TRANSITION MODELING AND ECONOMETRIC CONVERGENCE TESTS BY PETER C. B. PHILLIPS and DONGGYU SUL COWLES FOUNDATION PAPER NO. 1216 COWLES FOUNDATION FOR… [602146]

TRANSITION MODELING AND ECONOMETRIC
CONVERGENCE TESTS

BY

PETER C. B. PHILLIPS and DONGGYU SUL

COWLES FOUNDATION PAPER NO. 1216

COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
YALE UNIVERSITY
Box 208281
New Haven, Connecticut 06520-8281

2006

http://cowles.econ .yale.edu/

http://www.econometricsociety.org/
Econometrica , Vol. 75, No. 6 (November, 2007), 1771–1855
TRANSITION MODELING AND ECONOMETRIC
CONVERGENCE TESTS
PETER C. B. P HILLIPS
Cowles Foundation for Research in Economics, Yale University, New Haven, CT 06520-8281,
U.S.A.; University of Auckland, New Zealand; and University of York, York, U.K.
DONGGYU SUL
University of Auckland, Auckland, New Zealand
The copyright to this Article is held by the Econometric Society. It may be downloaded,
printed and reproduced only for educational or research purposes, including use in course
packs. No downloading or copying may be done for any commercial purpose without the
explicit permission of the Econometric Society. For such commercial purposes contactthe Office of the Econometric Society (contact information may be found at the websitehttp://www.econometricsociety.org or in the back cover of Econometrica ). This statement must
the included on all copies of this Article that are made available electronically or in any otherformat.

Econometrica , Vol. 75, No. 6 (November, 2007), 1771–1855
TRANSITION MODELING AND ECONOMETRIC
CONVERGENCE TESTS
BYPETER C. B. P HILLIPS AND DONGGYU SUL1
A new panel data model is proposed to represent the behavior of economies in tran-
sition, allowing for a wide range of possible time paths and individual heterogeneity.
The model has both common and individual specific components, and is formulated as
a nonlinear time varying factor model. When applied to a micro panel, the decompo-
sition provides flexibility in idiosyncratic behavior over time and across section, while
retaining some commonality across the panel by means of an unknown common growth
component. This commonality means that when the heterogeneous time varying idio-
syncratic components converge over time to a constant, a form of panel convergence
holds, analogous to the concept of conditional sigma convergence. The paper providesa framework of asymptotic representations for the factor components that enables the
development of econometric procedures of estimation and testing. In particular, a sim-
ple regression based convergence test is developed, whose asymptotic properties are
analyzed under both null and local alternatives, and a new method of clustering panels
into club convergence groups is constructed. These econometric methods are applied
to analyze convergence in cost of living indices among 19 U.S. metropolitan cities.
K
EYWORDS : Club convergence, relative convergence, common factor, convergence,
logtregression test, panel data, transition.
1.INTRODUCTION
IN THE P AST DECADE , the econometric theory for dynamic panel regressions
has developed rapidly alongside a growing number of empirical studies involv-
ing macro, international, regional, and micro economic data. This rapid devel-
opment has been stimulated both by the availability of new data sets and by the
recognition that panels help empirical researchers to address many new eco-nomic issues. For example, macro aggregated panels such as the Penn WorldT able (PWT) data have been used to investigate growth convergence and eval-uate the many diverse determinants of economic growth. Durlauf and Quah
(1999 ) and Durlauf, Johnson, and T emple ( 2005 ) provided excellent overviews
of this vast literature and the econometric methodology on which it depends.
Similarly, micro panel data sets such as the PSID have been extensively used toanalyze individual behavior of economic agents across section and over time;see Ermisch ( 2004 )a n dH s a i o( 2003 ) for recent overviews of micro panel re-
search. A pervasive finding in much of this empirical panel data research is the
importance of individual heterogeneity. This finding has helped researchers to
build more realistic models that account for heterogeneity, an example beingthe renewed respect in macroeconomic modeling for micro foundations that
1Our thanks go to two referees and a co-editor for helpful comments on an earlier version.
Phillips acknowledges partial research support from a Kelly Fellowship and the NSF under Grant
SES 04-142254 and SES 06-47086.
1771

1772 P . C. B. PHILLIPS AND D. SUL
accommodate individual heterogeneity; see Browning, Hansen, and Heckman
(1999 ), Krusell and Smith ( 1998 ), Givenen ( 2005 ), and Browning and Carro
(2006 ).
Concerns about capturing heterogeneous agent behavior in economic theory
and modeling this behavior in practical work have stimulated interest in theempirical modeling of heterogeneity in panels. One popular empirical modelinvolves a common factor structure and idiosyncratic effects. Early economet-ric contributions of this type analyzed the asymptotic properties of commonfactors in asset pricing models (e.g., Chamberlain and Rothschild ( 1983 )a n d
Connor and Korajczyk ( 1986 ,1988 )). Recent studies have extended these fac-
tor models in several directions and developed theory for the determination ofthe number of common factors and for inference in panel models with nonsta-tionary common factors and idiosyncratic errors (e.g., Bai ( 2003 ,2004 ), Bai and
Ng ( 2002 ,2006 ), Stock and Watson ( 1999 ), Moon and Perron ( 2004 ), Phillips
and Sul ( 2006 )). There is much ongoing work in the econometric development
of the field to better match the econometric methods to theory and to the needs
of empirical research.
T o illustrate some of the issues, take the simple example of a single factor
model
X
it=δiµt+/epsilon1it/commaori (1)
whereδimeasures the idiosyncratic distance between some common factor µt
and the systematic part of Xit. The econometric interpretation of µtin appli-
cations may differ from the prototypical interpretation of a “common factor”
or aggregate element of influence in micro or macro theory. The factor µt
may represent the aggregated common behavior of Xit, but it could also be
any common variable of influence on individual behavior, such as an interestrate or exchange rate. The model then seeks to capture the evolution of theindividual X
itin relation to µtby means of its two idiosyncratic elements: the
systematic element ( δi) and the error ( /epsilon1it).
The present paper makes two contributions in this regard. First, we extend
(1) in a simple manner by allowing the systematic idiosyncratic element to
evolve over time, thereby accommodating heterogeneous agent behavior andevolution in that behavior by means of a time varying factor loading coefficientδ
it/periodoriWe further allow δitto have a random component, which absorbs /epsilon1itin
(1) and allows for possible convergence behavior in δitover time in relation to
the common factor µt/commaoriwhich may represent some relevant aggregate variable
or possible representative agent behavior. The new model has a time varyingfactor representation
X
it=δitµt/commaori (2)
where both components δitandµtare time varying and there may be some spe-
cial behavior of interest in the idiosyncratic element δitover time. As discussed

TRANSITION MODELING AND CONVERGENCE TESTS 1773
in Section 4, we model the time varying behavior of δitin semiparametric form
as
δit=δi+σiξitL(t)−1t−α/commaori (3)
whereδiis fixed,ξitis iid(0/commaori1)acrossibut weakly dependent over t,a n dL(t)
is a slowly varying function (like log t)f o rw h i c h L(t)→∞ ast→∞ (see
equation ( 24)). This formulation ensures that δitconverges to δifor allα≥0/commaori
which therefore becomes a null hypothesis of interest. If this hypothesis holds
andδi=δjfori/negationslash=j/commaorithe model still allows for transitional periods in which
δit/negationslash=δjt, thereby incorporating the possibility of transitional heterogeneity or
even transitional divergence across i. As shown later, further heterogeneity
may be introduced by allowing the decay rate αand slowly varying function
L(t) to be individual specific.
Such formulations accommodate some recent models of heterogeneous
agent behavior. For example, heterogeneous discount factor models typicallyassume that the heterogeneity is transient and that the discount factors be-
come homogeneous in the steady state (e.g., Uzawa ( 1968 ), Lucas and Stokey
(1984 ), Obstfeld ( 1990 ), Schmidt-Grohe and Uribe ( 2003 ), Choi, Mark, and
Sul ( 2006 )). In such cases, δ
itcontains information relating to these assumed
characteristics. The parameter of interest is then δit, and particular attention
is focused on its temporal evolution and convergence behavior /periodori
The second contribution of the paper addresses this latter issue and involves
the development of an econometric test of convergence for the time varying
idiosyncratic components. Specifically, we develop a simple regression basedtest of the hypothesis H
0:δit→δfor some δast→∞ . The approach has
several features that make it useful in practical work. First, the test does notrely on any particular assumptions concerning trend stationarity or stochastic
nonstationarity in X
itorµt. Second, the nonlinear form of the model ( 2)i s
sufficiently general to include a wide range of possibilities in terms of the time
paths for δitand their heterogeneity over i/periodoriBy focusing on δit, our approach
delivers information about the transition path of δitand allows for the impor-
tant case in practice where individual behavior may be transitionally divergent.
The remainder of the paper is organized into eight sections. Section 2mo-
tivates our approach in terms of some relevant economic examples of factormodels in macroeconomic convergence, labor income evolution, and stockprices. A major theme in our work is the analysis of long run equilibrium andconvergence by means of a transition parameter, h
it. This parameter is con-
structed directly from the data Xitand is a functional of δitthat provides a
convenient relative measure of the temporal evolution of δit. Under certain
regularity conditions, we show in Section 3thathithas an asymptotic repre-
sentation in a standardized form that can be usefully interpreted as a relativetransition path for economy iin relation to other economies in the panel.
Section 4introduces a new regression test of convergence and a procedure
for clustering panel data into clubs with similar convergence characteristics.

1774 P . C. B. PHILLIPS AND D. SUL
We call the regression test of convergence the log ttest because it is based on a
time series linear regression of a cross section variance ratio of the hiton logt.
This test is very easy to apply in practice, involving only a simple linear regres-
sion and a one-sided regression coefficient test with standard normal criticalvalues. The asymptotic properties of this test are obtained and a local asymp-totic power analysis is provided. The regression on which this test is based alsoprovides an empirical estimate of the speed of convergence. This section pro-vides a step by step procedure for practical implementation of this test and its
use as a clustering algorithm to find club convergence groups. An analysis of
the statistical properties of the convergence test and club convergence cluster-ing algorithm is given in the Appendix .
Section 5reports the results of some Monte Carlo experiments that evaluate
the performance of the convergence test in finite samples. The experimentsare set up to include some practically interesting and relevant data generatingprocesses.
Section 6contains an empirical application of our methods to test for con-
vergence in the cost of living across 19 metropolitan U.S. cities using consumerprice indices. The empirical results reveal no convergence in cost of living
among U.S. cities. Apparently, the cost of living in major metropolitan cities
in California is increasing faster than in the rest of the United States, while thecost of living in St. Louis and Houston is decreasing relative to the rest of theUnited States.
Section 7concludes the paper. The Appendices contain technical material
and proofs.
2.
TIME VAR YING FACTOR REPRESENTATION AND CONVERGENCE
Panel data Xitare often usefully decomposed as
Xit=git+ait/commaori (4)
wheregitembodies systematic components, including permanent common
components that give rise to cross section dependence, and aitrepresents tran-
sitory components. For example, the panel Xitcould comprise log national
income data such as the PWT , regional log income data such as the 48 con-
tiguous U.S. state log income data, regional log consumer price index data, orpersonal survey income data among many others. We do not assume any par-ticular parametric specification for g
itandaitat this point, and the framework
includes many linear, nonlinear, stationary, and nonstationary processes.
As it stands, the specification ( 4) may contain a mixture of both common
and idiosyncratic components in the elements gitandait. T o separate common
from idiosyncratic components in the panel, we may transform ( 4) to the form
of (2), namely
Xit=/parenleftbigggit+ait
µt/parenrightbigg
µt=δitµtfor alliandt/commaori (5)

TRANSITION MODELING AND CONVERGENCE TESTS 1775
whereµtis a single common component and δitis a time varying idiosyn-
cratic element. For example, if µtrepresents a common trend component in
the panel, then δitmeasures the relative share in µtof individual iat timet/periodori
Thus,δitis a form of individual economic distance between the common trend
component µtandXit. The representation ( 5) is a time varying factor model
of the form ( 2)i nw h i c h µtis assumed to have some deterministic or stochasti-
cally trending behavior that dominates the transitory component aitast→∞ .
Factoring out a common trend component µtin (5) leads naturally to specifi-
cations of the form ( 3)f o rδit. Additionally, under some regularity conditions
it becomes possible to characterize a limiting relative transition path for Xit,
as discussed in Section 3.W h e nb o t h gitandaitbehave like I(0)variables over
time, this limiting characterization is not as relevant and factoring such as ( 5)
with transition properties for δitlike those of ( 3) are less natural.2
The following examples illustrate how the simple econometric representa-
tion ( 5) usefully fits in with some micro- and macroeconomic models that are
commonly used in applied work.
Economic Growth : Following Parente and Prescott ( 1994 ), Howitt and
Mayer-Foulkes ( 2005 ), and Phillips and Sul (2006), and allowing for hetero-
geneous technology progress in a standard neoclassical growth model, log per
capita real income, log yit/commaorican be written as
logyit=logy∗
i+(logyi0−logy∗
i)e−βit+logAit=ait+logAit/commaori (6)
where log y∗
iis the steady state level of log per capita real effective income,
logyi0is the initial log per real effective capita income, βitis the time varying
speed of convergence rate, and log Aitis the log of technology accumulation
for economy iat timet/periodoriThe relationship is summarized in ( 6) in the terms ait
and logAit/commaoriwhereaitcaptures transitional components and log Aitincludes
permanent components. Within this framework, Phillips and Sul ( 2006 ) further
decomposed log Aitas
logAit=logAi0+γitlogAt/commaori
writing current technology for country iin terms of initial technology accu-
mulation, log Ai0/commaoriand a component, γitlogAt/commaorithat captures the distance of
countryitechnology from publicly available advanced technology, log At/commaoriat
timet/periodoriThe coefficient γitthat measures this distance may vary over time and
across country. If advanced technology log Atis assumed to grow at a constant
ratea/commaorithen
logyit=/parenleftbiggait+logAi0+γitlogAt
at/parenrightbigg
at=δitµt/commaori
2Nonetheless, when factor representations like ( 5)d oa r i s ei nt h e I(0)case, some related
modeling possibilities for the transition curves are available and these will be explored in later
work.

1776 P . C. B. PHILLIPS AND D. SUL
corresponding to ( 5)a n dδitmay be modeled according to ( 3). Phillips and
Sul called δitatransition parameter andµtacommon growth component .B o t h
components are of interest in this example. In the analysis of possible growth
convergence or divergence over time and in the study of heterogeneous tran-
sition paths across economies, the time varying component δitis especially im-
portant.
Labor Income : In labor economics (e.g., Katz and Autor ( 1999 ), Moffitt and
Gottschalk ( 2002 )), personal log real income log yitwithin a particular age
group is commonly decomposed into components of permanent income, git,
and transitory income, ait/commaoriso that
logyit=git+ait/periodori
T ypically, transitory income is interpreted as an idiosyncratic component and
permanent income is regarded as having some common (possibly stochastic)
trend component. Again, this model may be rewritten as in ( 5) by factoring
out the common stochastic trend component. The main parameter of interestthen becomes the time profile of the personal factor loading coefficient δ
it/periodori
The evolution of this parameter may then be modeled in terms of individualattributes and relevant variables, such as education, vocational training, or job
experience.
For example, gender wage differences might be examined by modeling wages
as logy
it=δitµt,w i t hδitsatisfying
δit→/braceleftbigg
δMfori∈M(male),
δFfori∈F(female),
whereµtrepresents a common overall wage growth component. Alternatively,
wages might be modeled as log yit=δMitµMt+δFitµFtwith possibly distinct
male and female growth components µMtandµFtthat both influence overall
wage growth but with coefficients satisfying
δMit→δM/commaoriδFit→0f o r i∈M/commaori (7)
δMit→0/commaoriδFit→1f o r i∈F/periodori
Then log yit=(δMit(µMt/µFt)+δFit)µFt=δitµt/commaoriin which case the transition
coefficient δitmay diverge if trend wage growth is higher for males than fe-
males. In either case, we can model wages as a single common factor without
loss of generality within each convergent subgroup, and overall convergence
and divergence may be assessed in terms of the time evolution of δit.
Importantly also, by analyzing subgroup-convergent behavior among the
idiosyncratic transition coefficients δit, one may locate the sources of diver-
gence in a panel. Suppose, for instance, that wage inequality arises because of

TRANSITION MODELING AND CONVERGENCE TESTS 1777
gender differences as well as certain other factors. By identifying convergence
clubs in the wage transition coefficients and analyzing the characteristics ofthese clubs, the sources of wage inequality may be identified empirically.
Stock Price Factor Modeling : Models with a time varying factor structure
have been popular for some time in finance. For example, Fama and French(1993 ,1996 ) modeled stock returns R
itas
Rit=γ1/commaoriitθ1t+γ2/commaoriitθ2t+γ3/commaoriitθ3t+/epsilon1it/commaori (8)
where the θstare certain “common” determining factors for stock returns,
while the γs/commaoriitare time varying factor loading coefficients that capture the in-
dividual effects of the factors. It has often been found convenient in applied
research to assume that the time varying loading coefficients are constant over
short time periods. Ludvigson and Ng ( 2007 ), for instance, recently estimated
the number of common factors in a model of the form ( 8) based on time invari-
ant factor loadings. On the other hand, Adrian and Franzoni ( 2005 ) relaxed the
assumption and attempted to estimate time varying loadings by means of the
Kalman filter under the assumption that the factor loadings follow an AR (1)
specification.
Alternatively, as in Menzly, Santos, and Veronesi ( 2002 ), we may model
stock prices, Xit/commaoriinstead of stock returns in ( 8) with multiple common factors,
writing
Xit=J/summationdisplay
j=1δj/commaoriitµjt+eit=/parenleftBiggJ/summationdisplay
j=1δj/commaoriitµjt
µ1t+eit
µ1t/parenrightBigg
µ1t+eit=δitµt/commaori (9)
so that the time varying multiple common factor structure can be embedded
in the framework ( 5) of a time varying single common factor structure. If the
common trend elements in ( 9) are drifting I(1)variables of the form
µjt=mjt+t/summationdisplay
s=1ejsforj=1/commaori/periodori/periodori/periodori/commaoriJ/commaori withm1/negationslash=0/commaori
then
µjt
µ1t=mjt+/summationtextt
s=1/epsilon1js
m1t+/summationtextts=1/epsilon11s=mj
m1+op(1)
and we have
δit=J/summationdisplay
j=1δj/commaoriitmj
m1{1+op(1)}/commaoriµt=µ1t/periodori

1778 P . C. B. PHILLIPS AND D. SUL
Convergence occurs if δj/commaoriit→δj∀jast→∞ and then δit→/summationtextJ
j=1δj(mj/m 1)=
δ. It is not necessary to assume that there is a dominant common factor for this
representation to hold. Moreover, as in ( 7), we may have certain convergent
subgroups { Ga:a=1/commaori/periodori/periodori/periodori/commaoriA }of stocks for which δj/commaoriit→δa
jfori∈Gaand then
δit→δa=/summationtextJ
j=1δa
j(mj/m 1)fori∈Ga.I ns u c hc a s e s ,t h e Xitdiverge overall,
but the panel may be decomposed into Aconvergent subgroups. We will dis-
cuss how to classify clusters of convergent subgroups later in Section 4.
2.1. Long Run Equilibrium and Convergence
An important feature of the time varying factor representation is that it pro-
vides a new way to think about and model long run equilibrium. Broadly speak-
ing, time series macroeconomics presently involves two categories of analysis:
long run equilibrium growth on the one hand and short run dynamics on the
other. This convention has enabled extensive use of cointegration methods forlong run analysis and stationary time series methods for short run dynamic be-
havior. In the time varying factor model, the use of common stochastic trends
conveniently accommodates long run comovement in aggregate behavior with-out insisting on the existence of cointegration and it further allows for the mod-
eling of transitional effects. In particular, idiosyncratic factor loadings provide
a mechanism for heterogeneous behavior across individuals and the possibilityof a period of transition in a path that is ultimately governed by some commonlong run stochastic trend.
If two macroeconomic variables X
itandXjthave stochastic trends and are
thought to be in long run equilibrium /commaorithen the time series are commonly hy-
pothesized to be cointegrated and this hypothesis is tested empirically. Coin-
tegration tests are typically semiparametric with respect to short run dynamics
and rely on reasonably long time spans of data. However, in micro panels suchlong run behavior is often not empirically testable because of data limitations
that result in much shorter panels. In the context of the nonlinear factor model
(5), suppose that the loading coefficients δ
itslowly converge to δover time, but
the data available to the econometrician are limited. The difference between
two time series in the panel is given by Xit−Xjt=(δit−δjt)µt/periodoriIfµtis unit
root nonstationary and δit/negationslash=δjt, thenXitis generally not cointegrated with Xjt.
But since δitandδjtconverge to some common δast→∞/commaoriwe may think of
XitandXjtas being asymptotically cointegrated. However, even in this case, if
the speed of divergence of µtis faster than the speed of the convergence of δit/commaori
the residual (δit−δjt)µtmay retain nonstationary characteristics and standard
cointegration tests will then typically have low power to detect the asymptoticcomovement.
T o fix ideas, suppose
δ
it→/braceleftbigg
δafori∈Ga,
δbfori∈Gb,(10)

TRANSITION MODELING AND CONVERGENCE TESTS 1779
so that there is convergence within each of the two subgroups GaandGb.U n –
der model ( 3) for the transition coefficients, the following relation then holds
between series XitandXjtfori∈Gaandj∈Gb:
Xit−δa
δbXjt=/parenleftbigg
δit−δa
δbδjt/parenrightbigg
µt=/braceleftbigg
σiξit−δa
δbσjξjt/bracerightbiggµt
L(t)tα/periodori
Hence,Xit−δa/δbXjtisI(0)whenµt=Op(L(t)tα), and then any two se-
ries from subgroups GaandGbare cointegrated. For example, when µt=
µL(t)tα+/summationtextt
s=1ζsfor some stationary sequence ζs, then each individual series
Xitfollows a unit root process with nonlinear drift, and is cointegrated with
other series in Gawith cointegrating vector (1/commaori−1)and is cointegrated with
series inGbwith cointegrating vector (1/commaori−δa/δb). However, if L(t)−1t−αµtdi-
verges (e.g., when α=1/2a n dµt=Op(t)), then the series XitandXjtare not
cointegrated even though we have convergence ( 10) in subgroups GaandGb
whenα>0a n d
δit−δa
δbδjt→p0f o r i∈Ga/commaorij∈Gb/commaori
δit−δjt→p0f o r i/commaorij∈Ga/periodori
In effect, the speed of convergence is not fast enough to ensure cointegrated
behavior.
These examples show that for economists to analyze comovement and con-
vergence in the context of individual heterogeneity, and to analyze evolution
in the heterogeneity over time and across groups, some rather different econo-metric methods are needed. In particular, under these conditions, conventional
cointegration tests do not serve as adequate tests for convergence. Clearly, the
two hypotheses of cointegration and convergence are related but have distinct
features. As the above examples illustrate, even though there may be no em-
pirical support for cointegration between two series X
itandXjt, it does not
mean there is an absence of comovement or convergence between XitandXjt.
Accordingly, a simple but intuitive way to define “relative” long run equilib-
rium or convergence between such series is in terms of their ratio rather than
their difference or linear combinations. That is, relative long run equilibrium
exists among the Xitif
lim
k→∞Xit+k
Xjt+k=1f o r a l l iandj/periodori (11)
In the context of ( 5), this condition is equivalent to convergence of the factor
loading coefficients
lim
k→∞δit=δ/periodori (12)

1780 P . C. B. PHILLIPS AND D. SUL
On the other hand, if XitandXjtare cointegrated, then the ratio Xit/Xjttypi-
cally converges to a constant or a random variable, the former occurring when
the series have a nonzero deterministic drift.
2.2. Relative Transition
In the general case of ( 5), the number of observations in the panel is less than
the number of unknowns in the model. It is therefore impossible to estimatethe loading coefficients δ
itdirectly without imposing some structure on δitand
µt. Both parametric and nonparametric structures are possible. For example,
ifδitevolved according to an AR (1),w h i l eµtfollowed a random walk with a
drift, it would be possible to estimate both δitandµtby a filtering technique
such as the Kalman filter. Alternatively, as we show below, under some regu-larity conditions, it is possible to use a nonparametric formulation in which thequantities of interest are a transition function (based on δ
it) and a growth curve
(based on µt). Some further simplification for practical purposes is possible by
using a relative version of δitas we now explain.
Sinceµtis a common factor in ( 5), it may be removed by scaling to give the
relative loading or transition coefficient
hit=Xit
1
N/summationtextN
i=1Xit=δit
1
N/summationtextNi=1δit/commaori (13)
which measures the loading coefficient δitin relation to the panel average at
timet/periodoriWe assume that the panel average N−1/summationtextN
i=1δitand its limit as N→∞
differ from zero almost surely, so that hitis well defined by the construction
(13). In typical applications, Xit,µt,a n dδitare all positive, so the construc-
tion of this relative coefficient presents no difficulty in practice. Like δit/commaorihit
still traces out a transition path for economy i,b u tn o wd o e ss oi nr e l a t i o nt o
the panel average. The concept is useful in the analysis of growth convergence
and measurement of transition effects, as discussed in some companion em-
pirical work (Phillips and Sul ( 2006 )) where hitis called the relative transition
parameter .
Some properties of hitare immediately apparent. First, the cross sectional
mean of hitis unity by definition. Second, if the factor loading coefficients δit
converge to δ/commaorithen the relative transition parameters hitconverge to unity. In
this case, in the long run, the cross sectional variance of hitconverges to zero,
so that we have
σ2
t=1
NN/summationdisplay
i=1(hit−1)2→0a st→∞/periodori (14)
Later in the paper, this property will be used to test the null hypothesis of
convergence and to group economies into convergence clusters.

TRANSITION MODELING AND CONVERGENCE TESTS 1781
3.ASYMPTOTIC RELATIVE TRANSITION P ATHS
In many empirical applications, the common growth component µtwill have
both deterministic and stochastic elements, such as a unit root stochastic trend
with drift. In that case, µtis still dominated by a linear trend asymptotically.
In general, we want to allow for formulations of the common growth path µt
that may differ from a linear trend asymptotically, and a general specificationallows for the possibility that some individuals may diverge from the commongrowth path µ
t/commaoriwhile others may converge to it. These extensions involve some
technical complications that can be accommodated by allowing the functions to
be regularly varying at infinity (that is, they behave asymptotically like power
functions). We also allow for individual standardizations for Xit, so that ex-
pansion rates may differ, as well as imposing a common standardization for
µt. Appendix Aprovides some mathematical details of how these extensions
and standardizations can be accomplished so that the modeling framework is
more general. The present section briefly outlines the impact of these ideas andshows how to obtain a nonparametric formulation of the model ( 5)i nw h i c h
the quantities of interest are a nonparametric transition function δ(·)and a
growth curve µ(·)/periodori
In brief, we proceed as follows. Our purpose is to standardize X
itin (5)s o
that the standardized quantity approaches a limit function that embodies boththe common component and the transition path. T o do so, it is convenientto assume that there is a suitable overall normalization of X
itfor which we
may write equation ( 5) in the standardized form given by ( 15) below. Suppose
the standardization factor for XitisdiT=TγiWi(T)for some γi>0a n ds o m e
slowly varying function3Wi(T)/commaori so thatXitgrows for large taccording to the
power law tγiup to the effect of Wi(t)and stochastic fluctuations. We may
similarly suppose that the common trend component µtgrows according to
tγZ(t) for some γ>0a n dw h e r e Zis another slowly varying factor. Then we
may write
1
diTXit=1
TγiWi(T)/parenleftbiggait+git
µt/parenrightbigg
µt=δiT/parenleftbiggt
T/parenrightbigg
µT/parenleftbiggt
T/parenrightbigg
+o(1)/commaori (15)
where we may define the sample functions µTandbiTas
µT/parenleftbiggt
T/parenrightbigg
=/parenleftbiggt
T/parenrightbiggγZ(t
TT)
Z(T)andδiT/parenleftbiggt
T/parenrightbigg
=/parenleftbiggt
T/parenrightbiggγi−γWi(t
TT)Z(T)
Wi(T)Z(t
TT)/commaori (16)
as shown in Appendix A.
3That is,Wi(aT)/W i(T)→1a sT→∞ for alla> 0/periodoriFor example, the constant function,
log(T),a n d1/log(T)are all slowly varying functions.

1782 P . C. B. PHILLIPS AND D. SUL
Now suppose that t=[Tr]/commaorithe integer part of Tr/commaoriso thatris effectively the
fraction of the sample Tcorresponding to observation t/periodoriThen, for such values
oft/commaori(15) leads to the asymptotic characterization
1
diTXit∼δiT/parenleftbigg[Tr]
T/parenrightbigg
µT/parenleftbigg[Tr]
T/parenrightbigg
∼δiT(r)µT(r)/periodori (17)
In (17),µT(r)is the sample growth curve and δiT(r)is the sample transition
path (given Tobservations) for economy iat timeT/periodoriIt is further convenient to
assume that these functions converge in some sense to certain limit functions
asT→∞/periodoriFor instance, the requirement that δiTandµTsatisfy
µT(r)→pµ(r)/commaori δ iT(r)→pδi(r) uniformly in r∈[0/commaori1]/commaori (18)
where the limit functions µ(r) andδi(r)are continuous or, at least, piecewise
continuous, seems fairly weak. By extending the probability space in which the
functions δiTandµTare defined, ( 18) also includes cases where the functions
may converge to limiting stochastic processes.4The limit functions µ(r) and
δi(r)represent the common steady state growth curve and limiting transition
curve for economy i/commaorirespectively. Further discussion, examples, and some gen-
eral conditions under which the formulations ( 17)a n d( 18) apply are given in
Appendix A.
Combining ( 17)a n d( 18), we have the following limiting behavior for the
standardized version of Xit:
1
diTXit→pXi(r)=δi(r)µ(r)/periodori (19)
With this limiting decomposition, we may think about µ(r) as the limiting form
of the common growth path and about δi(r)as the limiting representation of
the transition path of individual ias this individual moves toward the growth
pathµ(r) . Representation ( 19) is sufficiently general to allow for cases where
individuals approach the common growth path in a monotonic or cyclical fash-
ion, either from below or above µ(r) .
T o illustrate ( 19), whenµtis a stochastic trend with positive drift, we have
the simple standardization factor diT=Tand then
T−1µt=[Tr]=m[Tr]
T+Op(T−1/2)→pmr
4For example, if µtis a unit root process, then under quite general conditions we have the
weak convergence T−1/2µ[Tr]=µT(r)⇒B(r) to a limit Brownian motion B(e.g., Phillips and
Solo ( 1992 )). After a suitable change in the probability space, we may write this convergence in
probability, just as in ( 18).

TRANSITION MODELING AND CONVERGENCE TESTS 1783
for some constant m> 0/periodoriSimilarly, the limit function δi(r)may converge to
δiasT→∞/periodoriCombining the two factors gives the limiting path Xi(r)=δimr
for individual i/commaoriso that the long run growth paths are linear across individuals.
When there is convergence across individuals, we have limit transition curves
δi(r)each with the property that δi(1)=δ, for some constant δ>0/commaoribut which
may differ for intermediate values (i.e., δi(r)/negationslash=δj(r)for some and possibly
allr<1). In this case, each individual may transition in its own way toward
a common limiting growth path given by the linear function X(r)=δmr.I n
this way, the framework permits a family of potential transitions to a commonsteady state.
Next we consider the asymptotic behavior of the relative transition parame-
ter. T aking ratios to cross sectional averages in ( 15) removes the common trend
µ
tand leaves the standardized quantity
hiT/parenleftbiggt
T/parenrightbigg
=d−1
iTXit
1
n/summationtextn
j=1d−1
jTXjt=δiT(t
T)
1
n/summationtextnj=1δjT(t
T)/commaori (20)
which describes the relative transition of economy iagainst the benchmark
of a full cross sectional average. Clearly, hiTdepends on nalso, but we omit
the subscript for simplicity because this quantity often remains fixed in the
calculations. In view of ( 18), we have
hiT/parenleftbigg[Tr]
T/parenrightbigg
→phi(r)=δi(r)
1
n/summationtextn
j=1δj(r)asT→∞/commaori (21)
and the function hi(r)then represents the limiting form of the relative transi-
tion curve for the individual i/periodori
For practical purposes of implementation when the focus of interest is long
run behavior in the context of macroeconomic data, it will often be prefer-
able to remove business cycle components first. Extending ( 5) to incorporate a
business cycle effect κit/commaoriwe can write
Xit=δitµt+κit/periodori
Smoothing methods offer a convenient mechanism for separating out the cycleκ
it/commaoriand we can employ filtering, smoothing, and regression methods to achieve
this. In our empirical work with macroeconomic data, we have used two meth-
ods to extract the long run component δitµt. The first is the Whittaker–
Hodrick–Prescott (WHP) smoothing filter.5The procedure is popular because
5Whittaker ( 1923 ) first suggested this penalized method of smoothing or “graduating” data
and there has been a large subsequent literature on smoothing methods of this type (e.g., see
Kitagawa and Gersch ( 1996 )). The approach has been used regularly in empirical work in time
series macroeconomics since the 1982 circulation of Hodrick and Prescott ( 1997 ).

1784 P . C. B. PHILLIPS AND D. SUL
of its flexibility, the fact that it requires only the input of a smoothing para-
meter, and does not require prior specification of the nature of the common
trendµtinXit. The method is also suitable when the time series are short. In
addition to the WHP filter, we employed a coordinate trend filtering method
(Phillips ( 2005 )). This is a series method of trend extraction that uses regres-
sion methods on orthonormal trend components to extract an unknown trendfunction. Again, the method does not rely on a specific form of µ
tand is ap-
plicable whether the trend is stochastic or deterministic.
The empirical results reported in our applications below were little changed
by the use of different smoothing techniques. The coordinate trend method has
the advantage that it produces smooth function estimates and standard errors
can be calculated for the fitted trend component. Kernel methods, rather than
orthonormal series regressions, provide another general approach to smooth
trend extraction and would also give standard error estimates. Kernel methods
were not used in our practical work here because some of the time series we use
are very short and comprise as few as 30 time series observations. Moreover,kernel method asymptotics for estimating stochastic processes are still largely
unexplored and there is no general asymptotic theory to which we may appeal,
although some specific results for Markov models have been obtained in work
by Phillips and Park ( 1998 ), Guerre ( 2004 ), Karlsen and Tjøstheim ( 2001 ), and
Wang and Phillips ( 2006 ).
Using the trend estimate ˆθ
it=/hatwidestbitµtfrom the smoothing filter, the estimates
ˆhit=ˆθit
1
n/summationtextn
i=1ˆθit(22)
of the transition coefficients hit=δit/(n−1/summationtextn
i=1δit)are obtained by taking ra-
tios to cross sectional averages. Assuming a common standardization6diT=dT
for simplicity and setting t=[Tr], we then have the estimate ˆhi(r)=ˆhi[Tr]of
the limiting transition curve hi(r)in (21). We can decompose the trend esti-
mate ˆθitas
ˆθit=θit+eit=/bracketleftbigg
δit+eit
µt/bracketrightbigg
µt/commaori (23)
whereeitis the error in the filter estimate of θit/periodoriSinceµtis the common
trend component, the condition eit/µt→p0 uniformly in iseems reasonable.7
6Alternatively, if the standardizations diTwere known (or estimated) and were incorporated
directly into the estimates ˆθit,t h e n ˆhit=ˆθit/(n−1/summationtextn
i=1ˆθit)would correspondingly build in the
individual standardization factors. Accordingly, ˆhitis an estimate of hit=hiT(t
T)a sg i v e ni n( 20).
7Primitive conditions under which eit/µt→p0 holds will depend on the properties of µtand
the selection of the bandwidth/smoothing parameter/regression number in the implementation

TRANSITION MODELING AND CONVERGENCE TESTS 1785
Then
ˆhi(r)=/bracketleftbig
δi[Tr]+ei[Tr]
µ[Tr]/bracketrightbig
1
n/summationtextn
i=1/bracketleftbig
δj[Tr]+ej[Tr]
µ[Tr]/bracketrightbig=δiT(t
T)
1
n/summationtextnj=1δjT(t
T)+op(1)
→pδi(r)
1
n/summationtextnj=1δj(r)=hi(r)/commaori
so that the relative transition curve is consistently estimated by ˆhi(r).
4.MODELING AND TESTING CONVERGENCE
A general theory for the calculation of asymptotic standard errors of fitted
curves of the type ˆhi(r)that allow for deterministic and stochastic trend com-
ponents of unknown form is presently not available in the literature and is be-
yond the scope of the present paper. Instead, we will confine ourselves to the
important special case where the trend function involves a dominating stochas-tic trend (possibly with linear or polynomial drift) and the transition coefficient
h
itis modeled semiparametrically. Our focus of attention is the development
of a test for the null hypothesis of convergence and an empirical algorithm of
convergence clustering.
As condition ( 12) states, under convergence, the cross sectional variation of
ˆhi(r)converges to zero as t→∞ . We note, however, that decreasing cross
sectional variation of ˆhi(r)does not in itself imply overall convergence. For
example, such decreasing cross sectional variation can occur when there is alocal convergence within subgroups and overall divergence. Such a situation isplotted in Figure 1.
T o design a statistical test for convergence, we need to take such possibilities
of local subgroup convergence into account. As discussed earlier, the approachwe use for this purpose is semiparametric and assumes the following general
form for the loading coefficients δ
it:
δit=δi+σitξit/commaoriσit=σi
L(t)tα/commaorit≥1/commaoriσi>0f o r a l l i/commaori (24)
where the components in this formulation satisfy the following conditions.
Some generalization of ( 24) is possible, including allowance for individual spe-
cific decay rates αiand slowly varying functions Li(t)that vary over i/periodoriThese
of the filter. In the case of the WHP filter, this turns on the choice of the smoothing parameter
(λ) in the filter and its asymptotic behavior as the sample size increases. For instance, if µtis
dominated by a linear drift and λ→∞ sufficiently quickly as T→∞ , then the WHP filter will
consistently estimate the trend effect. Phillips and Jin ( 2002 ) provided some asymptotic theory
for the WHP filter under various assumptions about λand the trend function.

1786 P . C. B. PHILLIPS AND D. SUL
FIGURE 1.—Stylized club convergence with two subgroups.
extensions are discussed later in Remark 6. Our theory is now developed under
(24) and the conditions below.
ASSUMPTION A1:ξitisiid(0/commaori1)with finite fourth moment µ4ξoverifor each
t,and is weakly dependent and stationary over twith autocovariance sequence
γi(h)=E(ξitξit+h)satisfying/summationtext∞
h=1h|γi(h)|<∞.Partial sums of ξitandξ2
it−1
overtsatisfy the panel functional limit laws
1√
T[Tr]/summationdisplay
t=1ξit⇒Bi(r) asT→∞ for alli/commaori (25)
1√
T[Tr]/summationdisplay
t=1(ξ2
it−1)⇒B2i(r) asT→∞ for alli/commaori (26)
whereBiandB2iare independent and form independent sequences of Brownian
motions with variances ωiiandω2ii,respectively ,overi.
ASSUMPTION A2: The limits
lim
N→∞N−1N/summationdisplay
i=1σ2
i=v2
ψ/commaorilim
N→∞N−1N/summationdisplay
i=1σ4
i=v4ψ/commaori
lim
N→∞N−1N/summationdisplay
i=1σ2
iωii=ω2
ξ/commaorilim
N→∞N−1N/summationdisplay
i=1σ4
iω2ii=ω2η/commaori
lim
N→∞N−2N/summationdisplay
i=2i−1/summationdisplay
j=1σ2
iσ2
j∞/summationdisplay
h=−∞γi(h)γj(h)/commaori lim
N→∞N−1N/summationdisplay
i=1δi=δ
all exist and are finite and δ/negationslash=0.

TRANSITION MODELING AND CONVERGENCE TESTS 1787
ASSUMPTION A3: Sums ofψit=σiξitandσ2
i(ξ2
it−1)overisatisfy the limit
laws
N−1/2N/summationdisplay
i=1σiξit⇒N(0/commaoriv2
ψ)/commaori (27)
N−1/2N/summationdisplay
i=1σ2
i(ξ2
it−1)⇒N(0/commaoriv4ψ(µ4ξ−1)) (28)
asN→∞ for allt/commaoriand the joint limit laws
T−1/2N−1/2T/summationdisplay
t=1N/summationdisplay
i=1σiξit⇒N(0/commaoriω2
ξ)/commaori (29)
T−1/2N−1/2T/summationdisplay
t=1N/summationdisplay
i=1σ2
i(ξ2
it−1)⇒N(0/commaoriω2η)/commaori (30)
T−1/2T/summationdisplay
t=1N−1N/summationdisplay
i=2i−1/summationdisplay
j=1σiσjξitξjt (31)
⇒N/parenleftBigg
0/commaorilim
N→∞N−2N/summationdisplay
i=2i−1/summationdisplay
j=1σ2
iσ2
j∞/summationdisplay
h=−∞γi(h)γj(h)/parenrightBigg
hold asN/commaoriT→∞/periodori
ASSUMPTION A4: The function L(t) in(24)is slowly varying (SV),increasing ,
and divergent at infinity .Possible choices for L(t) arelog(t+1),l o g2(t+1),or
log log(t+1).
Panel functional limit laws such as ( 25)a n d( 26) in Assumption A1are
known to hold under a wide set of primitive conditions and were explored
by Phillips and Moon ( 1999 ). These conditions allow for the variances ωii
to be random over i/commaorii nw h i c hc a s et h el i m i ti n( 25) is the mixture process
Bi(r)=ω1/2
iiVi(r),w h e r eViis standard Brownian motion. The central limit re-
sults ( 27)a n d( 28) hold under Assumptions A1and A2, and also for cases
where the components ξitare not identically distributed provided a uniform
moment condition, such as supiE(ξ4
it)<∞/commaoriholds. The joint limit laws ( 29)–
(31) are high level conditions that hold under primitive assumptions of the type
g i v e ni nP h i l l i p sa n dM o o n( 1999 ).
In Assumption A4, the slowly varying function L(t)→∞ ast→∞/periodoriIn ap-
plications, it will generally be convenient to set L(t)=log(t+1)or a similar
increasing slowly varying function. The presence of L(t) in (24) ensures that

1788 P . C. B. PHILLIPS AND D. SUL
δit→pδiast→∞ even when α=0/periodoriThus, when δi=δfor alli/commaorithe null hy-
pothesis of convergence is the weak inequality constraint α≥0/commaoriwhich is very
convenient to test. In view of the fact that δit→pδiast→∞ ,w ea l s oo b t a i n
a procedure for analyzing subgroup convergence. The presence of L(t) also
assists in improving power properties of the test, as we discuss below.
The conditions for convergence in the model can be characterized as
plim
k→∞δit+k=δif and only if δi=δandα≥0/commaori
plim
k→∞δit+k/negationslash=δif and only if δi/negationslash=δorα<0/periodori
Note that there is no restriction on αunder divergence when δi/negationslash=δ. However,
we have a particular interest in the case of divergence with δi/negationslash=δandα≥0/commaorias
this allows for the example considered in Figure 1where there is the possibility
of local convergence to multiple equilibria. This case is likely to be important
in empirical applications where there is evidence of clustering behavior, for ex-
ample, in individual consumption or income patterns over time. In such cases,
we may be interested in testing whether elements in a panel converge within
certain subgroups.
The remainder of this section develops an econometric methodology for
testing convergence in the above context and provides a step by step proce-dure for practical implementation. In particular, we show how to test the null
hypothesis of convergence, develop asymptotic properties of the test, includ-
ing a local power analysis, and provide an intuitive discussion of how the test
works. We also discuss a procedure for detecting panel clusters. Proofs and
related technical material are given in the Appendix .
4.1. A Regression Test of Convergence
The following procedure is a regression ttest of the null hypothesis of con-
vergence
H0:δi=δandα≥0/commaori
against the alternative HA:δi/negationslash=δfor alliorα<0.
Step 1: Construct the cross sectional variance ratio H1/Ht,w h e r e
Ht=1
NN/summationdisplay
i=1(hit−1)2/commaorihit=Xit
N−1/summationtextN
i=1Xit/periodori (32)
Step 2: Run the following regression and compute a conventional robust t
statistictˆbfor the coefficient ˆbusing an estimate of the long run variance of

TRANSITION MODELING AND CONVERGENCE TESTS 1789
the regression residuals:
log/parenleftbiggH1
Ht/parenrightbigg
−2l o gL(t)=ˆa+ˆblogt+ˆut/commaori (33)
fort=[rT]/commaori[rT]+1/commaori/periodori/periodori/periodori/commaoriT withr>0/periodori
In this regression we use the setting L(t)=log(t+1)and the fitted coefficient
of logtisˆb=2ˆα,w h e r e ˆαis the estimate of αinH0/periodoriNote that data for this
regression start at t=[rT]/commaorithe integer part of rTfor some fraction r>0. As
discussed below, we recommend r=0/periodori3.
Step 3: Apply an autocorrelation and heteroskedasticity robust one-sided t
test of the inequality null hypothesis α≥0u s i n g ˆband a HAC standard error.
At the 5% level, for example, the null hypothesis of convergence is rejected if
tˆb<−1/periodori65.
Under the convergence hypothesis, hit→1a n dHt→0a st→∞ for given
N/periodoriIn Appendix Bit is shown in ( 68)a n d( 71) thatHtthen has the logarithmic
form
logHt=−2l o gL(t)−2αlogt+2l o gvψN
δ+/epsilon1t/commaori (34)
with
/epsilon1t=1√
NηNt
v2
ψN−2
δ1
tαL(t)ψt+1
δ21
t2αL(t)2ψ2
t+Op/parenleftbigg1
N/parenrightbigg
/commaori (35)
wherev2
ψN=N−1(1−N−1)/summationtextN
i=1σ2
i→v2
ψasN→∞ ,ηNt=N−1/2/summationtextNi=1σ2
i(ξ2
it−
1),a n dψt=N−1/summationtextN
i=1σitξit.F r o m( 34) we deduce the simple regression equa-
tion
logH1
Ht−2l o gL(t)=a+blogt+ut/commaori (36)
whereb=2α,ut=−/epsilon1t, and the intercept a=logH1−2l o g(vψN/δ)=
−2l o gL(1)+u1does not depend on α.
Under convergence, log (H 1/Ht)diverges to ∞, either as 2 log L(t) when
α=0o ra s2 αlogtwhenα> 0/periodoriThus, when the null hypothesis H0ap-
plies, the dependent variable diverges whether α=0o rα>0/periodoriDivergence of
log(H 1/Ht)corresponds to Ht→0a st→∞ .T h u s ,H0is conveniently tested
in terms of the weak inequality null α≥0. Sinceαis a scalar, this null can be
tested using a simple one-sided ttest.
Under the divergence hypothesis HA, for instance, when δi/negationslash=δfor alli/commaori Ht
is shown in Appendix Bto converge to a positive quantity as t→∞ . Hence,

1790 P . C. B. PHILLIPS AND D. SUL
underHA/commaorithe dependent variable log (H 1/Ht)−2l o gL(t) diverges to −∞
in contrast to the null H0,u n d e rw h i c hl o g (H 1/Ht)diverges to ∞. The term
−2l o gL(t) in (36) therefore serves as a penalty that helps the test on the coef-
ficient of the log tregressor to discriminate the behavior of the dependent vari-
able under the alternative from that under the null. In particular, when α=0
andδi/negationslash=δfor some i/commaorithe inclusion of log L(t) produces a negative bias in the
regression estimate of bsince log tand−2l o gL(t) are negatively correlated /periodori
Thetstatistic for ˆbthen diverges to negative infinity and the test is consistent
even in this (boundary) case where α=0/periodori
Discarding some small fraction rof the time series data helps to focus at-
tention in the test on what happens as the sample size gets larger. The limit
distribution and power properties of the test depend on the value of r/periodoriOur
simulation experience indicates that r=0/periodori3 is a satisfactory choice in terms of
both size and power. Appendix Bprovides details of the construction of the
regression equation under the null and alternative, and derives the asymptoticproperties of the log ttest.
4.2. Asymptotic Properties of the logtConvergence Test
The following result gives the limit theory for the least squares estimate ˆbof
the slope coefficient bin the log tregression equation ( 36) and the associated
limit theory of the regression ttest under the null
H0.
THEOREM 1—Limit Theory under H0:Let the panel Xitdefined in (2)
have common factor µtand loading coefficients δitthat follow the generating
process (24)and satisfy Assumptions A1–A4.Suppose that the convergence hy-
pothesis H0holds and the regression equation (36)is estimated with time se-
ries data over t=[Tr]/commaori/periodori/periodori/periodori/commaoriT/commaori for some r>0/periodoriSuppose further that if α> 0,
T1/2/(T2αL(T)2N1/2)→0and ifα=0,T1/2/N→0asT/commaoriN→∞/periodori
(a)The limit distribution of ˆbis
√
NT(ˆb−b)⇒N(0/commaoriΩ2)/commaori (37)
whereΩ2=ω2
η/v4
ψ{(1−r)−(r
1−r)log2r}−1,ω2η=limN→∞1
N/summationtextN
i=1σ4
iω2ii,and
v2
ψ=limN→∞1
N/summationtextNi=1σ2
i.
(b)The limit distribution of the regression tstatistic is
tˆb=ˆb−b
sˆb⇒N(0/commaori1)/commaori
where
s2
ˆb=/hatwidestlvarr(ˆut)/bracketleftBiggT/summationdisplay
t=[Tr]/parenleftBigg
logt−1
T−[Tr]+1T/summationdisplay
t=[Tr]logt/parenrightBigg2/bracketrightBigg−1

TRANSITION MODELING AND CONVERGENCE TESTS 1791
and/hatwidestlvarr(ˆut)is a conventional HAC estimate formed from the regression residuals
ˆut=log(H 1/Ht)−2l o gL(t)−ˆa−ˆblogtfort=[Tr]/commaori/periodori/periodori/periodori/commaoriT .The estimate TNs2
ˆb
is consistent for Ω2asT/commaoriN→∞ .
REMARK 1: As shown in Appendix B, to avoid asymptotic collinearity in the
regressors (the intercept and log t), the regression ( 36) may be rewritten as
logH1
Ht−2l o gL(t)=a∗+blogt
T+ut/commaori (38)
wherea∗=a+blogT. The estimate ˆbthen has the form
ˆb−b=/parenleftBiggT/summationdisplay
t=[Tr]τtut/parenrightBigg/parenleftBiggT/summationdisplay
t=[Tr]τ2
t/parenrightBigg−1
/commaori
involving the demeaned regressor τt=(logt
T−logt
T)=logt−1
T−[Tr]+1×/summationtextT
t=[Tr]logt,w h e r e logt
T=1
T−[Tr]+1/summationtextTt=[Tr]logt
T.
REMARK 2: The convergence rate for ˆbunder the null of convergence is
O(√
NT) and is the same for all α≥0/periodoriAlthough the regression is based
on onlyO(T) observations (specifically, T−[Tr]observations), the conver-
gence rate is faster than√
Tbecause the dependent variable log Htinvolves
a cross section average ( 32)o v e rNobservations and this averaging affects
the order of the regression error ut=−/epsilon1t, as is apparent in ( 35). In par-
ticular, the leading term of /epsilon1tisOp(N−1/2)when the relative rate condition
T1/2/(T2αL(T)2N1/2)→0h o l d sf o r α> 0o rw h e nT1/2
N→0h o l d si f α=0.
These rate conditions require that Ndoes not pass to infinity too slowly rela-
tive toT/periodoriOtherwise the limit distribution ( 37) involves a bias term, as discussed
in Appendix B.
REMARK 3: The quantity Ω2
u=ω2η/v4
ψin the asymptotic variance formula
is the limit of a cross section weighted average of the long run variances ω2ii
ofηit=ξ2
it−1/periodoriAppendix Bshows how this average long run variance can
be estimated by a standard HAC estimate, such as the truncated kernel es-
timate /hatwidestlvarr(ˆut)=/summationtextM
l=−M1
T−[Tr]/summationtext
[Tr]≤t/commaorit+l≤Tˆutˆut+lgiven in ( 93) and formed in
the usual way from the residuals ˆutwith bandwidth (truncation) parameter M/periodori
Of course, other kernels may be used and the same asymptotics apply for stan-
dard bandwidth expansion rates for Msuch asM√
T+1
M→0/commaorias discussed in
Appendix B.

1792 P . C. B. PHILLIPS AND D. SUL
FIGURE 2.—The precision curve (1−r)−(r
1−r)log2r.
REMARK 4: The precision of the estimate ˆbis measured by the reciprocal of
Ω2and depends on the factor
(1−r)−/parenleftbiggr
1−r/parenrightbigg
log2r→/braceleftbigg
1/commaorir→0,
0/commaorir→1.
So the asymptotic variance of ˆbdiverges as r→1/commaoricorresponding to the fact
that the fraction of the sample used in the regression goes to zero in this event.
The precision curve is graphed in Figure 2.
REMARK 5: We call the one-sided regression ttest based on tˆbthe logttest.
T o test the hypothesis b=2α≥0, we fit the regression model ( 36), or equiv-
alently ( 38), overt=[Tr]/commaori/periodori/periodori/periodori/commaoriT and compute the tstatistictb=ˆb/sˆb/periodoriAs the
following result shows, this test is consistent against alternatives where the idio-syncratic components δ
itdiverge (i.e., when b=2α<0) as well as alternatives
where the δitconverge, but to values δithat differ across i/periodoriBoth cases seem
important in practical applications and it is an advantage of the log ttest that
it is consistent against both.
REMARK 6—Models with Heterogeneous Decay Rates αi/commaoriLi(t):A s i n d i –
cated earlier, the framework based on ( 24) may be extended by allowing for
individual specific decay rates in the loading coefficients. In such cases, boththe rate parameter αand the slowly varying function L(t) m a yv a r yo v e r i/periodoriRe-
laxation of ( 24) in this way may be useful in some applications where there is
greater heterogeneity across the population in terms of temporal responses to
the common trend effect. For instance, some individuals may converge faster
than others. Alternatively, in cases where there are subgroups in the popula-
tion, there may be heterogeneity in the temporal responses among the differentgroups. T o accommodate these extensions, we may replace ( 24) with the model
δ
it=δi+σitξit/commaoriσit=σi
Li(t)tαi/commaorit≥1/commaoriσi>0f o r a l l i/commaori (39)

TRANSITION MODELING AND CONVERGENCE TESTS 1793
where under the null hypothesis H0the decay rates αi≥0∀iand where each of
the slowly varying functions has the form Li(t)=logβitforβi>0a n dtlarge,
thereby satisfying Assumption A4. This specification should be sufficiently gen-
eral to include most cases of practical interest, the most important extension
being to allow for individual rate effects αi. T o extend our asymptotic develop-
ment to apply under ( 39), it is convenient to assume that the rates αi/commaoriβiand
the standard error σiare drawn from independent distributions with smooth
densities fα(α)/commaorif β(β)/commaorif σ(σ)supported over α∈[a/commaoriA]for some a≥0/commaoriA> 0/commaori
andβ∈[b/commaoriB]for some b/commaoriB> 0a n dσ∈[0/commaori∞). In this more general frame-
work, we still have hit→p1a n dHt→p0a st→∞ under the convergence
hypothesis H0. Specifically, we have
hit−1=ψit−ψt
δ+ψt/commaoriψit=σiξit
Li(t)tαi/commaori
where
ψt=N−1N/summationdisplay
i=1σiξit
Li(t)tαi
has mean zero and, assuming/integraltext∞
0σ2fσ(σ)dσ < ∞,
N−1N/summationdisplay
i=1σ2
i
(log2βit)t2αi→/integraldisplay∞
0σ2fσ(σ)dσ/integraldisplayB
bfβ(β)dβ
log2βt/integraldisplayA
afα(α)dα
t2α
asN→∞ by the strong law of large numbers. Using integration by parts, the
following asymptotic expansions are obtained for large t:
/integraldisplayB
bfβ(β)dβ
log2βt=fβ(b)
(log2bt)(log logt){1+op(1)}
and
/integraldisplayA
afα(α)dα
t2α=fα(a)
t2alogt{1+op(1)}/periodori
Thenψt=Op(N−1/2t−2a(log2bt)−1(log logt)−1)and
Ht=N−1N/summationdisplay
i=1(ψit−ψt)2
(δ+ψt)2=N−1/summationtextN
i=1ψ2
it
δ2/braceleftbigg
1+op/parenleftbigg1
N1/2t2αlog1+2bt/parenrightbigg/bracerightbigg
=/integraltext∞
0σ2fσ(σ)dσf α(a)fβ(β)
t2a(log2bt)(log logt)δ2{1+op(1)}/commaori

1794 P . C. B. PHILLIPS AND D. SUL
since
N−1N/summationdisplay
i=1ψ2
it=N−1N/summationdisplay
i=1σ2
iξ2
it
Li(t)2t2αi
=N−1N/summationdisplay
i=1σ2
i
Li(t)2t2αi+N−1N/summationdisplay
i=1σ2
i(ξ2
it−1)
Li(t)2t2αi
=/integraltext∞
0σ2fσ(σ)dσf α(a)fβ(β)
t2a(log2bt)(log logt){1+op(1)}/periodori
It follows that
logHt=log/braceleftbigg/integraltext∞
0σ2fσ(σ)dσf α(a)fβ(β)
t2a(log2bt)(log logt)δ2/bracerightbigg
{1+op(1)} (40)
=log/braceleftbigg
δ−2/integraldisplay∞
0σ2fσ(σ)dσf α(a)fβ(β)/bracerightbigg
−2alogt
−log{(log2bt)(log logt)}+op(1)
=2l o gν
δ−2alogt−2l o gL(t)+op(1)
withν=/integraltext∞
0σ2fσ(σ)dσf α(a)fβ(β) andL(t)2=(log2bt)(log logt).T h u s ,
apart from the particular form of the slowly varying function L(t) ,l o gHthas
the same specification for large tas (34) for the homogeneous decay rate case.
Note that in ( 40) the term involving log thas coefficient −2a,w h e r eais the
lower bound of the support of the decay rates αi. Corresponding to ( 40), we ob-
tain a regression equation with the same leading systematic form as the model
(36) given above. The regression equation is, in fact, identical to ( 36) when the
slowly varying component in ( 39) is homogeneous across iand only the rate
effectsαiare heterogeneous. The approach to testing convergence using the
logtregression ( 40) can therefore be applied when the model is of the more
general form ( 39), allowing for heterogeneity in the decay rates across the pop-
ulation. In such cases the coefficient of the log tregressor is the lower bound
of the decay rates across the population under the null. Under the alternativehypothesis of nonconvergence, when we allow for heterogeneity in the decay
ratesα
i,w em a yh a v e αi<0f o rs o m e iandαi>0 for other i.I ns u c hc a s e s ,
there may be the possibility of subgroup convergence among those individuals
with positive αiand this may be tested using the clustering algorithm described
below.
THEOREM 2—T est Consistency Under HA:Suppose the alternative hypothe-
sisHAholds and the other conditions of Theorem 1apply .

TRANSITION MODELING AND CONVERGENCE TESTS 1795
(a)Ifα≥0andδi∼iid(δ/commaoriσ2
δ)withσ2
δ>0,then
ˆb→p0/commaorit ˆb=ˆb
sˆb→− ∞/commaori
asT/commaoriN→∞ .
(b)Ifγ=−α>0,δi=δfor alli,andTγ−1/2/(√
NL(T)) +1
T+1
N→0,then
√
NTL(T)
Tγ(ˆb−b)⇒2
δN(0/commaoriQ2
ξ)/commaori
where
Q2
ξ=ω2
ξ/bracketleftbigg/integraldisplay1
r/braceleftbigg
logs−1
1−r/integraldisplay1
rlogpdp/bracerightbigg2
s2γds/bracketrightbigg
×/bracketleftbigg
(1−r)−/parenleftbiggr
1−r/parenrightbigg
log2r/bracketrightbigg−1
andtˆb=ˆb/sˆb→− ∞ .
(c)Ifγ=−α>0,δi=δfor alli,and√
NL(T)T−γ+T−1+N−1→0,then
ˆb→p0/commaorit ˆb=ˆb
sˆb→− ∞/periodori
In all cases the test is consistent .
REMARK 7: The alternative hypothesis under (a) involves δi∼iid(δ/commaoriσ2
δ)so
thatδi/negationslash=δfor alli.A si sc l e a rf r o mt h e proof of Theorem 2,i ti ss u f fi c i e n t
for the result to hold that δi/negationslash=δfori∈G/commaorisome subgroup of the panel, and
forNG=#{i∈G}/commaorithe number of elements in G/commaorito be such that NG/N→
λ>0a sN→∞/periodoriT est consistency therefore relies on the existence of enough
economies with different δi/periodoriThe condition will be satisfied, for instance, in
cases like that shown in Figure 1where there are two convergence clubs with
membership proportions λand 1 −λ. Obviously, the convergence null does
not hold in this case but the cross sectional variation of the relative transition
parameters measured by Htmay well decrease over time. Calculation reveals
that
lim
N→∞
t→∞Ht=λ(1−λ)(δA−δB)2
(λδA+(1−λ)δB)2:=HAB/commaori
whereδA=limi∈A/commaorit→∞δitandδB=limi∈B/commaorit→∞δitfor two subgroups GAandGB
with membership shares λ=limN→∞(NGA/N) and 1 −λ=limN→∞(NGB/N).
ClearlyHABwill be close to zero when the group means δAandδBare close.

1796 P . C. B. PHILLIPS AND D. SUL
REMARK 8: In part (a) of Theorem 2,ˆb→p0. The heuristic explanation is
that, when α≥0a n dδi∼iid(δ/commaoriσ2
δ),Httends to a positive constant, so that the
dependent variable in ( 36) behaves like −2l o gL(t) for large t/periodoriSince log L(t)
is the log of a slowly varying function and grows more slowly than log t,i t s
regression coefficient on log tis expected to be zero. More specifically, the
regression of −2l o gL(t) on log(t
T)produces a slope coefficient that is negative
and tends to zero like −2
logT/commaorias is shown in ( 102) in the Appendix .S i n c esˆb→p
0 also and at a faster rate, the tratiotˆbthen diverges to negative infinity and
the test is consistent.
REMARK 9: In part (b), ˆbis consistent to b=2α<0 ,b u ta tar e d u c e dr a t e
of convergence. The test is again consistent because tˆb=(ˆb−b)/s ˆb+(b/s ˆb)→
−∞ by virtue of the sign of b.
REMARK 10: In part (c), we again have ˆb→p0. In this case, the δithave
divergent behavior and Ht=Op(N). Hence, in the time series regression ( 36),
the dependent variable behaves like −2l o gL(t) for large t/commaoriand the slope co-
efficient is negative and tends to zero like −2/logT,j u s ta si np a r t( a ) .
It is also interesting to analyze the local asymptotic properties of the log t
test. The following result analyzes the asymptotic consistency of the test for
local departures from the null of the form
HLA:δi∼iid(δ/commaoric2T−2ω)/periodori (41)
Such departures measure deviations from the null H0in terms of a distance
|δi−δ|that is local to zero and of magnitude Op(T−ω)for some parameter
ω> 0/periodoriThis local consistency result turns out to be useful in the clustering
algorithm developed below.
THEOREM 3—Local Asymptotic Consistency: Suppose the local alternative
hypothesis HLAholds and the other conditions of Theorem 1apply .
(a)Under (41)withω≤α,ˆb→p0andtˆb=ˆb
sˆb→− ∞ asT/commaoriN→∞ .The
test is consistent and the rate of divergence of tˆbisO((logT)T1/2/M1/2)for all
choices of bandwidth M≤T.
(b)Under the local alternative (41)withω>α and when (T2(ω−α))/(√
N×
L(T)2)→0asT/commaoriN→∞ ,
ˆb−b=−c2
v2
ψNh(r)L(T)2
T2(ω−α){1+op(1)}→p0/commaori
tˆb→/braceleftbigg
∞/commaori forb=2α>0,
−∞/commaoriforb=2α=0,

TRANSITION MODELING AND CONVERGENCE TESTS 1797
where
h(r)=/bracketleftbigg/integraldisplay1
r/braceleftbigg
logs−1
1−r/integraldisplay1
rlogpdp/bracerightbigg
s2αds/bracketrightbigg
×/bracketleftbigg
(1−r)−/parenleftbiggr
1−r/parenrightbigg
log2r/bracketrightbigg−1
/periodori
REMARK 11: In the proof of part (a), it is shown that
ˆb=−2
logT+Op/parenleftbigg1
log2T/parenrightbigg
/commaorit ˆb=−2
logT×Op/parenleftbigg(log2T)T1/2
M1/2/parenrightbigg
/commaori
and so both ˆband thetratiotˆbhave the same asymptotic behavior as in the
case of fixed alternatives of the form δi∼iid(δ/commaoriσ2
δ)considered in part (a) of
Theorem 2. The reason for this equivalence is that under ( 41) the idiosyncratic
effects have the form
δit=δi+σitξit=δ+ξi
Tω+σiξit
L(t)tα/commaori
where the ξiare iid(0/commaoric2).W h e nω≤α, the final term is of smaller order
thanδi=δ+ξi/Tωand so the log tregression has the same discriminatory
power to detect the departure of δifromδas it does in the case where the
δiare iid(δ/commaoriσ2
δ). We say that the test is locally consistent in the sense that it is
consistent against local departures from the null of the form ( 41).
REMARK 12: When ω>α> 0/commaorithe test has negligible power to detect al-
ternatives of the form ( 41). Sinceω>α /commaori this is explained by the fact that the
alternatives are closer to the null than the convergence rate, so they elude
detection. However, when ω>α =0/commaorithe convergence rate of the idiosyn-
cratic effects δitis 1/L(t) and is slower than any power rate. In this case, re-
markably the test is consistent, although the divergence rate of the statistic is
onlyOp(T1/2/M1/2), which diverges whenM
T→0 (i.e., for standard bandwidth
choices in HAC estimation). The consistency is explained by the fact that, eventhough the alternatives δ
i/negationslash=δare still very close to the null in ( 41), the rate of
convergence of δitis so slow that the test is able to detect the local departures
from the null.
REMARK 13: Theorem 3may be interpreted to include the case where there
are additional individual effects in the formulation of the nonlinear factormodel. For instance, suppose the panel X
itinvolves an additive effect so that
X∗
it=Xit+ai=/parenleftbiggai
µt+δit/parenrightbigg
µt=δ∗
itµt

1798 P . C. B. PHILLIPS AND D. SUL
with
δ∗
it=δi+ai
µt+σiξit
L(t)tα:=δ∗i+σiξit
L(t)tα/periodori
Suppose the additive effect is common and ai=afor alli.T h e ni f δi=δfor
alliand if the common trend µt=Op(tθ)withθ>α ,w eh a v e
δ∗
it=δ+σiξit
L(t)tα+op/parenleftbigg1
L(t)tα/parenrightbigg
/commaori
in which case Theorem 1holds when α>0/periodoriIfδi=δfor alliandai∼iid(0/commaoric2)/commaori
then the model is equivalent to that considered in Theorem 3, in which case the
presence of the individual effects may be detectable, depending on the relative
magnitudes of the decay parameters θandα.
REMARK 14: Appendix Bprovides some discussion of the choice of the
slowly varying function L(t) in terms of the induced asymptotic power proper-
ties. It is shown there that among choices such as L(t)=logt,l o g2t=log logt,
and log3t=log log log t,a n dw h e r e tis large, the choice L(t)=logtis pre-
ferred in terms of asymptotic power. This choice was also found to work wellin simulations and is recommended in practice.
4.3. Club Convergence and Clustering
Rejection of the null of convergence does not imply there is no evidence
of convergence in subgroups of the panel. Many possibilities exist as we moveaway from a strict null of full panel convergence. Examples include the possibleexistence of convergence clusters around separate points of equilibria or steadystate growth paths, as well as cases where there may be both convergence clus-ters and divergent members in the full panel. If there are local equilibria orclub convergence clusters, then it is of substantial interest to be able to iden-tify these clusters, determine the number of clusters, and resolve individualsinto respective groups. In the empirical growth literature, the great diversity ineconomic performance across countries has made searching for convergenceclubs a central issue. For example, Canova ( 2004 ), Canova and Marcet ( 1995 ),
Durlauf and Johnson ( 1995 ), and Quah ( 1996 ,1997 ) all attempt to classify and
identify convergence clubs.
Perhaps the simplest case for empirical analysis occurs when subgroups can
be suitably categorized by identifying social or economic characteristics. Forexample, gender, education, age, region, or ethnicity could be identifying at-tribute variables. Under clustering by such covariates, convergence patternswithin groups may be conducted along the lines outlined above using log tre-
gressions. For instance, if the convergence null for individual consumption be-havior in a particular region (or age group) were rejected and it was suspected

TRANSITION MODELING AND CONVERGENCE TESTS 1799
that gender or ethnicity differences were a factor in the rejection log tconver-
gence tests could be rerun for different panels subgrouped according to gen-
der and ethnicity to determine whether convergence was empirically supported
within these subgroups.
Alternatively, if convergence subgroups can be determined by an empirical
clustering algorithm, then it becomes possible to subsequently explore links be-
tween the empirical clusters and various social and economic characteristics.In this case, the club convergence grouping becomes a matter for direct em-
pirical determination. A simple algorithm based on repeated log tregressions
is developed here to provide such an empirical approach to sorting individualsinto subgroups.
T o initiate the procedure, we start with the assumption that there is a “core
subgroup” G
Kwith convergence behavior, that this subgroup contains at least
Kmembers, and that the subgroup is known. Below we discuss a method for
detecting the initial core subgroup. Next we consider adding an additional in-
dividual ( K+1, say) to GK. T o assess whether the new individual belongs to
GK, we perform a log ttest. IfK+1b e l o n g st o GK, the point estimate of bin
the logttest will not be significantly negative and the null hypothesis will be
supported in view of Theorem 1. Otherwise, the point estimate of bwill de-
pend on the size of Kand the extent of the deviation from the null. T o see this,
setδi=δAfori=1/commaori/periodori/periodori/periodori/commaoriK andδi=δBfori=K+1/periodoriThe variation of δiin
the augmented subgroup is given by
σ2=1
K+1K+1/summationdisplay
i=1(δi−¯δ)2=K
(K+1)2(δA−δB)2/commaori (42)
where
¯δ=1
K+1K+1/summationdisplay
i=1δi=KδA
K+1+δB
K+1/periodori
AsK→∞ ,σ2=O(K−1)→0a n d ¯δ→δA.
An asymptotic analysis of club convergence patterns in such cases can be
based on local alternatives of the form
δi∼iidN(δ/commaoric2K−1)/periodori (43)
Appendix Cprovides such an analysis. It is shown there that when c2>0a n d
K/T2α→0a sT→∞/commaorithe procedure is consistent in detecting departures
of the form ( 43) for all bandwidth choices M≤T/periodoriGiven that σ2=O(K−1)
in (42), this analysis also covers the case where δi=δAfori=1/commaori/periodori/periodori/periodori/commaoriK and
δK+1=δB/negationslash=δA/periodoriOn the other hand, when δi=δAfori=1/commaori/periodori/periodori/periodori/commaoriK +1, the null
hypothesis holds for N=K+1a n dtˆb=(ˆb−b)/s ˆb⇒N(0/commaori1),a si nT h e o r e m 1.

1800 P . C. B. PHILLIPS AND D. SUL
WhenT2α/K→0, the alternatives ( 43) are very close to the null, relative
to the convergence rate except when α=0. This case is analogous to case (b)
of Theorem 3and as that theorem shows, the test is inconsistent and unable
to detect the departure from the null when α> 0. However, when α=0, the
convergence rate is slowly varying under the null, and Theorem 3shows that
the test is in fact consistent against local alternatives of the form ( 43). In effect,
although the alternatives are very close (because Kis large), the convergence
rate is slow (slower than any power rate) and this suffices to ensure that the
test is consistent as T→∞ .
We now suggest the following method of finding a core subgroup GK.W h e n
there is evidence of multiple club convergence as T→∞ ,t h i si su s u a l l ym o s t
apparent in the final time series observations. We therefore propose that the
panel be clustered initially according to the value of the final time series ob-
servation (or some average of the final observations). After ordering in this
way, size ksubgroups, Gk={1/commaori/periodori/periodori/periodori/commaorik }for{k=2/commaori/periodori/periodori/periodori/commaoriN }, may be constructed
based on panel members with the khighest final time period observations.
Within each of these subgroups, we may conduct log tregression tests for con-
vergence, denoting by tkthe test statistic from this regression using data from
Gk. Next, we choose k∗to maximize tkover all values for which tk>c for
k=2/commaori/periodori/periodori/periodori/commaoriN and where cis some critical value /periodoriA precise algorithm based on
these ideas is contained in the following step by step procedure to determinethe clustering pattern and to provide a stopping rule for the calculations.
Step 1:Last Observation Ordering . Order individuals in the panel according
to the last observation in the panel. In cases where there is substantial time
series volatility in X
it/commaorithe ordering may be done according to the time series
average, (T−[Ta])−1/summationtextT
t=[Ta]+1Xit, over the last fraction ( f=1−a)o ft h e
sample (for example, f=1/3o r1/2).
Step 2:Core Group Formation . Selecting the first khighest individuals in
the panel to form the subgroup Gkfor some N>k ≥2, run the log tregres-
sion and calculate the convergence test statistic tk=t(Gk)for this subgroup /periodori
Choose the core group size k∗by maximizing tkoverkaccording to the crite-
rion:
k∗=arg max
k{tk}subject to min {tk}>−1/periodori65/periodori (44)
The condition min {tk}>−1/periodori65 plays a key role in ensuring that the null hy-
pothesis of convergence is supported for each k/periodoriHowever, for each kthere
is the probability of a type II error. Choosing the core group size so that
k∗=arg maxk{tk}then reduces the overall type II error probability and helps
ensure that the core group Gk∗is a convergence subgroup with a very low false

TRANSITION MODELING AND CONVERGENCE TESTS 1801
inclusion rate.8Our goal is to find a core convergence group in this test and
then proceed in Step 3to evaluate additional individuals for membership of
this group. If there is a single convergence club with all individuals included,
then the size of the convergence club is N; when there are two or more conver-
gence clubs, each club necessarily has membership less than N. If the condition
min{tk}>−1/periodori65 does not hold for k=2, then the highest individual in Gkcan
be dropped from each subgroup and new subgroups G2j={2/commaori/periodori/periodori/periodori/commaorij }can be
formed for 2 ≤j≤N. The step can be repeated with test statistics tj=t(G 2j).
If the condition min {tj}>−1/periodori65 is not satisfied for the first j=2, the step may
be repeated again, dropping the highest individuals in Gjand proceeding as
before. If the condition does not hold for all such sequential pairs, then weconclude that there are no convergence subgroups in the panel. Otherwise, we
have found a core convergence subgroup, which we denote G
k∗.
Step 3:Sieve Individuals for Club Membership .L e tGc
k∗be the complemen-
tary set to the core group Gk∗. Adding one individual in Gck∗a tat i m et ot h e
k∗core members of Gk∗, run the log ttest. Denote the tstatistic from this re-
gression as ˆt. Include the individual in the convergence club if ˆt>c ,w h e r ecis
some chosen critical value. We will discuss the choice of the critical value below
and in the Monte Carlo section. Repeat this procedure for the remaining in-dividuals and form the first subconvergence group. Run the log ttest with this
first subconvergence group and make sure t
ˆb>−1/periodori65 for the whole group. If
not, raise the critical value, c/commaorito increase the discriminatory power of the log t
test and repeat this step until tˆb>−1/periodori65 with the first subconvergence group.
Step 4:Stopping Rule . Form a subgroup of the individuals for which ˆt<c in
Step 3. Run the log ttest for this subgroup to see if tˆb>−1/periodori65 and this cluster
converges. If so, we conclude that there are two convergent subgroups in thepanel. If not, repeat Steps 1–3on this subgroup to determine whether there is a
smaller subgroup of convergent members of the panel. If there is no kin Step 2
for which t
k>−1/periodori65, we conclude that the remaining individuals diverge.
The application in Section 6provides practical details and an illustration
of the implementation of this algorithm. T able IV, in particular, lays out the
sequence of steps involved in a specific application where there are multipleclusters.
8We might consider controlling the critical value based on the distribution of the max k{tk}
statistic over the cross section. However, since this distribution changes according to the true
size and composition of the actual convergence subgroup (which is unknown), this approach isnot feasible. Instead, the max
ktkrule is designed to be conservative in its selection of the core
subgroup so that the false inclusion rate is small. Note that the rule ( 44) is used to determine
only the membership of this core group. Subsequently, we apply individual log tregression tests
to assess membership of additional individuals. The performance of this procedure is found to be
very satisfactory in simulations that are reported in Section 5.

1802 P . C. B. PHILLIPS AND D. SUL
5.MONTE CARLO EXPERIMENTS
The simulation design is based on the data generating process (DGP)
Xit=δitµt/commaoriδit=δi+δ0
it/commaori (45)
δ0it=ρiδ0it−1+/epsilon1it/commaoriVa r(/epsilon1it)=σ2
iL(t+1)−2t−2α
fort=1/commaori/periodori/periodori/periodori/commaoriT andL(t+1)=log(t+1), so the slowly varying function
L(t+1)−1is well defined for all t≥1. We set /epsilon1it∼iidN(0/commaoriσ2
iL(t+1)−2t−2α)
andρi∼U[0/commaoriρ]forρ=0/periodori5/commaori0/periodori9. T o ensure that δit≥0f o ra l liandt,w ec o n –
trol the range of σiby setting σi∼U[0/periodori02/commaori0/periodori28]so that the 97.5% lower confi-
dence limit for δitatt=1 is greater than zero and then discard any trajectories
that involve negative realizations. The simulation treats δiandρias random
variables drawn from the cross section population, so that for each iteration
new values are generated. Since the common component µtcancels out in the
application of our procedure, there is no need to specify a parametric formforµ
t.
The logtregression procedure relies on input choices of the initiating sam-
ple fraction rand the slowly varying function L(t) . In simulations, we explored
a variety of possible choices; the full Monte Carlo results that consider theseare available online.
9Here we briefly report the results of varying these inputs
and make some recommendations for applied work.
First we review the effects of varying the initiating sample fraction r.V a r i –
ousNandTcombinations are considered under the DGP ( 45)a n da v e r a g e
rejection rates over these combinations were computed under the null hypoth-esis to assess the impact on test size. When α=0, test size is close to nominal
size for sample sizes N/commaoriT≥50 andr≥0/periodori3. When the decay rate αis small
and nonzero, for example when α∈(0/commaori0/periodori4), the rejection rate decreases as r
increases for given NandT. The rejection rate also decreases rapidly as α
increases. Of course, when α> 0, asymptotic theory shows that test size con-
v e r g e st oz e r os i n c et h e tstatistic diverges to positive infinity as N/commaoriT→∞ in
this case. Further, when rincreases, test power declines because the effective
sample size is smaller, which reduces discriminatory power. Thus, since αis
unknown, practical considerations suggest choosing a value of rfor which size
will be accurate when αis close to zero, for which size is not too conservative
whenαis larger, and for which power is not substantially reduced by the ef-
fective sample size reduction. The simulation results indicate that r∈[0/periodori2/commaori0/periodori3]
achieves a satisfactory balance. When Tis small or moderate ( T≤50, say),
r=0/periodori3 seems to be a preferable choice to secure size accuracy in the test for
smallα,a n dw h e n Tis large ( T≥100, say), the choice r=0/periodori2 seem satisfac-
tory in terms of size and this choice helps to raise test power. Panels A and Bof Figure 3provide a visualization of the effects of different choices of ron
actual size when α=0a n dα=0/periodori1 for various T.
9http://homes.eco.auckland.ac.nz/dsul013/working/power_size.xls .

TRANSITION MODELING AND CONVERGENCE TESTS 1803
FIGURE 3.—Effects of ron rejection rates under H0/commaoriρi∈(0/commaori0/periodori9)/commaorifor nominal size 5%.
Next, we review the effects of varying the choice of L(t) in the formula-
tion of the log tregression equation. T o do so, we standardize ( 45) so that
Va r(/epsilon1it)=σ2
it−2α. Hence, the true model for δit(and the DGP used in the
simulation) does not involve a slowly varying function, whereas the fitted log t
regression involves some slowly varying function L(t) . The inclusion of L(t)
in the regression then plays the role of a penalty function and we consider
the effects of the presence of this variable in the form of the four functionsL(t)=logt,2 l o gt,l o g l o gt,a n d2 l o g l o g t, calculating the corresponding re-
jection rates for the tests in each case. Note that when α=0 in the true DGP ,
convergence no longer holds and the test diverges to negative infinity, as inthe asymptotics of Theorem 2.F o rα> 0/commaoritest size should converge to zero
asN/commaoriT→∞/commaorijust as in the correctly specified case. However, simulations
show that for very small α(α=0/periodori01, say) there are substantial upward size dis-
tortions when L(t)=2l o gt/commaoriwhereas the test is conservative for L(t)=logt,
2l o gl o gt,a n dl o g l o g t. T est power is reduced when L(t)=2l o gl o gtand
log logtin comparison with L(t)=logt. Hence, among these possibilities, the
function L(t)=logtproduces the least size distortion and the best test power
asNandTincrease. A full set of simulation results is available online.
10
Based on these experiments, we recommend setting r=0/periodori3 and suggest
L(t)=logtfor the slowly varying function in the log tregressions. These set-
tings are used in the remaining experiments.
For the remaining simulations we set T=10/commaori20/commaori30/commaori40 andN=50/commaori100/commaori
200. Since the size of the test is accurate to two decimal place when α=0
and power is close to unity for moderately large T(T≥50), these results are
not reported here. The number of replications was R=2000/periodoriWe consider the
following four cases.
10http://homes.eco.auckland.ac.nz/dsul013/working/logtcomp.xls .

1804 P . C. B. PHILLIPS AND D. SUL
Case 1:Pure Convergence . T o check the size of the test, we set δi=1f o ra l l
iandα=0/periodori01/commaori0/periodori05/commaori0/periodori1/commaori0/periodori2. When α>0/periodori2/commaorithe test size is zero for all Tand
N/commaoriconfirming that the limit theory is accurate for small TandNin this case.
T o measure the bias in the estimate of the speed of convergence, ˆb, we used
α=0/periodori05, 0/periodori1, and 0/periodori5.
Case 2:Divergence .W es e tδi∼U[1/commaori2].
Case 3:Club Convergence . We considered two equal sized convergence clubs
in the panel with numbers S1=S2=50 and overall panel size N=100. For the
first panel, we set δ1=1, and for the second panel, we set δ2=[1/periodori1/commaori1/periodori2/commaori1/periodori5]to
allow for different distances between the convergence clubs.
Case 4:Sorting Procedure . T wo convergence clubs as in Case 3,b u tw i t h
δ1=1a n dδ2=1/periodori2/periodoriWe consider various convergence rates with α=
(0/periodori01/commaori0/periodori05/commaori0/periodori1/commaori0/periodori2)and with ρ=0/periodori5/periodori
Ta b l e Igives the actual test size. The nominal size is fixed to be 5%. When the
speed of convergence parameter αis very close to zero, there is size distortion
for small T. However, this distortion diminishes quickly when Tincreases or as
αincreases. As αincreases, the test becomes conservative. Rejection rates in
the test are expected to go to zero when α>0a sT/commaoriN→∞ since the limit the-
ory in Theorem 1shows that the regression tstatistic (centered on the origin)
diverges to positive infinity in this case.
TABLE I
SIZE OF THE logtTEST(5% N OMINAL SIZE)
ρ∈[0/commaori0/periodori5] ρ∈[0/commaori0/periodori9]
TN α =0/periodori01α=0/periodori05α=0/periodori1α=0/periodori2α=0/periodori01α=0/periodori05α=0/periodori1α=0/periodori2
10 50 0.30 0.21 0.13 0.04 0.25 0.18 0.10 0.03
10 100 0.40 0.26 0.12 0.02 0.32 0.22 0.10 0.01
10 200 0.56 0.32 0.12 0.01 0.41 0.24 0.08 0.00
20 50 0.15 0.09 0.03 0.00 0.13 0.07 0.03 0.00
20 100 0.18 0.08 0.01 0.00 0.14 0.05 0.01 0.00
20 200 0.23 0.07 0.01 0.00 0.17 0.04 0.00 0.00
30 50 0.09 0.05 0.01 0.00 0.10 0.04 0.01 0.00
30 100 0.14 0.03 0.00 0.00 0.10 0.02 0.00 0.00
30 200 0.14 0.02 0.00 0.00 0.10 0.01 0.00 0.00
40 50 0.09 0.03 0.01 0.00 0.07 0.03 0.00 0.00
40 100 0.09 0.02 0.00 0.00 0.08 0.02 0.00 0.00
40 200 0.11 0.01 0.00 0.00 0.08 0.01 0.00 0.00

TRANSITION MODELING AND CONVERGENCE TESTS 1805
TABLE II
MEAN VALUES OF THE ESTIMATED SPEED OF CONVERGENCE
ρ∈[0/commaori0/periodori5] ρ∈[0/commaori0/periodori9]
TN b =0/periodori1 b=0/periodori2 b=1/periodori0 b=0/periodori1 b=0/periodori2 b=1/periodori0
10 50 −0/periodori09 0.00 0.84 −0/periodori06 0.05 0.96
10 100 −0/periodori11 0.00 0.84 −0/periodori06 0.05 0.96
10 200 −0/periodori10 0.00 0.84 −0/periodori06 0.05 0.95
20 50 0 /periodori01 0.12 0.93 0 /periodori06 0.15 1.06
20 100 0 /periodori02 0.12 0.93 0 /periodori06 0.17 1.06
20 200 0 /periodori02 0.12 0.93 0 /periodori05 0.16 1.06
30 50 0 /periodori06 0.16 0.96 0 /periodori07 0.18 1.04
30 100 0 /periodori05 0.15 0.95 0 /periodori07 0.18 1.04
30 200 0 /periodori05 0.15 0.96 0 /periodori07 0.18 1.04
40 50 0 /periodori07 0.17 0.97 0 /periodori08 0.18 1.02
40 100 0 /periodori07 0.16 0.97 0 /periodori08 0.18 1.02
40 200 0 /periodori07 0.17 0.97 0 /periodori08 0.18 1.02
Ta b l e IIshows the mean values of ˆb/periodoriWhenαis small, there is a some-
what mild downward bias for small T/commaoriwhich arises from the correlation be-
tween the log tregressor and the second order terms in ut. The direction of
the correlation is negative since in the expansion, the second order term of
ut,L(t)−2t−2αψ2
t/δ2, is negatively correlated with L(t) . The bias is dependent
on the size of Tandαrather than N, just as the asymptotic theory predicts,
whenαis small. This downward bias quickly disappears for larger Tor asα
increases.
Ta b l e IIIshows the power of the test without size adjustment. For Case 2,
the power becomes 1, irrespective of the values of α,T,a n dN.F o rC a s e 3,
the logttest distinguishes well whether there is club convergence or not, even
with small Tandα/commaoriexcept when δ1is very close to δ2/periodoriForδ1−δ2=0/periodori1/commaorithe
rejection rate is more than 50% with α=0/periodori01 forT=10/commaoriand increases rapidly
asTorNgrows.
Figure 4shows how the empirical clustering procedure suggested in the pre-
vious section works. Overall the results are encouraging. Panels A and B in
Figure 4display the size and power of the clustering test across various critical
values with α=0a n dα=0/periodori2, respectively. When α>0, the size of the clus-
tering test—measuring the failure rate of including convergence members in
the correct subconvergence club—goes to zero asymptotically since tˆbtends to
positive infinity under the null of convergence as T→∞ .
As asymptotic theory predicts, the size of the clustering test goes to zero as T
increases in this case. Meanwhile, the power of the clustering test—the success
rate in excluding nonconvergence members from the correct subconvergenceclub—goes to unity asymptotically regardless of the critical values used. How-
ever, in finite samples, test power is less than unity and, as larger critical values

1806 P . C. B. PHILLIPS AND D. SUL
TABLE III
THEPOWER OF THE logtTEST(5% T EST)
δ1=1
TNα δ i−U[1/commaori2] δ2=1/periodori5 δ2=1/periodori2 δ2=1/periodori1
10 50 0.01 1.00 1.00 0.93 0.57
10 100 0.01 1.00 1.00 0.99 0.77
10 200 0.01 1.00 1.00 1.00 0.92
20 50 0.01 1.00 1.00 0.99 0.61
20 100 0.01 1.00 1.00 1.00 0.82
20 200 0.01 1.00 1.00 1.00 0.97
30 50 0.01 1.00 1.00 1.00 0.69
30 100 0.01 1.00 1.00 1.00 0.89
30 200 0.01 1.00 1.00 1.00 0.99
40 50 0.01 1.00 1.00 1.00 0.78
40 100 0.01 1.00 1.00 1.00 0.95
40 200 0.01 1.00 1.00 1.00 1.00
10 50 0.05 1.00 1.00 0.93 0.59
10 100 0.05 1.00 1.00 1.00 0.77
10 200 0.05 1.00 1.00 1.00 0.91
20 50 0.05 1.00 1.00 0.99 0.64
20 100 0.05 1.00 1.00 1.00 0.81
20 200 0.05 1.00 1.00 1.00 0.96
30 50 0.05 1.00 1.00 1.00 0.73
30 100 0.05 1.00 1.00 1.00 0.91
30 200 0.05 1.00 1.00 1.00 0.99
40 50 0.05 1.00 1.00 1.00 0.84
40 100 0.05 1.00 1.00 1.00 0.96
40 200 0.05 1.00 1.00 1.00 1.00
are employed in the selection procedure, we do find higher power in the test.
Panels C and D show the sum of the type I and II errors in this procedureagainst various significance levels when α=0a n dα=0/periodori2/commaorirespectively. As T
increases, the size and the type II error of the clustering test both go to zero.There is some trade-off between the type I and II errors, and in finite samples,the power gain by using higher significance level seems to exceed the size loss.Hence, for both cases α=0a n dα=0/periodori2, the use of a sign test (that is, a test
in which the critical value is zero at the 50% significance level) minimizes thesum of the type I and II errors for small T(that is,T=20/commaori50 in panel C). For
larger values of T(T=100/commaori200 in panel C), a lower nominal significance level
minimizes the sum of the two errors when α=0 (in panel C, these nominal
significance levels are 40% for T=100 and 20% for T=200, and these cases
are marked in the chart). When α=0/periodori2/commaorithe sign test minimizes the sum of the
type I and type II errors for all values of T,a si sc l e a ri np a n e lD .

TRANSITION MODELING AND CONVERGENCE TESTS 1807
FIGURE 4.—Effect of nominal critical value choice on test performance.
Figure 5shows the finite sample performance of the core group selection
procedure based on the max ktkrule. Panel A in Figure 5shows the false inclu-
sion rates of nonconvergence members into a core group. As Tincreases or as
αincreases, the max trule appears to sieve individuals very accurately. Even
whenα=0, more than 99% of the time the max trule does not include any
nonconvergence member into a core group when T≥100. Panel B in Figure 5
shows the size of the core groups selected for various values of αandT.A sα
andTincrease, the size of the core group increases steadily and approaches
the true size of the convergence club ( 51) for some configurations.
6.EMPIRICAL APPLICATION TO CALCULATING THE COST OF LIVING
We provide an empirical application to illustrate the usefulness of the time
varying nonlinear factor model and the operation of the log tregression test
for convergence and clustering. The example shows how to calculate a proxy

1808 P . C. B. PHILLIPS AND D. SUL
FIGURE 5.—Performance of the max trule in core group selection.
for cost of living indices by using 19 consumer price indices (CPI’s) for U.S.
metropolitan areas. Measuring the cost of living by statistical indices has been
a long-standing problem of econometrics that has many different contributionsand much controversy.
11A number of commercial web sites now provide vari-
ous online cost of living indices. From a strict economic perspective, the mostappropriate calculations for cost of living indices take account of the changing
basket of commodities and services over time as well as nonconsumer price
information such as local taxation, health and welfare systems, and economic
infrastructure; while relevant, such matters are beyond the scope of many stud-
ies, including the present analysis. Here we constrain ourselves to working
with cost of living indices obtained directly from commonly available consumer
price information for 19 different metropolitan areas.
Our goal is to measure the relative cost of living across various metropolitan
areas in the United States and to illustrate our empirical approach by exam-ining evidence for convergence in the cost of living. We use the relative tran-
sition parameter mechanism to model individual variation, writing individual
city CPI as
logP
o
it=δo
itlogPo
t+eit/commaori (46)
where log Po
itis the log CPI for the ith city, log Po
tis the common CPI trend
across cities, and eitcontains idiosyncratic business cycle components. The em-
pirical application is to 19 major metropolitan U.S. cities from 1918 to 2001.
Appendix Dgives a detailed description of the data set.
It is well known that consumer price indices cannot be used to compare the
cost of living across U.S. cities because of a base year problem. For example, if
11See the Journal of Economic Perspectives , 12, issue 1, for a recent special issue dealing with
cost of living indices.

TRANSITION MODELING AND CONVERGENCE TESTS 1809
the base year were taken to be the last time period of observation, then the CPI
indices would seem to converge because the last observations are identical. T oavoid such artificial forms of convergence, we take the first observation as thebase year and rewrite the data as log P
it=log(Po
it/Po
i1)=logPo
it−logPo
i1, from
which we obtain log Pit=[δo
it−δo
i1(logPo
1/logPo
t)+(eit−ei1)/logPo
t]logPo
t=
δitlogPo
t. The common price index Po
tusually has a trend component, so that
we have log Po
t=Op(tα)for some α>0. For instance, if log Po
tfollows a ran-
dom walk with drift, we have log Po
t=a+logPo
t−1+/epsilon1t=at+/summationtextt
s=1/epsilon1s=Op(t).
Then log Po
1/logPo
t=op(1)and(eit−ei1)/logPo
t=op(1)for large t, so the
impact of the initial condition on δitdisappears as t→∞/commaoriand more rapidly
the stronger the trend (or larger α). Cyclical effects are also of smaller magni-
tude asymptotically. Of course, these effects may be smoothed out using other
techniques such as various filtering devices.
Figure 6shows the cross sectional maximum, minimum, and median of the
period-by-period log consumer price indices across the 19 U.S. cities. Due to
the base year initialization, the CPI’s in 1918 are identical, but the initial ef-fects seem to have dissipated in terms of the observed dispersion in Figure 6
within two decades. T o avoid the base year effect in our own calculations, wediscard the first 42 annual observations. The relative transition parameters for8 major metropolitan cities over the subsequent period 1960–2000 are plottedin Figure 7after smoothing the CPI’s using the WHP filter. The transition pa-
rameter curves provide relative cost of living indices across these metropolitan
areas.
As is apparent in Figure 7, San Francisco shows the highest cost of living;
Seattle is in second place. Chicago has the median cost of living among the
19 cities at the end of the sample and Atlanta has the lowest cost of living,
FIGURE 6.—Min, max, and median of consumer price indices.

1810 P . C. B. PHILLIPS AND D. SUL
FIGURE 7.—Relative transition curves (relative cost of living).
again with little transition. Also apparent is that the cost of living indices in
Houston and St. Louis have declined relatively since 1984, while those in New
Y ork, Seattle, and San Francisco have increased. The estimated equation for
the overall log tregression with r=1/3i s
logH1
Ht−2l o gl o gt=0/periodori904−0/periodori98 logt/commaori
(14/periodori3)(−51/periodori4)
which implies that the null hypothesis of convergence in the relative cost ofliving is clearly rejected at the 5% level.
Next, we investigate the possibility of club convergence in cost of living in-
dices among cities. Following the steps suggested in the previous section, we
order the CPI’s based on the last time series observation (Step 1)a n dd i s –
play them in the first column of T able IV. Note that for further convenience
(based on the convergence results we obtain below), we changed the order
between New Y ork (NYC) and Cleveland (CLE) metro. Based on this order-
ing, we choose San Francisco as the base city in the ordering, run the log t
regression by adding further cities one by one, and calculate the tstatistics
until the tstatistic is less than −1/periodori65 (Step 2). Proceeding in this way, we
find that t
k=6/periodori1/commaori−0/periodori7/commaori1/periodori4/commaoriand−7/periodori8f o rk={1/commaori2},{1/commaori2/commaori3},{1/commaori2/commaori3/commaori4},a n d
{1/commaori2/commaori3/commaori4/commaori5}, respectively. When we add Minnesota, the tkstatistic becomes
tk=−7/periodori8 and we stop adding cities. The tkstatistics are maximized for the
groupk={1/commaori2}and so the core group is taken to be San Francisco and Seat-
tle. Next, working from this core group, we add one city at a time and print outitststatistic in the third column of T able IV. We use the 50% critical value (or
sign test), based on our findings in the Monte Carlo experiments. Only when

TRANSITION MODELING AND CONVERGENCE TESTS 1811
TABLE IV
CLUB CONVERGENCE OF COST OF LIVING INDICES AMONG 19 U.S. M ETROPOLITAN CITIES
LastT
Ordertvaluelogt
Te s tlogt
Te s t Name Step 1 Step 2 Club Name Step 1 Step 2 Club
1S F O B a s e Core
Core
0/periodori7

tS1=0/periodori71tS2=8/periodori18
2S E A 6.1 S1 tSc
2=−0/periodori68
4N Y C 1 /periodori4
3C L E −0/periodori7−0/periodori7
−51/periodori0
−12/periodori2
−2/periodori4
−3/periodori7
−14/periodori9
−28/periodori8
−12/periodori0
−35/periodori6
−46/periodori9
−50/periodori3
−124/periodori4
−16/periodori7
−134/periodori6
−116/periodori5
−20/periodori7

S
c
1tSc
1=−54/periodori6CLE Base CoreS2
S2 5M I N −7/periodori8M I N 1.0 Core
6 LAX LAX −1.7 −1.7Sc
2
7P O R P O R 5 . 3 S2
S2
S28 BOS BOS 13.9
9C H I C H I 6 . 1
10 BAL BAL −19.9Sc
2
11 PHI PHI 7.6 S2
12 PIT PIT −1/periodori6
−18/periodori1
−34/periodori6
−4/periodori9
−12/periodori3
−28/periodori0
−14/periodori1
−67/periodori2Sc
2
13 CIN CIN Sc
2
14 STL STL Sc
2
15 DET DET Sc
2
16 WDC WDC Sc
2
17 HOU HOU Sc
2
18 KCM KCM Sc
2
19 ATL ATL Sc
2
New Y ork (NYC) is added to the core group, is the tstatistic still positive. The
logtregression with these three cities gives a tstatistic of 0 /periodori71, and the null hy-
pothesis of convergence cannot be rejected. Hence the first convergence club,
S1, includes SFO, SEA, and NYC.
For the remaining 16 cities ( Sc
1), the log ttest rejects the null of convergence
even at the 1% level ( tSc
1=−54/periodori6). Hence, we further analyze the data for
evidence of club convergence among these 16 cities. Repeating the same pro-cedure again, we find the next core group as Cleveland (CLE) and Minneapo-
lis/St. Paul (MIN), and select 4 other cities (POR, BOS, CHI, and PHI) for the
second subgroup, S
2.T h el o g ttest with these 6 cities does not reject the null
of convergence ( tS2=8/periodori2). Further, the log ttest with the remaining 10 cities
does not reject the null either ( tSc
2=−0/periodori68) at the 5% level. Hence with the
last group, there is rather weak evidence for convergence.
Figure 8shows the relative transition parameters with the cross sectional
means of the three convergence clubs. The transition curves indicate that thethree clubs show some mild evidence of convergence until around 1982, butthat after this there is strong evidence of divergence. In sum, the evidence isthat the relative cost of living across 19 major U.S. metropolitan areas doesnot appear to be converging over time. However, there is some evidence ofrecent convergence clustering among three different metropolitan subgroups:

1812 P . C. B. PHILLIPS AND D. SUL
FIGURE 8.—Relative transition curves across clubs.
one with a very high cost of living, one with a moderate cost of living, and one
that is relatively less expensive than the other two groups.
7.CONCLUSION
This paper has proposed a new mechanism for modeling and analyzing eco-
nomic transition behavior in the presence of common growth characteristics.
The model is a nonlinear factor model with a growth component and a timevarying idiosyncratic component that allows for quite general heterogeneity
across individuals and over time. The formulation is particularly useful in mea-
suring transition toward a long run growth path or individual transitions overtime relative to some common trend, representative, or aggregate variable.
The formulation also gives rise to a simple and convenient time series regres-
sion test for convergence. This log tconvergence test further provides the basis
for a stepwise clustering algorithm that is proposed for finding convergence
clusters in panel data and analyzing transition behavior between clusters. The
tests have good asymptotic properties, including local discriminatory power,and are particularly easy to apply in practice. Simulations show that the pro-
posed log ttest and the clustering algorithm both work very well for values of
TandNthat are common in applied work. The empirical application reveals
some of the potential of these new procedures for practical work.
Some extensions of the procedures seem worthwhile to pursue in later work.
In particular, the procedures are developed here for panels of a scalar vari-able and will need to be extended when there are many variables. For exam-
ple, to analyze issues of convergence and clustering in the context of potential
relationships between two panel variables such as personal expenditure andincome, the concepts and methods in the paper must be modified, possibly by

TRANSITION MODELING AND CONVERGENCE TESTS 1813
working with panel regression residuals or through panel vector autoregression
and error correction formulations.
In addition, our procedures have been developed primarily for cases where
there is a single common growth factor. The approach has the advantage that itis applicable regardless of the form of the generating mechanism for this com-mon factor. In practical work, of course, there may be several relevant factors.
General factor-analytic techniques are designed to address situations where
there may be multiple factors and where the number of factors is unknown.These techniques have received attention in past econometric work (e.g., Sar-
gent and Sims ( 1997 )) and in recent large multidimensional panel modeling
(e.g., Bai and Ng ( 2002 )). In such work, investigators may use data based meth-
ods to search for the number of commonalities. In the present approach, there
is a presumption that one growth factor dominates the commonalities for large
tunder the null hypothesis of convergence. But, as seen in some of the exam-
ples of Section 2and the empirical application to city cost of living, sometimes
the alternative hypothesis will be more relevant in practice. The methods de-veloped here continue to apply in such situations, allowing for different sub-
group behavior in which multiple factors may indeed be present, manifestingthemselves in the form of the individual or subgroup transition effects.
Cowles Foundation for Research in Economics, Yale University, Box 208281,
New Haven, CT 06520-8281, U.S.A.; University of Auckland, New Zealand; and
University of York, York, U.K.; peter.phillips@yale.edu
and
Dept. of Economics, University of Auckland, Auckland, New Zealand.
Manuscript received October, 2006; final revision received April, 2007.
APPENDIX A: S TANDARDIZING GROWTH COMPONENTS
This appendix analyses how the growth components in the decomposition
Xit=ait+git=/parenleftbiggait+git
µt/parenrightbigg
µt=δitµt
may be standardized to yield the transition and growth curves discussed in Sec-
tion 3.W el e tt→∞ and characterize the limiting behavior of the components
δitandµt.
We first proceed as if the growth components were nonstochastic. Suppose
git=fi(t)is regularly varying at infinity with power exponent γi(e.g., see
Seneta ( 1976 ) for a discussion of regularly varying functions) so that
fi(t)=tγiWi(t)/commaori (47)

1814 P . C. B. PHILLIPS AND D. SUL
whereWi(t)is slowly varying at infinity, namely Wi(λt)/W i(t)→1a st→∞
for allλ>0. For example, we might have Wi(t)=logt,l o g2t/commaorior log log t.S i m –
ilarly, let µtbe regularly varying at infinity with power exponent γ>0 so that
µt=tγZ(t) (48)
for some slowly varying function Z(t) . The regular variation requirement
means that fi(t)andµtboth behave asymptotically very much like power func-
tions for large t. In the simplest case where the common growth component is
a linear drift (i.e., µt=t)a n dgit/t→mfor alliast→∞ , there is growth con-
vergence and we have γi=γ=1a n dWi(t)=Z(t)=1. Conditions ( 47)a n d
(48) allow for a much wider variety of asymptotic behavior, including the pos-
sibility that individual ieconomy’s growth may deviate from the common path
(whenγi/negationslash=γ) and that there may be a slowly varying component in the growth
path. For example, if γ=0a n dZ(t)=logt/commaorithenµtevolves logarithmically
withtand growth is therefore slower than any polynomial rate.
Sett=[Tr]for some r>0 representing the fraction of the overall sample T
corresponding to observation t. Then under ( 47),
T−γigit=T−γi[Tr]γiWi(Tr)
Wi(T)Wi(T)∼rγiWi(T) (49)
and
T−γµt=T−γ[Tr]γZ(Tr)
Z(T)Z(T) ∼rγZ(T)/periodori
We deduce from this asymptotic behavior and ( 5) that
T−γiXit=ait+git
Tγi=ait
Tγi+git
Tγi∼rγiWi(T)/commaori
T−γµt∼rγZ(T) =µ(r)Z(T)/commaori
whereµ(r)=rγ/periodoriWriting, as in ( 5),
/parenleftbiggait+git
µt/parenrightbigg
µt=δitµt/commaori
we then have
1
Tγi/parenleftbiggait+git
µt/parenrightbigg
µt=ait
Tγi+git
TγiTγ
µt/parenleftbiggµt
Tγ/parenrightbigg
=o(1)+git
TγiTγ
µt/parenleftbiggµt
Tγ/parenrightbigg
∼{rγi−γJi(T)}{rγZ(T)}
=δJ
iT(r)µZT(r)/commaori

TRANSITION MODELING AND CONVERGENCE TESTS 1815
where the ratio Ji(T)=Wi(T)/Z(T) is also slowly varying at infinity. Thus, the
functions δJ
iT(r)andµZT(r)are regularly varying and behave asymptotically like
the power functions rγi−γandrγ, at least up to slowly varying factors.
Next set diT=TγiJi(T)Z(T) =TγiWi(T), so that the slowly varying compo-
nents are factored into the standardization. Then, for t=[Tr],w eh a v e
1
diTXit=1
TγiJi(T)Z(T)/parenleftbiggait+git
µt/parenrightbigg
µt
=ait
TγiWi(T)+git
TγiWi(T)/parenleftbiggTγZ(T)
µt/parenrightbigg/parenleftbiggµt
TγZ(T)/parenrightbigg
=o(1)+git
TγiWi(T)/parenleftbiggTγZ(T)
µt/parenrightbigg/parenleftbiggµt
TγZ(T)/parenrightbigg
=o(1)+δiT/parenleftbiggt
T/parenrightbigg
µT/parenleftbiggt
T/parenrightbigg
(50)
∼δiT(r)µT(r)/periodori (51)
In (50), we define
µT/parenleftbiggt
T/parenrightbigg
=µt
TγZ(T)=µt
tγZ(t)/parenleftbiggtγZ(t)
TγZ(T)/parenrightbigg
=/parenleftbiggt
T/parenrightbiggγZ(t
TT)
Z(T)/commaori (52)
and in a similar manner,
δiT/parenleftbiggt
T/parenrightbigg
=/parenleftbiggt
T/parenrightbiggγi−γWi(t
TT)Z(T)
Wi(T)Z(t
TT)/periodori (53)
Then, for t=[Tr],w eh a v e
δiT(r)→δi(r)=rγi−γ(54)
and
µT(r)→µ(r)=rγ/periodori (55)
Relations ( 51)–(55) lead to a nonstochastic version of the stated result ( 18).
For a stochastic version, we may continue to assume that the standardized rep-
resentation ( 51) applies with an op(1)error uniformly in t≤Tand require
that
δiT(r)→pδi(r)=rγi−γ/commaori
µT(r)→pµ(r)=rγ

1816 P . C. B. PHILLIPS AND D. SUL
uniformly in r∈[0/commaori1], so that the limit transition function δi(r)and growth
curveµ(r) are nonrandom functions.
More generally, the limit functions δi(r)andµ(r) may themselves be sto-
chastic processes. For example, if the common growth component µtin logyit
is a unit root stochastic trend, then by standard functional limit theory (e.g.,
Phillips and Solo ( 1992 )) on a suitably defined probability space
T−1/2µ[Tr]=µT(r)→pB(r) (56)
for some Brownian motion B(r) .I np l a c eo f( 47), suppose that fi(t)=git/µt
is stochastically regularly varying at infinity in the sense that fi(t)continues to
follow ( 47) for some power exponent γi,b u tw i t h Wi(t)stochastically slowly
varying at infinity, that is, Wi(λt)/W i(t)→p1a st→∞ for allλ>0. Then, in
place of ( 49)w eh a v e
1
Tγigit
µt=[Tr]γi
TγiWi(Tr)
Wi(T)Wi(T)∼rγiWi(T)/periodori
SettingdiT=Tγi+1/2Wi(T) andt=[Tr], and working in the same probability
space where ( 56)h o l d s ,w eh a v e
d−1
iTXit=ait
Tγi+1/2Wi(T)+1
TγiWi(T)/parenleftbigggit
µt/parenrightbigg/parenleftbiggµt√
T/parenrightbigg
=op(1)+δiT(r)µT(r)→pδi(r)B(r)
withδi(r)=rγi. In this case the limiting common trend function is the stochas-
tic process µ(r)=B(r) and the transition function is the nonrandom function
δi(r)=rγi.
APPENDIX B: A SYMPTOTIC PROPERTIES OF THE logtCONVERGENCE TEST
B.1. Derivation of the logtRegression Equation
We proceed with the factor model ( 2) and the semiparametric representa-
tion(24), written here as
δit=δi+σitξit=δi+σiξit
L(t)tα:=δi+ψit
L(t)tα(57)
for some σi>0/commaorit≥1 and where the various components satisfy Assump-
tions A1–A4.F r o m( 27)w eh a v e
ψNt:=√
Nψt=1√
NN/summationdisplay
i=1ψit⇒N(0/commaoriv2
ψ)=ξψt/commaorisay, (58)

TRANSITION MODELING AND CONVERGENCE TESTS 1817
wherev2
ψ=plimN→∞N−1/summationtextN
i=1ψ2
it=limN→∞N−1/summationtextN
i=1σ2
i.S oψt=Op(N−1/2)
and
ψ2
t=N−1ψ2Nt=N−2N/summationdisplay
i=1ψ2it+N−2/summationdisplay
i/negationslash=jψitψjt (59)
=N−2N/summationdisplay
i=1σ2
i+N−2N/summationdisplay
i=1σ2
i(ξ2
it−1)+2N−2N/summationdisplay
i=2i−1/summationdisplay
j=1ψitψjt
=Op(N−1)/periodori
From ( 2) and the definition of hit,w eh a v e
hit−1=δit−1
N/summationtext
iδit
1
N/summationtext
iδit=δi−¯δ+(ψit−ψt)/(L(t)tα)
¯δ+ψt/(L(t)tα)/commaori (60)
where ¯δ=N−1/summationtext
iδi. Under the null H0of a homogeneous common trend
effect, we have δi=δfor alliandδ/negationslash=0 in view of Assumption A2.T h e n
hit−1=1
L(t)tαψit−ψt
δ+ψt/(L(t)tα)/commaori
(hit−1)2=(ψit−ψt)2
ψ2
t+L(t)2t2αδ2+2δL(t)tαψt/commaori
and
Ht=1
NN/summationdisplay
i=1(hit−1)2=1
N/summationtextN
i=1(ψit−ψt)2
ψ2
t+L(t)2t2αδ2+2δL(t)tαψt/periodori (61)
Letσ2
ψt=N−1/summationtextN
i=1(ψit−ψt)2=N−1/summationtextNi=1ψ2
it−ψ2t, so that by Assumptions A2
andA3and ( 59)w eh a v e
σ2
ψt=N−1N/summationdisplay
i=1σ2
iξ2
it−N−2N/summationdisplay
i=1σ2
i−N−2N/summationdisplay
i=1σ2
i(ξ2
it−1) (62)
−2N−2N/summationdisplay
i=2i−1/summationdisplay
j=1ψitψjt
=N−1
N2N/summationdisplay
i=1σ2
i+N−1
N2N/summationdisplay
i=1σ2
i(ξ2
it−1)−2
N2N/summationdisplay
i=2i−1/summationdisplay
j=1ψitψjt
=v2
ψN+N−1/2ηNt−N−1η2Nt/commaori

1818 P . C. B. PHILLIPS AND D. SUL
wherev2
ψN=N−1(1−N−1)/summationtextN
i=1σ2
i→v2
ψasN→∞ ,ηNt=N−1/2(1−
N−1)/summationtextN
i=1σ2
i(ξ2
it−1),a n dη2Nt=2N−1/summationtextNi=2/summationtexti−1
j=1ψitψjt.I nv i e wo f( 28), we
have
ηNt⇒N(0/commaoriv4ψ(µ4ξ−1)):=ξ2ψt/commaorisay, as N→∞/commaori
so thatηNt=Op(1)asN→∞ . Further, since the limit variate ξ2ψtdepends
on{(ξ2
it−1)}∞
i=1, it retains the same dependence structure over tas(ξ2
it−1).
Indeed, expanding the probability space in a suitable way, we may write ηNt=
ξ2ψt+op(1),a n dp a r t i a ls u m so v e r tsatisfy a functional law
T−1/2[Tr]/summationdisplay
t=1ξ2ψt⇒V2(r)/commaori
whereV2is Brownian motion with variance lim N→∞N−1/summationtextN
i=1σ4
iω2ii/commaoriwhich is
a sequential functional convergence version of ( 30). Assumption A3requires
that the following central limit law hold jointly as both N/commaoriT→∞ :
T−1/2N−1/2T/summationdisplay
t=1N/summationdisplay
i=1σ2
i(ξ2
it−1)⇒N/parenleftBigg
0/commaorilim
N→∞N−1N/summationdisplay
i=1σ4
iω2ii/parenrightBigg
/periodori (63)
Primitive conditions for this result may be developed along the lines of Phillips
and Moon ( 1999 ). Note also that, in view of ( 30),N−1T−1/summationtextT
t=1/summationtextNi=1σ2
i(ξ2
it−1)
has an asymptotic mean squared error of order O(N−1T−1), so that
T−1T/summationdisplay
t=1σ2
ψt=v2
ψN+Op(N−1/2T−1/2)/periodori (64)
Finally, we observe that in view of the independence of the ξitacrossi,i tf o l –
lows by standard weak convergence arguments that
η2Nt=2N−1N/summationdisplay
i=2i−1/summationdisplay
j=1ψitψjt⇒2/integraldisplay1
0Ut(r)dU t(r)/commaori (65)
whereUt(r)is a Brownian motion with variance v2
ψ=limN→∞N−1/summationtextN
i=1σ2
i.
Thus,η2Nt=Op(1)asN→∞ . Further, in view of ( 31), we have the joint
convergence
T−1/2T/summationdisplay
t=1N−1N/summationdisplay
i=2i−1/summationdisplay
j=1ψitψjt
⇒N/parenleftBigg
0/commaorilim
N→∞N−2N/summationdisplay
i=2i−1/summationdisplay
j=1σ2
iσ2
j∞/summationdisplay
h=−∞γi(h)γj(h)/parenrightBigg

TRANSITION MODELING AND CONVERGENCE TESTS 1819
asN/commaoriT→∞ .
We now proceed with the derivation of the regression equation for Ht.U n –
derH0,w ec a nw r i t e
Ht=σ2
ψt
ψ2
t+L(t)2t2αδ2+2δL(t)tαψt(66)
=/parenleftbigg1
L(t)2t2α/parenrightbiggσ2
ψt/δ2
1+L(t)−2t−2αψ2t/δ2+2L(t)−1t−αψt/δ
and
H1=σ2
ψ1
ψ2
1+L(1)2δ2+2δL(1)ψ1/commaori
which is independent of α/periodoriLet logH1=h1/periodoriT aking logs yields
logH1
Ht=logH1−logHt=h1−logHt (67)
and, using ( 62), we have
logHt=log/bracketleftbiggv2
ψN+N−1/2ηNt−N−1η2Nt
δ2/bracketrightbigg
−2l o gL(t)−2αlogt (68)
−log/braceleftbigg
1+L(t)−2t−2αψ2
t
δ2+2L(t)−1t−αψt
δ/bracerightbigg
=−2l o gL(t)−2αlogt+log/braceleftbiggv2
ψN
δ2/bracerightbigg
+/epsilon1t/commaori
where
/epsilon1t=log/bracketleftbigg
1+N−1/2ηNt
v2
ψN−η2Nt
Nv2
ψN/bracketrightbigg
(69)
−log/braceleftbigg
1+L(t)−2t−2αψ2t
δ2+2L(t)−1t−αψt
δ/bracerightbigg
/periodori
Even ifα=0/commaoriwe can still expand the logarithm in the second term of the above
expression for /epsilon1tsince the slowly varying factor L(t)−1→0 for large t.D e fi n e
λt=L(t)−1t−αψt/δandζt=λ2t+2λt/commaoriand using the expansion log (1+ζt)=
ζt−1
2ζ2
t+o(ζ3
t)and ( 59), we get
log(1+ζt)=λ2
t+2λt−1
2(λ2t+2λt)2+op(L(t)−3t−3αψ3t)

1820 P . C. B. PHILLIPS AND D. SUL
=2λt−1
2λ4
t−2λ3t−λ2t+op(L(t)−3t−3αψ3t)
=2L(t)−1t−αψt
δ−L(t)−2t−2αψ2
t
δ2+Op/parenleftbigg1
L(t)3t3αN3/2/parenrightbigg
/commaori
so that since N→∞ ,
/epsilon1t=log/braceleftbigg
1+N−1/2ηNt
v2
ψN−η2Nt
Nv2
ψN/bracerightbigg
−2L(t)−1t−αψt
δ+L(t)−2t−2αψ2t
δ2(70)
+Op/parenleftbigg1
L(t)3t3αN3/2/parenrightbigg
=/braceleftbigg
N−1/2ηNt
v2
ψN+εNt/bracerightbigg
−2L(t)−1t−αψt
δ+L(t)−2t−2αψ2t
δ2(71)
+Op/parenleftbigg1
L(t)3t3αN3/2/parenrightbigg
/commaori
where
εNt=−η2Nt
Nv2
ψN−1
2Nη2
Nt
v4
ψN+Op(N−3/2) (72)
=−1
2NE(ξ2
2ψt)
v4
ψN−η2Nt
Nv2
ψN−1
2Nξ2
2ψt−E(ξ2
2ψt)
v4
ψN+Op(N−3/2)/periodori
Expressions ( 67)a n d( 68) lead to the empirical regression equation
logH1
Ht−2l o gL(t)=a+blogt+ut/commaori (73)
where
a=h1−2l o gvψN
δ/commaorib=2α/commaori u t=−/epsilon1t/periodori (74)
Fort≥[Tr]andr>0, we may write
L(t)−1t−α=1
TαL(T)L(T)
L(t)1
(t
T)α=1
TαL(T)1
(t
T)α{1+o(1)}
and then
ut=−1√
N1
v2
ψNηNt−εNt+2
δ1
tαL(t)ψt−1
δ21
t2αL(t)2ψ2
t (75)

TRANSITION MODELING AND CONVERGENCE TESTS 1821
+Op/parenleftbigg1
L(t)3t3αN3/2/parenrightbigg
=−1√
N1
v2
ψNηNt−εNt+2
δ/bracketleftbigg{1+o(1)}
TαL(T)/bracketrightbigg1
(t
T)αψt
−1
δ2/bracketleftbigg{1+o(1)}
T2αL(T)2/bracketrightbigg1
(t
T)2αψ2
t+Op/parenleftbigg1
L(t)3t3αN3/2/parenrightbigg
/periodori
Sinceψt=Op(N−1/2),εNt=Op(N−1)/commaoriandL(T) →∞ , the first term of ( 75)
dominates the behavior of the regression error utwhenα≥0.
B.2. Proof of Theorem 1
In developing the limit theory, it is convenient to modify the regression equa-
tion ( 73) to avoid the singularity in the sample moment matrix that arises from
the presence of an intercept and log tin (73). Phillips ( 2007 ) provided a dis-
cussion and treatment of such issues in quite general regressions with slowly
varying regressors that includes cases such as ( 73). It is simplest to transform
to the equation
logH1
Ht−2l o gL(t)=a∗+blogt
T+ut/commaori (76)
wherea∗=a+blogT. This transformation clearly does not affect the estima-
tor ofb.
Define the demeaned regressor
τt=/parenleftbigg
logt
T−logt
T/parenrightbigg
/commaori
where logt
T=1
T−[Tr]+1/summationtextT
t=[Tr]logt
T. Then, empirical regression of ( 76)o v e rt=
[Tr]/commaori[Tr]+1/commaori/periodori/periodori/periodori/commaoriT for some r>0y i e l d s
ˆb−b=/summationtextTt=[Tr]τtut/summationtextTt=[Tr]τ2
t/periodori
Note that
T/summationdisplay
t=[Tr]τ2
t=T/summationdisplay
t=[Tr]/parenleftbigg
logt
T−logt
T/parenrightbigg2
(77)
=T/braceleftbigg/integraldisplay1
r/parenleftbigg
logs−1
1−r/integraldisplay1
rlogpdp/parenrightbigg2
ds+o(1)/bracerightbigg

1822 P . C. B. PHILLIPS AND D. SUL
=T/braceleftbigg/integraldisplay1
rlog2sds−1
1−r/parenleftbigg/integraldisplay1
rlogpdp/parenrightbigg2
+o(1)/bracerightbigg
=T/braceleftbigg
(1−r)−/parenleftbiggr
1−r/parenrightbigg
log2r+o(1)/bracerightbigg
by Euler summation and direct evaluation of the integral
/integraldisplay1
rlog2sds−1
1−r/parenleftbigg/integraldisplay1
rlogpdp/parenrightbigg2
=(1−r)−/braceleftbiggr
1−r/bracerightbigg
log2r/periodori
Using ( 75)w eh a v e
ˆb−b=−1
v2
ψN1√
NT/summationtextT
t=[Tr]τtηNt+1
T/summationtextTt=[Tr]τtεNt
T−1/summationtextTt=[Tr]τ2
t(78)
+2
δ1
T/summationtextTt=[Tr]τtψt/(tαL(t))
T−1/summationtextTt=[Tr]τ2
t−1
δ21
T/summationtextTt=[Tr]τtψ2
t/(t2αL(t)2)
T−1/summationtextT
t=[Tr]τ2
t
+Op/parenleftbigg1
L(T)3T3αN3/2/parenrightbigg
/periodori
Next observe that
1√
TT/summationdisplay
t=[Tr]τtηNt=1√
T1√
NT/summationdisplay
t=[Tr]τtN/summationdisplay
i=1σ2
i(ξ2
it−1) (79)
=1√
T√
NN/summationdisplay
i=1T/summationdisplay
t=[Tr]/parenleftbigg
logt
T−logt
T/parenrightbigg
σ2
i(ξ2
it−1)
⇒N/parenleftbigg
0/commaoriω2
η/integraldisplay1
r/parenleftbigg
logs−1
1−r/integraldisplay1
rlogpdp/parenrightbigg2
ds/parenrightbigg
=N/parenleftbigg
0/commaoriω2η/braceleftbigg
(1−r)−/parenleftbiggr
1−r/parenrightbigg
log2r/bracerightbigg/parenrightbigg
in view of ( 29), where ω2
η=limN→∞N−1/summationtextN
i=1σ4
iω2iiandω2iiis the long
run variance of ξ2
it. Also noting the fact that E(ξ2
2ψt)is constant over tand/summationtextT
t=[Tr]τt=0, we find that
1
TT/summationdisplay
t=[Tr]τtεNt=−1
v2
ψN1
NTT/summationdisplay
t=[Tr]τtη2Nt (80)

TRANSITION MODELING AND CONVERGENCE TESTS 1823
−1
2v4
ψN1
NTT/summationdisplay
t=[Tr]τt(ξ2
2ψt−E(ξ2
2ψt))+Op(N−3/2)
=Op/parenleftbigg1√
TN+1
N3/2/parenrightbigg
/commaori
since both T−1/2/summationtextT
t=[Tr]τtη2NtandT−1/2/summationtextT
t=[Tr]τt(ξ2
2ψt−E(ξ2
2ψt))areOp(1).
Further,
√
TN
TT/summationdisplay
t=[Tr]τt1
tαL(t)ψt (81)
=1√
TNN/summationdisplay
i=1T/summationdisplay
t=[Tr]τt1
tαL(t)ψit
=1√
TNN/summationdisplay
i=1T/summationdisplay
t=[Tr]τt/bracketleftbigg1
TαL(T)L(T)
L(t)1
(t
T)αψit/bracketrightbigg
=/braceleftbigg1
TαL(T){1+op(1)}/bracerightbigg
×1√
TNN/summationdisplay
i=1T/summationdisplay
t=[Tr]/parenleftbigg
logt
T−logt
T/parenrightbigg/parenleftbiggt
T/parenrightbigg−α
ψit
∼1
TαL(T)N/parenleftbigg
0/commaoriω2
ψ/integraldisplay1
r/braceleftbigg/parenleftbigg
logs−1
1−r/integraldisplay1
rlogpdp/parenrightbigg2
s−2αds/bracerightbigg/parenrightbigg
=Op/parenleftbigg1
TαL(T)/parenrightbigg
and, when α>0, we have
1
δ2√
TN
TT/summationdisplay
t=[Tr]τt1
t2αL(t)2ψ2
t
=1
δ2√
TN
TT/summationdisplay
t=[Tr]τt/bracketleftbigg1
t2αL(t)2/bracketrightbigg
×/braceleftBigg
N−2N/summationdisplay
i=1σ2
i+N−2N/summationdisplay
i=1σ2
i(ξ2
it−1)+N−2N/summationdisplay
i/commaorij=1
i/negationslash=jψitψjt/bracerightBigg

1824 P . C. B. PHILLIPS AND D. SUL
=√
TN
δ21
TT/summationdisplay
t=[Tr]τt/bracketleftbigg1
T2αL(T)2/parenleftbiggL(T)
L(t)/parenrightbigg21
(t
T)2α/bracketrightbiggv2
ψN
N{1−N−1}
+1
δ2N1√
T
×T/summationdisplay
t=[Tr]τt/bracketleftbigg1
T2αL(T)2/parenleftbiggL(T)
L(t)/parenrightbigg21
(t
T)2α/bracketrightbigg1√
NN/summationdisplay
i=1σ2
i(ξ2
it−1)/commaori
1
δ2N1/21√
TT/summationdisplay
t=[Tr]τt/bracketleftbigg1
T2αL(T)2/parenleftbiggL(T)
L(t)/parenrightbigg21
(t
T)2α/bracketrightbigg1
NN/summationdisplay
i/commaorij=1
i/negationslash=jψitψjt
=v2
ψN√
T{1+o(1)}
δ2√
NT2αL(T)21
TT/summationdisplay
t=[Tr]τt/parenleftbiggt
T/parenrightbigg−2α
(82)
+{1+o(1)}
δ2NT2αL(T)21√
TT/summationdisplay
t=[Tr]τt/parenleftbiggt
T/parenrightbigg−2α1√
NN/summationdisplay
i=1σ2
i(ξ2
it−1)
+{1+o(1)}
δ2N1/2T2αL(T)21√
TT/summationdisplay
t=[Tr]τt/parenleftbiggt
T/parenrightbigg−2α/parenleftBigg
1
NN/summationdisplay
i/commaorij=1
i/negationslash=jψitψjt/parenrightBigg
(83)
=v2
ψ
δ2T1/2
T2αL(T)2N1/2
×/integraldisplay1
r/braceleftbigg/parenleftbigg
logs−1
1−r/integraldisplay1
rlogpdp/parenrightbigg
s−2αds/bracerightbigg
{1+op(1)}
+Op/parenleftbigg1
T2αL(T)2√
N/parenrightbigg
/commaori (84)
since
1√
TT/summationdisplay
t=[Tr]τt/parenleftbiggt
T/parenrightbigg−2α1√
NN/summationdisplay
i=1σ2
i(ξ2
it−1)=Op(1)/commaori
1√
TT/summationdisplay
t=[Tr]τt/parenleftbiggt
T/parenrightbigg−2α/parenleftBigg
1
NN/summationdisplay
i/commaorij=1
i/negationslash=jψitψjt/parenrightBigg
=Op(1)

TRANSITION MODELING AND CONVERGENCE TESTS 1825
in view of ( 30)a n d( 31). Whenα=0, it is apparent that
T/summationdisplay
t=[Tr]τt/parenleftbiggt
T/parenrightbigg−2α
v2
ψ=T/summationdisplay
t=[Tr]τtv2
ψ=0
in line ( 82) of the earlier argument, in which case the first term of ( 84)i sz e r o
and the second term dominates, giving
1
δ2√
TN
TT/summationdisplay
t=[Tr]τt1
L(t)2ψ2
t=Op/parenleftbigg1
L(T)2√
N/parenrightbigg
/periodori (85)
From ( 78)w eh a v e
√
NT(ˆb−b) (86)
=−1
v2
ψN1√
T/summationtextT
t=[Tr]τtηNt+√
N√
T/summationtextTt=[Tr]τtεNt
T−1/summationtextTt=[Tr]τ2
t
+2
δ√
N√
T/summationtextTt=[Tr]τtψt/(tαL(t))
T−1/summationtextTt=[Tr]τ2
t−1
δ2√
N√
T/summationtextTt=[Tr]τtψ2
t/(t2αL(t)2)
T−1/summationtextT
t=[Tr]τ2
t
+Op/parenleftbigg √
T
L(T)3T3αN/parenrightbigg
/commaori
and it then follows from ( 80)–(84) that when α>0,
√
NT(ˆb−b)=−1
v2
ψN1√
T/summationtextTt=[Tr]τtηNt
T−1/summationtextTt=[Tr]τ2
t+Op/parenleftbigg1
N1/2+√
T
N/parenrightbigg
(87)
+Op/parenleftbigg1
TαL(T)/parenrightbigg
+Op/parenleftbiggT1/2
T2αL(T)2N1/2/parenrightbigg
+Op/parenleftbigg √
T
L(T)3T3αN/parenrightbigg
⇒1
v2
ψN/parenleftbigg
0/commaoriω2
η/braceleftbigg
(1−r)−/parenleftbiggr
1−r/parenrightbigg
log2r/bracerightbigg−1/parenrightbigg
asT/commaoriN→∞ ,p r o v i d e d T1/2/(T2αL(T)2N1/2)→0. Whenα=0, we have, using
(85),
√
NT(ˆb−b)=−1
v2
ψN1√
T/summationtextT
t=[Tr]τtηNt
T−1/summationtextTt=[Tr]τ2
t+Op/parenleftbigg1
N1/2+√
T
N/parenrightbigg

1826 P . C. B. PHILLIPS AND D. SUL
+Op/parenleftbigg1
L(T)/parenrightbigg
+Op/parenleftbigg1
L(T)2√
N/parenrightbigg
+Op/parenleftbigg√
T
L(T)3N/parenrightbigg
and precisely the same limit theory as ( 87) applies provided T1/2/N→0.
I tf o l l o w st h a ti nb o t hc a s e sw eh a v e√
NT(ˆb−b)⇒N(0/commaoriΩ2),w h e r e
Ω2=ω2
η
v4
ψ/braceleftbigg
(1−r)−/parenleftbiggr
1−r/parenrightbigg
log2r/bracerightbigg−1
/commaori (88)
ω2η=limN→∞N−1/summationtextN
i=1σ4
iω2ii,a n dv2
ψ=limN→∞N−1/summationtextNi=1σ2
i,a n dw h e r e ω2iiis
the long run variance of ξ2
it. This gives the required result (a).
When the relative rate condition T1/2/(T2αL(T)2N1/2)→0 does not hold,
the third term of ( 86) enters into the limit theory as a bias term. In particular,
we have
√
NT/braceleftbigg
(ˆb−b)+1
δ21
T/summationtextTt=[Tr]τtψ2
t/(t2αL(t)2)
T−1/summationtextT
t=[Tr]τ2
t/bracerightbigg
=−1
v2
ψN1√
T/summationtextTt=[Tr]τtηNt+√
N√
T/summationtextTt=[Tr]τtεNt
T−1/summationtextTt=[Tr]τ2
t+Op/parenleftbigg √
T
L(T)3T3αN/parenrightbigg
=−1
v2
ψN1√
T/summationtextT
t=[Tr]τtηNt
T−1/summationtextTt=[Tr]τ2
t+Op/parenleftbigg1
N1/2+√
T
N/parenrightbigg
+Op/parenleftbigg √
T
L(T)3T3αN/parenrightbigg
and using ( 87)w eh a v e
√
NT/braceleftbigg
(ˆb−b)+1
δ21
T/summationtextTt=[Tr]τtψ2
t/(t2αL(t)2)
T−1/summationtextT
t=[Tr]τ2
t/bracerightbigg
⇒N(0/commaoriΩ2)/commaori
provided√
T
N→0. In this case, there is an asymptotic bias of the form
−v2
ψ
δ21
T2αL(T)2N/integraldisplay1
r/braceleftbigg/parenleftbigg
logs−1
1−r/integraldisplay1
rlogpdp/parenrightbigg
s−2αds/bracerightbigg
in the estimation of b.T h i sb i a si so f O(T−2αL(T)−2N−1)and will generally
be quite small when α> 0 .T h eb i a si sz e r ow h e n α=0 because/integraltext1
r(logs−
1
1−r/integraltext1
rlogpdp)ds =0, explaining the milder rate condition in this case.
B.3. Asymptotic Variance Formula
Since the regressor in ( 76) is deterministic, we may consistently estimate the
asymptotic variance Ω2in a simple way by estimating the long run variance of

TRANSITION MODELING AND CONVERGENCE TESTS 1827
utusing the least squares residuals ˆut. In particular, we may use the variance
estimate
V(ˆb)=/hatwidestlvarr(ˆut)/bracketleftBiggT/summationdisplay
t=[Tr]τ2
t/bracketrightBigg−1
/commaori
where /hatwidestlvarr(ˆut)is a consistent estimate of N−1(δ2/v2
ψ)2ω2
η.T oc o n s t r u c t
/hatwidestlvarr(ˆut)we may use a conventional HAC estimate, as we now show.
We start by working directly with utand its autocovariance sequence. From
(75)w eh a v e
√
Nut=−ηNt
v2
ψN+2{1+o(1)}
TαL(T)1
(t
T)α√
Nψt+op(1) (89)
=−1
v2
ψN1√
NN/summationdisplay
i=1σ2
i(ξ2
it−1)+op(1)
=−1
v2
ψN1√
NN/summationdisplay
i=1σ2
iηit+op(1):= −wt
v2
ψN+op(1)/commaori (90)
whereηit=ξ2
it−1, whose long run variance is ω2ii. The serial autocovariances
of the leading term wt=1√
N/summationtextN
i=1σ2
iηitareE(wtwt+l)=N−1/summationtextNi=1σ4
iE(ηit×
ηit+l),a n da sM→∞ , it follows that
M/summationdisplay
l=−ME(wtwt+l)=1
NN/summationdisplay
i=1σ4
iM/summationdisplay
l=−ME(ηitηit+l)
=1
NN/summationdisplay
i=1σ4
i/braceleftBigg∞/summationdisplay
l=−∞E(ηitηit+l)+o(1)/bracerightBigg
→lim
N→∞N−1N/summationdisplay
i=1σ4
iω2ii/commaori
where lim N→∞N−1/summationtextN
i=1σ4
iω2ii. Contributions to the long run variance of√
Nutfrom the second and higher order terms of ( 89)a r eo fO(L(T)−2T−2α)
fort≥[Tr]andr>0. Hence, the long run variance of√
Nutis given by
lvar(√
Nut)=lim
N→∞1
v4
ψN1
NN/summationdisplay
i=1σ4
iω2ii=ω2
η
v4
ψ:=Ω2
usay.

1828 P . C. B. PHILLIPS AND D. SUL
Thus, the asymptotic variance formula ( 88)i s
Ω2=Ω2
u
{(1−r)−(r
1−r)log2r}2/periodori
The denominator can be directly calculated or estimated in the usual manner
with the moment sum of squares
T/summationdisplay
t=[Tr]τ2
t∼T/braceleftbigg
(1−r)−/parenleftbiggr
1−r/parenrightbigg
log2r/bracerightbigg
/commaori
a ss h o w ni n( 77) above. The numerator,
Ω2
u=lim
N→∞δ4
v4
ψN1
NN/summationdisplay
i=1σ4
iω2ii/commaori
is the limit of a weighted average of long run variances. As we next investigate,
it may be estimated using a long run variance estimate with the residuals of the
regression, namely ˆΩ2
u=/hatwidestlvarr(ˆut).
B.4. Estimation of the Weighted Average Long Run Variance
The sample serial covariances of the leading term wtof the regression error
ut(using the available observations in the regression from t=[Tr]/commaori/periodori/periodori/periodori/commaoriT )
have the form
1
T/summationdisplay
[Tr]≤t/commaorit+l≤Twtwt+l (91)
=1
NN/summationdisplay
i/commaorij=1σ2
iσ2
j1
T/summationdisplay
[Tr]≤t/commaorit+l≤Tηitηjt+l
=1
NN/summationdisplay
i=1σ4
i1
T/summationdisplay
[Tr]≤t/commaorit+l≤Tηitηit+l
+1√
T/bracketleftBigg
1√
NN/summationdisplay
i=1σ2
i/braceleftBigg
1√
NTN/summationdisplay
j/negationslash=iσ2
j/summationdisplay
[Tr]≤t/commaorit+l≤Tηitηjt+l/bracerightBigg/bracketrightBigg
=1
NN/summationdisplay
i=1σ4
i/braceleftbigg1
T/summationdisplay
[Tr]≤t/commaorit+l≤Tηitηit+l/bracerightbigg
+Op/parenleftbigg1√
T/parenrightbigg
/periodori

TRANSITION MODELING AND CONVERGENCE TESTS 1829
By virtue of the usual process of HAC estimation for M→∞ asT→∞ ,w e
have
M/summationdisplay
l=−M/braceleftbigg1
T/summationdisplay
[Tr]≤t/commaorit+l≤Tηitηit+l/bracerightbigg
→p(1−r)ω2
2ii/commaori
where the factor (1−r)reflects the fact that only the fraction 1 −rof the time
series data is used in the regression. For MsatisfyingM√
T+1
M→0a sT→∞ ,
we find from ( 91) and standard HAC limit theory that
M/summationdisplay
l=−M1
T/summationdisplay
[Tr]≤t/commaorit+l≤Twtwt+l (92)
=1
NN/summationdisplay
i=1σ4
i/braceleftBiggM/summationdisplay
l=−M1
T/summationdisplay
[Tr]≤t/commaorit+l≤Tηitηit+l/bracerightBigg
+op(1)
=1
NN/summationdisplay
i=1σ4
i{(1−r)ω22ii+op(1)}+op(1)
→p(1−r)lim
N→∞N−1N/summationdisplay
i=1σ4
iω2ii/periodori
Since the scaled regression residuals√
Nˆutconsistently estimate the quantities
−wt/v2
ψNin (90), we correspondingly have
/hatwidestlvar(√
Nˆut)→p(1−r)1
v4
ψNlim
N→∞N−1N/summationdisplay
i=1σ4
iω2ii=(1−r)Ω2
u/periodori
If we use a standardization of 1 /(T−[Tr]+1)rather than 1 /Tin the sample
serial covariances in ( 92), we have
M/summationdisplay
l=−M1
T−[Tr]+1/summationdisplay
[Tr]≤t/commaorit+l≤Twtwt+l→pΩ2
u/commaori
and the corresponding estimate (where the subscript rsignifies the use of the
scaling factor 1 /(T−[Tr]+1)in the sample covariance formulae)
/hatwidestlvarr(√
Nˆut)=M/summationdisplay
l=−MN
T−[Tr]+1/summationdisplay
[Tr]≤t/commaorit+l≤Tˆutˆut+l→pΩ2u/periodori (93)

1830 P . C. B. PHILLIPS AND D. SUL
The same behavior is observed for other HAC estimates constructed with dif-
ferent lag kernels.
Then the asymptotic variance estimate of ˆbis
s2
ˆb=/hatwidestlvarr(ˆut)/bracketleftBiggT/summationdisplay
t=[Tr]τ2
t/bracketrightBigg−1
=1
N/hatwidestlvarr(√
Nˆut)/bracketleftBiggT/summationdisplay
t=[Tr]τ2
t/bracketrightBigg−1
∼1
NTΩ2
u/braceleftbigg
(1−r)−/parenleftbiggr
1−r/parenrightbigg
log2r/bracerightbigg−1
=1
NTΩ2/commaori
andNTs2
ˆb→pΩ2asN/commaoriT→∞ . Accordingly, the tratiotˆb=(ˆb−b)/s ˆb⇒
N(0/commaori1)and result (b) follows.
B.5. Proof of Theorem 2
(a)α≥0. We assume δi∼iid(δ/commaoriσ2
δ)and let ¯δ=N−1/summationtextN
i=1δi. Under this
alternative, we have, from ( 60),
hit−1=δi−¯δ+(ψit−ψt)/(L(t)tα)
¯δ+ψt/(L(t)tα)/commaori
so that
Ht=σ2
¯δ21+(σ2
ψt/σ2)L(t)−2t−2α+2(σδψt/σ2)L(t)−1t−α
1+L(t)−2t−2αψ2
t/¯δ2+2L(t)−1t−αψt/¯δ/commaori (94)
whereσ2=N−1/summationtext(δi−¯δ)2→pσ2
δasN→∞ ,
σδψt=N−1N/summationdisplay
i=1(δi−¯δ)(ψit−ψt)=N−1/2ςNt+Op(N−1)/commaori (95)
whereςNt=N−1/2/summationtextN
i=1(δi−¯δ)σiξit=Op(1),a n dσ2
ψt=v2
ψN+N−1/2ηNt+
Op(N−1)from ( 62) above. Note that Ht→σ2/¯δ2>0a sT/commaoriN→∞ .
Ta k i n g l o g s i n ( 94) and assuming σ2>0 (otherwise the null hypothesis
holds), we have
logHt=2l o gσ
¯δ+/epsilon1t/commaori
where
/epsilon1t=log{1+(σ2
ψt/σ2)L(t)−2t−2α+2(σδψt/σ2)L(t)−1t−α} (96)
−log{1+L(t)−2t−2αψ2
t/¯δ2+2L(t)−1t−αψt/¯δ}/periodori

TRANSITION MODELING AND CONVERGENCE TESTS 1831
The generating process for log (H 1/Ht)therefore has the form, under the al-
ternative,
logH1
Ht−2l o gL(t)=logH1−2l o gσ
¯δ−2l o gL(t)−/epsilon1t/commaori (97)
while the fitted regression is
logH1
Ht−2l o gL(t)=ˆa∗+ˆblog/parenleftbiggt
T/parenrightbigg
+residual, (98)
so that
ˆb=−/summationtextT
t=[Tr]τt{2l o gL(t)+/epsilon1t}
/summationtextT
t=[Tr]τ2
t/periodori (99)
Note that t=Tafor some a>0a n df o r L(t)=logt,w eh a v e
logL(t)=logL(Ta) =log{logT+loga}=log/bracketleftbigg
logT/braceleftbigg
1+loga
logT/bracerightbigg/bracketrightbigg
(100)
=log logT+logt
T
logT−1
2log2t
T
log2T+O/parenleftbigglog3t
T
log3T/parenrightbigg
/commaori
giving
logL(t)−logL(t)=logt
T−logt
T
logT−1
2log2t
T−log2t
T
log2T+O/parenleftbigg1
log3T/parenrightbigg
(101)
and
/summationtextT
t=[Tr]τtlogL(t)
/summationtextT
t=[Tr]τ2
t(102)
=/summationtextT
t=[Tr]τt{logL(t)−logL(t)}
/summationtextT
t=[Tr]τ2
t
=1
logTT−1/summationtextT
t=[Tr]τt/parenleftbig
logt
T−logt
T/parenrightbig
T−1/summationtextT
t=[Tr]τ2
t
−1
2l o g2TT−1/summationtextT
t=[Tr]τt/parenleftbig
log2t
T−log2t
T/parenrightbig
T−1/summationtextT
t=[Tr]τ2
t+O/parenleftbigg1
log3T/parenrightbigg
=1
logT

1832 P . C. B. PHILLIPS AND D. SUL
−1
2l o g2T
×/integraltext1
r/parenleftbig
logs−1
1−r/integraltext1
rlogpdp/parenrightbig/parenleftbig
log2s−1
1−r/integraltext1
rlog2pdp/parenrightbig
ds
/integraltext1
r/parenleftbig
logs−1
1−r/integraltext1
rlogpdp/parenrightbig2ds
+O/parenleftbigg1
log3T/parenrightbigg
=1
logT−g(r)
2l o g2T+O/parenleftbigg1
log3T/parenrightbigg
/commaori
where
g(r)=2(logr)r2−(log3r)r+2(log2r)r−2(logr)r−4(1−r)2
(1−r)2−rlog2r/periodori
Hence, under the alternative, we have
ˆb=−/summationtextT
t=[Tr]τt/epsilon1t/summationtextTt=[Tr]τ2
t−2/summationtextTt=[Tr]τt{logL(t)−logL(t)}
/summationtextTt=[Tr]τ2
t(103)
=−/summationtextTt=[Tr]τt/epsilon1t/summationtextTt=[Tr]τ2
t−2
logT+g(r)
log2T+Op/parenleftbigg1
log3T/parenrightbigg
/periodori
Next consider/summationtextTt=[Tr]τt/epsilon1t/periodoriNote that σδψt=N−1/2ςNt+Op(N−1)=Op(N−1/2)
from ( 95),ψt=Op(N−1/2)from ( 58), andσ2
ψt=v2
ψN+N−1/2ηNt+Op(N−1)
from ( 62), where v2
ψN=N−1/summationtextNi=1σ2
i→v2
ψasN→∞ andηNt=N−1/2×/summationtextNi=1σ2
i(ξ2
it−1)=Op(1). Hence, expanding ( 96)f o rt≥[Tr]withr>0, we
get
/epsilon1t=σ2
ψt
σ21
L(t)2t2α+2σδψt
σ21
L(t)tα−2
¯δ1
L(t)tαψt+Op/parenleftbigg1
NL(T)2T2α/parenrightbigg
(104)
=v2
ψN
σ21
L(t)2t2α+1
σ2N−1/2ηNt
L(t)2t2α+2σδψt
σ21
L(t)tα−2
¯δ1
L(t)tαψt
+Op/parenleftbigg1
NL(T)2T2α/parenrightbigg
=v2
ψN
σ21
L(t)2t2α+Op/parenleftbigg1√
NL(T)Tα/parenrightbigg
/periodori

TRANSITION MODELING AND CONVERGENCE TESTS 1833
It follows that
1
TT/summationdisplay
t=[Tr]τt/epsilon1t (105)
=v2
ψN
σ21
L(T)2T1+2αT/summationdisplay
t=[Tr]τtL(T)2
L(t)2/parenleftbiggt
T/parenrightbigg−2α
+Op/parenleftbigg1√
NL(T)Tα/parenrightbigg
=v2
ψN
σ21
L(T)2T1+2αT/summationdisplay
t=[Tr]τt/parenleftbiggt
T/parenrightbigg−2α
{1+o(1)}+Op/parenleftbigg1√
NL(T)Tα/parenrightbigg
=v2
ψN
σ21
L(T)2T2α/integraldisplay1
r/braceleftbigg
logs−1
1−r/integraldisplay1
rlogpdp/bracerightbigg
s−2αds{1+o(1)}
+Op/parenleftbigg1√
NL(T)Tα/parenrightbigg
=v2
ψN
σ2r∗(α)
L(T)2T2α+Op/parenleftbigg1√
NL(T)Tα/parenrightbigg
/commaori
where
r∗(α)=/integraldisplay1
r/braceleftbigg
logs−1
1−r/integraldisplay1
rlogpdp/bracerightbigg
s−2αds (106)
=

(2α+2rαlogr−rlogr−2αr+2r2−2αα+r1−2αlogr
−2r1−2ααlogr−2r1−2αα)
/((2αr−r+1−2α)(2α−1))withα/negationslash=1
2/commaori
(lnr)r−2r+2+lnr
rwithα=1
2/periodori
Hence, from ( 103)a n d( 77)w eo b t a i n
ˆb=−1
T/summationtextT
t=[Tr]τt/epsilon1t
1
T/summationtextTt=[Tr]τ2
t−2
logT+g(r)
log2T+Op/parenleftbigg1
log3T/parenrightbigg
(107)
=−/parenleftbiggv2
ψN
σ2/parenrightbigg/parenleftbiggr∗(α)
{(1−r)−(r
1−r)log2r}/parenrightbigg/parenleftbigg1
L(T)2T2α/parenrightbigg
−2
logT
+g(r)
log2T+Op/parenleftbigg1
log3T/parenrightbigg

1834 P . C. B. PHILLIPS AND D. SUL
=−/parenleftbiggv2
ψN
σ2/parenrightbigg/parenleftbiggr(α)
L(T)2T2α/parenrightbigg
−2
logT+g(r)
log2T+Op/parenleftbigg1
log3T/parenrightbigg
/commaori
where
r(α)=r∗(α)
(1−r)−(r
1−r)log2r/periodori (108)
Whenα≥0a n dL(T) =logT, the second term in ( 107) dominates and we
have
ˆb=−2
logT+g(r)
log2T−/parenleftbiggv2
ψN
σ2/parenrightbigg/parenleftbiggr(α)
L(T)2T2α/parenrightbigg
+Op/parenleftbigg1
log3T/parenrightbigg
/periodori (109)
Thus, ˆb→p0 in this case. Heuristically, this outcome is explained by the fact
thatHttends to a positive constant, so that the dependent variable in ( 98)b e –
haves like −2l o gL(t) for large t.S i n c el o g L(t) is the log of a slowly varying
function at infinity, its regression coefficient on log tis expected to be zero.
More particularly, the regression of −2l o gL(t) on a constant and log (t
T)pro-
duces a slope coefficient that is negative and tends to zero like −2
logT,a ss h o w n
in (102).
Next consider the standard error of ˆbunder the alternative. Writing the
residual in ( 98)a sˆut, the long run variance estimate has the typical form
/hatwidestlvarr(√
Nˆut)=M/summationdisplay
l=−MN
T−[Tr]+1/summationdisplay
[Tr]≤t/commaorit+l≤Tˆutˆut+l/periodori (110)
In view of ( 97)a n d( 98), and with L(t)=logt,w ed e d u c ef r o m( 101)a n d( 102)
that
ˆut=−(/epsilon1t−¯/epsilon1)−2{logL(t)−logL(t)}−ˆb/braceleftbigg
logt
T−logt
T/bracerightbigg
(111)
=−(/epsilon1t−¯/epsilon1)−2logt
T−logt
T
logT−ˆb/braceleftbigg
logt
T−logt
T/bracerightbigg
+Op/parenleftbigg1
log3T/parenrightbigg
=−v2
ψN
σ2/braceleftbigg1
L(t)2t2α−1
L(T)2T2α1−r1−2α
(1−r)(1−2α)/bracerightbigg
+/braceleftbiggv2
ψN
σ2r(α)
L(T)2T2α−g(r)
log2T/bracerightbigg/braceleftbigg
logt
T−logt
T/bracerightbigg
+Op/parenleftbigg1
log3T/parenrightbigg
/commaori

TRANSITION MODELING AND CONVERGENCE TESTS 1835
using ( 107) and because, from ( 104),
1
T−[Tr]T/summationdisplay
t=[Tr]/epsilon1t (112)
=v2
ψN
σ21
T−[Tr]T/summationdisplay
t=[Tr]1
L(t)2t2α+Op/parenleftbigg1√
NL(T)Tα/parenrightbigg
=v2
ψN
σ21
L(T)2T1+2α1
1−rT/summationdisplay
t=[Tr]/parenleftbiggt
T/parenrightbigg−2α
+Op/parenleftbigg1√
NL(T)Tα/parenrightbigg
=v2
ψN
σ21
L(T)2T2α1
1−r/integraldisplay1
rs−2αds+Op/parenleftbigg1√
NL(T)Tα/parenrightbigg
=v2
ψN
σ21
L(T)2T2α1−r1−2α
(1−r)(1−2α)+Op/parenleftbigg1√
NL(T)Tα/parenrightbigg
/periodori
In view of ( 111), we have, for |l|≤M,M
T→0/commaoriandt≥[Tr]withr>0,
ˆut+l=−(/epsilon1t+l−¯/epsilon1)−ˆb/braceleftbigg
logt+l
T−logt
T/bracerightbigg
−2{logL(t+l)−logL(t)}
=−v2
ψN
σ2/braceleftbigg1
L(t)2t2α[1+o(1)]−1
L(T)2T2α1−r1−2α
(1−r)(1−2α)/bracerightbigg
+/braceleftbiggv2
ψN
σ2r(α)
L(T)2T2α−g(r)
log2T/bracerightbigg/braceleftbigg
logt
T[1+o(1)]−logt
T/bracerightbigg
+Op/parenleftbigglogL(T)√
NL(T)Tα/parenrightbigg
+Op/parenleftbigg1
log3T/parenrightbigg
=ˆut{1+o(1)}/periodori
Then
1
T−[Tr]/summationdisplay
[Tr]≤t/commaorit+l≤Tˆutˆut+l (113)
=1
T−[Tr]/summationdisplay
[Tr]≤t≤Tˆu2
t{1+o(1)}

1836 P . C. B. PHILLIPS AND D. SUL
=1
T−[Tr]v4
ψN
σ4/summationdisplay
[Tr]≤t≤T/braceleftbigg1
L(t)2t2α−1
L(T)2T2α1−r1−2α
(1−r)(1−2α)/bracerightbigg2
+1
T−[Tr]/braceleftbiggv2
ψN
σ2r(α)
L(T)2T2α−g(r)
log2T/bracerightbigg2/summationdisplay
[Tr]≤t≤T/braceleftbigg
logt
T−logt
T/bracerightbigg2
+2
T−[Tr]v2
ψN
σ2/braceleftbiggv2
ψN
σ2r(α)
L(T)2T2α−g(r)
log2T/bracerightbigg/summationdisplay
[Tr]≤t≤Tlogt
T−logt
T
L(t)2t2α
+Op/parenleftbigg1
log4T/parenrightbigg
=Op/parenleftbigg1
log4T/parenrightbigg
/commaori
uniformly in lwhenL(T)=logTandα≥0/periodoriHence, ( 110) becomes
/hatwidestlvarr(√
Nˆut)=M/summationdisplay
l=−MN
T−[Tr]+1/summationdisplay
[Tr]≤t/commaorit+l≤Tˆu2
t{1+o(1)}
=Op/parenleftbiggNM
log4T/parenrightbigg
and so
s2
ˆb=/hatwidestlvarr(ˆut)/bracketleftBiggT/summationdisplay
t=[Tr]τ2
t/bracketrightBigg−1
=Op/parenleftbiggM
(log4T)T/parenrightbigg/bracketleftBigg
1
TT/summationdisplay
t=[Tr]τ2
t/bracketrightBigg−1
(114)
=Op/parenleftbiggM
(log4T)T/parenrightbigg
/periodori
Using ( 109)a n d( 114), we see that the tratiotˆbhas the asymptotic behavior,
under the alternative,
tˆb=ˆb
sˆb=−2
logT÷Op/parenleftbiggM1/2
(log2T)T1/2/parenrightbigg
(115)
=−2
logT×Op/parenleftbigg(log2T)T1/2
M1/2/parenrightbigg
→− ∞
for allα≥0 and all bandwidth choices M≤T. It follows that the test is consis-
tent. The divergence rate is O((logT)T1/2/M1/2).

TRANSITION MODELING AND CONVERGENCE TESTS 1837
(b)α< 0. We consider the case where α< 0a n dδi=δfor alli.T h ec a s e
α< 0a n dδi/negationslash=δfor allimay be treated in the same way as case (a) and is
therefore omitted. Set γ=−α>0.
Whenδi=δfor alli,(73)a n d( 74) continue to hold but with α<0a n d
/epsilon1t=log/bracketleftbigg
1+N−1/2δ2
v2
ψNηNt/bracketrightbigg
(116)
−log{1+L(t)−2t−2αψ2
t/δ2+2L(t)−1t−αψt/δ}+Op(N−1)/periodori
IfTγ/√
N→0, then both logarithmic terms of /epsilon1tmay still be expanded as
T/commaoriN→∞ , but now the second term dominates rather than the first. Thus, in
place of ( 75), we get
ut=−/epsilon1t=2
δtγ
L(t)ψt+Op/parenleftbiggT2γ
L(t)2N+1√
N/parenrightbigg
/commaori (117)
where
ψt=1
NN/summationdisplay
i=1ψit=1
NN/summationdisplay
i=1σiξit=Op(N−1/2)/periodori
The limit theory proceeds as in the proof of Theorem 1,b u tw en o wh a v e
√
NTL(T)
Tγ(ˆb−b)
=2
δ(√
NL(T))/(√
TTγ)/summationtextT
t=[Tr]τt(tγ/L(t))ψ t
T−1/summationtextTt=[Tr]τ2
t+op(1)
=2
δ1√
NT/summationtextTt=[Tr]/summationtextNi=1τt(t
T)γσiξit
T−1/summationtextTt=[Tr]τ2
t+op(1)
⇒2
δN/parenleftBigg
0/commaoriω2
ξ/integraltext1
r/braceleftbig
logs−1
1−r/integraltext1
rlogpdp/bracerightbig2s2γds
(1−r)−(r
1−r)log2r/parenrightBigg
using Assumptions A2andA3,w h e r eω2
ξ=limN→∞N−1/summationtextN
i=1σ2
iωiiand where
1
TT/summationdisplay
t=[Tr]τ2
t/parenleftbiggt
T/parenrightbigg2γ
→/integraldisplay1
r/braceleftbigg
logs−1
1−r/integraldisplay1
rlogpdp/bracerightbigg2
s2γds/periodori
Thus, ifTγ−1/2/(√
NL(T)) →0,ˆbis still consistent, but at a reduced rate in
comparison with the null and provided γ=−αis not too large.

1838 P . C. B. PHILLIPS AND D. SUL
The behavior of the estimated standard error can be obtained in a similar
manner to the derivation under the null, given above. In particular, in view of
(117),
E(utut+l)=4
N2δ2N/summationdisplay
i=1σ2
itγ(t+l)γE(ξitξit+l)
L(t)L(t +l)
=4
N2δ2N/summationdisplay
i=1σ2
it2γE(ξitξit+l)
L(t)2t2α/parenleftbigg
1+O/parenleftbiggM
T/parenrightbigg/parenrightbigg
/commaori
where |l|<M andM
T→0a sT→∞ .S i n c eω2
i=/summationtext∞
l=−∞E(ξitξit+l),w eh a v e
M/summationdisplay
l=−ME(utut+l)=4
N2δ2N/summationdisplay
i=1σ2
iM/summationdisplay
l=−Mt2γE(ξitξit+l)
L(t)2(1+O(M
T))
∼4
N2δ2N/summationdisplay
i=1σ2
it2γω2
ii
L(t)2(1+o(1))
=4
δ2t2γ
NL(t)2ω2
ξ(1+o(1))/periodori
The sample quantity is
1
T−[Tr]/summationdisplay
[Tr]≤t/commaorit+l≤Tutut+l
=4
N2δ2N/summationdisplay
i=1σ2
i1
T−[Tr]/summationdisplay
[Tr]≤t/commaorit+l≤Tt2γξitξit+l
L(t)2(1+o(1))
=4
N2δ2N/summationdisplay
i=1σ2
i/braceleftbigg1
T−[Tr]/summationdisplay
[Tr]≤t/commaorit+l≤Tt2γE(ξitξit+l)
L(t)2(1+o(1))
+1
T−[Tr]/summationdisplay
[Tr]≤t/commaorit+l≤Tt2γ(ξitξit+l−Eξitξit+l)
L(t)2(1+o(1))/bracerightbigg
=4
N2δ2N/summationdisplay
i=1σ2
iE(ξitξit+l)/parenleftBigg
1
T−[Tr]T/summationdisplay
Tr1
L(t)2t2α/parenrightBigg
{1+op(1)}
=4
N2δ2N/summationdisplay
i=1σ2
iE(ξitξit+l)1
T−[Tr]T2γ
L(T)2T/summationdisplay
Tr(t
T)2γ
(L(t)2)/(L(T)2)
×{1+op(1)}

TRANSITION MODELING AND CONVERGENCE TESTS 1839
=4
N2δ2T2γ
L(T)2/integraltext1
rs2γds
1−rN/summationdisplay
i=1σ2
iE(ξitξit+l){1+op(1)}
=4
N2δ2T2γ
L(T)21−r1+2γ
(1+2γ)(1−r)N/summationdisplay
i=1σ2
iE(ξitξit+l){1+op(1)}/commaori
which gives
M/summationdisplay
l=−M1
T−[Tr]/summationdisplay
[Tr]≤t/commaorit+l≤Tutut+l
=4
N2δ2T2γ
L(T)21−r1+2γ
(1+2γ)(1−r)N/summationdisplay
i=1σ2
iM/summationdisplay
l=−ME(ξitξit+l){1+op(1)}
=4
N2δ2T2γ
L(T)21−r1+2γ
(1+2γ)(1−r)N/summationdisplay
i=1σ2
iω2ii{1+op(1)}
=4
Nδ2T2γ
L(T)21−r1+2γ
(1+2γ)(1−r)ω2
ξ(1+op(1))/periodori
Similarly, we find that
/hatwidestlvarr(√
Nˆut)=M/summationdisplay
l=−MN
T−[Tr]+1/summationdisplay
[Tr]≤t/commaorit+l≤Tˆutˆut+l
=4
δ2T2γ
L(T)21−r1+2γ
(1+2γ)(1−r)ω2
ξ(1+op(1))/commaori
and then
s2
ˆb=/hatwidestlvarr(ˆut)/bracketleftBiggT/summationdisplay
t=[Tr]τ2
t/bracketrightBigg−1
=1
NT/hatwidestlvarr(√
Nˆut)/bracketleftBigg
1
TT/summationdisplay
t=[Tr]τ2
t/bracketrightBigg−1
=4
NTδ2T2γ
L(T)21−r1+2γ
(1+2γ)(1−r)ω2
ξ/braceleftbigg
(1−r)−/parenleftbiggr
1−r/parenrightbigg
log2r/bracerightbigg−1
×(1+op(1))/periodori

1840 P . C. B. PHILLIPS AND D. SUL
It follows that
tˆb=ˆb
sˆb=(ˆb−b)
sˆb+b
sˆb=b
sˆb+Op(1)
and so, under the alternative with b=2α<0, we have
b
sˆb=/parenleftbigg
2α/braceleftbigg
(1−r)−/parenleftbiggr
1−r/parenrightbigg
log2r/bracerightbigg1/2/parenrightbigg
/slashBig/braceleftbigg4
N2Tδ2T2γ
L(T)21−r1+2γ
(1+2γ)(1−r)ω2
ξ/bracerightbigg1/2
→− ∞/commaori
confirming consistency of the test in this case. The divergence rate is O(L(T) ×
T1/2−γN).
(c) Finally, consider the case where γ=−αis such that Tγ/(√
NL(T)) →
∞.A g a i n ,( 73) holds so that
logH1
Ht−2l o gL(t)=a+blogt+ut (118)
and the second term of ( 116)d o m i n a t e s ,b u tn o w ,f o r t≥[Tr],w eh a v e
ut=−/epsilon1t=log/braceleftbigg
1+L(t)−2t2γψ2
t
δ2+2L(t)−1tγψt
δ/bracerightbigg
+Op(N−1/2)
=log/braceleftbiggt2γ
L(t)2N(√
Nψt)2
δ2/bracerightbigg
+Op/parenleftbigg√
NL(T)
Tγ+1√
N/parenrightbigg
=−2l o gL(t)+2γlogt−logN−logδ2+log(√
Nψt)2+op(1)
=−2l o gL(t)+2γlogt−logN−logδ2+log{ξψt+op(N−1/2)}2
+op(1)
=−2l o gL(t)+2γlogt+AN+ξwt+op(1)/commaori
whereξwt=logξ2
ψt−E{logξ2
ψt}andAN=E{ξwt}−logN−logδ2. Hence, ( 118)
is equivalent to
logH1
Ht−2l o gL(t)=aN+wt/commaoriwt=−2l o gL(t)+ξwt+op(1)/commaori (119)
whereaN=a+ANand the term in log tdrops out because 2 γ=−2α=−b/periodori
The error ( wt)i ne q u a t i o n( 119) therefore diverges to −∞ asT→∞ and
aN/commaoriAN=O(−logN)→− ∞ asN→∞/periodoriThis behavior is consistent with the
fact (easily deduced from ( 61)) thatHt=Op(N) in this case.

TRANSITION MODELING AND CONVERGENCE TESTS 1841
In view of ( 119), the fitted regression ( 98) behaves like a regression of
−2l o gL(t) on logt, so that just as in case (a) and ( 109)a b o v e ,w eh a v e
ˆb=/summationtextT
t=[Tr]τt{−2l o gL(t)+ξwt+op(1)}
/summationtextTt=[Tr]τ2
t
=−2
logT+O/parenleftbigg1
log2T/parenrightbigg
/periodori
Soˆb→p0/commaorias in case (a).
Next consider the standard error. When L(t)=logt,
logL(t)−logL(t)=logt
T−logt
T
logT+O/parenleftbigg1
log2T/parenrightbigg
from ( 101), so that
ˆut=−(ξwt−¯ξw)−2{logL(t)−logL(t)}−ˆb/braceleftbigg
logt
T−logt
T/bracerightbigg
=−(ξwt−¯ξw)−2logt
T−logt
T
logT+2
logT/braceleftbigg
logt
T−logt
T/bracerightbigg
+O/parenleftbigg1
log2T/parenrightbigg
=−(ξwt−¯ξw)+O/parenleftbigg1
log2T/parenrightbigg
/periodori
Assuming the long run variance of ξwtexists and writing ω2
ξω=/summationtext∞
k=−∞E(ξwt×
ξwt+k),w eh a v e
/hatwidestlvarr(ˆut)=M/summationdisplay
l=−M1
T−[Tr]+1/summationdisplay
[Tr]≤t/commaorit+l≤Tˆutˆut+l=ω2
ξω{1+op(1)}
and then
s2
ˆb=/hatwidestlvarr(ˆut)/bracketleftBiggT/summationdisplay
t=[Tr]τ2
t/bracketrightBigg−1
=ω2
ξω
T/bracketleftBigg
1
TT/summationdisplay
t=[Tr]τ2
t/bracketrightBigg−1
{1+op(1)}
=ω2
ξω
T/braceleftbigg
(1−r)−/parenleftbiggr
1−r/parenrightbigg
log2r/bracerightbigg−1
/periodori

1842 P . C. B. PHILLIPS AND D. SUL
Hence
tˆb=ˆb
sˆb
=/braceleftbigg
−2
logT+O/parenleftbigg1
log2T/parenrightbigg/bracerightbigg
÷/braceleftbiggω2
ξω
T/braceleftbigg
(1−r)−/parenleftbiggr
1−r/parenrightbigg
log2r/bracerightbigg−1/bracerightbigg1/2
=−2√
T
logT/braceleftbig
(1−r)−(r
1−r)log2r/bracerightbig1/2
ωξω{1+o(1)}→− ∞
and again the test is consistent. The divergence rate is O(T1/2/logT).
B.6. Proof of Theorem 3
(a) Under the local alternative ( 41), we have
δi∼iid(δ/commaoric2T−2ω)forα≥ω> 0/periodori (120)
Under this alternative the DGP for log (H 1/Ht)has the same form as in case
(a) in the proof of Theorem 2,b u tw i t h ¯δ=δand
σ2=N−1/summationdisplay
(δi−δ)2=c2T−2ω{1+Op(N−1/2)}=Op/parenleftbigg1
T2ω/parenrightbigg
/commaori (121)
σδψt=N−1N/summationdisplay
i=1(δi−δ)(ψit−ψt)=N−1/2T−ωςTNt{1+op(1)} (122)
=Op(N−1/2T−ω)
asN/commaoriT→∞/periodoriThus, as in ( 94), we have
Ht=1
δ2σ2+σ2
ψtL(t)−2t−2α+2(σδψt)L(t)−1t−α
1+L(t)−2t−2αψ2
t/δ2+2L(t)−1t−αψt/δ(123)
=1
δ2σ2+σ2
ψtL(t)−2t−2α+2(σδψt)L(t)−1t−α
1+L(t)−2t−2αψ2t/δ2+2L(t)−1t−αψt/δ
=Op(T−2ω)/commaori
sinceω≤α. It follows that Ht→p0a sT→∞ fort≥[Tr]andr>0, as the
model leads to behavior in Htlocal to that under the null hypothesis. More

TRANSITION MODELING AND CONVERGENCE TESTS 1843
explicitly, we have, using ( 97), (96), and ( 123),
logH1
Ht−2l o gL(t)=logH1−2l o gσ
δ−2l o gL(t)−/epsilon1t (124)
=logH1−2l o gc
δ+2ωlogT−2l o gL(t)−/epsilon1t/commaori
where
/epsilon1t=log/braceleftbigg
1+/parenleftbiggσ2
ψt
σ2/parenrightbigg
L(t)−2t−2α+2/parenleftbiggσδψt
σ2/parenrightbigg
L(t)−1t−α/bracerightbigg
(125)
−log/braceleftbigg
1+L(t)−2t−2αψ2
t
¯δ2+2L(t)−1t−αψt
¯δ/bracerightbigg
=log/braceleftbigg
1+/parenleftbiggσ2
ψt
c2/parenrightbigg
T2ωL(t)−2t−2α+2/parenleftbiggσδψt
c2/parenrightbigg
T2ωL(t)−1t−α/bracerightbigg
−log/braceleftbigg
1+L(t)−2t−2αψ2t
δ2+2L(t)−1t−αψt
δ/bracerightbigg
=/parenleftbiggσ2
ψt
c2/parenrightbigg
T2ωL(t)−2t−2α+Op/parenleftbigg1
L(T)4T4(α−ω)+1√
NL(T)T2(α−ω)/parenrightbigg
/periodori
The fitted regression is again
logH1
Ht−2l o gL(t)=ˆa†+ˆblog/parenleftbiggt
T/parenrightbigg
+residual/commaori (126)
where now ˆa†=logH1−2l o gc
δ+2ωlogTand, as in ( 103),
ˆb=−/summationtextT
t=[Tr]τt{2l o gL(t)+/epsilon1t}
/summationtextTt=[Tr]τ2
t
=−/summationtextTt=[Tr]τt/epsilon1t/summationtextTt=[Tr]τ2
t−2
logT+g(r)
log2T+Op/parenleftbigg1
log3T/parenrightbigg
/periodori
Next,
1
TT/summationdisplay
t=[Tr]τt/epsilon1t=v2
ψN
c21
L(T)2T1+2(α−ω)T/summationdisplay
t=[Tr]τtL(T)2
L(t)2/parenleftbiggt
T/parenrightbigg−2α
{1+op(1)} (127)
=v2
ψN
c21
L(T)2T2(α−ω)

1844 P . C. B. PHILLIPS AND D. SUL
×/integraldisplay1
r/braceleftbigg
logs−1
1−r/integraldisplay1
rlogpdp/bracerightbigg
s−2αds{1+op(1)}
=v2
ψN
c2r∗(α)
L(T)2T2(α−ω){1+op(1)}/commaori
wherer∗(α)=/integraltext1
r{logs−1
1−r/integraltext1
rlogpdp}s−2αdsi sg i v e ni n( 106)a b o v e .W ed e –
duce that
ˆb=−2
logT+g(r)
log2T−v2
ψN
c2r(α)
L(T)2T2(α−ω)+Op/parenleftbigg1
log3T/parenrightbigg
/commaori (128)
wherer(α) is given in ( 108), so that ˆb→p0a sT/commaoriN→∞ . The result is there-
fore comparable to that under case (a) of Theorem 2.
Next consider the standard error of ˆbunder the local alternative. Writing the
residual in ( 126)a sˆut, the long run variance estimate has typical form
/hatwidestlvarr(√
Nˆut)=M/summationdisplay
l=−MN
T−[Tr]+1/summationdisplay
[Tr]≤t/commaorit+l≤Tˆutˆut+l/periodori
In view of ( 125), (127), and ( 128), we have
ˆut=−(/epsilon1t−¯/epsilon1)−2{logL(t)−logL(t)}−ˆb/braceleftbigg
logt
T−logt
T/bracerightbigg
=−(/epsilon1t−¯/epsilon1)−2logt
T−logt
T
logT−ˆb/braceleftbigg
logt
T−logt
T/bracerightbigg
+Op/parenleftbigg1
log2T/parenrightbigg
=−v2
ψN
c2T−2ω/braceleftbigg1
L(t)2t2α−1
L(T)2T2α1−r1−2α
(1−r)(1−2α)/bracerightbigg
+/braceleftbiggv2
ψN
c2r(α)
L(T)2T2(α−ω)−g(r)
log2T/bracerightbigg/braceleftbigg
logt
T−logt
T/bracerightbigg
+Op/parenleftbigg1
log2T/parenrightbigg
/commaori
using ( 112). Then, for |l|≤M,M
T→0, andt≥[Tr]withr>0, we have, as in
case (a) of the proof of Theorem 2,
ˆut+l=−(/epsilon1t+l−¯/epsilon1)−ˆb/braceleftbigg
logt+l
T−logt
T/bracerightbigg
−2{logL(t)−logL(t)}
=−v2
ψN
c2T−2ω/braceleftbigg1
L(t)2t2α[1+o(1)]−1
L(T)2T2α1−r1−2α
(1−r)(1−2α)/bracerightbigg

TRANSITION MODELING AND CONVERGENCE TESTS 1845
+/braceleftbiggv2
ψN
c2r(α)
L(T)2T2α−g(r)
log2T/bracerightbigg/braceleftbigg
logt
T[1+o(1)]−logt
T/bracerightbigg
+Op/parenleftbigg1√
NL(T)Tα/parenrightbigg
+Op/parenleftbigg1
log2T/parenrightbigg
=ˆut{1+o(1)}/commaori
so that, just as in ( 113), we find
1
T−[Tr]/summationdisplay
[Tr]≤t/commaorit+l≤Tˆutˆut+l
=1
T−[Tr]/summationdisplay
[Tr]≤t≤Tˆu2
t{1+o(1)}
=1
T−[Tr]v4
ψN
c4T−4ω/summationdisplay
[Tr]≤t≤T/braceleftbigg1
L(t)2t2α−1
L(T)2T2α1−r1−2α
(1−r)(1−2α)/bracerightbigg2
+1
T−[Tr]/braceleftbiggv2
ψN
c2r(α)
L(T)2T2(α−ω)−g(r)
log2T/bracerightbigg2
×/summationdisplay
[Tr]≤t≤T/braceleftbigg
logt
T−logt
T/bracerightbigg2
−2
T−[Tr]v2
ψN
c2T−2ω/braceleftbiggv2
ψN
c2r(α)
L(T)2T2(α−ω)−g(r)
log2T/bracerightbigg
×/summationdisplay
[Tr]≤t≤Tlogt
T−logt
T
L(t)2t2α+Op/parenleftbigg1
log4T/parenrightbigg
=Op/parenleftbigg1
log4T/parenrightbigg
/commaori
uniformly in l,a n dw h e n L(T) =logTandα≥ω> 0. The remainder of the
proof follows that of case (a) in the proof of Theorem 2. In particular, we have
/hatwidestlvarr(√
Nˆut)=Op/parenleftbiggNM
log4T/parenrightbigg
/commaori
s2
ˆb=/hatwidestlvarr(ˆut)/bracketleftBiggT/summationdisplay
t=[Tr]τ2
t/bracketrightBigg−1
=Op/parenleftbiggM
(log4T)T/parenrightbigg
/commaori

1846 P . C. B. PHILLIPS AND D. SUL
and
tˆb=ˆb
sˆb=−2
logT÷Op/parenleftbiggM1/2
(log2T)T1/2/parenrightbigg
=−2
logT×Op/parenleftbigg(log2T)T1/2
M1/2/parenrightbigg
→− ∞
for allα≥ω> 0 and all bandwidth choices M≤T. Again, the divergence rate
isOp((logT)T1/2/M1/2). Thus, the test is consistent against all local alterna-
tives of the form ( 120)w i t hω≤α.
(b) When ω>α/commaori the alternative involves
δi∼iid(δ/commaoric2T−2ω)forω>α ≥0/commaori (129)
so thatδi=δ+Op(T−ω)=δ+op(T−α)and the alternatives are closer to the
null than in case (a). Now we have
σ2=N−1N/summationdisplay
i=1(δi−δ)2=c2T−2ω{1+Op(N−1/2)}=op/parenleftbigg1
T2α/parenrightbigg
/commaori
σδψt=N−1N/summationdisplay
i=1(δi−δ)(ψit−ψt)=N−1/2T−ωςTNt{1+op(1)}
=op(N−1/2T−α)/commaori
and ( 123) becomes
Ht=1
δ2σ2+σ2
ψtL(t)−2t−2α+2(σδψt)L(t)−1t−α
1+L(t)−2t−2αψ2
t/δ2+2L(t)−1t−αψt/δ
=1
δ2σ2
ψtL(t)−2t−2α+2(σδψt)L(t)−1t−α+c2T−2ω{1+Op(N−1/2)}
1+L(t)−2t−2αψ2t/δ2+2L(t)−1t−αψt/δ
=1
L(t)2t2α1
δ2
×σ2
ψt+c2T−2ωt2αL(t)2{1+Op(N−1/2)}+2(σδψt)L(t)tα
1+L(t)−2t−2αψ2
t/δ2+2L(t)−1t−αψt/δ/commaori
so that the behavior of Htis asymptotically the same as under the null (cf.
(66)). T aking logarithms, we have
logHt=−2l o gL(t)−2αlogt

TRANSITION MODELING AND CONVERGENCE TESTS 1847
+log/braceleftbigg1
δ2/bracketleftbigg
v2
ψN+N−1/2ηNt+Op(N−1)
+c2T−2ωt2αL(t)2{1+Op(N−1/2)}+Op/parenleftbiggL(T)
N1/2T(ω−α)/parenrightbigg/bracketrightbigg/bracerightbigg
−log/braceleftbigg
1+L(t)−2t−2αψ2
t
δ2+2L(t)−1t−αψt
δ/bracerightbigg
=−2l o gL(t)−2αlogt+2l o gvψN
δ+/epsilon1t/commaori
where
/epsilon1t=log/bracketleftbigg
1+N−1/2ηNt
v2
ψN+c2
v2
ψNT−2ωt2αL(t)2{1+Op(N−1/2)}
+Op/parenleftbiggL(T)
N1/2T(ω−α)/parenrightbigg/bracketrightbigg
−log/braceleftbigg
1+L(t)−2t−2αψ2t
δ2+2L(t)−1t−αψt
δ/bracerightbigg
+Op(N−1)/periodori
These formulae lead to the empirical regression equation
logH1
Ht−2l o gL(t)=a+blogt+ut/commaori
wherea=h1−2l o g(vψN/δ),b=2α,a n dut=−/epsilon1t,j u s ta si n( 73)a n d( 74), but
now
ut=−1√
NηNt
v2
ψN−c2
v2
ψNT−2ωt2αL(t)2{1+Op(N−1/2)} (130)
+2
δ1
tαL(t)ψt+Op/parenleftbigg1
L(T)2T2αN+L(T)
N1/2T(ω−α)+1
N/parenrightbigg
=−c2
v2
ψNt2α
T2ωL(t)2{1+Op(N−1/2)}−1√
NηNt
v2
ψN
+Op/parenleftbigg1
N1/2L(T)Tα+L(T)
N1/2T(ω−α)/parenrightbigg
/periodori
The first term of ( 130) dominates the behavior of the regression error utwhen
ω>α ≥0a n dT2(ω−α)/(√
NL(T)2)→0a sT/commaoriN→∞ . It follows that
ˆb−b=/summationtextT
t=[Tr]τtut/summationtextTt=[Tr]τ2
t

1848 P . C. B. PHILLIPS AND D. SUL
=−c2
v2
ψN1
T2ω1√
NT/summationtextT
t=[Tr]τtt2αL(t)2
T−1/summationtextT
t=[Tr]τ2
t−1
v2
ψN1√
NT/summationtextT
t=[Tr]τtηNt
T−1/summationtextT
t=[Tr]τ2
t
+2
δ1
T/summationtextT
t=[Tr]τtψt/(tαL(t))
T−1/summationtextT
t=[Tr]τ2
t+Op/parenleftbigg1
L(t)2T2αN/parenrightbigg
/commaori
and since
1
TT/summationdisplay
t=[Tr]τtt2αL(t)2
=L(T)2
T1−2αT/summationdisplay
t=[Tr]τt/parenleftbiggt
T/parenrightbigg2a
{1+o(1)}
=T2αL(T)2/integraldisplay1
r/braceleftbigg
logs−1
1−r/integraldisplay1
rlogpdp/bracerightbigg
s2αds{1+o(1)}/commaori
we have
ˆb−b=−c2
v2
ψNh(r)L(T)2
T2(ω−α){1+o(1)}/commaori (131)
where
h(r)=/integraltext1
r/braceleftbig
logs−1
1−r/integraltext1
rlogpdp/bracerightbig
s2αds
(1−r)−(r
1−r)log2r/periodori
Sinceω>α ,ˆb→pbandˆbis consistent.
Next, the regression residual has the form
ˆut=−(/epsilon1t−¯/epsilon1)−(ˆb−b)/braceleftbigg
logt
T−logt
T/bracerightbigg
=−c2
v2
ψNL(T)2/braceleftbiggt2α
T2ω−1
T2(ω−α)1−r1+2α
2α+1/bracerightbigg
{1+Op(N−1/2)}
+/braceleftbiggc2
v2
ψNh(r)L(T)2
T2(ω−α)/bracerightbigg/braceleftbigg
logt
T−logt
T/bracerightbigg
/commaori
since
1
TT/summationdisplay
t=[Tr]t2α=1
T1−2αT/summationdisplay
t=[Tr]/parenleftbiggt
T/parenrightbigg2a
=T2α/integraldisplay1
rs2αds{1+o(1)}

TRANSITION MODELING AND CONVERGENCE TESTS 1849
=T2α1−r1+2α
2α+1/periodori
As earlier, we find ˆut+l=ˆut{1+o(1)}for|l|≤M, and then
1
T−[Tr]/summationdisplay
[Tr]≤t/commaorit+l≤Tˆutˆut+l
=1
T−[Tr]/summationdisplay
[Tr]≤t≤Tˆu2
t{1+o(1)}
=1
T−[Tr]c4
v4
ψNL(T)4/summationdisplay
[Tr]≤t≤T/braceleftbiggt2α
T2ω−1
T2(ω−α)1−r1+2α
2α+1/bracerightbigg2
×{1+Op(N−1/2)}
+L(T)4
T−[Tr]/braceleftbiggc2
v2
ψNh(r)
T2(ω−α)/bracerightbigg2/summationdisplay
[Tr]≤t≤T/braceleftbigg
logt
T−logt
T/bracerightbigg2
−2
T−[Tr]c2
v2
ψNc2
v2
ψNh(r)L(T)4
T4(ω−α)/summationdisplay
[Tr]≤t≤T/braceleftbigg
logt
T−logt
T/bracerightbigg/parenleftbiggt
T/parenrightbigg2α
×{1+Op(N−1/2)}
=Op/parenleftbiggL(T)4
T4(ω−α)/parenrightbigg
uniformly in |l|≤M. Hence,
/hatwidestlvarr(ˆut)=M/summationdisplay
l=−M1
T−[Tr]+1/summationdisplay
[Tr]≤t/commaorit+l≤Tˆu2
t{1+o(1)}
=Op/parenleftbiggML(T)4
T4(ω−α)/parenrightbigg
and so
s2
ˆb=/hatwidestlvarr(ˆut)/bracketleftBiggT/summationdisplay
t=[Tr]τ2
t/bracketrightBigg−1
=Op/parenleftbiggML(T)4
T4(ω−α)T/parenrightbigg/bracketleftBigg
1
TT/summationdisplay
t=[Tr]τ2
t/bracketrightBigg−1
=Op/parenleftbiggML(T)4
T1+4(ω−α)/parenrightbigg
/periodori

1850 P . C. B. PHILLIPS AND D. SUL
It follows that
tˆb=ˆb
sˆb=/bracketleftbigg
b−c2
v2
ψNh(r)L(T)2
T2(ω−α){1+o(1)}/bracketrightbigg
×Op/parenleftbiggT1/2+2(ω−α)
M1/2L(T)2/parenrightbigg
→/braceleftbigg
∞/commaori forb=2α>0,
−∞/commaoriforb=2α=0.
Thus, when α> 0, the test has no power to detect alternatives of the form
(129), whereas when ω>α =0, the test is consistent. In both cases, the alter-
nativesδi/negationslash=δare close to the null because ω>α ,b u tw h e n α=0, the rate
of convergence of δitis slow (at a slowly varying rate) and the test is therefore
able to detect the local departures from the null.
B.7. Power and the Choice of the L(t) Function
This section provides a short discussion on the choice of the L(t) function.
Since the class of possible L(t) functions is vast, it is convenient to consider
the restricted class of logarithmic and higher order logarithmic functions
L(t)=logktfor integer k≥1/commaori (132)
where log1t=logt/commaorilog2t=log(logt), and so on. Since our concern is with sit-
uations where tis large in the regression asymptotics, L(t) andL(t)−1are both
well defined. Note that kcan be any positive integer, but we confine attention
below to the primary cases of interest where k=1/commaori2. Higher order cases can
be deduced by recursion.
From ( 132), we can rewrite ( 100)a n d( 102)a s
logL(t)=logL/parenleftbigg
Tt
T/parenrightbigg
=logk/braceleftbigg
logT+logt
T/bracerightbigg
=logk/bracketleftbigg
logT/braceleftbigg
1+logt
T
logT/bracerightbigg/bracketrightbigg
=

log
2T+logt
T
logT−1
2log2t
T
log2T+O/parenleftbigglog3t
T
log3T/parenrightbigg
/commaorik =1,
log/bracketleftbigg
log2T+logt
T
logT−1
2log2t
T
log2T+O/parenleftbigglog3t
T
log3T/parenrightbigg/bracketrightbigg
/commaorik=2,
=

log
2T+logt
T
logT−1
2log2t
T
log2T+O/parenleftbigglog3t
T
log3T/parenrightbigg
/commaorik =1,
log3T+logt
T
logTlog2T+O/parenleftbigglog2t
T
log2Tlog2T/parenrightbigg
/commaorik=2,

TRANSITION MODELING AND CONVERGENCE TESTS 1851
extending ( 101). Correspondingly, under the alternative α≥0a n dδi∼
iid(δ/commaoriσ2
δ)considered in case (a) of Theorem 2, the choice of L(t) affects the
bias formula in ( 109). Since
/summationtextT
t=[Tr]τtlogL(t)
/summationtextTt=[Tr]τ2
t=

1
logT+O/parenleftbigg1
log2T/parenrightbigg
/commaorik =1,
1
logTlog2T+O/parenleftbigg1
log2Tlog2T/parenrightbigg
/commaorik=2,
we find that
ˆb=

−2
logT+Op/parenleftbigg1
log2T/parenrightbigg
/commaorik =1,
−2
logTlog2T+O/parenleftbigg1
log2Tlog2T/parenrightbigg
/commaorik=2.
Proceeding as in the proof of Theorem 2(case (a)) we find that
s2
ˆb=

Op/parenleftbiggM
(log4T)T/parenrightbigg
/commaorik=1,
Op/parenleftbiggM
(log4
2T)T/parenrightbigg
/commaorik=2,
and then
tˆb=ˆb
sˆb=

−2
logT×Op/parenleftbigg(log2T)T1/2
M1/2/parenrightbigg
/commaorik=1,
−2
logT×Op/parenleftbigg(log2
2T)T1/2
M1/2/parenrightbigg
/commaorik=2.
So, the divergence rate and discriminatory power of the log ttest reduce as we
changeL(t) from log tto log2t=log logt. The test is still consistent for k=2,
provided
MlogT
Tlog22T→0/periodori
APPENDIX C: A SYMPTOTIC PROPERTIES OF THE CLUSTERING PROCEDURE
Section 4develops a clustering procedure based on augmenting a core panel
withKindividuals where δi=δAfori=1/commaori/periodori/periodori/periodori/commaoriK with additional individuals
one at a time for which δK+1=δB, say. This appendix provides an asymptotic

1852 P . C. B. PHILLIPS AND D. SUL
analysis of that procedure. We assume that the size of the core group K→∞
asN→∞ . The variation of the δiis then
σ2=1
K+1K+1/summationdisplay
i=1(δi−¯δ)2=K
(K+1)2(δA−δB)2=O(K−1)/commaori (133)
where
¯δ=1
K+1K+1/summationdisplay
i=1δi=KδA
K+1+δB
K+1=δA+O(K−1)
as in ( 42)a n dσ2depends on K. More generally, we can consider a panel with
idiosyncratic coefficients
δi∼iid(δ/commaoric2K−1)/commaori whereK
T2α→0a n d α>0/commaori (134)
so thatKis small relative to T/periodoriIn this case, σ2=c2K−1/commaorianalogous to ( 133).
In the same way as in ( 124)a n d( 125), under this alternative the DGP for
log(H 1/Ht)has the form
logH1
Ht−2l o gL(t)=logH1−2l o gc
δ+logK−2l o gL(t)−/epsilon1t/commaori
where
/epsilon1t=/parenleftbiggσ2
ψt
c2/parenrightbigg
KL(t)−2t−2α+Op/parenleftbiggK2
L(T)4T4α+K√
NL(T)T2α/parenrightbigg
/periodori
The fitted regression can now be written as
logH1
Ht−2l o gL(t)=ˆa†+ˆblog/parenleftbiggt
T/parenrightbigg
+ˆut/commaori
where ˆa†=logH1−2l o gc
δ+logKand, as in ( 128)b u tw i t h K/T2α→0, we
find that
ˆb=−2
logT+g(r)
log2T−v2
ψN
c2r(α)K
L(T)2T2α+Op/parenleftbigg1
log3T/parenrightbigg
/periodori
Proceeding as in the proof of Theorem 3, we find that
s2
ˆb=/hatwidestlvarr(ˆut)/bracketleftBiggT/summationdisplay
t=[Tr]τ2
t/bracketrightBigg−1
=Op/parenleftbiggM
(log4T)T/parenrightbigg

TRANSITION MODELING AND CONVERGENCE TESTS 1853
and
tˆb=ˆb
sˆb=−2
logT÷Op/parenleftbiggM1/2
(log2T)T1/2/parenrightbigg
=−2
logT×Op/parenleftbigg(log2T)T1/2
M1/2/parenrightbigg
→− ∞
for allα>0/commaoriforKsatisfying K/T2α→0/commaoriand all bandwidth choices M≤T/periodori
The test is therefore consistent against local alternatives of the form ( 134).
In view of ( 133), this includes the case where δi=δAfori=1/commaori/periodori/periodori/periodori/commaoriK with
δK+1=δB/negationslash=δA. On the other hand, when δi=δAfori=1/commaori/periodori/periodori/periodori/commaoriK , the null
hypothesis holds for N=Kandtˆb=(ˆb−b)/s ˆb⇒N(0/commaori1)as in Theorem 1.
WhenT2α/K→0, the alternatives ( 134) are very close to the null, relative
to the convergence rate except when α=0. This case may be treated as in
the proof of case (b) of Theorem 3. Accordingly, the test is inconsistent and
unable to detect the departure from the null when α>0. However, when α=0
and the convergence rate is slowly varying under the null, the test is consistent
against local alternatives of the form ( 134) just as in case (b) of Theorem 3.
In effect, although the alternatives are very close (because Kis large), the
convergence rate is so slow (slower than any power rate) and this suffices toensure the test is consistent as T→∞ .
APPENDIX D: D
ATA FOR THE COST OF LIVING INDEX EXAMPLE
Data Source :B u r e a uo fL a b o rS t a t i s t i c s
Data : 19 U.S. Cities CPI
Time Period : 1918–2001 (84 annual observations)
List of Cities : New Y ork (NYC), Philadelphia (PHI), Boston (BOS), Cleve-
land (CLE), Chicago (CHI), Detroit (DET), Washington, DC (WDC). Bal-
timore (BAL), Houston (HOU), Los Angeles (LAX), San Francisco (SFO),
Seattle (SEA), Portland (POR), Cincinnati (CIN), Atlanta (ATL), St. Louis(STL), Minneapolis/St. Paul (MIN), Milwaukee (KCM)
REFERENCES
ADRIAN ,T . , AND F. F RANZONI (2005): “Learning about Beta: Time-V arying Factor Loadings,
Expected Returns, and the Conditional CAPM,” Mimeo, Federal Reserve Bank of New Y ork.
[1777 ]
BAI, J. (2003): “Inferential Theory for Factor Models of Large Dimensions,” Econometrica , 71,
135–172. [1772 ]
(2004): “Estimation Cross-Section Common Stochastic T rends in Nonstationary Panel
Data,” Journal of Econometrics , 122, 137–183. [1772 ]
BAI,J . , AND S. N G(2002): “Determining the Number of Factors in Approximate Factor Models,”
Econometrica , 70, 191–221. [1772 ,1813 ]

1854 P . C. B. PHILLIPS AND D. SUL
(2006): “Evaluating Latent and Observed Factors in Macroeconomics and Finance,”
Journal of Econometrics , 131, 507–537. [1772 ]
BROWNING ,M . , AND J. C ARRO (2006): “Heterogeneity and Microeconometrics Modelling,”
Mimeo, University of Copenhagen. [ 1772 ]
BROWNING , M., L. P . H ANSEN ,AND J. J. H ECKMAN (1999): “Micro Data and General Equilib-
rium Models,” in Handbook of Macroeconomics , Vol. 1A. New Y ork: Elsevier, Chap. 8. [ 1772 ]
CANOVA , F . (2004): “T esting for Convergence Clubs: A Predictive Density Approach,” Interna-
tional Economic Review , 45, 49–77. [1798 ]
CANOVA ,F . , AND A. M ARCET (1995): “The Poor Stay Poor: Non-Convergence Across Countries
and Regions,” Discussion Paper 1265, CEPR. [ 1798 ]
CHAMBERLAIN ,G . , AND M. R OTHSCHILD (1983): “ Arbitrage, Factor Structure and Mean–
V ariance Analysis in Large Asset Markets,” Econometrica , 51, 1305–1324. [1772 ]
CHOI,C . ,N .M ARK,AND D. S UL(2006): “Endogenous Discounting, the World Savings Gluts and
U.S. Current Account,” Mimeo, University of Auckland. [ 1773 ]
CONNOR ,G . , AND R. A. K ORAJCSYK (1986): “Performance Measurement with Arbitrage Pricing
Theory: A New Framework for Analysis,” Journal of Financial Economics , 15, 373–394. [1772 ]
(1988): “Risk and Return in Equilibrium APT: Application of a New T est Methodology,”
Journal of Financial Economics , 21, 255–290. [1772 ]
DURLAUF ,S .N . , AND P. A . J OHNSON (1995): “Multiple Regimes and Cross-Country Growth
Behaviour,” Journal of Applied Econometrics , 10, 365–384. [1798 ]
DURLAUF ,S .N . , AND D. T . Q UAH (1999): “The New Empirics of Economic Growth,” in Hand-
book of Macroeconomics , Vol. 1A. New Y ork: Elsevier, Chap. 4. [ 1771 ]
DURLAUF ,S .N . ,P .A .J OHNSON ,AND J. T EMPLE (2005): “Growth Econometrics,” Mimeo, Uni-
versity of Wisconsin. [ 1771 ]
ERMISCH , J. (2004): “Using Panel Data to Analyze Household and Family Dynamics,” in Re-
searching Social and Economic Change: The Uses of Household Panel Studies ,e d .b yD .R o s e .
London and New Y ork: Routledge, 230–249. [ 1771 ]
FAMA ,E .F . , AND K. R. F RENCH (1993): “Common Risk Factors in the Returns on Stocks and
Bonds,” Journal of Financial Economics , 33, 3–56. [1777 ]
(1996): “Multifactor Explanations of Asset Pricing Anomalies,” Journal of Finance , 51,
55–84. [1777 ]
GIVENEN , F . (2005): “ An Empirical Investigation of Labor Income Processes,” Mimeo, University
of Rochester. [ 1772 ]
GUERRE , E. (2004): “Design-Adaptive Pointwise Nonparametric Regression Estimation for Re-
current Markov Time Series,” Mimeo, Université Pierre et Marie Curie. [ 1784 ]
HODRICK ,R . , AND E. P RESCOTT (1997): “Post War Business Cycles: An Empirical Investigation,”
Journal of Money Credit and Banking , 29, 1–16. [1783 ]
HOWITT ,P . , AND D. M AYER -FOULKES (2005): “R&D, Implementation and Stagnation: A Schum-
peterian Theory of Convergence Clubs,” Journal of Money, Credit and Banking , 37, 147–177.
[1775 ]
HSIAO , C. (2003): Analysis of Panel Data (Second Ed.). Cambridge, U.K.: Cambridge University
Press. [1771 ]
KARLSEN ,H .A . , AND D. T JØSTHEIM (2001): “Nonparametric Estimation in Null Recurrent Time
Series,” The Annals of Statistics , 29, 372–416. [1784 ]
KATZ,L .F . , AND D. H. A UTOR (1999): “Changes in the Wage Structure and Earnings Inequality,”
inHandbook of Labor Economics , Vol. 3A. New Y ork: Elsevier, Chap. 26. [ 1776 ]
KITAGAWA ,G . , AND W. G ERSCH (1996): Smoothness Priors Analysis of Time Series .N e wY o r k :
Springer Verlag. [1783 ]
KRUSELL ,P . , AND A. A. S MITH (1998): “Income and Wealth Heterogeneity in the Macroecon-
omy,” Journal of Political Economy , 106, 867–896. [1772 ]
LUCAS ,R .E .J R.,AND N. L. S TOKEY (1984): “Optimal Growth with Many Consumers,” Journal
of Economic Theory , 32, 139–171. [1773 ]

TRANSITION MODELING AND CONVERGENCE TESTS 1855
LUDVIGSON ,S .C . , AND S. N G(2007): “The Empirical Risk-Return Relation: A Factor Analysis
Approach,” Journal of Financial Economics , 83, 171–222. [1777 ]
MENZL Y ,L . ,T .S ANTOS ,AND P. V ERONESI (2002): “The Time Series of the Cross Section of Asset
Prices,” Working Paper 9217, NBER. [ 1777 ]
MOFFITT ,R .A . , AND P. G OTTSCHALK (2002): “T rends in the T ransitory V ariance on Earnings in
the United States,” Economic Journal , 112, C68–C73. [1776 ]
MOON ,H .R . , AND B. P ERRON (2004): “T esting for a Unit Root in Panels with Dynamic Factors,”
Journal of Econometrics , 122, 81–126. [1772 ]
OBSTFELD , M. (1990): “Intertemporal Dependence, Impatience, and Dynamics,” Journal of Mon-
etary Economics , 26, 45–75. [1773 ]
PARENTE ,S .L . , AND E. C. P RESCOTT (1994): “Barriers to T echnology Adoption and Develop-
ment,” Journal of Political Economy , 102, 298–321. [1775 ]
PHILLIPS , P . C. B. (2005): “Challenges of T rending Time Series Econometrics,” Mathematics and
Computers in Simulation , 68, 401–416. [1784 ]
(2007): “Regression with Slowly V arying Regressors and Nonlinear T rends,” Economet-
ric Theory , 23, 557–614. [1821 ]
PHILLIPS ,P .C .B . , AND S. J IN(2002): “Limit Behavior of the Whittaker (HP) Filter,” Mimeo,
Yale University. [ 1785 ]
PHILLIPS ,P .C .B . , AND H. R. M OON (1999): “Linear Regression Limit Theory for Nonstationary
Panel Data,” Econometrica , 67, 1057–1111. [1787 ,1818 ]
PHILLIPS ,P .C .B . , AND J. PARK (1998): “Nonstationary Density Estimation and Kernel Autore-
gression,” Cowles Foundation Discussion Paper 1181. [ 1784 ]
PHILLIPS ,P .C .B . , AND V. S OLO (1992): “ Asymptotics for Linear Processess,” The Annals of Sta-
tistics , 20, 971–1001. [1782 ,1816 ]
PHILLIPS ,P .C .B . , AND D. S UL(2003): “The Elusive Empirical Shadow of Growth Convergence,”
Discussion Paper 1398, Cowles Foundation, Yale University.
(2006): “Economic T ransition and Growth,” Mimeo, University of Auckland. [ 1772 ,
1775 ,1780 ]
QUAH, D. (1996): “Empirics for Economic Growth and Convergence,” European Economic Re-
view, 40, 1353–1375. [1798 ]
(1997): “Empirics for Growth and Distribution: Polarization, Stratification and Conver-
gence Clubs,” Journal of Economic Growth , 2, 27–59. [1798 ]
SARGENT ,T .J . , AND C. A. S IMS(1977): “Business Cycle Modeling Without Pretending to Have
T oo Much a priori Economic Theory,” in New Methods in Business Cycle Research ,e d .b yC .A .
Sims. Minneapolis: Federal Reserve Bank of Minneapolis, 45–109. [ 1813 ]
SCHIMDT -GROHÉ ,S . , AND M. U RIBE (2003): “Closing Small Open Economy Models,” Journal of
International Economics , 61, 163–185. [1773 ]
SENETA , E. (1976): Regularly Varying Functions , Lecture Notes in Mathematics, Vol. 408. Berlin:
Springer. [1813 ]
STOCK ,J .H . , AND M. W . W ATSON (1999): “Diffusion Indexes,” Working Paper 6702, NBER.
[1772 ]
WANG ,Q . , AND P. C . B . P HILLIPS (2006): “ Asymptotic Theory for Local Time Density Estimation
and Nonparametric Cointegrating Regression,” Working Paper, Yale University. [ 1784 ]
WHITTAKER , E. T . (1923): “On a New Method of Graduation,” Proceedings of the Edinburgh
Mathematical Association , 78, 81–89. [1783 ]
UZAWA , H. (1968): “Time Preference and the Penrose Effect in a T wo-Class Model of Economic
Growth,” Journal of Political Economy , 77, 628–652. [1773 ]