Nelson and Plosser Revisited:
A Re-Examination using OECD Panel
Data
Christophe Hurlin
November 2004
1
1
MOTIVATIONS
Twenty two years after the seminal paper by Nelson and
Plosser (1982), why testing the presence of a unit root in
the same macroeconomic series by using a panel of OECD
countries?
• Testing the homogeneity of the unit root result in an international panel framework
• Technical reason: the power deficiencies of pure time
series-based tests for unit roots and cointegration, even
for the ’’new generation’’ of powerful procedures of test
(ADF-GLS in Elliott, Rothenberg and Stock, 1996, MaxADF in Leybourne, 1995).
• A new generation of panel unit root tests allows to take
into account the genuine international dimension of a panel.
– These tests relax the restrictive assumption of crosssectional independence.
– The international, sectoral or regional co-movements of
the economic series, largely documented since for instance Backus and Kehoe (1992).
2
The cross-sectional independence assumption in panel
unit root tests
• Two generations of panel unit root tests can now be distinguished (Hurlin et Mignon, 2004)
• The common feature of first generation tests is the restriction that all cross-sections are independent.
• Under this independence assumption the Lindberg-Levy
central limit theorem or other central limit theorems can
be applied to derive the asymptotic normality of panel
test statistics.
• Levin and Lin (1992, 1993); Levin, Lin and Chu (2002);
Choi (2001); Im, Pesaran and Shin (1997, 2003).
• However, this cross-sectional independence assumption
is quite restrictive in many empirical applications.
=⇒ More generally, this assumption raises the issue
of the validity of the panel approach in macroeconomic,
finance or international finance.
3
MESSAGES
=⇒ Testing the unit root in a panel with international
shocks or international dependences is not economically
equivalent to a collection of individual time series tests
or to test unit root in panel under the cross-sectional independence assumption.
=⇒ Parallel between the dichotomy cross-sectional independence versus correlations in panel unit root tests
and the dichotomy general versus partial equilibriums
in macroeconomics.
=⇒ So, it is not only a technical problem of power and
size to know if (i) the unit root must be tested in panel or
in time series and (ii) if it is necessary to consider crosssectional dependent processes. This a genuine economic
issue linked to the importance of international, regional,
sectorial or individual dependencies in the dynamics.
For these reasons, we propose here a re-examination of
the seminal work of Nelson and Plosser for the OECD, based
on panel unit root tests without and with cross-sectional dependencies.
4
A second generation of panel unit root tests
• The second generation panel unit root tests relax the
cross-sectional independence assumption.
• The first issue is to specify the cross-sectional dependencies, since as pointed out by Quah (1994), individual
observations in a cross-section have no natural ordering.
• The second problem is that the usual t-statistics unit root
tests have limit distributions that are dependent in a very
complicated way upon various nuisance parameters defining correlations across individual units.
• We distinguish two groups of tests: the first group tests
are based on a dynamic factor model or an error-component
model . The cross-sectional dependency is then due to the
presence of one or more common factors or to a random
time effect.
• The tests of the second group are defined by opposition
to these specifications based on common factor or time
effects. In this group, some specific or more general specifications of the cross-sectional correlations are proposed.
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=⇒ In this paper, these various panel unit root tests are applied to OECD panel databases for the same 14 macroeconomic and financial variables as those considered in Nelson
and Plosser (1982).
=⇒ The period considered is 1950-2000 : the issue of
breakpoint
=⇒ Our results highlight the importance of (i) the heterogeneous specification of the model and (ii) the crosssectional independence assumption.
=⇒ For some macroeconomic variables generally considered as non-stationary, such as the real GDP for instance, the
null of unit root is strongly rejected for our OECD sample
with first generation tests. On the contrary, when the international cross-correlations are taken into account in the
dynamic analysis, the results are clearly more in favour of
the unit root for all the considered variables, including the
unemployment rate.
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2
2.1
FIRST GENERATION UNIT ROOT TESTS
Im, Pesaran and Shin unit root tests
• The well-known IPS test (1997, 2003) is now avalaible
under usual software (Eviews 5.0) Their model with individual effects and no time trend is:
pi
[
∆yit = αi + ρiyi,t−1 +
β i,z ∆yi,t−z + εit
(1)
z=1
• The null hypothesis is defined as H0 : ρi = 0 for all i =
1, ..N and the alternative hypothesis is H1 : ρi < 0 for
i = 1, ..N1 and ρi = 0 for i = N1 + 1, .., N, with 0 <
N1 ≤ N .
• Their test is based on the (augmented) Dickey-Fuller statistics
averaged
across groups. Let tiT (pi, β i) with β i =
β i,1, .., β i,pi denote the t-statistic for testing unit root in
the ith country, the IPS statistic is:
N
1 [
t barNT =
tiT (pi, β i)
(2)
N i=1
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• Under the crucial assumption of cross-sectional independence, the statistic t barN T is shown to sequentially
converge to a normal distribution when T tends to infinity, followed by N . A similar result is conjectured when
N and T tend to infinity while the ratio N/T tends to a
finite non-negative constant.
• In order to propose a standardization of the t-bar statistic, IPS have to compute the values of E [tiT (pi, β i)] and
V ar [tiT (pi, β i)].
• Two solutions can be considered: the first one is based on
the asymptotic moments E (η) and V ar (η). The corresponding standardized t-bar statistic is denoted Zt bar .
• The second solution is to carry out the standardization
of the t-bar statistic using the means and variances of
tiT (pi, 0) evaluated
by simulations under the null ρi = 0. l
k
√
SN
−1
N t barNT − N
i=1 E [tiT (pi , 0)| ρi = 0]
t
Wtbar =
SN
−1
N
i=1 V ar [tiT (pi , 0)| ρi = 0]
d
−→ N (0, 1)
T,N→∞
• Although the tests Ztbar and Wtbar are asymptotically equivalent, simulations show that the Wtbar statistic performs
much better in small samples.
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RESULTATS
• If we consider the standardized statistic Wtbar , the unit
root hypothesis is not rejected for 8 macroeconomic variables out of 14 at a 5% significance level: nominal GDP,
real per capita GDP, employment GDP deflator, consumer
prices, velocity, bond yield and common stock prices.
• Except for the nominal GDP, the results are robust to the
use the standardized statistic Ztbar based on asymptotic
moments instead of Wtbar .
• More surprising, except for the nominal GDP and the unemployment rate, the results are also robust when we consider the statistic ZtDF
bar based on the average of Dickey
Fuller individuals statistics.
• The results are globally robust to the specification of deterministic component.
• Special care need to be exercised when interpreting the
results of the 6 variables for which the null hypothesis is
rejected. Due to the heterogeneous nature of the alternative, rejection of the null hypothesis does not necessarily
imply 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 observed
significant 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 continuous, the corresponding p-values, denoted pi, are uniform
(0, 1) variables. Consequently, under the assumption of
cross-sectional independence, the statistic proposed by
Maddala and Wu (1999) and defined as:
N
[
PM W = −2
log (pi)
(3)
i=1
has a chi-square distribution with 2N degrees of freedom, when T tends to infinity and N is fixed.
• For large N samples, Choi (2001) proposes a similar standardized
√statistic:
−1
SN
N N PMW − E [−2 log (pi)]
log (pi) +
s
ZM W =
= − i=1 √
N
V ar [−2 log (pi)]
(4)
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RESULTATS TESTS DE FISHER
• The results confirm our previous conclusions. If we consider the PM W test at a 5% significant level, we do not
reject the unit root for 7 out of 14 variables.
• The only difference with the IPS results is for the nominal
GDP, for which we reject the null here. This is precisely
the only variable for which the two IPS standardized statistics, Wtbar and Ztbar , do not give the same conclusions.
• Except for the real per capita GDP, the conclusions are
identical with the Choi’s standardized statistic.
• The results are globally robust to the specification of
the deterministic component, except for industrial production, nominal GNP and money.
• Contrary to the IPS tests, the Fisher tests lead to the rejection of the null for unemployment rate in a model with
time trends. This point clearly indicates the ambiguity of
the non stationarity analysis for this variable.
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CONCLUSION PART I
• With the panel unit root tests based on the cross-sectional
independence assumption, the conclusions on the non
stationarity of OECD macroeconomic variables are no
clear-cut.
• The unit root hypothesis is strongly rejected for 4 macroeconomic variables (real GDP, wages, real wages and
money stocks), which are generally considered as non stationary for the most of OECD countries.
• The non stationarity is also rejected for the unemployment rate, but in this case, it is not surprising given the
times series results (Nelson and Plosser, 1982).
• The non stationarity is robust to the choice of the test
and 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 assumption of cross-sectional independence used to derive the
asymptotic normality of the test statistics?
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• Obviously, these first generation tests are likely to yield
biased results if applied to panels with a cross-sectional
dependency.
• First intuition: Maddala and Wu (1999) =⇒ important
size 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
3.1
A SECOND GENERATION UNIT ROOT TESTS
Tests based on factor structure
• For all these tests, the idea is to shift data into two unobserved components: one with the characteristic that is
strongly cross-sectionally correlated and one with the characteristic that is largely unit specific.
• The testing procedure is always the same and consists in
two main steps: in a first one, data are de-factored, and in
a second step, panel unit root test statistics based on defactored data and/or common factors are then proposed.
• These statistics do not suffer from size distortions as those
which affect the standard tests based the cross-sectional
independence assumption when common factors exist in
the 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 of
integration of series.
• yit = a deterministic component + common component expressed as a factor structure + error idiosyncratic.
• Instead of testing for the presence of a unit root directly in yit, Bai and Ng propose to test the common
factors and the idiosyncratic components separately.
• For that, Bai and Ng have to use a decomposition method
of the data which is robust to the degree of integration
of the common or idiosyncratic components.
• Bai and Ng accomplish this by estimating factors on firstdifferenced data and cumulating these estimated factors.
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• Let us consider a model with individual effects and no
time trend:
yit = αi + λiFt + eit
(6)
where Ft is a r × 1 vector of common factors and λi is a
vector of factor loadings.
• The corresponding model in first differences is:
∆yit = λi ft + zit
where zit = ∆eit and ft = ∆Ft with E (ft) = 0.
(7)
• The common factors in ∆yit are estimated by the principal component method. Let us denote fet these estimates,
ei the corresponding loading factors and zeit the estimated
λ
residuals.
• Then, the ’differencing and re-cumulating’ estimation
procedure is based on the cumulated variables defined
as:
t
t
[
[
fˆms
êit =
zeis
(8)
F̂mt =
s=2
s=2
for t = 2, .., T, m = 1, .., r and i = 1, .., N.
• Bai and Ng test the unit root hypothesis in the idiosyncratic component eit and in the common factors Ft
with the estimated variables F̂m t and êi t.
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• To test the non stationarity of the idiosyncratic component
∆êit = δ i,0êi, t−1 + δ i,1∆êi, t−1 + .. + δ i,p∆êi,t−p + µit (9)
Let ADFeec (i) be the ADF t-statistic for the idiosyncratic
component of the ith country.
• The asymptotic distribution of ADFeec (i) coincides with
the Dickey Fuller distribution for the case of no constant.
=⇒ Therefore, a unit root test can be done for each
idiosyncratic component of the panel.
=⇒ The great difference with unit root tests based on
the pure time series is that the common factors, as global
international trends or international business cycles for
instance, have been withdrawn from data.
• Example : real GDP (table ??). For 12 countries, the
conclusions of both tests are opposite at a 5% significant
level: for 8 countries, the ADF tests on the initial series
lead to reject the null, whereas the idiosyncratic component is founded to be non-stationary.
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• Individual time series tests have the same low power as
those based on initial series =⇒ pooled tests are also
proposed.
• These tests are similar to the first generation ones.
However, the great difference is that the estimated idiosyncratic components êi,t are asymptotically independent across units
• Let denote pcee (i) the p-value of the ADFeec (i) test, this
statistic is:S
N
c
−
(i)] − N
d
c
i=1 log [pe
√ e
Zee =
−→ N (0, 1) (10)
T,N→∞
N
• RESULTATS
(1) At a 5% significant level, the non stationarity of idiosyncratic components is not rejected only for 6 out
of 14 variables (industrial production, employment,
consumer prices, real wages, velocity and common stock
prices).
(2) It implies that if the macroeconomic series are nonstationary, this property seems to be more due to the
common factors, as international business cycles or
growth 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 a
model with an intercept.
∆F̂1t = c + γ i,0 F̂1,t−1 + γ i,1∆F̂1,t−1 + .. + γ i,p∆F̂1,t−p + vit
(11)
The corresponding ADF t-statistic, denoted ADFFec , has
the same limiting distribution as the Dickey Fuller test for
the constant only case.
• If there are more than one common factors (r > 1), Bai
and Ng test the number of common independent stochastic trends in these common factors, denoted r1.
• If r1 = 0 it implies that there are N cointegrating vectors
for N common factors, and that all factors are I(0).
• Bai and Ng(2004) propose two statistics based on the r
demeaned estimated factors F̂m t for m = 1, .., m. These
statistics are similar to those proposed by Stock and Watson (1988).
1
In the first working paper (Bai and Ng, 2001), the procedure was the same whatever the number of common
factors and was only based on ADF tests.
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RESULTATS
• There is only one common factor in real GDP and in
real per capita GDP, which can be analyzed as an international stochastic growth factor. For both variables,
this common factor is found to be non stationary.
• For all the others variables, the estimated number of common factors ranges from 2 to 4. Whatever the test used,
MQc or MQf , the number of common stochastic trends
is always equal to the number of common factors, as reported on tables ?? and ??.
• These results are robust to the choice of the number of
common factors.
=⇒ All the macroeconomic series are I(1) as in NP
(1982)
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3.2
Other Approaches: Chang nonlinear IV unit root
tests
• There is a second approach to model the cross-sectional
dependencies, which is more general than those based on
dynamic factors models or error component models.
• It consists in imposing few or none restrictions on the covariance matrix of residuals. O’Connell (1998), Maddala
and Wu (1999), Taylor and Sarno (1998), Chang (2002,
2004).
• Such an approach raises some important technical
problems. With cross-sectional dependencies, the usual
Wald type unit root tests based on standard estimators
have limit distributions that are dependent in a very
complicated way upon various nuisance parameters
defining correlations across individual units. There
does not exist any simple way to eliminate these nuisance parameters.
21
One solution consists in using the instrumental variable
(IV thereafter) to solve the nuisance parameter problem due
to cross-sectional dependency.
=⇒ Chang (2002).
Her testing procedure is as follows.
(1) In a first step, for each cross-section unit, she estimates
the autoregressive coefficient from an usual ADF regression using the instruments generated by an integrable
transformation of the lagged values of the endogenous
variable.
(2) She then constructs N individual t-statistic for testing the
unit root based on these N nonlinear IV estimators. For
each unit, this t-statistic has limiting standard normal distribution under the null hypothesis.
(3) In a second step, a cross-sectional average of these individual unit test statistics is considered, as in IPS.
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Let us consider the following ADF model:
pi
[
∆yit = αi + ρiyi,t−1 +
β i,j ∆yi,t−j + εit
(12)
2 j=1
where εit are i.i.d. 0, σ εi across time periods, but are allowed to be cross-sectionally dependent.
• To deal with this dependency, Chang uses the instrument
generated by a nonlinear function F (yi,t−1) of the lagged
values yi,t−1.
• This function F (.) is called the Instrument Generating Function (IGF thereafter). ItU must be a regularly
∞
integrable function which satisfies −∞ xF (x) dx = 0.
• This assumption can be interpreted as the fact that the
nonlinear instrument F (.) must be correlated with the
regressor yi,t−1.
• Under the null, the nonlinear IV estimator of the parameter ρi, denoted e
ρi, is defined as:
l−1
k
−1
e
ρi = F (yl,i) yl,i − F (yl,i) Xi (Xi Xi) Xi yl,i
k
l
−1
F (yl,i) εi − F (yl,i) Xi (Xi Xi) Xi εi (13)
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=⇒ Chang shows that the t-ratio used to test the unit
root hypothesis, denoted Zi , asymptotically converges to
a standard normal distribution if a regularly integrable
function is used as an IGF.
e
ρ
d
Zi = i −→ N (0, 1) for i = 1, ..N
(14)
T
→∞
σ
eeρi
=⇒ This asymptotic Gaussian result is very unusual and
entirely due to the nonlinearity of the IV.
• Chang provides several examples of regularly integrable
IGFs. In our application, we consider three functions in
order to assess the sensitivity of the results to the choice
of the IGF.
IGF1(x) = x exp (−ci |x|)
where ci ∈ R is determined by ci = 3 T −1/2s−1 (∆yit)
where s2 (∆yit) is the sample standard error of ∆yit.
IGF2(x) = I(|x| < K)
IGF3(x) = I(|x| < K) ∗ x
The IV estimator constructed from the IGF2 function is
simply the trimmed OLS estimator based on observations
in the interval [−K, K] .
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RESULTATS
• The results are clear. The SN statistics based on the instrument generating functions IGF2 and IGF3 provide
strong evidence in favor of the unit root. The null is not
rejected for all the considered variables and the corresponding p-values are always very close to one.
• When the first instrument generating function IGF1 is
used, the results are also in favor of the unit root hypothesis 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 property
of the main macroeconomic and financial indicators.
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1.Panel Unit Root Tests
First Generation
Cross-sectional independence
1. Nonstationarity tests Levin and Lin (1992, 1993)
Levin, Lin and Chu (2002)
Harris and Tzavalis (1999)
Im, Pesaran and Shin (1997, 2002, 200
Maddala and Wu (1999)
Choi (1999, 2001)
2- Stationarity tests
Hadri (2000)
Second Generation
Cross-sectional dependencies
1- Factor structure
Bai and Ng (2001, 2004)
2- Other approaches
Moon and Perron (2004a)
Phillips and Sul (2003a)
Pesaran (2003)
Choi (2002)
O’Connell (1998)
Chang (2002, 2004)
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