A Generalized Nonlinear IV Unit Root Test for Panel Data with Cross-Sectional Dependence

Shaoping Wang

School of Economics, Huazhong University of Science and Technology,WuHan, China

Peng Wang

Department of Economics,New York University, New York, NY, U.S.A.

Jisheng Yang

School of Economics, Huazhong University of Science and Technology,WuHan, China

Zinai Li

School of Economics and Management, Tsinghua University, Beijing, China

Introduction

• Panel unit root test: • Survey by Hurlin and Mignon (2004)

• Assume cross sectional independence• Levin and Lin (1992,1993); Levin, Lin and Chu (2

002); Harris and Tzavalis (1999); Im, Pesaran and Shin (1997, 2003); Maddala and Wu (1999); Choi (1999,2001)

• Panel unit root test : Assuming cross sectional dependence

• Flôres, Preumont and Szafarz (1995): The test statistics are based on a seemingly unrelated regres

sion (SUR), and follow some nonstandard asymptotic distributions.

• Tayor and Sarno (1998): They propose a Wald-type test, but the asymptotic distributi

on of the test statistics is unknown. • Breitung and Das (2004): They apply the ADF test to pooled samples with robust stan

dard errors.

Introduction

• Panel unit root test : Assumming cross sectional dependence

• Bai and Ng (2001, 2004); Moon and Perron (2004); Phillips and Sul (2003):

They all assume that the dependence of the cross-sectional units comes from some common factors, and employ the principal components method to eliminate the common factors (hence the correlation of cross-sectional units), and then apply the ADF type test.

Introduction

• Panel unit root test : Assumming cross sectional dependence

• Choi (2002)

He models the cross-sectional dependency by time-invariant common factors, and employs the demeaning or detrending method developed by Elliott, Rothenberg and Stock (1996) to eliminate the common factors, and then applies the combining

p-value test of Choi (2001).

Introduction

Motivation

• Chang (2002)• Bai and Ng (2004)

This Paper

A Two Step Test

Improved the performance

This paper, following Bai & Ng (2004) and Chang (2002),(1) apply Bai & Ng’s common factor method to eliminate the

cross-sectional dependency;(2) employ Chang’s NIV estimation to the treated data to

form the test which convergence to the standard normal distribution.

Bai & Ng (2002, 2004)’s Model

ittiit eFy '

tt uLCFL )()1(

ititii eLLW )1)(( j

j j LCLC

0)(

They model the cross-sectional dependency by common factor:

Ft is the r×1 common factor among individuals

After eliminating the cross-sectional dependency by the method Of principal component , and then employ the combining p-valuetest of ADF statistics. The statistics has the normal standard limitingDistribution.

Chang’s Model

• (1)

• : coefficient on the lagged dependent variable

• : error term which follows the AR(p) process:

(2)

• : lag operator

• : autoregressive coefficient

• : some integer that is known and fixed

, 1it i i t ity y u 1, ,i N Tt ,1

itu

ititi uLW )(

p

k

kki

i zzW1 ,1)(

L},...,2,1;,...,2,1,{ , Nipkki

p

i

Model Assumptions

• To ensure the AR(p) process in (2) is invertible• Assumption 1: for all and

• To restrict the distribution of error term• Assumption 2: Denote

• (1) are independent and identically distributed and its distribution is absolutely continuous with respect to Lebesgue measure

• (2) has mean zero and covariance matrix• (3) satisfies for some and has a • characteristic function that satisfies for some

0)( zW i 1z 1, ,i N

)',...,( 1 NttNt

TtNt ,...,2,1,

Nt )( ijNt }|{| lN

tE 4l 0)(lim

0

NIV Estimation

• OLS estimation: • Under , the asymptotic distribution of obtaine

d from (3) is asymmetric, and not the usual t-distribution

• NIV estimation:• as instrument for , where is some fu

nction satisfying

Assumption 3: is regularly integrable and satisfy

0H ˆi

t

, 1i ty )( 1, tiyG G

)(xG

0)(xxG

• Under assumption 1-3, Chang draw the key result: as

and are asymptotically uncorrelated regardless of the cross sectional dependence

• And the test statistic

has a limiting standard normal distribution

N

iN it

NS

1

1

Chang Test

T

it j

t

0])(][)([1

1,1,

T

tjttiitti yGyGT

Findings about Chang (2002)

• The bigger the N is, the Smaller the correlation coefficient of cross-sectional units becomes.

• The test statistic does not fully follow the limiting standard normal distribution when the cross sectional dependence is strong.

• Chang test perform well in finite samples when when the cross sectional dependence is low.

NS

Our Test: A Two Step Test

Step1: eliminate the cross sectional dependence through the method of principal components suggested by Bai & Ng.

Step2: Apply Chang’s NIV estimation to the treated data, and form our test statistics which convergence to the standard normal distribution.

Model SettingAdopting the DGP in Bai and Ng (2004) to model the cross- s

ectional dependency by common factor:

or (3)

Where: Ft is the r×1 common factor among individuals. Error term has zero mean with covarianc

e matrix , for .

eit is the panel data after eliminating the common factor when the common factor and its loading vector are known.

)( ij

ittiiti zFyL ))(1(

ititi zLW )(

0ij ji

'1 ),,( Ntt

tiitit Fye ititi zeL )1(

We are interested to test:

• for all

VS

• for some

Hypothesis

:0H 1i 1, ,i N

:1H 1i i

Test procedure: Step one

• Under the null hypothesis:

• The differenced common component estimator of

is times the eigenvectors corresponding to the largest r eigenvalue of the matrix , the estimated loading matrix is given by . .

ittiit zFy

'2 ),,( tFF 1T

'))(( yy )1/()(),,(ˆ ''

1

TFyN

Step one (cont’d)

• the data with weak (or no) cross-sectional dependence

where can be set as 0.

• After eliminating the common factor, model (3) can be rewritten as:

(4)

• Model (4) is of the same form as Chang’s model (1), but with a different error term.

tiitit Fye ˆ

t

s st FF2

ˆ

1F

ititi

ititi

zLW

zeL

ˆˆ)(

,ˆˆ)1(

Step two

• Denote , , , ,

where .

We have the model with no (weak) cross-sectional

dependence

(5)

1,

,

1,

ˆ

ˆ

ˆ

Ti

pi

i

e

e

ei

iiiiii ee ˆˆˆˆ 1,

Ti

pi

i

e

e

ei

,

1,

ˆ

ˆ

ˆ

',

'1,

ˆ

ˆ

ˆ

Ti

pi

i

i

)ˆ,,ˆ( ,1,'

iptitiit ee

Ti

pi

i

i

,

1,

ˆ

ˆ

ˆ

Step Two (cont’d)

• Denote , The NIV estimator for (5) is:

• t-ratio of : , is the variance estimator of .

• Test statistics: .*

1

1ˆ ˆi

N

N iS t

N

i

i

i

Vt i

ˆ

1ˆˆ

iV i

11 )')('()'()ˆ( iiiiiii ZVZZVZrVar

*'1' )(ˆˆ

ˆ iiiii

ii yZVZr

)ˆ,ˆ(ˆ,ˆ),ˆ(ˆ1,1, iiiiii eVeGZ

The Distribution• Suppose that Assumption 1 - 3 and Bai & Ng (2004)’s

assumptions A-E hold. We have:

)1(ˆ 2/12/1pitit eTeT

)1()ˆ(][

1

2/1pitit

Tr

t

T

)1()ˆ()ˆ(2

212

2

1p

T

titit

T

t

eTeT

Then, under those assumptions, the t-ratio of auto-regression coefficient in (5) is similar to its asymptotic distribution in Chang’s model (1).

The Distribution• Theorem 1. Suppose that Assumption 1 - 3 and Bai & Ng (2004)’s

assumptions A-E hold. Under the null hypothesis of panel unit root, we obtain, as ,

for all and , where denote the correlation coefficient.

Theorem 2. Suppose that Assumption 1 - 3 Bai & Ng (2004)’s assumptions A-E hold. Under the null hypothesis of panel unit root and as , we obtain

Extend Theorem 1 and 2 to panel data models with individual intercept and/or time trend by de-meaning and/or de-trending schemes

T

ˆ (0,1)dt N

1, ,i N ˆ ˆ( , ) 0i j pcorr t t corr

T

)1,0(ˆ11

* NtN

SN

i dN i

Simulation• DGP with General Cross-sectional dependence

(6)

The covariance matrix of ,

, 1it i i t itz z ]4.0,2.0[~ Ui

, 1it i i t ity y z 1, ,i N Tt ,1

'),,( Ntitt

dependencetionalcrossstrongfor

dependencetionalcrossweakfor

ceindependentionalcrossfor

ij

ii

sec8.0

sec3.0

sec0

1

)( ij

DGP with General cross sectional dependence

• for size evaluation,

for power evaluation

• The number of common factor is set as 1 for eliminating the cross-sectional dependency by method of principal components.

1i]99.0,85.0[~ Ui

Simulation (cont’d)

DGP with General cross sectional dependence

Size no intercept and no linear trend

• The empirical sizes of BN test and our test in all cases are fairly close to the nominal sizes (we pick up 5%).

• The distortions of Chang test are more pronounced when the cross sectional dependence is high (e.g., 0.8).

N = 3 0 5 % te s t T = 5 0 T = 1 0 0

S N* 0 .0 6 7 0 .0 4 3

S N 0 .4 1 9 0 .4 0 8 8.0ij

B N 0 .0 8 2 0 .0 6 2

S N* 0 .0 6 6 0 .0 6 4

S N 0 .1 4 4 0 .1 4 1 3.0ij

B N 0 .0 7 3 0 .0 6 8

S N* 0 .0 8 7 0 .0 7 2

S N 0 .0 6 4 0 .0 6 0 0ij

B N 0 .0 7 1 0 .0 8 7

Simulation (cont’d)

• The three tests have reasonably good power in all designs and the power increases as N and T increase.

• The power of BN test is a little lower than the power of our test in some cases.

no intercept and no linear trend ~ Uniform[0.85, 0.99]. i

General cross sectional dependence

Power

N = 3 0 5 % te s t T = 5 0 T = 1 0 0

S N* 0 .9 3 6 0 .9 8 9

S N 0 .9 1 7 0 .9 9 7 8.0ij

B N 0 .8 6 7 0 .9 7 6

S N* 1 1

S N 0 .9 9 8 1 3.0ij

B N 0 .9 9 9 1

S N* 1 1

S N 1 1 0ij

B N 1 1

Simulation (cont’d)

Simulation DGP of Chang (2002)

• DGP of Chang (2002) is also based on (6) with the covariance matrix generated randomly.

• Size

• Both tests are very close to each other and approximate the nominal test sizes very well.

T=50 T=100

N tests 5% test 10% test 5% test 10% test

SN* 0.078 0.141 0.075 0.125 10

SN 0.077 0.134 0.076 0.124

SN* 0.085 0.143 0.075 0.125 20

SN 0.080 0.128 0.082 0.134

SN* 0.100 0.168 0.057 0.127 30

SN 0.069 0.116 0.068 0.111

no intercept and no linear trend

Simulation DGP of Chang (2002)

• Power

• Chang’s test is more powerful than our test when sample is small.

• The empirical power of two tests are high enough.

T=50 T=100

N tests 5% test 10% test 5% test 10% test

SN* 0.942 0.967 1 1 10

SN 0.978 0.992 1 1

SN* 0.999 1 1 1 20

SN 1 1 1 1

SN* 1 1 1 1 30

SN 1 1 1 1

no intercept and no linear trend

, 1

, 1

it i t it

it i i t it

it i i t it

y F e

e e

h

~ (0,1)tF iidN~ (1,1)i iidN'

1( , , )t t Nth h h The covariance matrix of , ( )h hij

~ [0.5,1.5]ii U

0,ij for i j

~ [0.2,0.4]i U

1i for size evaluation,

for power evaluation]99.0,85.0[~ Ui

Simulation DGP with one common factor

no intercept and no linear trend

The empirical sizes of two tests are almost identical and close to the nominal sizes.

Simulation (cont’d): Common factor dependence

Size

T=50

N tests 5% test 10% test

SN* 0.067 0.102 10

BN 0.063 0.101

SN* 0.073 0.121 20

BN 0.063 0.119

SN* 0.035 0.079 30

BN 0.048 0.078

no intercept and no linear trend

The empirical power of two tests are high enough.

Simulation (cont’d): Common factor dependence

Power

T=50

N tests 5% test 10% test

SN* 0.946 0.974 10

BN 0.864 0.917

SN* 0.996 0.998 20

BN 0.990 0.995

SN* 0.997 0.999 30

BN 0.994 0.997

• Idea: remove the cross sectional dependence before applying Chang test

• Method: first removing the cross-sectional dependence by method of principal component and then applying Chang NIV to the treated data

• Test Distribution: a limiting standard normal distribution under the null hypothesis of panel unit root

• Test Performance: (1) much better than Chang test when the cross sectional dependence is moderate to high; (2) as good as Chang test when the cross sectional dependence is low; (3) The finite sample performance is similar to that of BN test.

Conclusion

• Thank you!