotically
equally efficient, because of heteroskedasticity (σi2 = σ 2 ). We note that the asymptotic power of Ŵ1 and Ŵ2 does not depend on mother wavelet ψ(·). All
wavelets are asymptotically equally efficient by Bahadur’s criterion. This differs from the kernel method, where the choice of a kernel affects the asymptotic power of tests (Hong (1996)).
6. ADAPTIVE CHOICE OF FINEST SCALE
Theorem 1 implies that the choice of Ji is not important for asymptotic
normality of Ŵ1 and Ŵ2 . Both small and large Ji can be used However,
the choice of Ji may have significant impact on the power. Therefore, it is
desirable to choose Ji via suitable data-driven methods. We now develop a
data-driven method to select a finest scale. We first justify the use of a dataˆ For simplicity, we consider a common Jˆ for all i here. We
driven finest scale J.
ˆ
use Ŵc (J) and Ŵc (J) to denote the Ŵc tests using Jˆ and J respectively, where
c = 1 2.
We impose a condition on the smoothness of ψ̂(·) at 0.
ASSUMPTION 7: |ψ̂(z)| ≤ C|z|q for some q ∈ (0 ∞).
THEOREM 5: Suppose Assumptions 1–5 and 7 hold, and Jˆ is a data-driven
ˆ
finest scale with 2J /2J = 1 + oP (2−J/2 ) where J is a nonstochastic finest scale
2J
2
such that 2 /(n + T ) → 0 as n → ∞ and T → ∞ If {εit } in (2.1) is i.i.d.
p
p
d
ˆ − W1 (J) →
ˆ − W2 (J) →
ˆ →
0 W2 (J)
0Ŵ1 (J)
N(0 1), and
for each i, then W1 (J)
ˆ → N(0 1).
Ŵ2 (J)
d
1534
Y. HONG AND C. KAO
Thus, the use of Jˆ rather than J has no impact on the limit distribution of
ˆ and Ŵ2 (J)
ˆ provided that Jˆ converges to J at a suitable rate. The rate
Ŵ1 (J)
J
Jˆ
condition 2 /2 − 1 = oP (2−J/2 ) is mild. If 2J ∝ T 1/5 for example, we require
ˆ
2J /2J = 1 + oP (T −1/10 ) If J is fixed (e.g., J = 0), which occurs under H0 for our
p
ˆ
data-driven method below, the condition becomes 2J /2J → 1
So far very few data-driven methods to choose J are available in the literature. Walter (1994) and Hall and Patil (1996) consider some data-driven
methods in related but different contexts. They cannot be applied directly to
our tests. We now develop a data-driven Jˆ that can satisfy the condition of Theorem 5. We first derive the average asymptotic IMSE formula for {fˆi (·)}ni=1 which was not available in the literature. We impose an additional condition
on {vit }.
ASSUMPTION 8:
Assumption 7.
∞
h=−∞
|h|q |Ri (h)| ≤ C for all i, where q ∈ [1 ∞) is as in
This characterizes the smoothness of fi (·) It rules out long memory
processes. Under Assumption 8, we have a well-defined qth order generalized
spectral derivative of fi (ω):
(6.1)
f
(q)
i
(ω) ≡ (2π)
−1
∞
|h|q Ri (ω)e−ihω ω ∈ [−π π]
h=−∞
We also define a measure λq ∈ (0 ∞) of the smoothness of ψ̂(·) at 0:
(6.2)
λq ≡ −
(2π)2q+1
|ψ̂(z)|2
lim
1 − 2−2q z→0 |z|2q
For the Franklin wavelet (3.3), q = 2; for the second-order spline wavelet (3.4),
q = 3.
To state the next result, we define a pseudo spectral density estimator f¯i (·)
Ti
for fi (·) that is based on the unobservable series {vit }t=1
; namely,
Ji
2
j
(6.3)
f¯i (ω) ≡ (2π)−1 R̄i (0) +
ᾱijk Ψjk (ω)
ω ∈ [−π π]
j=1 k=1
where R̄i (h) ≡ Ti−1
Ti
t=h+1
vit vit−|h| and ᾱijk ≡ (2π)−1/2
Ti −1
h=1−Ti
R̄i (h)Ψ̂jk∗ (h).
THEOREM 6: Suppose Assumptions 1–8 hold, λq ∈ (0 ∞) Ji → ∞, 2Ji/Ti → 0
as Ti → ∞. Then (a) for each i Q(fˆi fi ) = Q(f¯i fi ) + oP (2Ji /Ti + 2−2qJi )
1535
SERIAL CORRELATION IN PANEL MODELS
and
EQ(f¯i fi ) =
π
2
(q)
2Ji +1 π 2
fi (ω) dω + 2−2q(Ji +1) λq
f (ω) dω
Ti −π
−π
J
−2qJi
i
+ o 2 /Ti + 2
(b) If in addition Ji = J for all i and Ti /T = ci then n−1
n
n−1 i=1 Q(f¯i fi ) + oP (2J /T + 2−2qJ ) and
n
−1
n
i=1
2J+1 −1 −1
EQ(f¯i fi ) =
ci
n
T
i=1
n
+ 2−2q(J+1) λq n−1
n
i=1
Q(fˆi fi ) =
π
−π
fi2 (ω) dω
n π
2
(q)
fi (ω) dω
i=1
−π
+ o(2J /T + 2−2qJ )
Theorem 6(a) gives the asymptotic IMSE of f¯i (·) and Theorem 6(b) gives
the average asymptotic IMSE of {f¯i (·)}ni=1 . They imply that the optimal convern
gence rates of Q(fˆi fi ) and n−1 i=1 Q(fˆi fi ) are the same as those of Q(f¯i fi )
n
and n−1 i=1 Q(f¯i fi ) respectively. Parameter estimation uncertainty in β̂ has
n
no impact on the optimal convergence rates of Q(fˆi fi ) and n−1 i=1 Q(fˆi fi )
The optimal finest scale J0 that minimizes the average asymptotic IMSE
of {f¯i (·)}ni=1 is
2J0 +1 = [2qλq ξ0 (q)T ]1/(2q+1) π
n π (q)
n
where ξ0 (q) ≡ i=1 −π [fi (ω)]2 dω/ i=1 ci−1 −π fi2 (ω) dω This is infeasible because ξ0 (q) is unknown under HA However we can use some estimator ξ̂0 (q) to obtain a plug-in finest scale Jˆ0 :
(6.4)
(6.5)
ˆ
2J0 +1 ≡ [2qλq ξ̂0 (q)T ]1/(2q+1) Because Jˆ0 is a nonnegative integer, we should use
1
ˆ
J0 ≡ max
(6.6)
log2 2qλq ξ̂0 (q)T − 1 0 2q + 1
where the square bracket denotes the integer part. We impose the following
condition on ξ̂0 (q).
ASSUMPTION 9: ξ̂0 (q) − ζ0 (q) = oP (T −δ ) for some constant ζ0 (q) where δ =
1/2(2q + 1) if ζ0 (q) ∈ [c C] and δ = 1/(2q + 1) if ζ0 (q) = 0.
1536
Y. HONG AND C. KAO
Note that the condition on ξ̂0 (q) is more stringent when ζ0 (q) = 0 than
when ζ0 (q) = 0 but for both cases the conditions are mild We do not require
p lim ξ̂0 (2) ≡ ζ0 (q) = ξ0 (q) where ξ0 (q) is as in (6.4) When (and only when)
ζ0 (q) = ξ0 (q) Jˆ0 in (6.6) will converge to the optimal J0 in (6.4).
COROLLARY 1: Suppose Assumptions 1–9 hold and Jˆ0 is given as in (6.6).
d
d
If {εit } in (2.1) is i.i.d. for each i, then Ŵ1 (Jˆ0 ) → N(0 1) and Ŵ2 (Jˆ0 ) → N(0 1).
We can use parametric or nonparametric (e.g., local linear smoothing) methods for estimator ξ̂0 (q). The former generally deliver a suboptimal finest scale,
but have less variation in finite samples. The latter will deliver an asymptotically optimal finest scale, but are subject to substantial variation. There are
also some methods (e.g., cross-validation and AIC) in the literature for selecting the truncation parameter. There is usually a trade-off between Type I and
Type II errors in choosing a specific method.
To obtain reasonable power while having a proper rejection probability under H0 for sample sizes often encountered in economics, we use a parametric
AR(pi ) model for each i:
(6.7)
v̂it = γi0 +
pi
γih v̂it−h + εit
(t = 1 Ti ; i = 1 n)
h=1
where v̂it ≡ 0 if t ≤ 0 In practice, one can use AIC or BIC to select pi for each i
Suppose γ̂i ≡ (γ̂i0 γ̂i1 γ̂ip ) is the OLS estimator of γi ≡ (γi0 γi1 γip ) We consider q = 2; an example is the Franklin wavelet in (3.3), whose λ2 =
π 4 /45 We have
2
π
n π
n
d2 ˆ
f (ω) dω
fˆi2 (ω) dω
(6.8)
(Ti /T )
ξ̂0 (2) ≡
2 i
dω
−π
−π
i=1
i=1
pi
where fˆi (ω) ≡ (2π)−1 |1 − h=1
γ̂ih e−ihω |−2 We note that ξ̂0 (2) satisfies Assumption 9 with q = 2 because for parametric AR(pi ) approximations, ξ̂0 (2) −
ζ0 (2) = OP ((nT )−1/2 ).
The performance of the plug-in method in (6.6) relies on the specification
in (6.7). To further improve the power, one could also consider a data-driven,
individual-specific Jˆi using the IMSE criterion of fi (·) in Theorem 6(a) Such
individual-specific Jˆi may more effectively capture spatial nonhomogeneity in
the degree of serial correlation across i However, they may have wide variations across i, leading to large Type I errors for the tests. A compromise is to
develop a data-driven Jˆc where c is an index for some suitable groups such as
regions and sectors where all individuals in the same group will have the same
finest scale.
SERIAL CORRELATION IN PANEL MODELS
1537
In fact, the IMSE criterion is more suitable for estimation than for testing.
A better criterion for testing is to maximize power or trade off between level
distortion and power improvement. This will, however, require higher-order
asymptotic analysis for our tests, which is beyond the scope of this paper and
should be pursued in future work. Simulation studies below show that the finest
scale chosen via (6.6)–(6.8) gives reasonable rejection probabilities under H0
and gives robust and good power, particularly when the spectrum has distinct
peaks or kinks.
7. MONTE CARLO EXPERIMENT
We now compare the performance of Ŵ1 and Ŵ2 with three existing tests
for serial correlation—Bhargava, Franzini, and Narendranathan’s (1982; BFN)
Durbin–Watson type test, Baltagi and Li’s (1995; BL) LM test,
(7.1)
n T
2
n
T
nT 2 2
v̂it v̂it−1
v̂it BL =
T − 1 i=1 t=1
i=1 t=1
and Bera, Sosa-Escudero, and Yoon’s (2001; BSY) modified LM test:
(7.2)
2 2
Ã
BSY = nT B̃ +
(T − 1) 1 −
T
T
2
where v̂it is the within residual,
n T
n
à ≡ 1 −
ũi IT ũi
ũ2it i=1
n
B̃ ≡
T
i=1 t=1
ũit ũit−1
i=1 t=1
n
T
ũ2it i=1 t=1
IT is a matrix of ones, and ũi ≡ (ũi1 ũiT ) is the OLS residual without random effects. All these tests consider balanced panels. Both BL and BSY have
an asymptotic χ21 distribution under H0 . In contrast, BFN converges to 2 under H0 , a degenerate distribution. Bhargava, Franzini, and Narendranathan
(1982, p. 436) suggest using a critical value of 2 at the 5% level. We find
that BFN rejects H0 up to 677%, 669%, and 648% at the 5% level, when
(n T ) = (25 32), (50 64), and (100 128) respectively. It seems that this test
cannot be used, so we drop it from comparison. For Ŵ1 and Ŵ2 because the
choice of ψ(·) is not important, we only use the Franklin wavelet in (3.3).
To examine the impact of the choice of J we consider J = 0 1 and the datadriven Jˆ0 in (6.6). We also compare Ŵ1 and Ŵ2 with the panel versions of
Hong’s (1996) kernel-based tests, K̂1 and K̂2 , which are obtained by gener-
1538
Y. HONG AND C. KAO
alizing Hong’s (1996) kernel method to model (2.1). They are based on an
individual-specific kernel spectral density estimator. We use the Daniell kernel k(z) = sin(πz)/πz z ∈ R which has the optimal power over a class of
kernels. We choose a data-driven bandwidth using a plug-in method similar to
that for Jˆ0 We consider the following two data generating processes (DGP): (a) DGP1 a static panel model: Yit = 5 + 5Xit + µi + εit , Xit = 5Xit−1 + ηit ηit ∼
i.i.d. U[−5 5] and (b) DGP2 a dynamic panel model: Yit = 5 + 5Yit−1 + µi +
εit For both DGPs, µi ∼ i.i.d. N(0 σµ2 ) Let τ measure the relative strength of
random effects (no random effect when τ = 0) and we choose a variety of τ
as in Baltagi, Chang, and Li (1992). To examine the probability of rejecting a
correct H0 , we set εit = zit where zit ∼ i.i.d. N(0 1) To examine the power, we
consider the following error processes:
AR(1) Alternatives:

i = 1 n
AR(1)a : εit = 2εit−1 + zit 



 AR(1)b : εit = −2εit−1 + zit i = 1 n
(7.3)

i = 1 n2 
 AR(1)c : ε = 2εit−1 + zit 

it
−2εit−1 + zit i = n2 + 1 n
ARMA(12 4) Alternatives:

i = 1 n
ARMA(12 4)a: εit = −3εit−12 + zit + zit−4 



b
 ARMA(12 4) : εit = 3εit−12 + zit − zit−4 i = 1 n
(7.4)

−3εit−12 + zit + zit−4 i = 1 n2 

c

:
ε
=
ARMA(12
4)

it
3εit−12 + zit − zit−4 i = n2 + 1 n
AR(1)a and AR(1)b are the full positive and negative AR(1) respectively.
We expect BL and BSY to have optimal power against them. Wavelet tests
have no advantages because these alternatives have a relatively flat spectrum.
AR(1)c is a mixed AR(1), where the first half individuals have a positive AR
coefficient, and the second half have a negative AR coefficient. On the other
hand, ARMA(12 4) can arise from monthly data; it has four distinct spectral peaks.
The top panel of Table I reports rejection probabilities under H0 with τ = 4
for DGP1 When n < T , the rejection probabilities of BSY are reasonable and
the best among all the tests. BL overrejects H0 , while Ŵ1 , Ŵ2 , K̂1 , and K̂2 underreject H0 but not excessively. For other values of τ (not reported), the rejection
probability of BSY is sensitive to the choice of τ displaying severe overrejections when τ is large. BL still overrejects H0 for all τ The Ŵ1 , Ŵ2 K̂1 , and K̂2
tests are robust to the choice of τ. The rejection probabilities of Ŵ1 and Ŵ2
TABLE I
PERCENTAGE OF REJECTIONS UNDER THE NULL HYPOTHESIS IN THE STATIC PANEL MODEL
(n, T )
Level
(5, 8)
10%
5%
(10, 16)
10%
5%
(25, 32)
10%
5%
(50, 64)
10%
5%
(5, 5)
10%
(10, 10)
5%
10%
5%
(25, 25)
10%
5%
(50, 50)
10%
5%
(8, 5)
10%
5%
(16, 10)
10%
5%
(32, 25)
10%
5%
(64, 50)
10%
5%
Asymptotic Critical Values
Bootstrapped Critical Values
Ŵ1 (0)
Ŵ1 (1)
Ŵ1 (Jˆ0 )
Ŵ2 (0)
Ŵ2 (1)
Ŵ2 (Jˆ0 )
K̂1
K̂2
BL
BSY
89 46
86 40
89 46
127 58
122 67
127 58
129 61
209 113
50 15
37 12
83
89
83
109
112
109
64
106
18
17
33
44
33
53
59
53
33
68
4
4
105
94
97
115
106
111
92
113
69
47
53
48
40
55
63
59
47
50
27
17
109
101
105
119
111
108
117
119
94
92
58
49
48
56
58
54
59
62
40
50
122
124
122
160
161
160
225
369
106
14
53
55
53
102
93
102
119
239
27
0
98 47
99 51
98 47
147 82
155 91
147 82
93 46
211 114
8
2
4
0
95
91
95
123
128
123
92
135
36
31
39
49
39
64
76
64
39
77
15
15
104
95
93
120
117
121
99
109
81
84
38
36
42
49
50
52
37
61
41
34
78
88
98
119
118
134
73
109
81
73
33
36
50
43
66
80
34
70
40
54
104
92
92
117
100
100
94
108
86
108
38
54
54
48
56
56
50
62
38
59
103
102
100
106
104
94
101
106
115
107
48
52
58
55
38
38
57
53
63
55
106
74
84
113
92
94
104
105
116
101
59
44
48
58
44
54
59
50
57
53
1539
Note: (a) DGP: Yit = 5 + 5Xit + µi + εit Xit = 5Xit−1 + ηit ηit ∼ i.i.d. U[−5 5] µi ∼ i.i.d. N(0 4) and εit ∼ i.i.d. N(0 1). (b) Ŵ1 , heteroskedasticity-consistent Franklin
wavelet-based test; Ŵ2 , heteroskedasticity-corrected Franklin wavelet-based test; Jˆ0 data-driven finest scale. (c) K̂1 , heteroskedasticity-consistent Daniell kernel-based test;
K̂2 heteroskedasticity-corrected Daniell kernel-based test. (d) BL, Baltagi–Li test; BSY, Bera, Sosa-Escudero, and Yoon test. (e) 1000 simulation replications; 500 bootstrap
replications.
SERIAL CORRELATION IN PANEL MODELS
Ŵ1 (0)
25 16 45 28 60 31 66 29
6
0 31 17 48 26 66 34
4
0 26 18 49 25 63 30
Ŵ1 (1)
18 13 33 20 51 28 58 27
3
0 20 13 37 21 58 35
2
1 17
7 35 12 48 21
Ŵ1 (Jˆ0 ) 18 13 33 19 63 40 79 40
6
0 20 13 45 25 78 43
4
1 17
7 41 17 69 38
Ŵ2 (0)
14
4 45 25 57 30 68 34
1
0 24 11 43 22 66 35
1
0 14
7 43 28 59 29
Ŵ2 (1)
10
4 28 14 46 19 60 30
0
0
8
3 29 14 54 26
0
0
7
3 24 17 42 22
Ŵ2 (Jˆ0 ) 10
4 30 11 47 23 82 37
1
0
8
3 30 16 72 36
1
0
7
3 26 13 63 37
K̂1
25 15 36 17 66 39 72 39
9
0 23 13 50 29 75 40
4
2 16
8 42 21 67 35
K̂2
14
4 28 12 58 26 80 38
1
0 12
2 33 20 75 34
1
0
7
2 31 17 71 32
BL
305 199 226 149 232 146 225 145 453 324 352 239 302 208 282 192 578 463 480 354 347 245 353 238
BSY
90 30 57 21 67 18 77 22 183 93 135 51 99 34 73 28 319 207 220 103 142 59 114 48
1540
Y. HONG AND C. KAO
are better when a smaller J or data-driven Jˆ0 is used. When n = T Ŵ1 , Ŵ2 K̂1 , and K̂2 all underreject H0 , but they improve when n = T increase. On the
other hand, BL overrejects severely while BSY performs well. When n > T
again, BSY has the best rejection probabilities if T > 10 but overrejects if T is
small BL still overrejects H0 for all sample sizes. The Ŵ1 , Ŵ2 K̂1 , and K̂2 tests
all underreject H0 but they are substantially improved when T > 25, especially
when data-driven Jˆ0 or J = 0 is used. The rejection probabilities of Ŵ1 and Ŵ2
are similar to those of K̂1 and K̂2 in all cases.
Table I indicates that when using asymptotic critical values, most tests do
not perform well when n T < 32. For small samples, we suggest using wild
bootstrap (e.g., Härdle and Mammen (1993), Davidson and Flachaire (2001),
Horowitz (2001)) as an alternative to asymptotic approximation. The bottom
panel of Table I reports bootstrap rejection probabilities, which are based
on 1000 replications and 500 bootstrap resamples. The Ŵ1 and Ŵ2 tests have
reasonable rejection probabilities in small samples for all cases (n < T , n = T ,
and n > T ). It appears that wild bootstrap can remedy the underrejection of
Ŵ1 and Ŵ2 using asymptotic critical values, especially when the sample size is
as small as 5 or 8. However, K̂1 and K̂2 overreject when (n T ) = (5 5). BL and
BSY perform better using wild bootstrap, but they still tend to underreject.
Table II reports the rejection probabilities under DGP2 , a dynamic panel
process. Again Ŵ1 , Ŵ2 K̂1 , and K̂2 all underreject using asymptotic critical values, but they all perform well using wild bootstrap, though they still underreject
compared to Table I. Unlike under DGP1 (a static panel process), BSY and BL
perform poorly using either asymptotic or bootstrap critical values, though BL
performs slightly better than BSY when n ≤ T . BSY overrejects for almost all
cases using bootstrap critical values.
The top panel of Table III first reports the Type I error corrected powers against AR(1) alternatives, with τ = 4 using empirical critical values,
which provides a fair comparison. Because empirical critical values are not
observable in practice, we also report power using bootstrap critical values. Under AR(1)a BSY is the most powerful, followed by BL. This is expected because both BSY and BL are optimal against AR(1). The Ŵ1 , Ŵ2 K̂1 , and K̂2 tests have nontrivial but substantially lower power when sample
sizes are small. This is because AR(1)a has a relatively flat spectrum and
the advantage of consistent testing is not displayed. Under AR(1)b , BL becomes the most powerful. Somewhat surprisingly, Ŵ1 , Ŵ2 K̂1 , and K̂2 have
rather high power and dominate BSY, perhaps due to the fact that a negative
AR(1) has a less smooth spectrum than a positive AR(1). For example, with
(n T ) = (10 16), the power of BSY is 63% while those of BL and Ŵ1 (0) are
714% and 471% respectively. Interestingly, both BSY and BL fail to detect
AR(1)c the mixed model. In contrast, Ŵ1 , Ŵ2 K̂1 , and K̂2 are very powerful
against AR(1)c , indicating that wavelet and kernel tests are rather effective in
TABLE II
PERCENTAGE OF REJECTIONS UNDER THE NULL HYPOTHESIS IN THE DYNAMIC PANEL MODEL
(n, T )
Level
(5, 8)
10%
5%
(10, 16)
10%
5%
(25, 32)
10%
5%
(50, 64)
10%
5%
(5, 5)
10%
5%
(10, 10)
10%
5%
(25, 25)
10%
5%
(50, 50)
10%
5%
(8, 5)
10%
5%
(16, 10)
10%
5%
(32, 25)
10%
5%
(64, 50)
10%
5%
Asymptotic Critical Values
2 13 11 46 23
2 11
4 25 13
2 11
4 28 11
0 10
3 34 20
0
4
2 26 13
0
4
2 32 11
3 16
7 37 22
0
8
2 34 21
98 48 18 21
3
62 298 198 999 997
Ŵ1 (0)
35
Ŵ1 (1)
20
Ŵ1 (Jˆ0 ) 35
Ŵ2 (0)
40
Ŵ2 (1)
35
Ŵ2 (Jˆ0 ) 40
K̂1
60
K̂2
35
BL
120
BSY
155
15 45
20 60
15 45
15 65
10 90
15 65
10 65
5 65
45 20
80 100
62
41
63
55
35
59
70
67
3
0
29
0
0
5
24
0
0
4
33
0
0
4
24
0
0
1
20
0
0
0
27
0
0
0
38
0
0
3
28
0
0
1
0 402 314 128
0 240 163 142
55 75
45 85
55 90
60 90
65 90
60 90
35 90
60 85
0
5
90 205
45 45 30 70
50 40 25 70
60 45 30 70
45 80 45 65
40 65 40 40
40 80 45 65
50 40 25 55
45 70 40 55
0 240 160 110
90 220 130 110
0 27 19 38 12
4
0 26 18 49 25 63 30
3 27 11 20
7
2
1 17
7 35 12 48 21
3 23 15 35 18
4
1 17
7 41 17 69 38
0 23 14 30 15
1
0 14
7 43 28 59 29
0 17
4 18
5
0
0
7
3 24 17 42 22
0 12
7 28
9
1
0
7
3 26 13 63 37
2 28 14 32 15
4
2 16
8 42 21 67 35
0 18
9 33 16
1
0
7
2 31 17 71 32
58 28 10 13
2 567 442 177 100 19
5
6
2
73 969 931 949 949 233 160 145 81 994 986 590 590
Bootstrapped Critical Values
30
40
30
25
25
25
30
35
15
25
120
85
120
95
115
95
85
75
25
150
20 55 25
30 80 35
20 55 25
10 75 25
20 100 40
10 75 40
20 70 35
15 85 30
60 30 15
75 210 110
110 65 30
5 40
85 30 20
5 40
100 55 30
5 40
95 50 50 30 50
100 40 25 15 45
120 50 50 30 50
105 55 20
5 30
110 65 25 15 35
15
0 335 205 140
240 120 195 110 115
20 55 35
15 65 35
20 55 35
15 65 30
20 65 30
15 65 30
15 70 30
15 55 30
60 35 15
50 230 135
80 40
110 45
90 45
105 60
95 65
100 65
75 50
110 65
5
0
305 160
1541
Notes: (a) DGP: Yit = 5 + 5Yit−1 + µi + εit µi ∼ i.i.d. N(0 4) and εit ∼ i.i.d. N(0 1). (b) Ŵ1 , heteroskedasticity-consistent Franklin wavelet-based test;
Ŵ2 , heteroskedasticity-corrected Franklin wavelet-based test; Jˆ0 data-driven finest scale. (c) K̂1 , heteroskedasticity-consistent Daniell kernel-based test; K̂2 heteroskedasticitycorrected Daniell kernel-based test. (d) BL, Baltagi–Li test; BSY, Bera, Sosa-Escudero, and Yoon test. (e) 1000 simulation replications; 500 bootstrap replications.
SERIAL CORRELATION IN PANEL MODELS
Ŵ1 (0)
5
Ŵ1 (1)
4
Ŵ1 (Jˆ0 )
5
Ŵ2 (0)
1
Ŵ2 (1)
0
Ŵ2 (Jˆ0 )
1
K̂1
6
K̂2
1
BL
166
BSY
129
TABLE III
Model 1
(n, T )
Level
(5, 8)
10%
5%
(10, 16)
10%
5%
Model 2
(25, 32)
10%
5%
(50, 64)
10%
5%
(5, 8)
10%
(10, 16)
5%
10%
5%
Model 3
(25, 32)
10%
1542
PERCENTAGE OF REJECTIONS UNDER THE AR(1) ALTERNATIVE IN THE STATIC PANEL MODEL
(50, 64)
5%
(5, 8)
(10, 16)
(25, 32)
(50, 64)
10%
5%
10%
5%
10%
5%
10%
5%
10%
5%
999
999
999
999
999
999
999
999
999
999
999
999
999
999
999
999
999
999
999
999
167
160
161
143
154
155
164
149
177
111
105
106
105
72
72
72
103
81
83
62
386
299
257
344
238
221
346
287
116
87
252
185
159
197
151
127
242
69
610
45
933
837
800
937
800
758
879
866
174
107
887
736
678
883
681
666
807
775
124
60
999
999
999
999
999
999
999
999
139
126
999
999
999
999
999
999
999
999
74
63
995
995
995
995
995
995
995
995
995
995
125
120
125
140
145
140
130
45
70
65
75
75
75
65
50
65
65
70
25
65
420
270
420
355
255
355
350
300
100
60
255
170
255
240
160
240
195
205
35
55
915
795
825
915
770
800
845
865
175
20
825
700
710
830
675
715
785
780
115
20
995
995
995
995
995
995
995
995
115
80
995
995
995
995
995
995
995
995
80
75
Empirical Critical Values
144 61 748
99 58 554
115 69 564
93 35 683
72 40 462
102 49 477
157 96 720
116 58 653
199 119 978
748 599 999
623
424
417
537
333
373
600
511
960
999
999
999
999
999
999
999
999
999
999
999
999
999
998
999
999
998
999
999
999
999
286
247
246
263
245
245
270
265
401
79
198
163
160
164
140
140
179
163
243
52
560
370
410
515
350
340
555
530
905
995
995
995
995
995
995
995
995
995
995
995
995
995
995
995
995
995
995
995
995
995
215
180
215
285
245
285
220
295
160
40
145
100
145
170
165
170
150
170
80
40
608
498
369
592
470
366
508
476
830
95
471
356
283
422
352
257
392
354
714
63
996
554
902
992
462
883
949
946
999
708
979
424
828
977
333
840
908
905
999
592
Bootstrapped Critical Values
Ŵ1 (0) 25 20 90 30
Ŵ1 (1) 25 15 75 40
Ŵ1 (Jˆ0 ) 25 20 75 40
Ŵ2 (0) 40
5 65 40
Ŵ2 (1) 35 15 60 30
Ŵ2 (Jˆ0 ) 40
5 60 30
K̂1
30 20 90 40
K̂2
50 10 90 45
BL
15
0 110 40
BSY
205 195 670 665
700
495
535
645
445
480
700
700
950
995
620
490
960
585
485
955
450
510
715
80
465
335
920
470
335
930
340
345
580
80
995
930
995
985
930
995
995
995
995
995
970
880
995
970
885
995
995
995
995
995
995
995
995
995
995
995
995
995
995
995
Notes: (a) DGP: Yit = 5 + 5Xit + µi + εit Xit = 5Xit−1 + ηit ηit ∼ i.i.d. U[−5 5] µi ∼ i.i.d. N(0 4) and Model 1: εit = 2εit−1 + zit , i = 1 n; Model 2: εit =
−2εit−1 + zit , i = 1 n; and Model 3: εit = 2εit−1 + zit , i = 1 n/2 and εit = −2εit−1 + zit i = n/2 + 1 n where zit ∼ i.i.d. N(0 1). (b) Ŵ1 , heteroskedasticityconsistent Franklin wavelet-based test; Ŵ2 , heteroskedasticity-corrected Franklin wavelet-based test; Jˆ0 data-driven finest scale. (c) K̂1 , heteroskedasticity-consistent Daniell
kernel-based test; K̂2 heteroskedasticity-corrected Daniell kernel-based test. (d) BL, Baltagi–Li test; BSY, Bera, Sosa-Escudero, and Yoon test. (e) 200 simulation replications;
200 bootstrap replications.
Y. HONG AND C. KAO
Ŵ1 (0) 36 14
Ŵ1 (1) 43 14
Ŵ1 (Jˆ0 ) 42 14
Ŵ2 (0) 36
9
Ŵ2 (1) 56 14
Ŵ2 (Jˆ0 ) 56 14
K̂1
47 16
K̂2
38 15
BL
35 11
BSY
307 209
SERIAL CORRELATION IN PANEL MODELS
1543
capturing inhomogeneous serial correlations across individuals. The bottom
panel of Table III reports bootstrap power. Due to the extensive computation
involved, we use 200 bootstrap resamples and 200 replications for bootstrap
power. Again, BSY is the best for AR(1)a and BL, Ŵ1 , Ŵ2 K̂1 , and K̂2 have
low power when T < 32. We also observe that Ŵ1 and Ŵ2 are much more
powerful than K̂1 and K̂2 under AR(1)b for (n T ) = (10 16) though the advantages diminish as the sample sizes increase. For all AR(1) alternatives, the
choice of J has significant impact on Ŵ1 and Ŵ2 and J = 0 gives the best power
for Ŵ1 and Ŵ2 against various AR(1) alternatives The data-driven Jˆ0 delivers
reasonable and robust power in all cases.
The top panel of Table IV reports the Type I error corrected powers against
ARMA(12 4) using empirical critical values. All tests have no power when
the sample size is (10 16) since the effective sample size in fact is (10 4).
This is because we remove the first 12 time series observations for each i.
Hence the effective samples for the results reported are (10 4), (25 20),
(50 52). Ŵ1 (Jˆ0 ) and Ŵ2 (Jˆ0 ) have the best power and dominate K̂1 , K̂2 BL,
and BSY against all three ARMA(12 4). This is apparently due to the fact
that ARMA(12 4) has four sharp spectral peaks, which can be more effectively captured by wavelets than kernels. This confirms our prediction that
wavelet-based tests are powerful in capturing spectral modes/peaks in small
and finite samples. The bottom panel of Table IV reports bootstrap powers.
Both Ŵ1 (Jˆ0 ) and Ŵ2 (Jˆ0 ) perform the best and clearly dominate K̂1 and K̂2
for most cases. We note that the clear dominance of Ŵ1 and Ŵ2 over K̂1 and
K̂2 is reduced if bootstrap rather than empirical critical values are used. The
choice of J has significant impact on the power either using empirical or bootstrap critical values of Ŵ1 and Ŵ2 Data-driven Jˆ0 gives better power than
J = 0 1 Apparently due to the seasonal patterns of ARMA(12 4), the choice
of J = 0 1 yields little or no power for Ŵ1 and Ŵ2 against ARMA(12 4)b and
ARMA(12 4)c . In contrast, Jˆ0 is able to adapt to the unknown serial correlation pattern and gives robust and high power. This highlights the value of our
data-driven finest scale Jˆ0 .
We also conduct a simulation study on the power under a dynamic panel
model (DGP2 ) which we do not report in the paper for space. The relative
ranking between our tests and other tests under a dynamic model remains
similar to that under a static model, and in fact the dominance of our waveletbased tests over the kernel-based tests against ARMA(12 4) error alternatives becomes more striking. For example, the Type I error corrected powers
of Ŵ1 , Ŵ2 K̂1 , and K̂2 are 673%, 828%, 457%, and 583% respectively when
(n T ) = (10 16). On the other hand, the relative ranking between BL and
BSY is reversed: unlike under a static model, BSY now has poor power while
BL is most powerful against a positive or negative (but not mixed) AR(1) error
alternative.
TABLE IV
Model 1
(n, T )
Level
(10, 16)
10%
(25, 32)
5%
10%
5%
Model 2
(50, 64)
10%
5%
(10, 16)
10%
(25, 32)
5%
10%
5%
1544
PERCENTAGE OF REJECTIONS UNDER THE ARMA(12, 4) ALTERNATIVE IN THE STATIC PANEL MODEL
Model 3
(50, 64)
10%
5%
(10, 16)
10%
(25, 32)
5%
(50, 64)
10%
5%
10%
5%
Empirical Critical Values
0
0
10
0
0
0
0
0
27
0
0
0
0
0
0
0
0
0
0
0
181
463
432
161
420
348
350
277
291
95
119
466
299
95
301
262
241
191
177
47
418
915
968
442
913
968
930
941
194
66
334
841
952
314
841
956
903
915
123
29
0
0
1
0
0
0
0
0
20
0
0
0
0
0
0
0
0
0
0
0
38
1
344
27
0
165
116
73
86
339
10
0
228
7
81
289
59
32
50
237
73
0
955
57
0
955
958
961
62
229
39
0
955
28
0
944
950
944
35
148
0
0
0
0
0
0
0
0
20
0
0
0
0
0
0
0
0
0
0
0
98
108
371
79
81
289
230
163
162
210
57
60
258
40
35
196
240
102
98
124
220
293
943
229
260
944
948
944
147
171
146
197
936
132
201
939
937
939
72
103
45
0
970
45
0
970
975
975
30
75
40
20
40
100
80
100
15
60
10
5
10
5
10
55
25
55
5
20
5
5
135
115
340
130
160
360
280
370
80
15
55
70
255
60
95
275
195
240
35
15
290
335
945
225
300
945
955
955
125
50
145
235
940
150
215
925
940
935
65
50
Bootstrapped Critical Values
Ŵ1 (0)
Ŵ1 (1)
Ŵ1 (Jˆ0 )
Ŵ2 (0)
Ŵ2 (1)
Ŵ2 (Jˆ0 )
K̂1
K̂2
BL
BSY
30
20
30
55
35
55
15
30
10
5
5
0
5
30
10
30
0
10
5
5
270
570
550
275
610
605
470
505
170
0
165
425
425
180
480
455
335
375
85
0
395
925
965
420
935
980
920
925
165
10
300
855
950
300
870
960
900
895
105
5
30
20
30
90
65
90
15
55
15
10
10
5
10
45
35
45
0
30
5
10
60
0
160
65
0
155
210
230
45
50
10
0
90
35
0
105
125
155
15
50
115
0
975
80
0
970
975
975
70
85
Notes: (a) DGP: Yit = 5 + 5Xit + µi + εit Xit = 5Xit−1 + ηit ηit ∼ i.i.d. U[−5 5] µi ∼ i.i.d. N(0 4) and Model 1: εit = −3εit−2 + zit + zit−4 , i = 1 n; Model 2:
εit = 3εit−2 + zit + zit−4 , i = 1 n; and Model 3: εit = −3εit−2 + zit + zit−4 , i = 1 n/2 and εit = 3εit−2 + zit + zit−4 , i = n/2 + 1 n where zit ∼ i.i.d. N(0 1).
(b) Ŵ1 , heteroskedasticity-consistent Franklin wavelet-based test; Ŵ2 , heteroskedasticity-corrected Franklin wavelet-based test; Jˆ0 data-driven finest scale. (c) K̂1 ,
heteroskedasticity-consistent Daniell kernel-based test; K̂2 heteroskedasticity-corrected Daniell kernel-based test. (d) BL, Baltagi–Li test; BSY, Bera, Sosa-Escudero, and Yoon
test. (e) 200 simulation replications; 200 bootstrap replications.
Y. HONG AND C. KAO
Ŵ1 (0)
Ŵ1 (1)
Ŵ1 (Jˆ0 )
Ŵ2 (0)
Ŵ2 (1)
Ŵ2 (Jˆ0 )
K̂1
K̂2
BL
BSY
SERIAL CORRELATION IN PANEL MODELS
1545
8. CONCLUSION
We have proposed a class of generally applicable wavelet-based consistent
tests for serial correlation in static and dynamic panel models. Wavelets are
powerful for detecting serial correlation where the spectrum has peaks or
kinks. The new tests have a convenient limit N(0 1) distribution, which is not
affected by parameter estimation uncertainty, even if regressors contain lagged
dependent variables or deterministic/stochastic trending variables. Our tests
do not require an alternative model, and are consistent against serial correlation of unknown form even in the presence of substantial inhomogeneity in
serial correlation across individuals. They are applicable to unbalanced heterogeneous panel models with one way or two way error components. A datadriven method is developed to select the smoothing parameter—the finest
scale in wavelet estimation, making our tests entirely operational in practice.
Simulation shows that our tests perform well in finite samples.
Dept. of Economics and Dept. of Statistical Science, Cornell University, 424
Uris Hall, Ithaca, NY 14850, U.S.A., and Department of Economics, Tsinghua
University, Beijing 100084, China; yh20@cornell.edu
and
Dept. of Economics and Center for Policy Research, Syracuse University, 426
Eggers Hall, Syracuse, NY, U.S.A.; cdkao@maxwell.syr.edu.
Manuscript received October, 2000; final revision received February, 2004.
APPENDIX A: MATHEMATICAL APPENDIX
To prove Theorems 1–6 and Corollary 1, we will use the following lemma.
LEMMA A.1: Suppose Assumptions 1 and 2 hold, and let bJi (h m) be as in (3.15). Then for any
Ji Ti ∈ Z+ and a bounded constant C that does not depend on i, Ji and Ti (i) bJi (h m) is real-valued, bJi (0 m) = bJi (h 0) = 0, and bJi (h m) = bJi (m h);
Ti −1 Ti −1 υ
(1+υ)(Ji +1)
(ii)
for 0 ≤ υ ≤ 12 ;
m=1 h |bJi (h m)| ≤ C2
h=1
Ti −1 Ti −1
2
(Ji +1)
(iii)
[ m=1 |bJi (h m)|] ≤ C2
;
h=1
Ti −1 Ti −1 Ti −1
2
Ji +1
(iv)
;
h1 =1
h2 =1 [
m=1 |bJi (h1 m)bJi (h2 m)|] ≤ C(Ji + 1)2
Ti −1
Ji +1
Ji +1 Ji +1
− 1)| ≤ C[(Ji + 1) + 2 (2 /Ti )(2τ−1) ] where τ is in Assump(v) | h=1 bJi (h h) − (2
tion 2; Ti −1 Ti −1 2
Ji +1
(vi) | h=1
− 1)| ≤ C[(Ji + 1)2 + 2Ji +1 (2Ji +1 /Ti )(2τ−1) ];
m=1 bJi (h m) − 2(2
(vii) sup1≤hm≤Ti −1 |bJi (h m)| ≤ C(Ji + 1);
Ti −1
(viii) sup1≤h≤Ti −1 m=1
|bJi (h m)| ≤ C(Ji + 1).
PROOF OF LEMMA A.1: This lemma extends Lee and Hong (2001, Lemma A.1), who consider the case where Ji ≡ J → ∞ as Ti ≡ T → ∞ See Hong and Kao (2002) for a detailed
proof.
Q.E.D.
PROOF OF THEOREM 1: We shall show Theorems A.1–A.3.
1546
Y. HONG AND C. KAO
THEOREM A.1: Let α̂ijk and ᾱijk be defined as in (3.12) and (6.3), and VnT ≡
Ji 2j
p
−1/2 n
2
2
Vi0 is as in (3.15). Then VnT
i=1 2πTi
j=0
k=1 (α̂ijk − ᾱijk ) → 0.
n
i=1
σi8 Vi0 , where
−1/2 n
THEOREM A.2: Put MnT ≡ ni=1 σi4 Mi0 where Mi0 is as in (3.14). Then VnT
( i=1 2πTi ×
Ji 2j
d
2
ᾱ
−
M
)
→
N(0
1).
nT
j=0
k=1 ijk
p
−1/2
THEOREM A.3: Let M̂ and Vˆ be defined as in (3.15). Then VnT
(M̂ − MnT ) → 0 and
p
Vˆ /VnT → 1.
PROOF OF THEOREM A.1: Because α̂2ijk − ᾱ2ijk = (α̂ijk − ᾱijk )2 + 2(α̂ijk − ᾱijk )ᾱijk we shall show
Propositions A.1 and A.2 below under the conditions of Theorem 1.
Q.E.D.
−1/2
PROPOSITION A.1: VnT
−1/2
PROPOSITION A.2: VnT
n
i=1 2πTi
n
i=1 2πTi
Ji 2j
j=0
k=1 (α̂ijk
Ji 2j
j=0
k=1 (α̂ijk
−1/2
1/2
− ᾱijk )2 = OP [VnT
+ (n−1 + T −1 )VnT
].
p
− ᾱijk )ᾱijk → 0.
PROOF OF PROPOSITION A.1: By the definition of v̂it in (2.3), we have v̂it = ṽit − X̃it (β̂ − β)
where X̃it and ṽit are as in Assumption 5. We note that under H0 {vit } in (2.4) coincides with the
true errors {εit } in (2.1), and so is i.i.d. for each i and {vit } and {vls } are independent for all i = l
and all t s
Putting R̃i (h) ≡ T −1 Tt=|h|+1 ṽit ṽit−|h| , we write
(A.1)
R̂i (h) − R̃i (h) = (β̂ − β) Γ˜ixx (h)(β̂ − β) − (β̂ − β) Γ˜ixv (h) − (β̂ − β) Γ˜ivx (h)
≡
3
ξ̂ci (h)
c=1
T i
where Γ˜ixx (h) ≡ Ti−1 t=|h|+1
X̃it X̃it−|h|
and Γ˜ixv (h) and Γ˜ivx (h) are as in Assumption 5.
Next, recalling the definition of R̄i (h) as in (6.3), we can write
(A.2)
Ti
R̃i (h) − R̄i (h) = Ti−1
−v̄i·vit − v̄i·ṽit−|h| − vit v̄·t−|h| − v̄·t ṽit−|h| + v̄vit + v̄ṽit−|h|
t=|h|+1
≡
9
ξ̂ci (h)
c=4
Given (A1) and (A2), we have α̂ijk − ᾱijk = (2π)−1/2
(A.3)
n
i=1
2πTi
9
Ti −1
c=1
h=1−Ti
ξ̂ci (h)Ψ̂jk (h) It follows that
2 n
T −1
Ji
Ji
2j
2j
9
i
ξ̂ci (h)Ψ̂jk (h)
(α̂ijk − ᾱijk )2 ≤ 25
Ti
j=0 k=1
c=1
≡ 25
9
i=1
j=0 k=1
h=1−Ti
Âc c=1
−1/2
p
We shall show that VnT Âc → 0 for 1 ≤ c ≤ 9 We first consider Â1 From (A.1), we have
|ξ̂1i (h)| ≤ β̂ − β2 Γ˜ixx (h) ≤ β̂ − β2 Γ˜ixx (0) Let bJi (h m) be defined as in Lemma A.1.
1547
SERIAL CORRELATION IN PANEL MODELS
Then we have
n
i −1 T
i −1
T
−1/2
−1/2 Ti
bJi (h m)ξ̂1i (h)ξ̂1i (m)
VnT |Â1 | = VnT i=1
h=1 m=1
≤
−1/2
β̂ −
VnT
4
β
n
1/2 Ti −1 Ti −1
Ti
i=1
b2Ji (h m)
h=1 m=1
n
1/2
Ti3 Γ˜ixx (0)4
i=1
= OP (n−3/2 )
given Lemma A.1(vi), VnT ≤ C ni=1 2Ji +1 , Assumptions 3–5, Ti ≤ CT and σi2 ∈ [c C].
Next, we consider the second term Â2 in (A.3). Recalling Γixv (h) ≡ p limTi →∞ Γ˜ixv (h) we have
ξ̂2i (h) = (β̂ − β) Γixv (h) + (β̂ − β) [Γ˜ixv (h) − Γixv (h)] It follows that
2
T −1
Ji
2j
n
(A.4)
Ti
Γ
(h)
Ψ̂
(h)
Â2 ≤ 2β̂ − β2
ixv
jk
i=1
j=0 k=1
h=1−T
T −1
2 +
[Γ˜ixv (h) − Γixv (h)]Ψ̂jk (h)
h=1−T
≡ 2β̂ − β M̂1 + 2β̂ − β M̂2 say
π
−1
We now consider M̂1 in (A.4). Let Λxv
×
ijk ≡ −π fixv (ω)Ψjk (ω) dω, where fxv (ω) ≡ (2π)
∞
∞
xv
−ijω
−1/2
Then Λijk = (2π)
h=−∞ Γxv (h)e
h=−∞ Γixv (h)Ψ̂jk (h) by Parseval’s identity, and
Ti −1
1/2 xv
Γ
(h)
Ψ̂
(h)
=
(2π)
Λ
+
Γ
ixv
jk
|h|≥Ti ixv (h)Ψ̂ij (h) It follows by the Cauchy–Schwarz
ijk
h=1−Ti
inequality that
2
(A.5)
M̂1 ≤ 2
n
2
Ji
2
j
2πTi
i=1
2
Λxv
ijk + 2
j=0 k=1
n
Ti
|h|≥Ti
i=1
Ji
2
j
Γixv (h)2
|Ψ̂jk (h)|2
j=0 k=1 |h|≥Ti
¯
= O(nT ) + o[nT (2J /T )2τ ] = O(nT )
Ji
¯
given 2J /T → 0 where J¯ ≡ max1≤i≤n (Ji ). Here, we have used the facts that (a)
j=0 ×
2j
∞
xv 2
−1
2
k=1 Λijk = (2π)
h=−∞ Γixv (h) ≤ C by Parseval’s identity and Assumption 5;
i 2j
(b) |h|≥Ti Γixv (h)2 = o(T −1 ) given Assumption 5 and Ti = ci T ≥ cT ; and (c) Jj=0
k=1 ×
Ji
2τ−1
2
j 2
2τJi
/Ti
given Assumption 2.
h≥|Ti | |Ψ̂jk (h)| ≤
j=0
|h|≥Ti |ψ̂(2πh/2 )| ≤ C2
For the second term M̂2 in (A.4), we have
(A.6)
β̂ − β2 M̂2
n
≤ β̂ − β2
Ti −1 Ti −1
Ti
i=1
−1
= OP (nT )
|bJi (h m)|Γ˜ixv (h) − Γixv (h)Γ˜ixv (m) − Γixv (m)
h=1 m=1
n T
i −1 T
i −1
|bJi (h m)|
i=1 h=1 m=1
= OP [(nT )−1 VnT ]
given Lemma A.1(ii), VnT ≤ C Ti=1 2Ji +1 , and Assumption 5. Combining (A.4)–(A.6) yields
−1/2
−1/2
−1/2
−1/2
1/2
−1 1/2
Â3 = OP (VnT
+ (nT )−1 VnT
)
VnT Â2 = OP (VnT + (nT ) VnT ) Similarly, we have VnT
1548
Y. HONG AND C. KAO
Now we consider the term Â4 in (A.3). By the Cauchy–Schwarz inequality and the fact that
under H0 {vit } coincides with {εit } and so is i.i.d. with Evit8 ≤ C for each i, we have E(v̄i·2 |Ti−1 ×
T i
T i
−1
−2
t=h+1 vit ||Ti
t=m+1 vit |) ≤
nCTi J +1 for h m > 0 It follows from Markov’s inequality,
Lemma A.1(ii), and VnT ≤ C i=1 2 i that
(A.7)
−1/2
Â4
VnT
≤
−1/2
VnT
n
i=1
=
Ti −1 Ti −1
Ti
Ti
|bJi (h m)|v̄i·2 Ti−1
h=1 m=1
t=h+1
Ti
−1 vit Ti
vit t=m+1
1/2
OP (T −1 VnT )
−1/2
Similarly, we have VnT Â5 = OP (T −1 VnT )
Next, for the term Â6 in (A.3), noting that vit and v̄·t−h are independent for h > 0 under H0 Ti
T i
we have E(|v̄·t−h v̄·t−m ||Ti−1 t=h+1
vit ||Ti−1 t=m+1
vit |) ≤ C(nTi )−1 for h m > 0 by the Cauchy–
−1/2
1/2
8
Schwarz inequality and Evit ≤ C It follows that VnT Â6 = OP (n−1 VnT ) Similarly, we have
−1/2
−1 1/2
VnT Â7 = OP (n VnT )
i
i
Finally, given E(v̄2 |Ti−1 Tt=h+1
vit ||Ti−1 Tt=m+1
vit |) ≤ Cn−1 Ti−2 for h m > 0 under H0 we
1/2
p
−1/2
1/2
−1/2
have VnT
Âc = OP [(nT )−1 VnT
Âc → 0 for all 1 ≤ c ≤ 9 given
] for c = 8 9 We have shown VnT
max1≤i≤n 22(Ji +1) /(n2 + T ) → 0 Proposition A.1 then follows from (A.3).
Q.E.D.
PROOF OF PROPOSITION A.2: Recalling α̂ijk − ᾱijk = (2π)−1/2
can write
(A.8)
n
9
Ti −1
c=1
h=1−Ti
ξ̂ci (h)Ψ̂jk (h), we
n
T −1
Ji
Ji
2j
2j
9
i
ξ̂ci (h)Ψ̂jk (h) ᾱijk
2πTi
(α̂ijk − ᾱijk )ᾱijk =
Ti
i=1
j=0 k=1
c=1
≡
9
i=1
j=0 k=1
h=1−Ti
δ̂c c=1
p
−1/2
We shall show VnT δ̂c → 0 for 1 ≤ c ≤ 9 First, we have
(A.9)
−1/2
−1/2
VnT |δ̂1 + δ̂8 + δ̂9 | ≤ VnT (Â1 + Â8 + Â9 )1/2
n
i=1
= OP [n
−3/4
1/4
VnT
Ji
2
1/2
j
2πTi
ᾱ2ijk
j=0 k=1
+ (VnT /nT ) ]
1/2
n
Ji 2j
−1
−1
2
2
where VnT
×
j=0
k=1 ᾱijk = OP (1) by Lemma A.1(v) and E ᾱijk ≤ CTi
i=1 2πTi
Ti−1 −1
2
|
ψ̂
(2πh)|
.
jk
h=1−Ti
Next, we consider the second term δ̂2 in (A8). We write
(A.10)
δ̂2 = (β̂ − β)
n
i=1
Ji
2
j
Ti
j=0 k=1
Ti −1
Γixv (h)Ψ̂jk (h)
h=1−Ti
Ti −1
+
[Γ˜ixv (h) − Γixv (h)]Ψ̂jk (h) ᾱijk
h=1−Ti
≡ (β̂ − β) M̂3 + (β̂ − β) M̂4 say.
SERIAL CORRELATION IN PANEL MODELS
1549
For the first term M̂3 noting that {ᾱijk } is a zero-mean sequence independent across i we obtain
E M̂32
=
n
T −1 T −1
i
i
Ti2 E i=1
≤
n
σi4 Ti
h=1 m=1
T −1
i
i=1
2
bJi (h m)Γixv (h)R̄i (m)
2
Γixv (h)
h=1
=O T
n
sup
1≤h≤Ti −1
T −1
i
2
b2Ji (h m)
m=1
(Ji + 1)2 = O(T VnT )
i=1
−1/2
given Assumption 5, Lemma A.1(viii), and VnT ≤ C ni=1 2Ji +1 . It follows that VnT
(β̂ −
β) M̂3 = OP (n−1/2 ) by Chebyshev’s inequality For the second term M̂4 in (A.10) we have
Ji 2j
−1/2
−1/2
1/2 1/2
2 1/2
VnT |(β̂ − β) M̂4 | ≤ VnT β̂ − βM̂2 ( ni=1 2πTi j=0
= OP [(nT )−1 VnT ] where
k=1 ᾱijk )
−1/2
M̂2 = OP [(nT )−1 VnT ] as shown in (A.6). It follows from (A.10) that VnT
δ̂2 = OP (n−1/2 +
−1/2
−1 1/2
−1/2
−1 1/2
(nT ) VnT ) Similarly, we have VnT δ̂3 = OP (n
+ (nT ) VnT )
Ti −1 Ti −1
n
We now consider δ̂4 in (A.8).
Write
δ̂
=
Ti h=1
4
i=1
m=1 bJi (h m)ξ̂4i (h)R̄i (m) Using
n J +1
Lemma A.1(ii) and VnT ≤ C i=1 2 i , we can obtain
E δ̂24 =
n
n Ti −1 Ti −1 Ti −1 Ti −1
Ti Tl
bJi (h1 m1 )bJl (h2 m2 )
h1 =1 h2 =1 m1 =1 m2 =1
i=1 l=1
≤ CT −1
T −1 T −1
n
i
i
i=1
× E[ξ̂4i (h1 )R̄i (m1 )ξ̂4l (h2 )R̄l (m2 )]
2
2
n T −1 T −1
i
i
−2
|bJi (h m)| + CT
|bJi (h m)|
h=1 m=1
J¯
i=1 h=1 m=1
≤ C(2 /T )VnT + CT
−2
2
VnT
where the first inequality follows from the facts that (a) |E[ξ̂4i (h1 )R̄i (m1 )ξ̂4i (h2 )R̄i (m2 )]| ≤
CTi−3/2 Tl−3/2 , and (b) for i = l |E[ξ̂4i (h1 )R̄i (m1 )ξ̂4l (h2 )R̄l (m2 )]| ≤ CTi−2 Tl−2 which can be
shown by exploiting the facts that under H0 , {vit } coincides with {εit } and so is i.i.d. with
E(vit8 ) ≤ C for each i and {vit } and {vlt } are mutually independent for i = l. Hence, we have
−1/2
¯
1/2
−1/2
VnT δ̂4 = OP (2J2 /T 1/2 + VnT /T ) by Chebyshev’s inequality Similarly, we can obtain VnT δ̂5 =
¯
1/2
1/2
J/2
OP (2 /T + VnT /T )
Ti −1 Ti −1
Next, we consider δ̂6 Write δ̂6 = ni=1 Ti h=1
m=1 bJi (h m)ξ̂6i (h)R̄i (m) where ξ̂6i (h) =
Ti
n
−1
Ti
v
as
in
(A.2).
Then
using
Lemma
A.1(ii)
and VnT ≤ C i=1 2Ji +1 , we can obtain
v̄
it
·t−h
t=h+1
E δ̂26 =
n
n Ti −1 Ti −1 Ti −1 Ti −1
Ti Tl
i=1 l=1
≤ Cn−1
bJi (h1 m1 )bJl (h2 m2 )
h1 =1 h2 =1 m1 =1 m2 =1
T −1 T −1
n
i
i
i=1
J¯
× E[ξ̂6i (h1 )R̄i (m1 )ξ̂6l (h2 )R̄l (m2 )]
n T −1 T −1
2
2
i
i
−2
|bJi (h m)| + Cn
|bJi (h m)|
h=1 m=1
≤ C(2 /T )VnT + Cn
i=1 h=1 m=1
−2
2
VnT
1550
Y. HONG AND C. KAO
where for the first inequality, we have used the facts that (a) |E[ξ̂6i (h1 )R̄i (m1 )ξ̂6i (h2 )R̄i (m2 )]|
≤ CTi−2 n−1 ; (b) for i = l |E[ξ̂6i (h1 )R̄i (m1 )ξ̂6l (h2 )R̄l (m2 )]| ≤ CTi−1 Tl−1 n−2 which can be shown
by exploiting the i.i.d. property of {vit } and the independence between {vit } and {vlt } for i = l
¯
−1/2
under H0 via tedious algebra. It follows by Chebyshev’s inequality and 2J /n → 0 that VnT δ̂6 =
p
p
¯
¯
1/2
−1/2
1/2
1/2
1/2
J/2
J/2
OP (2 /T + VnT /n) → 0 Similarly, we have VnT δ̂7 = OP (2 /T + VnT /n) → 0 We have
p
−1/2
shown VnT
δ̂c → 0 for 1 ≤ c ≤ 9 given max1≤i≤n 22(Ji +1 )/(n2 + T ) → 0 Proposition A.2 then
follows from (A.8).
Q.E.D.
PROOF OF THEOREM A.2: Recalling the definition of ᾱijk in (6.3), we can write ni=1 2πTi ×
Ji 2j
n
T −1
T −1
n
2
j=0
k=1 ᾱijk =
i=1 Ti
h=1
m=1 bJi (h m)R̄i (h)R̄i (m) ≡
i=1 (Âi + B̂1i − B̂2i − B̂3i ) where
Ti −1 Ti −1
Âi ≡ 2Ti−1
bJi (h m)
h=1 m=1
bJi (h m)
h=1 m=1
B̂2i ≡ Ti−1
T −1
T −1 T −1
T −1 (by symmetry of bJi (· ·))
Ti
vit2 vit−h vit−m t=1
bJi (h m)
h=1 m=1
B̂3i ≡ Ti−1
vit vit−j vis vis−m
t=2 s=1
Ti −1 Ti −1
B̂1i ≡ Ti−1
Ti t−1
Ti
h
vit vit−h vis vis−m t=1 s=m+1
bJi (h m)
h=1 m=1
Ti
m
vit vit−h vis vis−m t=1 s=1
Note again that under H0 , {vit } coincides with {εit } and so is i.i.d. for each i and {vit } and {vls }
are independent for i = l and all t s.
Q.E.D.
i 2j
−1/2 n
−1/2 n
2
PROPOSITION A.3: VnT
( i=1 2πTi Jj=0
k=1 ᾱijk − MnT ) = VnT
i=1 Âi + oP (1).
Next, we decompose Âi into the terms with t − s > qi and t − s ≤ qi , for some integer qi ∈ Z+ :
T t−q −1 T
Ti −1 Ti −1
t−1
i
i
i
−1
(A.11)
bJi (h m)
+
Âi = 2Ti
vit vit−h vis vis−m
h=1 m=1
t=qi +2
s=1
t=2 s=max(t−qi 1)
≡ B̂i + B̂4i Furthermore, we decompose
q q
qi
Ti t−qi −1
n−1
n−1
n−1 i i
bJi (h m)
B̂i = 2Ti−1
(A.12)
+
+
vit vit−h vis vis−m
h=1 m=1
h=1 m=qi +1
h=qi +1 m=1
t=qi +2
s=1
≡ Ûi + B̂5i + B̂6i ¯
PROPOSITION A.4: Suppose Assumptions 1 and 2 hold, 22J /T → 0 qi ≡ qi (Ti ) → ∞,
−1/2 n
qi /2Ji → ∞, and qi2 /Ti → 0 where J¯ ≡ max1≤i≤n (Ji ) If {vit } is i.i.d. for each i then VnT
i=1 Âi =
−1/2 n
Û
VnT
+
o
(1).
P
i=1 i
−1/2
PROPOSITION A.5: Under the conditions of Proposition A.4, VnT
n
i=1 Ûi
d
→ N(0 1).
1551
SERIAL CORRELATION IN PANEL MODELS
Propositions A.3–A.5 and the Slutsky theorem imply Theorem A.2 We now prove Propositions A.3–A.5.
PROOF OF PROPOSITION A.3: Recall the definition of MnT in Theorem A.2. We obtain
n
Ji
2
j
2πTi
i=1
ᾱ2ijk
− MnT
=
j=0 k=1
n
Âi +
i=1
n
n
n
B̂2i −
B̂3i (B̂1i − σi4 Mi0 ) −
i=1
i=1
i=1
p
p
−1/2 n
−1/2 n
−1/2
We shall show (a) VnT ( i=1 B̂1i − MnT ) → 0; (b) VnT
×
i=1 B̂2i → 0; and (c) VnT
n
p
B̂
→
0
i=1 3i
(a) Observe that B̂1i has a structure similar to B̂1n in Lee and Hong (2001). Following Lee
and Hong’s (2001) reasoning and using Lemma A.1(ii), we can obtain that for each i and for Ti
sufficiently large,
E(B̂1i − E B̂1i )2 ≤ CTi−1
T −1 T −1
i
i
2
|bJi (h m)|
h=1 m=1
¯
≤ C 2 (2J /T )
Ti −1 Ti −1
|bJi (h m)|
h=1 m=1
n
Because {B̂1i } is a random sequence independent across i and
i=1 E B̂1i = MnT we have
n
¯
n
E( i=1 B̂1i − MnT )2 = i=1 E(B̂1i − E B̂1i )2 = O(VnT 2J /T ) given Lemma A.1(ii) and VnT ≤
¯
−1/2 n
C ni=1 2Ji +1 Hence, by Chebyshev’s inequality and 22J /T → 0 we have VnT
( i=1 B̂1i − MnT ) =
¯
OP [(2J /T )1/2 ] = oP (1).
2
≤ CTi−1 ×
(b) Next, we consider B̂2i Following Lee and Hong (2001), we have E B̂2i
Ti −1 Ti −1
3
[ h=1 m=1 |bJi (h m)|] Then by the fact that B̂2i is a zero-mean random sequence inde
2
pendent across i, Lemma A.1(ii), and VnT ≤ C ni=1 2Ji +1 , we have E( ni=1 B̂2i )2 = ni=1 E B̂2i
=
p
−1/2 n
2J¯
2J¯
B̂
→
0
by
Chebyshev’s
inequality
and
2
/T
→
0
O[(2 /T )VnT ] Hence, VnT
i=1 2i
p
−1/2
B̂3i → 0
(c) By reasoning similar to (b) we can obtain VnT
Q.E.D.
PROOF OF PROPOSITION A.4: Given (A.11) and (A.12), we have Âi = Ûi + B̂4i + B̂5i + B̂6i It
p
−1/2 n
suffices to show VnT
i=1 B̂ci → 0 for c = 4 5 6 (a) We first consider B̂4i in (A.11) From Lee
and Hong (2001, Proof of Theorem 1), we have for each i and for Ti sufficiently large,
2
E B̂4i
T −1 T −1
2
Ti −1 Ti −1
i
i
¯
≤ C(qi /Ti )
|bJi (h m) ≤ C 2 (q̄2J /T )
|bJi (h m)|
h=1 m=1
h=1 m=1
where q̄ ≡ max1≤i≤n (qi ) and the last inequality follows from Lemma A.1(ii). Hence, using the
fact that {B̂4i } is a zero-mean random sequence independent across i, Lemma A.1(ii) and VnT ≤
¯
2
= O(VnT q̄2J /T ) This, Chebyshev’s inequality,
C ni=1 2Ji +1 , we have E( ni=1 B̂4i )2 = ni=1 E B̂4i
p
¯
−1/2
n
q̄2 /T → 0, and 22J /T → 0 imply VnT
i=1 B̂4i → 0
(b) Next, we consider B̂5i as in (A.12) By the definition of bJi (h m) the Cauchy–Schwarz
inequality, and Assumption 2, we obtain
2
E B̂5i
= σi4
T −1
T −1 h=1 m>qi
b2Ji (h m) ≤ C
Ji
|ψ̂(2πm/2j )|2 ≤ C 2 22τJi /qi2τ−1 j=0 m>qi
¯
2
Therefore, E( ni=1 B̂5i )2 = ni=1 E B̂5i
≤ C(2J /q0 )2τ−1 ni=1 2Ji where q0 ≡ min1≤i≤n (qi ) It fol−1/2 n
2τ−1
J¯
J¯
lows by Chebyshev’s inequality and 2 /q0 → 0 that VnT
] = oP (1)
i=1 B̂5i = OP [(2 /q0 )
1552
Y. HONG AND C. KAO
(c) Finally, we consider B̂6i in (A.12) Following Lee and Hong (2001, Proof of Theorem 1),
we obtain
2
E B̂6i
≤ C2
2τJi
/qi2τ−1
+
CTi−1
T −1 T −1
2
i
i
bJ (h m)
i
h=1 m=1
¯
≤ C2Ji (2Ji /q0 )2τ−1 + C(2J /T )
Ti −1 Ti −1
bJ (h m)
i
h=1 m=1
p
¯
−1/2 N
2τ−1
J¯
by Lemma A.1(ii) and Assumption 2. Thus, VnT
+ (2J /T )1/2 ] → 0 by
i=1 B̂6i = OP [(2 /q0 )
n J +1 J¯
2J¯
i
Chebyshev’s inequality, VnT ≤ C i=1 2 2 /q0 → 0, and 2 /T → 0. This completes the proof
of Proposition A.4.
Q.E.D.
T i
qi
PROOF OF PROPOSITION A.5: We write Ûi = Ti−1 t=q
Uit where Uit ≡ 2vit h=1
vit−h ×
i +2
qi
t−q −1
bJi (h m)Sit−qi −1 (m), and Sit−qi −1 (m) ≡ s=1i vis vis−m Then
Hit−qi −1 (h) Hit−qi −1 (h) ≡ m=1
Û = T̄t=q0 +2 Ut where T̄ ≡ max1≤i≤n (Ti ) Ut ≡ ni=1 Uit 1(qi ≤ t ≤ Ti ) and 1(·) is the indicator
function Put Ft ≡ ni=1 Fit where Fit is the sigma field generated by {vis s ≤ t}ni=1 Because {vit vit−h }
is independent of Hit−qi −1 (h) for 0 < h ≤ qi {Ut Ft−1 } is an adapted martingale difference sequence (m.d.s.), with
E Û 2 =
T̄
EUt2 =
t=q0 +2
n
T̄
E(Uit2 )1(qi ≤ t ≤ Ti ) = VnT [1 + o(1)]
t=q0 +2 i=1
given qi → ∞ qi /2Ji → ∞ q̄2 /T → 0. We apply Brown’s (1971) martingale limit theorem by
verifying his two conditions:
(a) var−2 (Û) T̄t=q0 +2 E{Ut2 1[|Ut | ≥ var1/2 (Û)]} → 0 for all > 0,
T̄
p
(b) var−2 (Û 2 )T −2 t=q0 +2 E(Ut2 |Ft−1 ) → 1
T̄
−2
4
We first verify (a) by showing VnT
t=q0 +2 EUt → 0 Given t {Uit } is a zero-mean independent
sequence across i, so we have EUt4 ≤ C[ ni=1 Ti−2 (EUit4 )1/2 1(qi ≤ t ≤ Ti )]2 Moreover, following Lee and Hong (2001, Proof of Theorem 1), we can obtain that for each i and for Ti sufqi qi
T̄
−2
2
4
−1
) → 0
ficiently large, EUit4 ≤ Ct 2 h=1
m=1 bJi (h m) It follows that VnT
t=1 EUt = O(T
Hence, condition (a) holds.
−2
E(Ũ 2 − E Û 2 ) → 0 where
Next, we verify condition (b) by showing VnT
Ũ 2 ≡
T̄
E(Ut2 |Ft−1 )
t=q0 +2
E(Ut2 |Ft−1 )
≡E
Ti−1
n
2
Uit 1(qi ≤ t ≤ Ti )
Ft−1
i=1
=
n
i=1
Ti−1 E(Uit2 |Fit−1 )1(qi ≤ t ≤ Ti )
1553
SERIAL CORRELATION IN PANEL MODELS
where the second equality follows from the facts that for each t {Uit } is a zero-mean random
sequence independent across i, and that for each i {Uit Fit−1 } is an m.d.s. Lee and Hong (2001)
show that for each i and for Ti sufficiently large,
T
2
T −1 T −1
2
i
i
i
E
[E(Uit2 |Fit−1 ) − EUit2 ] ≤ C(q̄/T )
|bJi (h m)| + C(Ji + 1)2Ji t=qi +2
h=1 m=1
It follows that
E(Ũ 2 − E Ũ 2 )2 =
n
E
i=1
Ti
2
[E(Uit2 |Fit−1 ) − EUit2 ]
2
≤ C(q̄/T )VnT
+ C(J¯ + 1)VnT
t=qi +2
i
for T sufficiently large, where the equality follows from the fact that { Tt=q
[E(Uit2 |Fit−1 ) −
i +2
2
EUit ]} is a zero-mean independent sequence across i and the inequality follows from Lem
¯
−2
ma A.1(ii) and VnT ≤ C ni=1 2Ji +1 . Hence, given q̄2 /T → 0 and 22J /T → 0 we have VnT
E(Ũ 2 −
n
−2
2 2
Ji
E Ũ ) = O(q̄/T ) + O[VnT
i=1 (Ji + 1)2 ] → 0 as n T → ∞ Thus, condition (b) holds, and so
−1/2
Û → N(0 1) by Brown’s theorem.
VnT
d
Q.E.D.
PROOF OF THEOREM A.3: (a) Recalling the definition of M̂ and MnT we have
(A.13)
M̂ − MnT =
Ti −1
n
[R̂i (0) − Ri (0)]2 + 2[R̂i (0) − Ri (0)]Ri (0)
bJi (h h)
i=1
h=1
≡ M̂5 + 2M̂6 Because R̂i (0) − Ri (0) = [R̂i (0) − R̃i (0)] + [R̃i (0) − R̄i (0)] + [R̄i (0) − Ri (0)] we can write
(A.14)
M̂5 ≤ 4
Ti −1
n
[R̂i (0) − R̃i (0)]2 + [R̃i (0) − R̄i (0)]2 + [R̄i (0) − Ri (0)]2
bJi (h h)
i=1
h=1
≡ 4(M̂51 + M̂52 + M̂53 )
Using (A1), the Cauchy–Schwarz inequality, Assumptions 3–5, Lemma A.1(v), and VnT ≤
C ni=1 2Ji +1 , we have
−1/2
−1/2
VnT M̂51 ≤ 4VnT β̂ − β2
n
β̂ − β2 Γ˜ixx (0)4 + Γ˜ixv (0)2 + Γ˜ivx (0)2
i=1
Ti −1
×
bJi (h h)
h=1
1/2 = OP (nT )−1 VnT
n J +1
Similarly, using VnT ≤ C i=1 2 i , the Cauchy–Schwarz inequality, Markov’s inequality, the i.i.d.
property of {vit } for each i and independence between {vit } and {vls } for all i = l under H0 we can obtain E[R̃i (0) − R̄i (0)]2 ≤ C(Ti−2 + n−1 Ti−1 ) It follows from Markov’s inequality, the
−1/2
Cauchy–Schwarz inequality, Lemma A.1(v), and VnT ≤ C ni=1 2Ji +1 that VnT M̂52 = OP [(T −2 +
1/2
−1
−1 −1
2
n T )VnT ] Using Markov’s inequality, E[R̄i (0) − Ri (0)] ≤ CTi Lemma A.1(v), and VnT ≤
−1/2
1/2
−1/2
1/2
C ni=1 2Ji +1 , we have VnT
M̂53 = OP (T −1 VnT
M̂5 = OP (T −1 VnT
) Hence, we have VnT
)=
oP (1) given (A.14) and VnT /T 2 → 0
1554
Y. HONG AND C. KAO
Next, we consider the second term M̂6 in (A.13). We write
(A.15)
M̂6 =
Ti −1
n
[R̂i (0) − R̃i (0)] + [R̃i (0) − R̄i (0)] + [R̄i (0) − Ri (0)] Ri (0)
bJi (h h)
i=1
h=1
≡ M̂61 + M̂62 + M̂63 Using (A.1) with h = 0 Assumptions 3–5, Lemma A.1(v), and VnT ≤ C
−1/2
−1/2
VnT
|M̂61 | ≤ VnT
β̂ − β2
n
n
i=1
2Ji +1 , we have
β̂ − βΓ˜ixx (0) + Γ˜ixv (0) + Γ˜ivx (0) Ri (0)
i=1
Ti −1
×
bJi (h h)
h=1
1/2 = OP (nT )−1/2 VnT
Also, using (A.2), E[R̃i (0) − R̄i (0)]2 ≤ C(Ti−2 + (nTi )−1 ), the Cauchy–Schwarz inequality,
n
−1/2
1/2
Markov’s inequality, Lemma A.1(ii), and VnT ≤ C i=1 2Ji +1 , we have VnT M̂62 = OP [T −1 VnT +
(nT )−1/2 VnT ]
Finally, noting that R̄i (0) − Ri (0) is a zero-mean sequence independent across i, we have
2
T −1 T −1
n
n T
i
i
i −1 T
i −1
¯
2
E M̂63
=
E[R̄i (0) − Ri (0)]2
bJi (h h) ≤ C 2 (2J /T )
bJi (h h)
i=1
h=1 m=1
where the last inequality follows by Lemma A.1(v).
¯
by Chebyshev’s inequality. Hence, (A.15) and 22J /T
p
−1/2
VnT (M̂ − MnT ) → 0.
p
(b) The proof for Vˆ /VnT → 1 is analogous to (a).
i=1 h=1 m=1
−1/2
¯
It follows that VnT M̂63 = OP (2J/2 /T 1/2 )
p
−1/2
→ 0 imply VnT M̂6 → 0. We have shown
Q.E.D.
PROOF OF THEOREM 2: To conserve space, we only show for W̃1 . The proof for W̃2 is
similar. Put M̃ ≡ ni=1 R̂2i (0)(2Ji +1 − 1) and V˜ ≡ 4 ni=1 R̂4i (0)(2Ji +1 − 1) Then we can write
W̃1 − Ŵ1 = Ŵ1 (Vˆ 1/2 /V˜ 1/2 − 1) + Vˆ −1/2 (M̂ − M̃)(Vˆ 1/2 /V˜ 1/2 ) where M̂ and Vˆ are as in (3.14).
p
p
−1/2
Because Ŵ1 = OP (1) by Theorem 1, VnT (M̂ − MnT ) → 0 and Vˆ /VnT → 1 by Theorem A.3, it
p
p
p
d
−1/2
suffices for W̃1 − Ŵ1 → 0 and W̃1 → N(0 1) if (a) VnT (M̃ − MnT ) → 0 and (b) V˜ /VnT → 1
We first show (a). Following reasoning analogous to the proof of Theorem A.3, we ob
p
−1/2
−1/2
0
0
0
(M̃ − MnT
) → 0 where MnT
≡ ni=1 σi4 (2Ji +1 − 1) It remains to show VnT
(MnT
−
tain VnT
2
Ji +1
ν
ν
MnT ) → 0 This follows from Lemma A.1(v), 2
= ai T n/T log2 (T ) → 0, and
n/T 2(2τ−1)−2(2τ−1/2)ν → 0 because
n
n
¯
−1/2
−1/2
0
VnT |MnT − MnT
| ≤ CVnT
(Ji + 1) + (2J /T )(2τ−1)
2Ji +1 → 0
i=1
Now we show
i=1
n
(b). Put
≡ i=1 σi8 (2Ji +1 − 1) Following reasoning
0 p
0
→ 1 It remains to show VnT /VnT
we can obtain V˜nT /VnT
0
VnT
analogous to the proof
→ 1 This follows from
of Theorem A.3,
Lemma A.1(vi) and Ji → ∞, because
n
n
−1
0
−1
2
(2τ−1)
Ji +1
J¯
VnT |VnT − VnT | ≤ CVnT
(Ji + 1) + (2 /T )
2
→ 0
i=1
i=1
1555
SERIAL CORRELATION IN PANEL MODELS
−1
where VnT
n
i=1 (Ji
+ 1)2 → 0 given VnT ≤ C
n
i=1
2Ji +1 and Ji → ∞
Q.E.D.
PROOF OF THEOREM 3: Recall the definition of M̂ and Vˆ as in (3.14). Following reasoning analogous to the proof of Theorem A.3, we can obtain M̂ = MnT [1 + oP (1)] and Vˆ =
i 2j
α̂2 + oP (1)
VnT [1 + oP (1)] It follows that (nA T )−1 Vˆ −1/2 Ŵ1 = (nA T )−1 ni=1 2πTi Jj=0
n k=1 ijk
n
given MnT ≤ C i=1 (2Ji +1 − 1) = O(VnT ), and VnT /nA T → 0 by (nA T )−1 i=1 2Ji +1 → 0 It remains to show Ji 2j
p
n
2
2
(a) (nA T )−1 i=1 2πTi j=0
k=1 (α̂ijk − αijk ) → 0; and
j
n
Ji
2
n
2
−1
(b) n−1
j=0
k=1 αijk = (nA T )
A
i=1 2πTi
i=1 2πci Q(fi fi0 ) + o(1) where αijk is defined
in (3.8)
We first show (a). Because
(A.16)
(nA T )−1
n
Ji
2
i
2πTi
i=1
(α̂2ijk − α2ijk )
j=0 k=1
= (nA T )−1
n
Ji
2
i
2πTi
i=1
[(α̂ijk − αijk )2 + 2(α̂ijk − αijk )αijk ]
j=0 k=1
it suffices to show that the first term in (A.16) vanishes in probability. That the second term in
(A.16) vanishes in probability then follows by the Cauchy–Schwarz inequality and the fact that
n
Ji 2j
2
2
(nA T )−1 i=1 2πTi j=0
k=1 αijk ≤ C supi∈NA Q(fi f0 ) ≤ C Noting α̂ijk − αijk = (α̂ijk − ᾱijk ) +
(ᾱijk − αijk ), we obtain
(A.17)
n
Ji
2
j
2πTi
i=1
(α̂ijk − αijk )2 ≤ 2
j=0 k=1
n
Ji
2
j
2πTi
i=1
[(α̂ijk − ᾱijk )2 + (ᾱijk − αijk )2 ]
j=0 k=1
≡ 2(M̂71 + M̂72 )
Following reasoning analogous to the proof of Proposition A.1, we can obtain
(A.18)
(nA T )−1 M̂71 = OP [(nA T )−1 + (nA T )−1 VnT ]
under Assumptions 1–6 and HA Note that we have obtained a slower rate for M̂71 under HA than
under H0 For the second term in (A.17), we further decompose
(A.19)
M̂72 ≤ 2
n
Ji
2
j
2πTi
i=1
[(ᾱijk − E ᾱijk )2 + (E ᾱijk − αijk )2 ] ≡ 2(M̂721 + M̂722 )
j=0 k=1
We now consider the first term in (A.19). We have sup1≤h≤Ti −1 var[R̄i (h)] ≤ CTi−1 , which follows from Assumption 6 and
Ti −1
var[R̄i (h)] = Ti−1
(1 − |l|/Ti )[R2i (l) + Ri (l − h)Ri (l + h) + κi (h l l + h)]
l=1−Ti
Cf. Hannan (1970, p. 209). Therefore, we have
M̂721 ≤
n
i=1
Ti −1 Ti −1
Ti
sup var[R̄i (h)]
1≤h≤Ti −1
h=1 m=1
|bJi (h m)| = O(VnT )
1556
Y. HONG AND C. KAO
For the second term in (A.19), noting that |E ᾱijk − αijk | ≤ (2π)−1/2 Ti−1
we have
∞
T −1
Ji
2j
n i
−1
2
2
2
M̂722 ≤
Ti
Ri (h)
h |Ψ̂jk (h)|
i=1 j=1 k=1
¯
= O (22J /T )
h=1−Ti
n
∞
h=−∞ |hRi (h)Ψ̂jk (h)|
h=−∞
2Ji +1 = o(VnT )
i=1
¯
given Assumption 2 and 22J /T → 0. It follows by Markov’s inequality that (nA T )−1 M̂72 =
OP [(nA T )−1 VnT ] This, (A.17), (A.18), and VnT /(nA T ) → 0 imply (a).
Next, we show (b). This follows because
n
(nA T )−1
2
∞ j
2πTi
i=1
α2ijk − (nA T )−1
n
j=0 k=1
2
∞ Ji
2
j
2πTi
i=1
α2ijk
j=0 k=1
j
≤ C sup
i∈NA
α2ijk → 0
j=Ji +1 k=1
as min1≤i≤n (Ji ) → ∞ and Qi (fj f0 ) =
∞ 2j
2
k=1 αijk
j=0
≤ C This completes the proof
for Ŵ1 .
Q.E.D.
PROOF OF THEOREM 4: Given Ti = ci T and 2Ji +1 = ai Tiν , we have 2Ji +1 = bi T ν where bi ≡
ai ciν When T → ∞ we have
n
n
0
−1
8 Ji +1
ν
−1
8
VnT ≡ n
σi (2
− 1) = T n
σi bi [1 + o(1)] = b̄T ν [1 + o(1)]
i=1
where b̄ ≡ n−1
n
i=1
0
σi8 bi It follows from Theorem 2 and Vˆ /VnT
→ 1 that
p
i=1
(nA T )−1 (b̄T ν )1/2 Ŵ1 = n−1
A
ci Q(fi fi0 ) + oP (1)
and
i∈NA
(nA T )−1 (b̄T ν )1/2 Ŵ2 = n−1
A
(b̄/σi8 bi )1/2 ci Qi + oP (1)
i∈NA
(c)
For c = 1 2 we put SnT
≡ −2 ln[1 − Φ(Ŵc )] where Φ(·) is the N(0 1) CDF Because ln[1 −
1 2
Φ(z)] = − 2 z [1 + o(1)] as z → +∞, we have
−2
(nA T )
(1)
b̄T ν SnT
n−1
A
=
2
ci Q(fi fi0 )
+ oP (1)
and
i∈NA
−2
(nA T )
(2)
b̄T ν SnT
=
n−1
A
2
(b̄/σi8 bi )1/2 ci Q(fi fi0 )
+ oP (1)
i∈NA
(1)
(2)
Suppose {Ti(1) }ni=1 and {Ti(2) }ni=1 are two sequences of sample sizes used for Ŵ1 and Ŵ2 re(2)
(1)
spectively so that Sn(1)
n(2) T (1) , and T (2) → ∞ Then Bahadur’s relative
(1) T (1) /Sn(2) T (2) → 1 as n
SERIAL CORRELATION IN PANEL MODELS
1557
efficiency of Ŵ1 to Ŵ2
n(2)
n(2) (2)
1
(2) (2)
( n(2)
Ti
i=1 ci )n T
BE(Ŵ1 : Ŵ2 ) ≡ lim i=1
=
lim
(1)
(1)
n
n
(1)
1
(1) (1)
( n(1)
i=1 Ti
i=1 ci )n T
n !
lim
−1
b̄/(σi8 bi )ci Q(fi fi0 ) (1+κ)/(3−ν)
n→∞ n
i=1
=
limn→∞ n−1 ni=1 ci Q(fi fi0 )
where the last equality follows from n(c) = γ[T (c) ]κ for c = 1 2. Hence, BE(Ŵ1 : Ŵ2 ) > 1 if ai is a
monotonically increasing function of Q(fi fi0 ) and ci = c (i.e., Ti = T ) for all i
Q.E.D.
ˆ the proof for Ŵ2 (J)
ˆ is similar. We write
PROOF OF THEOREM 5: We only consider Ŵ1 (J);
ˆ − Ŵ2 (J)
Ŵ1 (J)
n
2j
Jˆ −1/2
2
ˆ − M̂(J)] − [Vˆ (J)1/2 /Vˆ (J)1/2 − 1]Ŵ1 (J)
ˆ
2πTi
α̂ijk − [M̂(J)
= Vˆ (J)
i=1
j=J
k
ˆ −
Given Ŵ1 (J) = OP (1) by Theorem 1 and Vˆ (J)/VnT → 1 by Theorem A.3, it suffices for Ŵ1 (J)
ˆ 2j 2
p
p
d
−1/2 n
ˆ →
ˆ − M̂(J)]} →
Ŵ1 (J) → 0 and W1 (J)
N(0 1) if (a) V
{
2πTi J
α̂ − [M̂(J)
0 and
p
nT
i=1
j=J
k
ijk
p
(b) Vˆ (J)/Vˆ (J) → 1
We first show (a). Decompose
(A.20)
n
2
Jˆ j
2πTi
i=1
α̂2ijk
j=0 k=1
=
n
2
Jˆ j
2πTi
i=1
[(α̂ijk − ᾱijk )2 + ᾱ2ijk + 2(α̂ijk − ᾱijk )ᾱijk ]
j=0 k=1
≡ Ĝ1 + Ĝ2 + 2Ĝ3 For the first term in (A.20), we write Ĝ1 =
n
−1/2
i=1
p
ˆ
j
2πTi ( Jj=0 − Jj=0 ) 2k=1 (α̂ijk − ᾱijk )2 ≡
Ĝ11 − Ĝ12 By Proposition A.1, we have VnT Ĝ12 → 0 Next, for any given constants M > 0
and > 0 we have
ˆ
ˆ
P(Ĝ11 > ) ≤ P Ĝ11 > 2J/2 M|2J /2J − 1| ≤ + P 2J/2 M|2J /2J − 1| > ˆ
p
where the second term vanishes to 0 as n T → ∞ given 2J/2 |2J /2J − 1| → 0 For the first probˆ
−1/2
ability, given 2J/2 M|2J /2J − 1| ≤ we have for all n and all Ti sufficiently large, VnT
Ĝ11 ≤
J+1 2j
−1/2 n
−1/2
2 p
VnT
Ĝ1 = oP (1)
k=1 (α̂ijk − ᾱijk ) → 0 by Proposition A.1. Therefore, VnT
i=1 2πTi
j=0
Next, we consider Ĝ2 By symmetry of bJ (· ·) we write
(A.21)
Ĝ2 =
n
Ti −1
σi4
i=1
+2
[bJˆ(h h) − bJ (h h)] +
h=1
n
i=1
Ti
n T
i −1
[Ti−1 R̄2i (h) − σi4 ][bJˆ(h h) − bJ (h h)]
i=1 h=1
Ti −1 h−1
R̄i (h)R̄i (m)[bJˆ(h m) − bJ (h m)]
h=1 m=1
≡ Ĝ21 + Ĝ22 + 2Ĝ23 say.
1558
Y. HONG AND C. KAO
For the last term Ĝ23 in (A.21), we have for any constants M > 0 and > 0
−1/2
P(VnT |Ĝ23 | > )
−1/2
ˆ
ˆ
|Ĝ23 | > 2J/2 M|2J /2J − 1| ≤ + P 2J/2 M|2J /2J − 1| > ≤ P VnT
where, again, the second term vanishes to 0 as n T → ∞ We now show the first probability vanˆ
ishes. Put T̄ ≡ max1≤i≤n (Ti ) as before. Given 2J/2 M|2J /2J − 1| ≤ and the definition of aJ (h m)
as in (3.14), we have
[log2 2J (1+/M2J/2 )]
|aJˆ(h m) − aJ (h m)| = 2π
|cj (h m)ψ̂(2πh/2j )ψ̂∗ (2πm/2j )|
j=[log2 2J (1−/M2J/2 )]
where
2
j
(A.22)
−j
cj (h m) ≡ 2
e
i2π(m−h)k/2j
=
k=1
1
if m = h + 2j r for any integer r
0
otherwise.
Cf. Priestley (1981, p. 392, (6.19)). It follows that we have for all n and T sufficiently large,
n
T̄ −1
T̄ −1
E|Ĝ23 | ≤
E
1(h ≤ Ti )1(m ≤ Ti )Ti R̄i (h)R̄i (m)
h=1−T̄ m=1−T̄
[log2
× 2π
i=1
2J (1+/2J/2 M)]
|cj (h m)ψ̂(2πh/2j )ψ̂∗ (2πm/2j )|
j=[log2 2J (1−/2J/2 M)]
[log2 2J (1+/2J/2 M)]
≤ 2πn
×
1/2
j
2 2
j=[log2 2J (1−/2J/2 M)]
∞
T̄ −1
−j
|ψ̂(2πh/2 )|
j
h=1−T̄
|ψ̂(2πh/2 + 2πr)|
j
r=−∞
1/2
≤ Cn1/2 2J+1 /2J/2 M = O(VnT
/M)
−1/2
ˆ
where the second inequality follows given Assumption 2. Therefore, P(VnT |Ĝ23 | > 2J/2 M|2J /
p
−1/2
2J − 1| ≤ ) also vanishes to 0. Consequently, we have VnT Ĝ23 → 0 Similarly, we can also obtain
p
−1/2
VnT Ĝ22 → 0 and
−1/2 VnT
−1/2
ˆ − M̂(J)] = VnT
Ĝ21 − [M̂(J)
n
p
[σi4 − R̂2i (0)][bJˆ (h h) − bJ (h h)] → 0
i=1
n
−1
2
4
−1/2
where n
] given Assumptions 3–5, and that under H0 {vit }
i=1 [R̂i (0) − σi ] = OP [(nT )
−1/2
ˆ −
coincides with {εit } and so is i.i.d. for each i It follows from (A.21) that VnT {Ĝ2 − [M̂(J)
p
M̂(J)]} → 0
Next, by the Cauchy–Schwarz inequality, we have
−1/2
−1/2
−1/2
VnT |Ĝ3 | ≤ (VnT Ĝ1 )1/2 (VnT |Ĝ2 |)1/2
−1/4
1/4 = OP VnT
+ (T −1/2 + n−1/2 VnT
) oP (n1/4 ) = oP (1)
1559
SERIAL CORRELATION IN PANEL MODELS
p
−1/2
−1/2
Ĝ2 = n−1/2 VnT
by Proposition A.1 and the fact that n−1/2 VnT
(Ĝ21 + Ĝ22 + Ĝ23 ) → 0, as can be
shown using reasoning similar to that for Ĝ23 Result (a) then follows from (A.20).
p
p
ˆ nT →
ˆ Vˆ (J) = 1 + oP (1) it suffices to show Vˆ (J)/V
1 given Vˆ (J)/VnT → 1 by
(b) To show Vˆ (J)/
ˆ and VnT we can use reasoning analogous to that
Theorem A.3. Recalling the definitions of Vˆ (J)
for Ĝ23 to obtain
ˆ − VnT ]/VnT = V −1
[Vˆ (J)
nT
n
Ti −1 Ti −1
[R̂4i (0) − σi8 ]
i=1
p
[bJˆ(h m) − bJ (h m)] → 0
h=1 m=1
n
−1
p
8
4
−1/2
ˆ Vˆ (J) →
where we used the fact that n
] Thus, Vˆ (J)/
1 It foli=1 [R̂i (0) − σi ] = OP [(nT )
p
ˆ Vˆ (J) − 1]Ŵ1 (J) → 0 given Ŵ1 (J) = OP (1) by Theorem 1. Therefore, result (b)
lows that [Vˆ (J)/
p
d
ˆ − Ŵ1 (J) →
ˆ →
holds, and we have Ŵ1 (J)
0 and Ŵ1 (J)
N(0 1) This completes the proof. Q.E.D.
PROOF OF THEOREM 6: We shall prove for Theorem 6(b) only; the proof for Theorem 6(a) is
n
n
similar and simpler. (a) We first show n−1 i=1 Q(fˆi fi ) = n−1 i=1 Q(f¯i fi ) + oP (2J /T + 2−2qJ )
Write
(A.23)
n−1
n
[Q(fˆi fi ) − Q(f¯i fi )]
i=1
= n−1
n
Q(fˆi f¯i ) + 2n−1
i=1
n i=1
π
−π
[fˆi (ω) − f¯i (ω)][f¯i (ω) − fi (ω)] dω
≡ Q̂1 + 2Q̂2 For Q̂1 in (A.23), by Parseval’s identity, (A.18), and VnT ∝ n2J+1 we have
J 2
n (α̂ijk − ᾱijk )2 = OP [(nT )−1 + 2J /nT ] = oP (2J /T )
j
Q̂1 = n−1
i=1 j=0 k=1
as n T → ∞ Next, we have Q̂2 = oP (2J /T + 2−2qJ ) by the Cauchy–Schwarz inequality, Q̂1 =
n
oP (2J /T ), n−1 i=1 Q(f¯i fi ) = OP (2J /T + 2−2qJ ) which follows by Markov’s inequality and
n
−1
J
¯
n
/T + 2−2qJ ) The latter is to be shown below.
i=1 EQ(fi fi ) = O(2
n
−1
¯
(b) To compute n
i=1 EQ(fi fi ) we write
(A.24)
n−1
n
EQ(f¯i fi ) = n−1
i=1
n
EQ(f¯i E f¯i ) + n−1
i=1
n
Q(E f¯i fi )
i=1
We first consider the second term in (A.24). By the orthonormality of {Ψjk (·)} we obtain
(A.25)
n−1
n
∞ 2
n j
Q(E f¯i fi ) = n−1
i=1
i=1 j=J+1 k=1
J 2
n j
α2ijk + n−1
(E ᾱijk − αijk )2 i=1 j=0 k=1
For the first term in (A.24), using (3.9) and (A.22), we have
2
∞ j
α2ijk =
∞
∞
∞
R(h)R(m)cj (h m)ψ̂(2πh/2j )ψ̂∗ (2πm/2j )
j=J+1 h=−∞ m=−∞
j=J+1 k=1
=
∞
∞
∞
j=J+1 h=−∞ r=−∞
R(h)R(h + 2j r)ψ̂(2πh/2j )ψ̂∗ (2πh/2j + 2πr)
1560
Y. HONG AND C. KAO
We now evaluate the terms corresponding to r = 0 and r = 0 respectively. For r = 0 we have
∞
∞
R2i (h)|ψ̂(2πh/2j )|2 =
j=J+1 h=−∞
∞
(2π/2j )q
j=J+1
∞
|ψ̂(2πh/2j )|2 2 2
h Ri (h)
|2πh/2j |2q
h=−∞
∞
|ψ(z)|2 (2π/2j )q
z→∞ |z|2q
j=J+1
= lim
= λq 2−2q(J+1)
π
−π
∞
h2 R2i (h) [1 + o(1)]
h=−∞
2
(q)
fi (ω) dω [1 + o(1)]
where f (·) and λq are defined in (6.1) and (6.2), and o(1) is uniform in i and ω ∈ [−π π] For
the term corresponding to r = 0 it can be shown to be o(2−2q(J+1) ) It follows that for the first
term in (A.25),
(q)
∞ 2
n j
(A.26)
n−1
α2ijk = 2−2q(J+1) λ2q n−1
i=1 j=J+1 k=1
n i=1
π
−π
2
(q)
fi (ω) dω + o 2−2q(J+1) For the second term in (A.25), by Lemma A.1(vii) and Assumption 8 we have
J 2
n j
(A.27)
n−1
(E ᾱijk − αijk )2
i=1 j=0 k=1
J 2
n j
= n−1
Ti −1
i=1 j=0 k=1
≤ 4Cn−1
n
i=1
Ti−1
2
Ri (h)ψ̂jk (2πh)
|h|≥Ti
h=1−Ti
Ti −1 Ti −1
Ti−2
|h|Ri (h)ψ̂jk (2πh) +
|hRi (h)mRi (m)bJ (h m)| = O (J + 1)/T 2 h=1 m=1
Finally, we consider the first term in (A.24), the variance factor We write
n−1
n
EQ(f¯i E f¯i ) = n−1
i=1
Ti −1
n
Ti −1
bJi (h m) cov[R̄i (h) R̄i (m)]
i=1 h=1−Ti m=1−Ti
= n−1
Ti −1
n
Ti −1
bJi (h m)Ti−1
i=1 h=1−Ti m=1−Ti
η(l) + m
1−
Ti
l
× Ri (l)Ri (l + m − h) + Ri (l + m)Ri (l − h) + κi (l h m − h)
≡ Ω1nT + Ω2nT + Ω3nT say,
where η(l) = l if l > 0 η(l) = 0 if h − m ≤ l ≤ 0, and η(l) = −l + h − m if −(Ti − h) + 1 ≤ l ≤
h − m Cf. Priestley (1981, p. 326). Given Assumption 6 and Lemma A.1(vii), we have |Ω2nT | ≤
C(J + 1) and |Ω3nT | ≤ C(J + 1) For the first term Ω1nT , we can write
Ω1nT = n−1
Ti −1
n
bJi (h h)Ti−1
i=1 h=1−Ti
+ n−1
Ti −1 Ti −1
n
i=1 h=1−Ti |r|=1
h
1−
R2i (l)
Ti
l
bJi (h h + r)Ti−1
η(l) + h − r
1−
Ri (l)Ri (l + r)
Ti
l
SERIAL CORRELATION IN PANEL MODELS
= n−1
n
Ti−1 (2Ji +1 − 1)
i=1
∞
1561
R2i (h) + O[(J + 1)/T ]
h=−∞
where we have used Lemma A.1(v) for the first term, which corresponds
to h = m; the second
term corresponds to h = m and it is O[(J + 1)/T ] uniformly in i given ∞
h=−∞ |Ri (h)| ≤ C and
Lemma A.1(v) It follows that as J → ∞,
n
n 2J+1 −1 π 2
n−1
(A.28)
EQ(f¯i E f¯i ) =
fi (ω) dω + o(2J /T )
n
T
−π
i=1
i=1
Collecting (A.24)–(A.28) and J → ∞, we obtain
n
n n π
(q)
2
2J+1 −1 π 2
n−1
fi (ω) dω
EQ(fˆi fi ) =
fi (ω) dω + 2−2q(J+1)λq n−1
n
T
−π
−π
i=1
i=1
i=1
+ o(2J /T + 2−2qJ )
Q.E.D.
This completes the proof of Theorem 6.
PROOF OF COROLLARY 1: The result follows from Theorem 5 because Assumption 9 implies
ˆ
2J /2J − 1 = oP (T −1/(2(2q+1)) ) = oP (2−J/2 ) where the nonstochastic finest scale J is given by 2J+1 ≡
Q.E.D.
max{[2αλ2q ζ0 (q)T ]1/(2q+1) 0} The latter satisfies the conditions of Theorem 5.
REFERENCES
ANDERSON, T. W., AND C. HSIAO (1982): “Formulation and Estimation of Dynamic Models Using
Panel Data,” Journal of Econometrics, 18, 47–82.
ANDREWS, D. W. K. (1991): “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation,” Econometrica, 59, 817–858.
BALTAGI, B. H. (2002): Econometric Analysis of Panel Data, 2nd Edition. New York: John Wiley.
BALTAGI, B. H., AND Q. LI (1991): “A Joint Test for Serial Correlation and Random Individual
Effects,” Statistical and Probability Letters, 11, 277–280.
(1995): “Testing AR(1) Against MA(1) Disturbances in an Error Component Model,”
Journal of Econometrics, 68, 133–151.
BALTAGI, B. H., Y.-J. CHANG, AND Q. LI (1992): “Monte Carlo Results on Several New and
Existing Tests for the Error Component Model,” Journal of Econometrics, 54, 95–120.
BAHADUR, R. R. (1960): “Stochastic Comparison of Tests,” Annals of Mathematical Statistics, 31,
276–295.
BERA, A. K., W. SOSA-ESCUDERO, AND M. YOON (2001): “Tests for the Error Component Model
in the Presence of Local Misspecification,” Journal of Econometrics, 101, 1–23.
BINDER, M., C. HSIAO, AND M. H. PESARAN (1999): “Likelihood Based Inference for Panel
Vector Autoregressions: Testing for Unit Roots and Cointegration in Short Panels,” Working
Paper, Department of Economics, University of Southern California.
BIZER, D. S., AND S. DURLAUF (1992): “Testing the Positive Theory of Government Finance,”
Journal of Monetary Economics, 26, 123–141.
BHARGAVA, A., L. FRANZINI, AND W. NARENDRANATHAN (1982): “Serial Correlation and Fixed
Effects Model,” Review of Economic Studies, 49, 533–549.
BLUNDELL, R., AND S. BOND (1998): “Initial Conditions and Moment Restrictions in Dynamic
Panel Data Models,” Journal of Econometrics, 87, 115–143.
BOX, G., AND D. PIERCE (1970): “Distribution of Residual Autocorrelations in Autoregressive
Integrated Moving Average Time Series Models,” Journal of the American Statistical Association, 65, 1509–1526.
1562
Y. HONG AND C. KAO
BREUSCH, T. S., AND A. PAGAN (1980): “The Lagrange Multiplier Test and Its Applications to
Model Specification in Econometrics,” Review of Economic Studies, 47, 239–253.
BROWN, B. M. (1971): “Martingale Central Limit Theorems,” Annals of Mathematical Statistics,
42, 59–66.
CHOI, I. (2002): “Instrumental Variables Estimation of a Nearly Nonstationary Error Component
Model,” Journal of Econometrics, 109, 1–32.
DAUBECHIES, I. (1992): Ten Lectures on Wavelets. Philadelphia: SIAM.
DAVIDSON, R., AND E. FLACHAIRE (2001): “The Wild Bootstrap, Tamed at Last,” Working Paper,
Department of Economics, Queen’s University.
DONOHO, D. L., AND I. M. JOHNSTONE (1994): “Ideal Spatial Adoption by Wavelet Shrinkage,”
Biometrika, 81, 425–455.
(1995a): “Adapting to Unknown Smoothness via Wavelet Shrinkage,” Journal of the
American Statistical Association, 90, 1200–1224.
(1995b): “Wavelet Shrinkage: Asymptopia,” Journal of the Royal Statistical Society, Series B, 57, 301–369.
DONOHO, D. L., I. M. JOHNSTONE, I. M. KERKYACHARIAN, AND D. PICARD (1996): “Density
Estimation by Wavelet Thresholding,” Annals of Statistics, 24, 508–539.
DURBIN, J., AND G. S. WATSON (1951): “Testing for Serial Correlation in Least Squares Regression: II,” Biometrika, 38, 159–178.
DURLAUF, S. (1991): “Spectral Based Testing for the Martingale Hypothesis,” Journal of Econometrics, 50, 1–19.
FAN, J. (1996): “Test of Significance Based on Wavelet Thresholding and Neyman’s Truncation,”
Journal of American Statistical Association, 91, 674–688.
GAO, H.-Y. (1997): “Choice of Thresholds for Wavelet Shrinkage Estimate of the Spectrum,”
Journal of Time Series Analysis, 18, 231–251.
GEWEKE, J. (1981): “The Approximate Slopes of Econometric Tests,” Econometrica, 49,
1427–1442.
GRANGER, C. W. J. (1969): “Investigating Causal Relations by Econometric Models and CrossSpectral Methods,” Econometrica, 37, 424–438.
(1996): “Evaluation of Panel Models: Some Suggestions from Time Series,” Working
Paper, Department of Economics, University of California, San Diego.
GRANGER, C. W. J., AND P. NEWBOLD (1977): Forecasting Economic Time Series. New York: Academic Press.
HAHN, J., AND G. KUERSTEINER (2002): “Asymptotically Unbiased Inference for a Dynamic
Panel Model with Fixed Effects when Both n and T Are Large,” Econometrica, 70, 1639–1657.
HALL, P., AND P. PATIL (1996): “On the Choice of Smoothing Parameter, Threshold and Truncation in Nonparametric Regression by Nonlinear Wavelet Methods,” Journal of the Royal Statistical Society, 58, 361–377.
HANNAN, E. (1970): Multiple Time Series. New York: John Wiley.
HÄRDEL, W., AND E. MAMMEN (1993): “Comparing Nonparametric versus Parametric Regression Fits,” The Annals of Statistics, 21, 1926–1947.
HERNÁNDEZ, E., AND G. WEISS (1996): A First Course on Wavelets. Boca Raton: CRC Press.
HJELLVIK, V., Q. YAO, AND D. TJØSTHEIM (1998): “Linearity Testing Using Local Polynomial
Approximation,” Journal of Statistical Planning and Inference, 68, 295–321.
HONG, Y. (1996): “Consistent Testing for Serial Correlation of Unknown Form,” Econometrica,
64, 837–864.
HONG, Y., AND C. KAO (2002): “Wavelet-Based Testing for Serial Correlation of Unknown Form
in Panel Models,” Working Paper, Department of Economics and Department of Statistical
Science, Cornell University.
HONG, Y., AND J. LEE (2001): “One-Sided Testing for ARCH Effects,” Econometric Theory, 17,
1051–1981.
HOROWITZ, J. L. (2001): “The Bootstrap,” in Handbook of Econometrics, Vol. 5, ed. by
J. J. Heckman and E. E. Leamer. New York: Elsevier, Ch. 52, 3159–3228.
SERIAL CORRELATION IN PANEL MODELS
1563
HSIAO, C. (2003): Analysis of Panel Data. Cambridge: Cambridge University Press.
JENSEN, M. J. (2000): “An Alternative Maximum Likelihood Estimator of Long-Memory
Processes Using Compactly Supported Wavelets,” Journal of Economic Dynamics and Control,
24, 361–387.
KAO, C., AND M.-H. CHIANG (2000): “On the Estimation and Inference of a Cointegrated Regression in Panel Data,” Advances in Econometrics, 15, 179–222.
KAO, C., AND J. EMERSON (2004): “On the Estimation of a Linear Time Trend Regression with
a One-Way Error Component Model in the Presence of Serially Correlated Errors,” Journal of
Probability and Statistical Science, forthcoming.
LEE, J., AND Y. HONG (2001): “Testing for Serial Correlation of Unknown Form Using Wavelet
Methods,” Econometric Theory, 17, 386–423.
LI, Q., AND C. HSIAO (1998): “Testing Serial Correlation in Semiparametric Panel Data Models,”
Journal of Econometrics, 87, 207–237.
MASRY, E. (1994): “Probability Density Estimation from Dependent Observations Using Wavelet
Orthonormal Bases,” Statistics and Probability Letters, 21, 181–194.
(1997): “Multivariate Probability Density Estimation by Wavelet Methods: Strong Consistency and Rates for Stationary Time Series,” Stochastic Processes and Their Applications, 67,
177–193.
NEUMANN, M. H. (1996): “Spectral Density Estimation via Nonlinear Wavelet Methods for Stationary Non-Gaussian Time Series,” Journal of Time Series Analysis, 17, 601–633.
NEWEY, W., AND K. WEST (1994): “Automatic Lag Selection in Covariance Matrix Estimation,”
Reviews of Economic Studies, 61, 631–653.
PHILLIPS, P. C., AND H. R. MOON (1999): “Linear Regression Limit Theory for Nonstationary
Panel Data,” Econometrica, 67, 1057–1112.
PRIESTLEY, M. B. (1981): Spectral Analysis and Time Series. New York: Academic Press.
RAMSEY, J. (1999): “The Contribution of Wavelets to the Analysis of Economic and Financial
Data,” Philosophical Transactions Royal Society of London, Series A, 537, 2593–2606.
SPOKOINY, V. G. (1996): “Adaptive Hypothesis Testing Using Wavelets,” The Annals of Statistics,
24, 2477–2498.
WALTER, G. (1994): Wavelets and Other Orthogonal Systems with Applications. Boca Raton:
CRC Press.
WANG, Y. (1995): “Jump and Sharp Cusp Detection by Wavelets,” Biometrika, 82, 385–397.
WATSON, M. W. (1993): “Measures of Fit for Calibrated Models,” Journal of Political Economy,
101, 1011–1041.