nelson and plosser revisited: a re-examination using ......• their test is based on the...
TRANSCRIPT
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Nelson and Plosser Revisited:
A Re-Examination using OECD PanelData
Christophe Hurlin
November 2004
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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).
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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.
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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.
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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.
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=⇒ 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.
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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)
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• 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.
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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.
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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)
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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.
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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?
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• 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
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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.
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• 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.
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• 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.
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• 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.
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• 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.
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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.
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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)
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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.
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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.
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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)
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=⇒ 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] .
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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.
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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)
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