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 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


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.



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.



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.



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.



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



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.




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



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.




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



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




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:




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 +




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.




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)

)




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.




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.




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.




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.




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.



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.




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.




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.




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



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).



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 %.




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



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.




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 → ∞.



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.




(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 .




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 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.



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.



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



zit E[∫

WiZW′iZ ]

0 12

1 12

µi16Ik

(µi, t)115Ik



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


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.



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.



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 .



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)



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).





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).




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.




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).




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.


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