Nelson and Plosser Revisited:

A Re-Examination using OECD PanelData

Christophe Hurlin

November 2004

1

1 MOTIVATIONSTwenty two years after the seminal paper by Nelson andPlosser (1982), why testing the presence of a unit root inthe same macroeconomic series by using a panel of OECDcountries?

• Testing the homogeneity of the unit root result in an in-ternational panel framework

• Technical reason: the power deficiencies of pure timeseries-based tests for unit roots and cointegration, evenfor the ’’new generation’’ of powerful procedures of test(ADF-GLS in Elliott, Rothenberg and Stock, 1996, Max-ADF in Leybourne, 1995).

• A new generation of panel unit root tests allows to takeinto account the genuine international dimension of a panel.

– These tests relax the restrictive assumption of cross-sectional independence.

– The international, sectoral or regional co-movements ofthe economic series, largely documented since for in-stance Backus and Kehoe (1992).

2

The cross-sectional independence assumption in panelunit root tests

• Two generations of panel unit root tests can now be dis-tinguished (Hurlin et Mignon, 2004)

• The common feature of first generation tests is the re-striction that all cross-sections are independent.

• Under this independence assumption the Lindberg-Levycentral limit theorem or other central limit theorems canbe applied to derive the asymptotic normality of paneltest statistics.

• Levin and Lin (1992, 1993); Levin, Lin and Chu (2002);Choi (2001); Im, Pesaran and Shin (1997, 2003).

• However, this cross-sectional independence assumptionis quite restrictive in many empirical applications.

=⇒ More generally, this assumption raises the issueof the validity of the panel approach in macroeconomic,finance or international finance.

3

MESSAGES

=⇒ Testing the unit root in a panel with internationalshocks or international dependences is not economicallyequivalent to a collection of individual time series testsor to test unit root in panel under the cross-sectional in-dependence assumption.

=⇒ Parallel between the dichotomy cross-sectional in-dependence versus correlations in panel unit root testsand the dichotomy general versus partial equilibriumsin macroeconomics.

=⇒ So, it is not only a technical problem of power andsize to know if (i) the unit root must be tested in panel orin time series and (ii) if it is necessary to consider cross-sectional dependent processes. This a genuine economicissue linked to the importance of international, regional,sectorial or individual dependencies in the dynamics.

For these reasons, we propose here a re-examination ofthe seminal work of Nelson and Plosser for the OECD, basedon panel unit root tests without and with cross-sectional de-pendencies.

4

A second generation of panel unit root tests

• The second generation panel unit root tests relax thecross-sectional independence assumption.

• The first issue is to specify the cross-sectional depen-dencies, since as pointed out by Quah (1994), individualobservations in a cross-section have no natural ordering.

• The second problem is that the usual t-statistics unit roottests have limit distributions that are dependent in a verycomplicated way upon various nuisance parameters defin-ing correlations across individual units.

• We distinguish two groups of tests: the first group testsare based on a dynamic factor model or an error-componentmodel . The cross-sectional dependency is then due to thepresence of one or more common factors or to a randomtime effect.

• The tests of the second group are defined by oppositionto these specifications based on common factor or timeeffects. In this group, some specific or more general spec-ifications of the cross-sectional correlations are proposed.

5

=⇒ In this paper, these various panel unit root tests are ap-plied to OECD panel databases for the same 14 macroeco-nomic and financial variables as those considered in Nelsonand Plosser (1982).

=⇒ The period considered is 1950-2000 : the issue ofbreakpoint

=⇒Our results highlight the importance of (i) the het-erogeneous specification of the model and (ii) the cross-sectional independence assumption.

=⇒For some macroeconomic variables generally consid-ered as non-stationary, such as the real GDP for instance, thenull of unit root is strongly rejected for our OECD samplewith first generation tests. On the contrary, when the in-ternational cross-correlations are taken into account in thedynamic analysis, the results are clearly more in favour ofthe unit root for all the considered variables, including theunemployment rate.

6

2 FIRST GENERATION UNIT ROOT TESTS

2.1 Im, Pesaran and Shin unit root tests

• The well-known IPS test (1997, 2003) is now avalaibleunder usual software (Eviews 5.0) Their model with in-dividual effects and no time trend is:

∆yit = αi + ρiyi,t−1 +pi

z=1

βi,z∆yi,t−z + εit (1)

• The null hypothesis is defined as H0 : ρi = 0 for all i =1, ..N and the alternative hypothesis is H1 : ρi < 0 fori = 1, ..N1 and ρi = 0 for i = N1 + 1, .., N, with 0 <N1 ≤ N .

• Their test is based on the (augmented) Dickey-Fuller sta-tistics averaged across groups. Let tiT (pi,βi) with βi =βi,1, ..,βi,pi denote the t-statistic for testing unit root in

the ith country, the IPS statistic is:

t barNT =1

N

N

i=1

tiT (pi,βi) (2)

7

• Under the crucial assumption of cross-sectional inde-pendence, the statistic t barNT is shown to sequentiallyconverge to a normal distribution whenT tends to infin-ity, followed by N . A similar result is conjectured whenN and T tend to infinity while the ratio N/T tends to afinite non-negative constant.

• In order to propose a standardization of the t-bar statis-tic, IPS have to compute the values of E [tiT (pi, βi)] andV ar [tiT (pi,βi)].

• Two solutions can be considered: the first one is based onthe asymptotic moments E (η) and V ar (η). The corre-sponding standardized t-bar statistic is denoted Zt bar.

• The second solution is to carry out the standardizationof the t-bar statistic using the means and variances oftiT (pi, 0) evaluated by simulations under the null ρi = 0.

Wtbar =

√N t barNT −N−1 Ni=1E [tiT (pi, 0)| ρi = 0]

N−1 Ni=1 V ar [tiT (pi, 0)| ρi = 0]d−→

T,N→∞N (0, 1)

• Although the testsZtbar andWtbar are asymptotically equiv-alent, simulations show that the Wtbar statistic performsmuch better in small samples.

8

RESULTATS

• If we consider the standardized statistic Wtbar, the unitroot hypothesis is not rejected for 8macroeconomic vari-ables out of 14 at a 5% significance level: nominal GDP,real per capita GDP, employment GDP deflator, consumerprices, velocity, bond yield and common stock prices.

• Except for the nominal GDP, the results are robust to theuse the standardized statistic Ztbar based on asymptoticmoments instead ofWtbar.

• More surprising, except for the nominal GDP and the un-employment rate, the results are also robust when we con-sider the statistic ZDFt bar based on the average of DickeyFuller individuals statistics.

• The results are globally robust to the specification of de-terministic component.

• Special care need to be exercised when interpreting theresults of the 6 variables for which the null hypothesis isrejected. Due to the heterogeneous nature of the alterna-tive, rejection of the null hypothesis does not necessarilyimply that the non stationarity is rejected for all countries.

9

2.2 Fisher type unit root tests

• Idea: testing strategy based on combining the observedsignificant levels from the individual tests (p-values)=⇒Fisher (1932) type tests

• Choi (2001) and Maddala and Wu (1999).• Let us consider pure time series unit root test statistics

(ADF, ERS, Max-ADF etc.). If these statistics are contin-uous, the corresponding p-values, denoted pi, are uniform(0, 1) variables. Consequently, under the assumption ofcross-sectional independence, the statistic proposed byMaddala and Wu (1999) and defined as:

PMW = −2N

i=1

log (pi) (3)

has a chi-square distribution with 2N degrees of free-dom, when T tends to infinity and N is fixed.

• For largeN samples, Choi (2001) proposes a similar stan-dardized statistic:

ZMW =

√N N−1PMW −E [−2 log (pi)]

V ar [−2 log (pi)]= −

Ni=1 log (pi) +√

N(4)

10

RESULTATS TESTS DE FISHER

• The results confirm our previous conclusions. If we con-sider the PMW test at a 5% significant level, we do notreject the unit root for 7 out of 14 variables.

• The only difference with the IPS results is for the nominalGDP, for which we reject the null here. This is preciselythe only variable for which the two IPS standardized sta-tistics,Wtbar andZtbar, do not give the same conclusions.

• Except for the real per capita GDP, the conclusions areidentical with the Choi’s standardized statistic.

• The results are globally robust to the specification ofthe deterministic component, except for industrial pro-duction, nominal GNP and money.

• Contrary to the IPS tests, the Fisher tests lead to the re-jection of the null for unemployment rate in a model withtime trends. This point clearly indicates the ambiguity ofthe non stationarity analysis for this variable.

11

CONCLUSION PART I

• With the panel unit root tests based on the cross-sectionalindependence assumption, the conclusions on the nonstationarity of OECD macroeconomic variables are noclear-cut.

• The unit root hypothesis is strongly rejected for 4macro-economic variables (real GDP, wages, real wages andmoney stocks), which are generally considered as non sta-tionary for the most of OECD countries.

• The non stationarity is also rejected for the unemploy-ment rate, but in this case, it is not surprising given thetimes series results (Nelson and Plosser, 1982).

• The non stationarity is robust to the choice of the testand the choice of the standardization only for 6 variables(employment, GDP deflator, consumer prices, velocity,bond yield and common stock prices).⇒ So, we are far from the general results obtained by

Nelson and Plosser.

• Are these surprising results due to the restrictive assump-tion of cross-sectional independence used to derive theasymptotic normality of the test statistics?

12

• Obviously, these first generation tests are likely to yieldbiased results if applied to panels with a cross-sectionaldependency.

• First intuition: Maddala and Wu (1999) =⇒ importantsize distortions.

size = Pr [H1/H0 true] (5)

• Two solutions:– Adaptation of first generation unit root tests (Maddala

and Wu, 1999)

– Development of new tests

• How to specify these cross-sectional dependencies?– Metric of economic distance (Conley, GMM 1999)

– With a factor structure

– Others approaches

13

3 A SECOND GENERATION UNIT ROOT TESTS3.1 Tests based on factor structure

• For all these tests, the idea is to shift data into two un-observed components: one with the characteristic that isstrongly cross-sectionally correlated and one with the char-acteristic that is largely unit specific.

• The testing procedure is always the same and consists intwo main steps: in a first one, data are de-factored, and ina second step, panel unit root test statistics based on de-factored data and/or common factors are then proposed.

• These statistics do not suffer from size distortions as thosewhich affect the standard tests based the cross-sectionalindependence assumption when common factors exist inthe panel.

14

• In this context, the unit root tests by Bai and Ng (2001,2004) provide a complete procedure to test the degree ofintegration of series.

• yit = a deterministic component + common compo-nent expressed as a factor structure + error idiosyn-cratic.

• Instead of testing for the presence of a unit root di-rectly in yit, Bai and Ng propose to test the commonfactors and the idiosyncratic components separately.

• For that, Bai and Ng have to use a decomposition methodof the data which is robust to the degree of integrationof the common or idiosyncratic components.

• Bai and Ng accomplish this by estimating factors on first-differenced data and cumulating these estimated factors.

15

• Let us consider a model with individual effects and notime trend:

yit = αi + λiFt + eit (6)where Ft is a r × 1 vector of common factors and λi is avector of factor loadings.

• The corresponding model in first differences is:∆yit = λi ft + zit (7)

where zit = ∆eit and ft = ∆Ft with E (ft) = 0.

• The common factors in ∆yit are estimated by the princi-pal component method. Let us denote ft these estimates,λi the corresponding loading factors and zit the estimatedresiduals.

• Then, the ’differencing and re-cumulating’ estimationprocedure is based on the cumulated variables definedas:

F̂mt =t

s=2

f̂ms êit =t

s=2

zis (8)

for t = 2, .., T, m = 1, .., r and i = 1, .., N.

• Bai and Ng test the unit root hypothesis in the idio-syncratic component eit and in the common factors Ftwith the estimated variables F̂m t and êi t.

16

• To test the non stationarity of the idiosyncratic component∆êit = δi,0êi, t−1+ δi,1∆êi, t−1+ ..+ δi,p∆êi,t−p+µit (9)Let ADFce (i) be the ADF t-statistic for the idiosyncraticcomponent of the ith country.

• The asymptotic distribution of ADFce (i) coincides withthe Dickey Fuller distribution for the case of no constant.

=⇒ Therefore, a unit root test can be done for eachidiosyncratic component of the panel.

=⇒ The great difference with unit root tests based onthe pure time series is that the common factors, as globalinternational trends or international business cycles forinstance, have been withdrawn from data.

• Example : real GDP (table ??). For 12 countries, theconclusions of both tests are opposite at a 5% significantlevel: for 8 countries, the ADF tests on the initial serieslead to reject the null, whereas the idiosyncratic compo-nent is founded to be non-stationary.

17

• Individual time series tests have the same low power asthose based on initial series =⇒ pooled tests are alsoproposed.

• These tests are similar to the first generation ones.However, the great difference is that the estimated idio-syncratic components êi,t are asymptotically indepen-dent across units

• Let denote pce (i) the p-value of the ADFce (i) test, thisstatistic is:

Zce =− Ni=1 log [pce (i)]−N√

N

d−→T,N→∞

N (0, 1) (10)

• RESULTATS(1) At a 5% significant level, the non stationarity of idio-

syncratic components is not rejected only for 6 outof 14 variables (industrial production, employment,consumer prices, real wages, velocity and common stockprices).

(2) It implies that if the macroeconomic series are non-stationary, this property seems to be more due to thecommon factors, as international business cycles orgrowth trends, than to the idiosyncratic components.

18

Testing the non-stationarity of the common factors

• Bai and Ng (2004) distinguish two cases1.• When there is only one common factor among the N

variables (r = 1), they use a standard ADF test in amodel with an intercept.

∆F̂1t = c+γi,0 F̂1,t−1+γi,1∆F̂1,t−1+ ..+γi,p∆F̂1,t−p+vit(11)

The corresponding ADF t-statistic, denoted ADFcF

, hasthe same limiting distribution as the Dickey Fuller test forthe constant only case.

• If there are more than one common factors (r > 1), Baiand Ng test the number of common independent sto-chastic trends in these common factors, denoted r1.

• If r1 = 0 it implies that there areN cointegrating vectorsfor N common factors, and that all factors are I(0).

• Bai and Ng(2004) propose two statistics based on the rdemeaned estimated factors F̂m t form = 1, ..,m. Thesestatistics are similar to those proposed by Stock and Wat-son (1988).

1 In the first working paper (Bai and Ng, 2001), the pro-cedure was the same whatever the number of commonfactors and was only based on ADF tests.

19

RESULTATS

• There is only one common factor in real GDP and inreal per capita GDP, which can be analyzed as an inter-national stochastic growth factor. For both variables,this common factor is found to be non stationary.

• For all the others variables, the estimated number of com-mon factors ranges from 2 to 4. Whatever the test used,MQc orMQf , the number of common stochastic trendsis always equal to the number of common factors, as re-ported on tables ?? and ??.

• These results are robust to the choice of the number ofcommon factors.=⇒ All the macroeconomic series are I(1) as in NP

(1982)

20

3.2 Other Approaches: Chang nonlinear IV unit roottests

• There is a second approach to model the cross-sectionaldependencies, which is more general than those based ondynamic factors models or error component models.

• It consists in imposing few or none restrictions on the co-variance matrix of residuals. O’Connell (1998), Maddalaand Wu (1999), Taylor and Sarno (1998), Chang (2002,2004).

• Such an approach raises some important technicalproblems. With cross-sectional dependencies, the usualWald type unit root tests based on standard estimatorshave limit distributions that are dependent in a verycomplicated way upon various nuisance parametersdefining correlations across individual units. Theredoes not exist any simple way to eliminate these nui-sance parameters.

21

One solution consists in using the instrumental variable(IV thereafter) to solve the nuisance parameter problem dueto cross-sectional dependency.=⇒ Chang (2002).

Her testing procedure is as follows.

(1) In a first step, for each cross-section unit, she estimatesthe autoregressive coefficient from an usual ADF regres-sion using the instruments generated by an integrabletransformation of the lagged values of the endogenousvariable.

(2) She then constructsN individual t-statistic for testing theunit root based on theseN nonlinear IV estimators. Foreach unit, this t-statistic has limiting standard normal dis-tribution under the null hypothesis.

(3) In a second step, a cross-sectional average of these in-dividual unit test statistics is considered, as in IPS.

22

Let us consider the following ADF model:

∆yit = αi + ρiyi,t−1 +pi

j=1

βi,j∆yi,t−j + εit (12)

where εit are i.i.d. 0,σ2εi across time periods, but are al-lowed to be cross-sectionally dependent.

• To deal with this dependency, Chang uses the instrumentgenerated by a nonlinear functionF (yi,t−1) of the laggedvalues yi,t−1.

• This function F (.) is called the Instrument Generat-ing Function (IGF thereafter). It must be a regularlyintegrable function which satisfies ∞−∞ xF (x) dx = 0.

• This assumption can be interpreted as the fact that thenonlinear instrumentF (.)must be correlated with theregressor yi,t−1.

• Under the null, the nonlinear IV estimator of the parame-ter ρi, denoted ρi, is defined as:

ρi = F (yl,i) yl,i − F (yl,i) Xi (XiXi)−1Xiyl,i−1

F (yl,i) εi − F (yl,i) Xi (XiXi)−1Xiεi (13)

23

=⇒ Chang shows that the t-ratio used to test the unitroot hypothesis, denotedZi, asymptotically converges toa standard normal distribution if a regularly integrablefunction is used as an IGF.

Zi =ρiσρi

d−→T→∞

N (0, 1) for i = 1, ..N (14)

=⇒ This asymptotic Gaussian result is very unusual andentirely due to the nonlinearity of the IV.

• Chang provides several examples of regularly integrableIGFs. In our application, we consider three functions inorder to assess the sensitivity of the results to the choiceof the IGF.

IGF1(x) = x exp (−ci |x|)where ci ∈ R is determined by ci = 3T−1/2s−1 (∆yit)

where s2 (∆yit) is the sample standard error of ∆yit.IGF2(x) = I(|x| < K)

IGF3(x) = I(|x| < K) ∗ xThe IV estimator constructed from the IGF2 function issimply the trimmed OLS estimator based on observationsin the interval [−K,K] .

24

RESULTATS

• The results are clear. The SN statistics based on the in-strument generating functions IGF2 and IGF3 providestrong evidence in favor of the unit root. The null is notrejected for all the considered variables and the corre-sponding p-values are always very close to one.

• When the first instrument generating function IGF1 isused, the results are also in favor of the unit root hypothe-sis for 10 variables: the only exceptions are nominal GDP,GDP deflator, consumer prices, wages.

• However, it is important to note that Im and Pesaran (2003)found very large size distortions with this test.

• The non-stationarity appears to be a general propertyof the main macroeconomic and financial indicators.

25

1.Panel Unit Root Tests

First Generation Cross-sectional independence1. Nonstationarity tests Levin and Lin (1992, 1993)

Levin, Lin and Chu (2002)Harris and Tzavalis (1999)

Im, Pesaran and Shin (1997, 2002, 200Maddala and Wu (1999)

Choi (1999, 2001)2- Stationarity tests Hadri (2000)

Second Generation Cross-sectional dependencies

1- Factor structure Bai and Ng (2001, 2004)

Moon and Perron (2004a)

Phillips and Sul (2003a)Pesaran (2003)

Choi (2002)2- Other approaches O’Connell (1998)

Chang (2002, 2004)

26