Panel Unit Root and Cointegration Testsweb.nchu.edu.tw/~finmyc/Tim_panl_unit_p.pdf ·...
Transcript of Panel Unit Root and Cointegration Testsweb.nchu.edu.tw/~finmyc/Tim_panl_unit_p.pdf ·...
Panel Unit Roots Panel Cointegration Tests
Panel Unit Root and Cointegration Tests
Mei-Yuan Chen
Department of Finance
National Chung Hsing University
February 25, 2013
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Panel Unit RootsTesting for unit roots has become a standard procedure in
time series analyzes. For panel data, panel unit root tests have
been proposed by Levin and Lin (1992), Im, Pesaran and
Shin (1997), Harris and Tzavalis (1999), Madala and
Wu (1999), Choi (1999), Hadri (1999), and Levin, Lin and
Chu (2002). Originally, Bharagava et al. (1982) proposed
modified Durbin-Watson statistic based on fixed effect
residuals and two other test statistics on differenced OLS
residuals for random walk residuals in a dynamic model with
fixed effects. Boumahdi and Thomas (1991) generalized the
Dickey-Fuller (DF) test for unit roots in panel data.M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Various modified DF test statistics have been applied by
Breitung and Meyer (1994). Quah (1994) suggested a test for
unit root in a panel data without fixed effects when both N
and T go to infinity at the same rate such that N/T is
constant. Levin and Lin (1992) extended the model to allow
for fixed effects, individual deterministic trends and
heterogeneous serially correlated errors. Both T and N are
assumed to tend to infinity and N/T → 0. Phiilips and
Moon (1999b) pointed out the asymptotic properties of
estimators and tests proposed for nonstationary panels are
determined by the way of N and T tending to infinity.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
First, take one index, say N is fixed and T is allowed to
increase to infinity, an intermediate limit is obtained. Then by
letting N tend to infinity subsequently, a sequential limit
theory is established. Phillips and Moon (1999b) argued that
these sequential limits are easy to derive and are helpful in
extracting quick asymptotics. However, Phillips and
Moon (1999b) gave a simple example to illustrate how
sequential limits could mislead asymptotic results.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Second, Quah (1994) and Levin and Lin (1992) allowed N and
T to pass to infinity along a specific diagonal path in the two
dimensional array. The path can be determined by a
monotonically increasing functional relation of T = T (N).
Phillips and Moon (1999b) demonstrated that the limit theory
established by this assumption is dependent on the functional
relation T = T (N) and the assumed expansion path may not
provide an appropriate approximation for a given (T,N)
situation.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Third, a joint limit theory is established by allowing both N
and T tend to infinity simultaneously without placing specific
diagonal path restriction on the divergence. Phillips and
Moon (1999b) argued that, in general, joint limit theory is
more robust than either sequential limit or diagonal path limit.
The multi-index asymptotic theory in Phillips and
Monn (1999a, b) is applied to joint limits in which T,N → ∞and (T/N) → ∞, i.e., to situations where the time series
sample is large relative to the cross-section sample. However,
the general approach of Phillips and Moon (1999b) is also
applicable to situations (T/N) → 0 although different limit
results will usually be obtained.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Levin and Lin (1992) Tests
Levin and Lin (1992) TestsConsider the model
yit = ρiyit−1 + z′itγ + uit, i = 1, . . . , N, t = 1, . . . , T, (1)
where zit is the deterministic component and uit is a
stationary process. zit could be zero, one, the fixed effects, µi,
or fixed effects as well as a time trend, t. The Levin and
Lin (LL) tests assume that ρi = ρ for all i and are interesting
in testing the null hypothesis
H0 : ρ = 1
against the alternative hypothesis
Ha : ρ < 1.M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Levin and Lin (1992) Tests
The model in (1) can be written as, given ρi = ρ,
yit = δyit−1 + z′itγ + uit, i = 1, . . . , N, t = 1, . . . , T, (2)
where δ = ρ−1. Then the null of ρ = 1 is equivalent to δ = 0.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Levin and Lin (1992) Tests
Let ρ be the OLS estimator of ρ in (1) and define
zt =
z1t
z2t
...
zNt
=
1 µ1 t
1 µ2 t...
......
1 µN t
h(t, s) = z′t
(
T∑
t=1
ztz′t
)
zs,
uit = uit −T∑
s=1
h(t, s)uis
yit = yit −T∑
s=1
h(t, s)yis, (3)
then we have M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Levin and Lin (1992) Tests
and the corresponding t-statistic
tρ =(ρ− 1)
√
∑Ni=1
∑Tt=1 y
2i,t−1
se,
where
s2e =1
TN
N∑
i=1
T∑
t=1
u2it.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Levin and Lin (1992) Tests
Assume that there existing a scaling matrix DT and piecewise
continuous function Z(r) such that
D−1/2T
[Tr]∑
t=1
zt ⇒ Z(r)
uniformly for r ∈ [0, 1]. For a fixed N ,
1√N
N∑
i=1
1
T
T∑
t=1
yi,t−1uit ⇒1√N
N∑
i=1
∫
WiZdWiZ
and
1
N
N∑
i=1
1
T 2
T∑
t=1
y2i,t−1 ⇒1
N
N∑
i=1
∫
W 2iZ ,
as T → ∞, where WZ(r) =W (r)− [∫
WZ ′][∫
ZZ ′]−1Z(r)
is an L2 projection residual of W (r) on Z(r).M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Levin and Lin (1992) Tests
Assume that∫
WiZdWiZ and∫
W 2iZ are independent across i
and have finite moments. Then it follows that
1
N
N∑
i=1
∫
W 2iZ →p E
[∫
W 2iZ
]
1√N
N∑
i=1
(∫
WiZdWiZ − E
[∫
WiZdWiZ
])
⇒ N
(
0, var
(∫
WiZ
as N → ∞ by a law of large number and the Lindeberg-Levy
central limit theorem. By Levin and Lin (1992), we have
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Levin and Lin (1992) Tests
zit E[∫
WiZdWiZ] var[∫
WiZdWiZ ] E[∫
W 2iZ ] var[
∫
W 2iZ ]
0 0 12
12
13
1 0 13
12
?
µi−12
112
16
145
(µi, t)−12
160
115
116300
Using above results, Levin and Lin (1992) obtain the following
limiting distribution of√NT (ρ− 1) and tρ:
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Levin and Lin (1992) Tests
zit ρ
0√NT (ρ− 1) ⇒ N(0, 2) tρ ⇒
1√NT (ρ− 1) ⇒ N(0, 2) tρ ⇒
µi
√NT (ρ− 1) + 3
√N ⇒ N(0, 51/5)
√1.25tρ +
√
(µi, t)√N(T (ρ− 1) + 7.5) ⇒ N(0, 2895/112)
√
448/277(tρ +
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Levin and Lin (1992) Tests
Sequential limit theory, i.e., T → ∞ followed by N → ∞, is
used to derive the limiting distributions. In case uit is
stationary, the asymptotic distributions of ρ and tρ need to be
modified due to the presence of serial correlation.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Levin and Lin (1992) Tests
Harris and Tzavalis (1999) also derived unit root tests for (1)
with zit = 0, µi, or µi, t when the time dimension of
the panel, T , is fixed.
This is typical case for micro panel studies. The main results are
zit ρ
0√N(ρ− 1) ⇒ N
(
0, 2T (T−1)
)
µi
√N(
ρ− 1 + 3T+1
)
⇒ N(
0, 3(17T2−20T+17)
5(T−1)(T+1)3
)
(µi, t)√N(
ρ− 1 + 152(T+2)
)
⇒ N(
0, 15(193T2−728T+1147)
112(T+1)3(T−2)
)
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Levin and Lin (1992) Tests
Harris and Tzavalis (1999) also showed that the assumption
that T tends to infinity at a faster rate than N as in tests of
Levin and Lin (1992) rather than T fixed as in the case in
micro panels, yields tests which are substantially undersized
and have low power especially when T is small.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Levin and Lin (1992) Tests
Recently, tests of Levin and Lin (1992) have been
implemented to test the PPP hypothesis using panel data.
O’Connel (1998), however, showed that tests of Levin and
Lin (1992) suffered from significant size distortion in the
presence of correlation among contemporaneous
cross-sectional error terms. O’Connel highlighted the
importance of controlling for cross-sectional dependence when
testing for a unit root in panels of real exchange rates.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Levin and Lin (1992) Tests
Virtually all the existing nonstationary panel literature assume
cross-sectional independence. It is true that the assumption of
independence across i is rather strong, but it is needed in
order to satisfy the requirement of the Lindeberg-Levy central
limit theorem. Moreover, as pointed out by Quah (1994),
modeling cross-sectional dependence is involved because
individual observations in a cross-section have no natural
ordering. Driscoll and Kraay (1998) presented a simple
extension of common nonparametric covariance matrix
estimation techniques which are robust to very general forms
of spatial and temporal dependence as the time dimension
becomes large.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Levin and Lin (1992) Tests
Conley (1999) presented a spatial model of dependence among
agents using a metric of economic distance that provides
cross-sectional data with a structure similar to time-series
data. Conley proposed a generalized method of moment
(GMM) using such dependent data and a class of
nonparametric covariance matrix estimators that allow for a
general form of dependence characterized by economic
distance.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Levin and Lin (1992) Tests
It is worthy to mention that a method called PANIC (Panel
Analysis of Non-stationary in the Idiosyncratic and Common
components) has been suggested by Bai and Ng (2003) to
overcome the problem of cross-sectional dependence across i.
This method will be introduced later.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Im, Pesaran and Shin (1997) Tests
Im, Pesaran and Shin (1997) Tests
The tests of Levin and Lin (1992) are restrictive in the sense
that it requires ρ to be homogeneous across i. As
Maddala (1999) pointed out, the null may be fine for testing
convergence in growth among countries, but the alternative
restricts every country to converge at the same rate. Im,
Pesaran and Shin (1997) allow for heterosgeneous coefficient
of yit−1 and proposed an alternative testing procedure based
on the augmented DF tests when uit is serially correlated with
different serial correlation properties across cross-sectional
units, i.e., uit =∑pi
j=1 ψijuit−j + ǫit.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Im, Pesaran and Shin (1997) Tests
Substituting this uit in (1), we get
yit = ρiyit−1 +
pi∑
j=1
ψijyit−j + z′itγ + ǫit, i = 1, . . . , N, t = 1, . . . ,
The null hypothesis is
H0 : ρi = 1
for all i against the alternative hypothesis
Ha : ρi < 1
for at least one i.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Im, Pesaran and Shin (1997) Tests
The t-statistic suggested by Im, Pesaran and Shin (1997) is
defined as
t =1
N
N∑
i=1
tρi , (5)
where tρi is the individual t-statistic of testing H0 : ρi = 1 in
(4). It is known that for a fixed N ,
tρi ⇒∈10 WiZdWiZ
[∈10 W
2iZ ]
1/2= tiT (6)
as T → ∞. Im, Pesaran and Shin (1997) assume that tiT are
i.i.d. and have finite means and variances.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Im, Pesaran and Shin (1997) Tests
Then√N(
1N
∑Ni=1 tiT − E[tiT |ρi = 1]
)
√
var[tiT |ρi = 1]⇒ N(0, 1)
as N → ∞ by the Lindeberg-Levy central limit theorem.
Hence, the test statistic of Im, Pesaran and Shin (1997) has
the limiting distribution as
tIPS =
√N(t− E[tiT |ρi = 1])√
var[tiT |ρi = 1]⇒ N(0, 1)
as T → ∞ followed by N → ∞ sequentially. The values of
E[tiT |ρi = 1] and var[tiT |ρi = 1] have been computed by Im,
Pesaran and Shin (1997) via simulations for different values of
T and pi’s.M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Im, Pesaran and Shin (1997) Tests
The local power of tests of Levin and Lin (1992) and Im,
Pesaran and Shin (1997) has been investigated by
Breitung (2000). Breitung (2000) finds that those tests suffer
from a dramatic loss of power if individual specific trends are
included. Besides, the power of these tests is very sensitive to
the specification of the deterministic terms. McCoskey and
Selden (1998) applied the test of Im, Pesaran and Shin (1997)
for testing unit root for per capita national health care
expenditures and gross domestic product for a panel of OECD
countries and the null is rejected for both panels. Gerdtham
and Lothgren (1998) obtained different results by adding a
time trend into the regression model (4).
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Combining p-value Tests
Combining p-value Tests
Let Gi,Tibe a unit root test statistic for the i-th group in (1)
and assume that as Ti → ∞, Gi,Ti⇒ Gi. Let pi be the
p-value of a unit root test for cross-section i, i.e.,
pi = 1− F (Gi,Ti), where F (·) is the distribution function of
the random variable Gi. Note that the null of unit root is not
rejected when the p-value is larger than 5 % if the significance
level is set at 95 %. In contrast, the null is rejected when
p-value is smaller than 5 %.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Combining p-value Tests
Maddala and Wu (1999) and Choi (1999) proposed a Fisher
type test:
P = −2N∑
i=1
ln pi (7)
which combines the p-value from unit root tests for each
cross-section i to test for unit root in panel data. P is
distributed as χ2 with 2N degrees of freedom as Ti → ∞ for
all N . When pi closes to 0 (null hypothesis is rejected), ln pi
closes to −∞ so that large value P will be found and then the
null hypothesis of existing panel unit root will be rejected. In
contrast, when pi closes to 1 (null hypothesis is not rejected),
ln pi closes to 0 so that small value P will be found and then
the null hypothesis of existing panel unit root will not rejected.M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Combining p-value Tests
Choi (1999) pointed out the advantages of the Fisher test: (1)
the cross-sectional dimension, N , can be either finite or
infinite, (2) each group can have different types of
nonstochastic and stochastic components, (3) the time series
dimension, T can be different for each i (imbalance panel
data), and (4) the alternative hypothesis would allow some
groups to have unit roots while others may not. A main
disadvantage involved is that the p-value have to be derived by
Monte Carlo simulations.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Combining p-value Tests
When N is large, Choi (199) also proposed a Z test,
Z =
1√N
∑Ni=1(−2 ln pi − 2)
2(8)
since E[−2 ln pi] = 2 and var[−2 ln pi] = 4. Assume pi’s are
i.i.d. and use the Lindeberg-Levy central limit theorem to get
Z ⇒ N(0, 1)
as Ti → ∞ followed by N → ∞.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Residual Based LM Test
Residual Based LM TestHadri (1999) proposed a residual based Lagrange
Multiplier (LM) test for the null that the time series for each i
are stationary around a deterministic trend against the
alternative of a unit root in panel data. Consider the following
model
yit = z′itγ + rit + ǫit (9)
where zit is the deterministic component, rit is a random walk
rit = rit−1 + uit
uit ∼ i.i.d.(0, σ2u) and ǫit is a stationary process.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Residual Based LM Test
(9) can be written as
yit = z′itγ + eit (10)
where
eit =
t∑
j=1
uij + ǫit.
Let eit be the residuals from the regression in (10) and σ2e be
the estimate of the error variance. Also, let Sit be the partial
sum process of the residuals,
Sit =
t∑
j=1
eij .
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Residual Based LM Test
Then the LM statistic is
LM =1N
∑Ni=1
1T 2
∑Tt=1 S
2it
σ2e
.
It can be shown that
LM →p E
[∫
WiZ
]
as T → ∞ followed by N → ∞ provided E[∫
W 2iZ ] <∞.
Also,√N(LM − E[
∫
W 2iZ ])
√
var[∫
W 2iZ ]
⇒ N(0, 1)
as T → ∞ followed by N → ∞.M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Panel Cointegration Tests
Entorf (1997) studied spurious fixed effects regressions when
the true model involves independent random walks with and
without drifts. For T → ∞ and N finite, the nonsense
regression phenomenon holds for spurious fixed effects
mmodels and inference based on t-value can be highly
misleading. Kao (1999) and Phillips and Moon (1999) derived
the asymptotic distributions of the least squares dummy
variable estimator and various conventional statistics from the
spurious regression in panel data.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Consider a spurious regression model for all i using panel data:
yit = x′itβ + z − it′γ + eit,
where
xit = xit−1 + ǫit,
and eit is I(1). The OLS estimator of β is
β =
[
N∑
i=1
T∑
t=1
xitx′it
]−1 [ N∑
i=1
T∑
t=1
xityit
]
, (11)
where yit is defined in (3) and
xit = xit −T∑
t=1
h(t, s)xis.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
It is known that if a time-series regression for a given i is
performed in model (11), the OLS estimator of β is spurious.
It is easy to see that
1
N
N∑
i=1
1
T 2
T∑
t=1
xitx′it →p E
[∫
WiZW′iZ
]
Ωǫ
and
1
N
N∑
i=1
1
T 2
T∑
t=1
xityit →p E
[∫
WiZW′iZ
]
Ωuǫ
where
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
zit E[∫
WiZW′iZ ]
0 12
1 12
µi16Ik
(µi, t)115Ik
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Then we have
β →p Ω−1ǫ Ωuǫ. (12)
(12) shows that the OLS estimator of β, β, is consistent for
its true value, Ω−1ǫ Ωuǫ. This is due to the fact that the noise,
eit, is as strong as the signal, xit, since both eit and xit are
I(1). In the panel regression (11) with a large number of
cross-sections, the strong noise of eit is attenuated by pooling
the data and a consistent estimate of β can be extracted. The
asymptotics of the OLS estimator are very different from those
of the spurious regression in pure time series. This has an
important consequence for residual-based cointegration tests
in panel data, because the null distribution of residual-based
cointegration tests depends on the asymptotics of the OLSM.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Kao Tests
Kao Tests
Kao (1999) presented two types of cointegration tests in panel
data, the DF and ADF types tests. The DF type tests from
Kao can be calculated from the estimated residuals in (11) as:
eit = ρeit−1 + vit, (13)
where
eit = yit − x′itβ.
The null of no cointegration is represented as H0 : ρ = 1.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Kao Tests
The OLS estimate of ρ and the t-statistic are given as
ρ =
∑Ni=1
∑Tt=1 eiteit−1
∑Ni=1
∑Tt=1 e
2it−1
and
tρ =(ρ− 1)
√
∑Ni=1
∑Tt=1 e
2it−1
se,
where
s2e =
∑Ni=1
∑Tt=1(eit − ρeit−1)
2
TN.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Kao Tests
Kao (1999) proposed the following four DF type tests by
assuming zit = µi:
DFρ =
√NT (ρ− 1) + 3
√N√
10.2,
DFt =√1.25tρ +
√1.875N,
DF∗ρ =
√NT (ρ− 1) + 3
√Nσ2
v
σ2
0v√
3 + 36σ4v
5σ4
0v
,
DF∗t =
tρ +√6Nσv
2σ0v√
σ2v
2σ2
0v
+ 3σ2v
10σ2
0v
,
where σ2v = Σu − ΣuǫΣ
−1ǫ and σ2
0v = Ωu − ΩuǫΩ−1ǫ .
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Kao Tests
While DFρ and DFt are based on the strong exogeneity of the
regressors and errors, DF∗ρ and DF∗
t are for the cointegration
with endogenous relationship between regressors and errors.
For the ADF tests, the following regression is considered
eit = ρeit−1 +
p∑
j=1
ψjeit−j + vit. (14)
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Kao Tests
With the null hypothesis of no cointegration, the ADF test
statistics can be constructed as:
ADF =tADF +
√6Nσu
2σ0v√
σ2
0v
2σ2v+ 2σ2
v
10σ2
0v
where tADF is the t-statistic of ρ in (14). The asymptotic
distributions of DFρ, DFt, DF∗ρ, DF
∗t and ADF converge to a
standard normal distribution N(0, 1).
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Residual Based LM Test
Residual Based LM Test
McCoskey and Kao (1998) derived a residual-based test for
the null of cointegration in panels. For the residual based test,
it is necessary to use an efficient estimation technique of
cointegrated variables. There are several methods have been
shown to be efficient asymptotically in literature. These
include the fully modified estimator of Phillips and
Hansen (1990) and the dynamic least squares estimator of
Saikkonen (1991) and Stock and Watson (1993).
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Residual Based LM Test
The model considered in McCoskey and Kao (1998) allows for
varying slopes and intercepts:
yit = αi + x′itβi + eit, (15)
xit = xit−1 + ǫit (16)
eit = rit + uit, (17)
rit = rit−1 + θuit,
where uit are i.i.d. (0, σ2u). The null hypothesis of
cointegration is equivalent to θ = 0.
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Residual Based LM Test
The test statistic proposed by McCoskey and Kao (1998) is
defined as:
LM =1N
1T 2
∑Tt=1 S
2it
σ2e
, (18)
where Sit is the partial sum process of the residuals,
Sit =t∑
j=1
eij
and σ2e is defined in McCoskey and Kao (1998).
M.-Y. Chen Panel Unit
Panel Unit Roots Panel Cointegration Tests
Residual Based LM Test
The asymptotic result for the test is:
√N(LM− µv) ⇒ N(0, σ2
v).
The moments, µv and σ2v , can be found through Monte Carlo
simulation. The limiting distribution of LM is then free of
nuisance parameters and robust to heteroskedasticity.
M.-Y. Chen Panel Unit