Convergence: Variation in Concept and Empirical Results
Nazrul Islam
Department of Economics
Emory University
First Draft: July 1996
Current Draft: June 1998
---------------------------------------------------------------------------------------------------------------I would like to thank all those who commented on earlier versions of this paper. Their
suggestions have led to significant improvements of this paper. All remaining errors and
shortcomings are mine.
Convergence: Variation in Concept and Empirical Results
1. Introduction
A central issue around which the recent growth literature has evolved is the issue of
convergence. Whether the poorer countries of the world are converging to the income levels of
the richer countries is, by itself, a question of paramount importance for human welfare.
However, the interest in the issue has been fueled further by the fact that convergence became
linked with the question, which kind of growth theory better describes the world. It has been
generally thought that convergence is an implication of the neo-classical growth theory (NCGT),
while the new growth theories (NGT) do not have this implication. Hence, it was supposed that
by testing for convergence, one could test which of these two classes of growth theory was more
valid. Given this link, it is hardly surprising that the convergence issue has engaged so numerous
and outstanding minds of the economics profession. This has led to a wide variety of concepts
and results, so much so that the overall situation has become somewhat confusing. This has not
been helped by the fact that there have been very few attempts to review this literature.
This paper is an attempt to fill that gap. It takes stock of this vast literature and tries to
synthesize the results. In doing so, it uses the logical-historical method. The literature on
convergence has unfolded over time responding to perceived logical inadequacies of the works
of the previous period. Hence, understanding of the logical-analytical points requires an
examination of how this literature has evolved over time.
As we shall see, there is a wide variety of ways in which convergence has been defined
and understood. This is evident from the following, often encountered, dichotomies:
(a) Convergence within an economy vs. convergence across economies;
(b) Convergence in terms of growth rate vs. convergence in terms of income level;
(c) β-Convergence vs. s -convergence;
(d) Unconditional (absolute) convergence vs. conditional convergence;
(e) Unconditional convergence vs. club-convergence;
(f) Income-convergence vs. TFP (total factor productivity)-convergence; and
(g) Deterministic convergence vs. stochastic convergence.
1
It is not that all these different interpretations of convergence were apparent from the
beginning. Research on convergence has proceeded through several stages, and it is only with
time that these different interpretations of convergence emerged and gained currency. From
methodological point of view, we can identify the following five different approaches to
convergence study:
(a) Informal cross-section approach,
(b) Formal cross-section approach,
(c) Panel approach,
(d) Time-series approach, and
(e) Distribution approach.
There is some correspondence between the convergence concepts, on the one hand, and
the methodologies used, on the other. However, this correspondence is not unique. Thus, for
example, informal and formal cross-section approaches, panel approach, and time-series
approach (in part) have all focused on β-convergence, either conditional or unconditional. These
approaches have generally dealt with convergence across economies and in terms of per capita
income level. The formal cross-section and panel approaches have also been employed to
examine club-convergence and TFP-convergence. The cross-section approach has even been
used to study σ -convergence. On the other hand, time series approach has been used to
investigate convergence both within an economy and across-economies, and this approach has
often proceeded from the notion of stochastic convergence. Finally, the distribution approach has
gone beyond investigating just σ -convergence and has focused on the entire shape of the
distribution and intra-distribution dynamics.
This variety of concepts and methodological approaches has led to a plethora of empirical
results. However, at a broad level, there is considerable agreement in the results. For example,
despite differences in approach and methodology, the finding of conditional β-convergence has
remained relatively robust. This has been true for both small sample of developed economies and
large, global sample. In fact, for the developed economies, researchers have often reported
unconditional convergence. Similarly, once it is remembered that σ -convergence research
generally focuses on unconditional convergence, it becomes clear that results regarding σ convergence largely agree with those regarding β-convergence. Evidence of σ -convergence is
2
found precisely in those small samples of developed economies for which there is also evidence
of unconditional β-convergence. On the other hand, in large global sample, neither unconditional
β-convergence nor σ -convergence are found to hold. Finally, time series analysis also has
produced evidence of conditional convergence, even though steady state variation considered in
this approach have generally been limited to time-invariant differences and trend breaks only.
Despite these agreements, at a more concrete level, convergence research has not
produced consensus estimates of the structural parameters of growth model. Two particular
parameters that have figured prominently in this regard are rate of convergence and elasticity of
output with respect to capital. Also, not all approaches to convergence research have been
equally concerned with values of structural parameters. From this point of view, convergence
studies can be classified into two broad groups. One group, comprised of the cross-section and
panel approaches, imposes structure on the data and produces estimates of the structural
parameters of growth model. The other group, comprised of the time series and distribution
approaches, tends to avoid structure and thus resembles reduced-form analysis of output data.
These latter approaches, therefore, do not produce structural parameter estimates and do not
answer questions concerning precise values of these parameters.
Given the differences in approach, sample, data, model, estimation technique, etc.,
consensus regarding the parameter values was, perhaps, not expected. Some generalities have,
nevertheless, emerged. It has been generally observed that the more differences in the steady
state of economies are controlled for (either by sample selection or by inclusion of relevant
variables in the regression), the higher is the resulting rate of convergence. An important
manifestation of this can be seen in the context of technological differences. It has been found
that much higher convergence rates result when the technology determinant of the steady state is
controlled for than when it is not. This also indicates that capital deepening and technological
diffusion, the two processes that jointly determine income-convergence, may not always play
symmetric role. Evidence indicates that while TFP-convergence has aided income-convergence
in small samples of developed economies, in large, global sample, this might not have been the
case. In the latter sample, there is strong evidence of capital deepening, but very large TFP
differences remain. However, TFP dynamics in the global sample are yet to be fully studied.
What are the implications of these convergence results? It is clear that the evidence of
conditional β -convergence does not necessarily mean that per capita income levels of all
3
countries are converging to the income level of the richer countries. Hence, from welfare point of
view, the implication of conditional convergence is rather hollow. In fact, conditional
convergence does not imply that the income levels of the countries are converging to any one
particular income level. A parallel of this can be seen in the fact that a conditional negative β
does not necessarily imply a decreasing σ . But, it is also true that an increasing σ does not
preclude the possibility of a negative β . This indicates the necessity of going beyond evidence of
conditional β -convergence and examining the evolution of the distribution as a whole and intradistribution dynamics. The distribution approach to convergence emphasizes these latter issues
and produces some important results. However, a fuller understanding of these results requires a
change in the focus of growth and convergence research.
What are the implications of the convergence results for the growth theory debate? The
most important development in this regard has been the introduction of the concept of
conditional convergence and establishment of the fact that convergence implication of NCGT is,
at best, conditional on differences in steady state and not absolute. This has helped reconcile
cross-country growth data with NCGT, and we have the overall result that NCGT cannot be
rejected on the basis of evidence from convergence research.1 This does not mean that the
conflict between NCGT and NGT has been fully resolved. What this means is that the test
between NCGT and NGT has to move on to other grounds and/or formulated differently than
just in terms of convergence.2 One difficulty in this regard is that, unlike NCGT, there is no
‘consensus’NGT. Commonality of different variants of NGT lies mainly in their common
rejection of exogenous steady state growth of NCGT. However, they vary widely with regard to
the source and workings of alternative, endogenous growth. It is therefore not surprising that
convergence discussion has revolved more around the question whether or not NCGT is rejected
than the question whether or not NGT is vindicated. Positive tests of NGT have been few and far
between.
However, the concept of conditional convergence has had two important consequences.
First, it has worked toward making NCGT and NGT observationally equivalent. This is
1
This has led to Barro’s recent remark: “It is surely an irony that one of the lasting contributions of endogenous growth theory is
that it stimulated empirical work that demonstrated the explanatory power of the neoclassical growth model.” (Barro 1997, p.
x)
2
For examples of such tests, see Jones (1995a, 1995b).
4
particularly true when the concept of conditional convergence is pushed to the extent that
countries can have not only different steady state level but also different steady state growth rate.
Convergence then becomes virtually an empty construct, and equilibrium data cannot effectively
discriminate between NCGT and NGT. This problem of observational equivalence has been
aggravated from the other end by the fact that, partly in response to the empirical evidence
supporting convergence, variants of NGT have now been put forward that themselves have
convergence implication. 3
The second consequence of the concept of conditional convergence has been that, to a
certain extent, it has diverted attention away from important issues of growth. This is because it
focuses on growth after controlling for the determinants of steady state and, thus, abstracts from
the important issue of the determination of the determinants. Yet, even from the NCGT
perspective, the long run income level of an economy depends on these determinants, and
improvement in them not only implies higher steady state level but also induces transitional
growth. If, in addition, the steady state growth rates are also allowed to be country-specific, then
there remains little useful that conditional convergence is worth. In view of this emptiness, it is
perhaps somewhat unfortunate that more research is devoted to correctly estimating the rate of
conditional convergence than to analyzing the steady state determinants.
This relative lack of attention to the determinants of steady state is also hindering fuller
understanding of the results produced by the distribution approach to convergence. Two main
results of this line of research are, first, that the cross-section distribution in large sample of
countries is not collapsing, and, second, that this distribution is becoming increasingly bi-modal.
These results are not incompatible with conditional β -convergence provided it is kept in mind
that steady state determinants can not only vary cross-sectionally but also change over time.
More research directed to the dynamics of these determinants (among which are investment,
fertility, technology, and institutions) can enable us to understand the underlying causes of
increasing bi-modality. This may also help in making current growth and convergence research
more practically relevant.
In understanding the dynamics of the steady state determinants, it is important to pay due
attention to the fact of interdependence of the economies of the world. One nagging problem
with convergence study is that models used to study it are geared to describe dynamics within an
3
For example, see Jones and Manuelli (1990).
5
economy. Yet convergence that we are more interested in is essentially an across-economy
process. The theoretical transition from this within-economy model to one that describes growth
in an interconnected world is yet to be fully made. Such a construct has to address crosseconomy processes concerning both factor accumulation and technological diffusion. While
international trade theory provides some perspectives on the former, there is lacking regarding
the latter. Some important beginnings have been made;4 however, much more needs to be done.
Also, theoretical advances that have been made in this regard are yet to be distilled into testable
empirical hypotheses. Thus, both theoretical and empirical works, focussed directly on the crosscountry processes, have become another important priority.
The discussion of this paper is organized as follows. In section-2, we discuss the link
between growth theory and the issue of convergence. Section-3 provides an introductory
description of different concepts of convergence. Section-4 reviews the initial evidence on
convergence based in informal cross-section regressions. The formal model-based equation that
has become the mainstay of convergence research is presented in section-5. Section-6 reviews
the cross-section results based on formal specifications. Section-7 discusses the panel approach
to convergence study. Time series approach to convergence analysis is reviewed in Section-8.
Section-9 discusses the distribution approach to convergence. Conclusions are drawn in section10. Most sections are provided with a summary at the end to help reader follow the discussion.
The literature on convergence is too vast to make an all-inclusive survey possible.
Accordingly, many works remain outside of the review here, which does not mean that these are
not important. Also, this paper is not meant to be a review of the entire new literature of growth.
It considers only that part of this literature, which focuses on convergence and its relationship
with growth theory.
2. Growth Theory and the Issue of Convergence
For a long time, the neoclassical growth theory (NCGT) has been the main paradigm for
discussion of economic growth. However, in the mid-eighties, two interrelated dissatisfactions
arose, both of which can be traced back to the NCGT-assumption of diminishing returns to
capital in the particular form of the Inada conditions. The first of these concerns the source of
4
An important recent contribution in this regard is by Ventura (1997). Other contributions in this direction include Barro,
Mankiw, and Sala-i-Martin (1995) and Barro and Sala-i-Martin (1997).
6
steady state growth. Because of Inada-type diminishing returns, steady state growth in NCGT
has to come from ‘outside,’ in the form of exogenous technological progress, for which there is
no within-model explanation. This specification of steady state growth, which Solow had
conceived as a short cut to capturing a more complicated process,5 no longer proved
satisfactory.6
The second dissatisfaction concerns NCGT’s ability to explain cross-country regularities of
growth. The specification of technological progress in NCGT is based on the following
assumptions: (a) no resources are needed to generate technological innovation, (b) everybody
equally benefits from it, and (c) nobody pays any compensation for benefiting from it. When
extended to a global setting, these assumptions lead to a convergence implication. If all
economies can share in technological progress equally, then they all, sooner or later, should grow
at a common rate given by the rate of exogenous technological progress. This gives a hypothesis
of convergence in terms of growth rate. However, a hypothesis of convergence in terms of per
capita income level is also ascribed to NCGT based on the following reasoning. Diminishing
returns imply higher marginal productivity of capital in a capital-poor country. Hence, with
similar savings rates, their economies will grow faster and eventually catch up with the richer
economies in terms of per capita income. In cross-country data, therefore, there should be a
negative correlation between the initial level of income and the subsequent growth rate.7
However, initial look at the Summers and Heston (1988, 1991) data set led to the claim that, for
large samples, the convergence hypothesis did not hold.8 This alleged non-conformity became
the second dissatisfaction with NCGT.
As Romer (1994) explained, these two dissatisfactions were also the two origins of NGT, and
they also influenced the initial course that NGT took. To the extent that the convergence
implication of NCGT is rooted in the assumption of diminishing returns to capital, NGT tried to
avoid convergence implication by moving away from this assumption. This is most evident in
the Ak-version of NGT, which opts for a straightforward replacement of diminishing returns by
5
See Solow (1994, 1997).
6
Actually, this dissatisfaction is not new. Development economists were long unhappy about the property of NCGT whereby the
long run growth rate was exogenous and could not be influenced by policies. This found expression in the contention of policyirrelevance against NCGT.
7
Note that this negative correlation is necessary for convergence in terms of both growth rate and per capita income.
8
See, for example, Baumol (1986) and Romer (1989a, 1989b). We shall consider this evidence in more detail shortly.
7
constant returns. In other initial versions of NGT, e.g., Romer (1986) and Lucas (1988), similar
goal is accomplished in more roundabout ways. The generic implication of these versions of
NGT is that economies starting out with lower levels of per capita capital stock have no inherent
reason to experience higher growth rate, hence no reason for convergence.
Convergence thus became a proving ground for testing NCGT versus NGT. It became a test
for diminishing returns to capital. In the Cobb-Douglas case, returns to capital is determined by
capital’s exponent. Convergence, therefore, became an issue of correct estimation of this
parameter. Empirical estimate of this elasticity parameter indicates whether private return from
capital differs from its social or aggregate return. To that extent that many of the initial variants
of NGT rely on externality, while NCGT does not, evidence of wedge between capital’s private
and social return has direct bearing on the growth theory debate. All these questions coalesced in
the debate over convergence, and it is because of these various ramifications that the
convergence debate has been raging so forcefully for such a long time.
3. Different Concepts of Convergence
A. Convergence Within vs. Convergence Across
Robert Solow (1970), in his exposition of growth theory, starts out by relating to the six
stylized facts about growth that were put forwarded by Kaldor (1971). Coming to the fifth and
sixth of these,9he pauses and makes the following comment:
“The remaining ‘stylized facts’are of a different kind, and will concern me less, because they
relate more to comparisons between different economies than to the course of events within any
one economy.” (p. 3; my italic)
It is somewhat ironic that one of the recent dissatisfactions with Solow model has been its
alleged failure to explain across- or between-country variation in growth rate and income level.
Historically, the main objective of the Solow model was to show that once factor substitution
was allowed, the economy could achieve stable dynamic equilibrium instead of suffering from
inherent instability that characterized previous growth models by Harrod and Domar. In NCGT,
no matter whether the economy starts off from a per capita capital stock that is lower or higher
8
than the equilibrium, the forces of the economy lead it to the equilibrium. Hence, this is indeed a
proposition of convergence, albeit within economy. Paradoxically, the concept of convergence
that arose and became associated with NCGT, was in the across-economy sense.
B. Convergence in Terms of Growth Rate vs. Convergence in Terms of Income
Level.
These two variants of the convergence hypothesis have already been discussed in section 2.
The hypothesis of convergence in terms of growth rate follows directly from an extension of the
NCGT-assumptions regarding technological progress to a global setting, and no additional
assumption is required. In contrast, the hypothesis of convergence in terms of per capita income
level requires additional assumptions, which are not innocuous. The distinction between
unconditional and conditional convergence is centered on these assumptions.
C. ß-Convergence vs. s -Convergence
The common methodology of investigating convergence in its across-economy sense has
been to run cross-country regression with subsequent growth as the dependent variable and
initial level as the explanatory variable.10 This set up is known as the growth-initial level
regression.11 The hypothesized negative correlation between initial level and subsequent growth
is supposed to be picked up by the coefficient (say, ß) of the initial income variable.
Convergence judged by the sign of ß came to be known as ß-convergence.12
However, researchers like Quah (1993a), Friedman (1994), and others have noted that
convergence is a proposition regarding the dispersion of the cross-sectional distribution of
income, and a negative ß from growth-initial level regression does not imply a reduction in this
dispersion. They point out that negative ß can be another example of the more general
phenomenon of reversion to mean, and, by reading convergence in it, growth researchers are
falling into Galton fallacy. According to this view, instead of judging indirectly and perhaps
9
The fifth of these stylized facts was that the growth rate of per capita output varied widely across countries, and the sixth, that
economies with high share of profits in income had higher investment to output ratios.
10
Either alone or along with other right hand side variables.
11
Sometimes these are also called Barro-regressions, referring to Barro (1991).
12
In this paper we shall use ß as the generic notation for the coefficient on the initial level variable in the growth-initial level
regressions. Note that negative ß can be interpreted as evidence of convergence in terms of both income level and growth rate.
9
erroneously, through the sign of ß, convergence should be judged directly by looking at the
dynamics of dispersion of income level and/or growth rate across countries. This gave rise to the
concept of s -convergence, s being the standard deviation of the corresponding distribution.13
However, despite the limitations above, researchers have continued to be interested in ßconvergence, in part, because it is a necessary condition of s -convergence, though not sufficient.
The other reason for continued interest in ß-convergence is that it provides answers regarding
structural parameters of growth models. In contrast, convergence research along the distribution
approach generally avoids structure and tends towards reduced form analysis.
D. Unconditional Convergence vs. Conditional Convergence
Proceeding from the Solow model and assuming a Cobb-Douglas production function of the
type Yt = K tα ( At Lt )1− α , the steady state level of per capita income, y ∗ , is given by
(1)
y ∗ = A0 e gt [s /(n + g + δ)]α /(1− α ) ,
where s is the saving rate, g and n are the assumed exponential growth rates of At and Lt ,
respectively.14 This shows clearly that the steady state income level of a country depends on the
vector θ, which has the following six elements, ( A0 , g , s, n, δ,α ).15 Unconditional convergence,
which is based on the assumption of a common steady state, implies that all elements of θ are
same across the economies considered. In terms of the growth-initial level regression, it therefore
implies that the sign of ß should be negative even if no other variable is included on the right
hand side. In contrast, the concept of conditional convergence emphasizes the possible
differences in steady state and hence requires that appropriate variables should be included on
the right hand side of the growth-initial level regression to proxy for these differences. However,
which of the different elements of θ should be allowed to vary and which not, is an important
issue in convergence research, as we shall see.
13
The expressions ß-convergence and s -convergence were first coined by Sala-i-Martin.
14
Other notations are standard: Yt is the income or output, K t and Lt are capital and labor inputs respectively, and At is the
shift parameter.
15
In the case of the Cass- Koopmans model, θ also has similar set of elements with s replaced by parameters for the rate of time
preference and the inter-temporal elasticity of substitution
10
E. Conditional Convergence vs. Club Convergence
The concept of club convergence can be traced back to Durlauf and Johnson (1995).
Recently Galor (1996) has provided a more explicit formulation of the concept. One property of
the standard NCGT is that the equilibrium is unique, and the usual notion of convergence
assumes this uniqueness. In case of unconditional convergence, there is only one equilibriumlevel to which all economies approach. In case of conditional convergence, equilibrium differs
by economy, and each particular economy approaches its own but unique equilibrium.
In contrast, the idea of club-convergence is based on models that are characterized by the
possibility of multiple equilibria.16 Which of these different equilibria an economy will be
reaching, depends on its initial position.17 This produces club-convergence: 18 different
groups/clubs approach different equilibrium depending on the common initial location (or some
other attribute) they share. Observationally, therefore, the notion of club-convergence can be
thought as an intermediate concept between conditional and unconditional convergence. This
may be illustrated using the vector θ above. Absolute convergence assumes that values of all six
of its elements are same across economies. Conditional convergence, on the other hand, can
allow values of all these elements to differ. Club convergence emphasizes the difference in
initial condition, say A0 .19 However, club convergence does not imply that each different value
of initial condition yields a different equilibrium. Rather, the idea is that there are ranges of
values of the initial condition that correspond to different equilibrium.
F. Income-convergence vs. TFP-convergence
Convergence research has generally dealt with convergence in terms of per capita
income, i.e., income convergence. However, income convergence can be the joint outcome of the
twin processes of capital deepening and technological catch-up. While most researchers have
16
For models with multiple equilibria, see, for example, Azariadis and Drazen (1990).
17
And, perhaps, some other attributes, though it is the difference in initial conditions that is emphasized.
18
Note that this use of the term ‘club’is different from the way it was used by Baumol (1986) when he coined the expression
‘convergence club.’In Baumol’s usage, economies had unique equilibrium, and ‘convergence club’denoted groups of similar
economies, and hence the same equilibrium.
19
However, definition of initial conditions may not be limited to
A0 .
11
focused on the parameters of capital deepening process, other researchers, like Dowrick and
Nguyen (1989), Dougherty and Jorgenson (1996, 1997), Wolff (1991), and Dollar and Wolff
(1994) have directed their attention to the process of technological catch-up. Since, total factor
productivity (TFP) is the closest measure of technology, these researchers have investigated
whether over time countries have come closer in terms of TFP levels. This has given rise to the
concept of TFP-convergence. Clearly, income convergence can get either accelerated or
thwarted depending on whether the initial TFP-differences narrow or widen over time. Note that
TFP-convergence can either be an independent query or a subsequent part of the general query
into income convergence.
G. Deterministic Convergence vs. Stochastic Convergence
The concept of stochastic convergence has been put forward by Bernard and Durlauf (1996),
Carlino and Mills (1993), Evans (1996), Evans and Karras (1996a), and others. The idea may be
expressed as follows. Two economies, i and j, are said to converge if the (per capita) output for
them, y i ,t and y j ,t satisfy the following condition:
(2)
lim k → ∞ E ( y i ,t + k − a ⋅y j , t + k | I t ) = 0 .
Note that with a = 1, it is a variant of the notion of unconditional convergence. This definition of
stochastic convergence is relatively unambiguous for a two-economy situation. This is not so
when convergence is considered in a sample consisting of more than two economies.
Researchers have differed in this regard. Some have taken deviations from a reference economy
as the measure of convergence in a multi-country situation. In this treatment, y it in equation (2)
is replaced by y1t , where 1 is the index for the reference country. Others have based their
analysis of convergence on deviations from the sample average. In this treatment, y it is replaced
by y t , the average for time t.
The notion of stochastic convergence is related with β-convergence. From the point of
view of the above formulation, β-convergence test whether, proceeding from y it > y jt , we have
(3)
lim k → ∞ E ( y i , t + k − y j , t + k | I t ) < y it − y jt .
12
Countries i and j converge between dates t and (t + T) if there is the tendency of output
differences to narrow over time, as formalized above. It is clear that β-convergence, in its acrosseconomy sense, is a precondition for stochastic convergence as defined above. The relationship
between the time series approach and β-convergence will be formulated more directly in section7. However, the formulations above already show that time series analysis can be used to
examine conditional convergence as well. For example, if a is different from 1 in equation (2),
we effectively have a situation of differing steady states. Thus, presence of an intercept term in
the time series analysis of deviations, as above, is equivalent to allowing time-invariant
difference in the steady states of the economies.
The above gives a brief introduction to the different concepts of convergence. We now move
on to the discussion of empirical evidence. Along the way, we will consider the econometric
formulations that have been used to investigate these different concepts of convergence. We
begin by considering evidence on β-convergence.
4. Initial Stage of Convergence Study
In the initial studies of convergence, the regression specifications were not formally derived
from theoretical models of growth. This does not mean that these studies did not have connection
with growth models. In fact, to the extent that this connection was less formal, some of these
works derived inspiration from several theoretical paradigms and, therefore, had multiple focus.
Although the concepts of unconditional and conditional convergence were not rigorously
formulated and distinguished yet, works of the initial stage produced evidence that, with
hindsight, can be attributed to both these notions of convergence.
A. Initial Evidence of Unconditional Convergence
Unconditional convergence was investigated in studies like Baumol (1986). The main part of
his analysis was based on a sample of 16 OECD countries for which long term data were
13
available from Maddison (1982). Baumol obtained a significant negative coefficient20 on the
initial income variable in a growth-initial level regression for these countries, which he took as
strong evidence of (unconditional) convergence. However, prodded by Romer, Baumol also
considered the relationship in an extended sample of 72 countries. In this larger sample,
however, he could not find evidence of convergence.21 Thus, Baumol’s study produced evidence
of both presence and absence of unconditional convergence, depending on the sample. He also
coined the expression ‘convergence-club.’Introspecting on the basis of the growth-initial level
scatter, he suggested that, while there was no convergence in the larger sample as a whole, there
existed ‘convergence clubs,’within which evidence of convergence could be seen.22 DeLong
(1988) later showed that Baumol’s finding of unconditional convergence in the 16-country
OECD sample suffered from selection bias.23 Nevertheless, Baumol’s finding of absence of
unconditional convergence in the larger sample of countries became an important point of
departure for further discussion of convergence.
B. Initial Evidence of Conditional Convergence
While Baumol’s study focussed on unconditional convergence, other studies, like Kormendi
and Meguire (1985) and Grier and Tullock (1989), provided evidence that can be interpreted as
of conditional convergence. Reflecting research interests of an earlier period, these studies were
also motivated by other issues, like inflation-output trade off, Philips curve relationship, etc.
Accordingly, the regressions in these works included variables representing these other
relationships. However, the basic neoclassical paradigm was preserved through the inclusion of
labor and (at least, in some of the regressions) capital variables and also of initial real per capita
GDP. In a sample of about fifty countries, Kormendi and Meguire’s regressions yielded negative
20
Note that the numerical magnitudes of ß from different studies are not directly comparable because of the differences in
regression specification.
21
Numerical results of this regression were not presented, but Baumol reported that it yielded ‘slightly positive slope,’indicating
a process of rather divergence.
22
One example was the OECD group, already considered. Another example, according to him, was the group of formerly
centrally planned countries. According to Baumol, such clubs consisted of countries which had certain degree of homogeneity
in ‘product mix and education’enabling them to share in the ‘public good properties of the innovations and investments of
other nations.’(p. 1080)
23
The proper criterion for sample selection for convergence study, DeLong pointed out, is ex-ante income level, and not ex-post.
In particular, he showed that, if, guided by the ex-ante criterion, Baumol’s OECD sample was modified slightly, the result of
unconditional convergence no longer held. Baumol largely accepted this criticism. See Baumol and Wolff (1988).
14
β,24 which could be taken as evidence of conditional convergence. Grier and Tullock (1989)
extended Kormendi and Meguire’s study, with larger sample size and longer sample period. This
allowed them to consider the issue of parameter stability across sub-samples and sub-periods.25
For the OECD sub-sample, they found the coefficient on the initial income variable to be
negative. For the larger ROW sub-sample, this coefficient turned out to be positive. However,
upon splitting ROW into (three) smaller samples, they found the sign of β to vary.
C. Initial Evidence on Convergence and the Growth Theory controversy
How did the initial studies of convergence relate their results to the growth theory
controversy? Some of these, for example, Baumol (1986) and Kormendi and Meguire (1985),
either pre-dated or were contemporaneous to the publication of pioneering NGT papers, and
hence, growth theory controversy was not very prominent for them. Those studies that were
subsequent to the advent of NGT did try to relate their results to growth theory dispute.
However, to the extent that their regression specifications were not formally linked with the
growth models, such efforts had to be limited.26 Thus, Grier and Tullock (1989), for example,
could make only the broad observation that their results were generally supportive of NCGT, and
that these results might show directions for further development of NGT.
However, from another point of view, absence of formal link between regression
specification and growth model saved these initial studies from the within-across tension of the
convergence concept. The interpretation of a negative initial income coefficient in these studies
could be entirely cross-sectional. It was taken as evidence showing that, other things held
constant, countries starting with lower levels of income grew faster.27 When specifications are
formally linked to growth models, it is not possible to limit the interpretation to such pure crosssectional and reduced form.
D. Convergence and Human Capital
24
Its value ranged from -0.0055 to -0.0091 in different specifications.
25
Their sample consisted of 113 countries, and they split it into an OECD (24 countries) and a Rest of the World (ROW,
comprised of the remaining 89 countries) samples. Time period considered was 1950-81.
26
Grier and Tullock (1989) were themselves quite keen about this limitation. For their own comment on this point, see (p. 260).
27
See for example of such interpretation Kormendi and Meguire (1985, p. 147)
15
In the studies of conditional convergence discussed above, variables are considered in
addition to those suggested by NCGT, and human capital does not figure among them. Yet we
know that, in order to relax the constraint of diminishing returns, NGT primarily relies on human
capital. It was, therefore, natural that human capital would soon appear in the empirical work on
growth and convergence. This starts with Barro (1991)’s seminal study, which is directly
inspired by the NCGT-NGT controversy and tries to look at the convergence issue from the
perspective of NGT.28
Accordingly, Barro starts by abandoning the standard neoclassical format and, instead,
emphasizes the simultaneity among growth, investment, and fertility.29 In view of this
simultaneity, he runs separate sets of regressions with growth, investment, and fertility as
dependent variables.30 Barro’s perspective is reflected in the fact that his basic regressions of
growth do not include physical capital and labor as explanatory variables. Instead, the variable of
focus is human capital, which now appears in all the regressions.31 These regressions show
human capital to be generally important, and Barro interprets this as vindication of the NGTemphasis on human capital.
To study convergence, Barro includes the initial income variable in his regressions. He starts
by reporting absence of unconditional convergence in a broad sample of 98 countries,32 and
interprets this as supportive of the NGT.33 However, he finds that when the initial measures of
28
Also, Barro was not an outsider to this debate. He embarked on this empirical work after already making his own contribution
to the development of NGT. In Barro (1990) he examined the role of government spending in the setting of an Ak-style model
of endogenous growth. Becker and Barro (1988) and Barro and Becker (1989) addressed the issue of fertility choice. These
works and his close association with the genesis of NGT, gave Barro quite a few propositions to test out, and he was quite
explicit that he was doing so ‘using recent theories of economic growth as a guide.’(p. 437)
29
Barro argues for this simultaneity on the basis of new theories of growth. In particular, he cites extensively the conclusions of
Rebelo (1991), Barro (1990), Romer (1990), Barro and Becker (1989) and Becker, Murphy and Tamura (1990).
30
Barro’s work is, thus, not limited to the convergence issue; instead it addresses a host of other issues related to growth. It,
therefore, to some extent, shares the characteristic of multiple focus that we mentioned earlier.
31
Other right hand side variables of Barro’s regressions include government consumption (reflecting his earlier interest in it), an
index of relative inflation, indicators of political stability and some regional and continent dummies. Although the inflation
variable is interpreted as measure of ‘market distortion,’its inclusion can as well be linked to the output-inflation literature, in
which Barro himself played a no small role. See for example, Barro (1976), (1977) and (1978). In fact, Barro has returned to
this issue, as can be seen in Barro (1997).
32
The correlation between the average growth rate of per capita real gross domestic product between 1960 and 1925 (GR6085)
and the 1960 value of real per capita GDP (GDP60) is reported to be positive 0.09.
33
“This finding accords with recent models, such as Lucas (1988) and Rebelo (1991), that assume constant returns to a broad
concept of reproducible capital, which includes human capital. In these models the growth rate of per capita product is
independent of the starting level of per capita product.” (p. 408)
16
human capital are included, β turns negative and significant.34 This negative β leads Barro to
conclude that the data support the convergence hypothesis in a “modified sense.”35 Inclusion of
human capital causes β to be negative in the regressions with investment as dependent variable
as well, and Barro interprets this too as ‘consistent with the convergence implication of the
neoclassical growth model.’36 To the extent that Barro’s growth regressions include control
variables other than initial income, his convergence in a ‘modified sense’can be interpreted as
conditional convergence. However, since investment and labor force growth rates do not appear
as control variables,37 which are the main conditioning variables from NCGT-point of view,
there may be some ambiguity regarding this interpretation. Towards the end of the paper,
however, Barro presents growth-initial level regression of more conventional (meaning,
neoclassical) format, i.e., inclusive of investment and population growth rates as controls.38 He
finds that βˆ obtained from this regression is also negative though little different in magnitude 39
from that obtained from earlier set of growth regressions which do not include investment and
fertility rates as controls.40
The evidence produced by the convergence studies with informal specifications can,
therefore, be summarized as follows: (a) absence of unconditional convergence in larger sample
of countries, (b) presence of absolute convergence in selected sample of countries,41 and (c)
presence of conditional convergence even in larger sample of countries.42 In addition, Barro’s
34
The numerical value of β in Barro’s growth regressions ranges from -0.0062 to -0.0111, with that for the base regression (no.
1) being -0.0075.
35
(Barro 1991, p. 409, our italics). We can see the concept of conditional convergence emerging here.
36
(Barro 1991, p. 427). With the ratio of private investment to GDP as the dependent variable, the coefficient on the initial
income is significant and ranges between -.0093 and -.0098.
37
Which, however, appear as the main controls in other pre-model studies of conditional convergence, as we saw to be the case
with Kormendi and Meguire (1985) or Grier and Tullock (1989).
38
The argumentation for the specification is, however, not neo-classical. Instead, Barro observes that NGT relationships among
growth, investment, and fertility imply that residuals from the growth regression will be positively related with those from
investment regression, and negatively related with residuals from fertility regression. He justifies the conventional format of
the growth regression as an alternative way of checking whether the stipulated relationships among the residuals are true.
39
It is again significant and ranges from -0.0068 to -0.0077, with the base case being -0.0077.
40
Barro interprets this as showing that the negative effect of the initial income on growth does not work through its effects on
investment or fertility. Instead, it works mainly through lower rate of return on investment. (p. 430)
41
The selection may be flawed, however.
42
Evidence on this point was yet ambiguous.
17
work established the importance of the role of human capital as a conditioning variable.43
However, the formal concept of conditional convergence did not arise yet, and, hence, there was
no attempt in these studies to portray the controls as determinants of steady state.44 The
interpretation of the initial income coefficient was limited to what is standard for a reduced form
regression coefficient, and the within-across tension did not arise. Since, the specification were
not formally linked with growth models, there was no attempt to recover structural parameters
from the estimated coefficients. For these developments, we had to wait till next stage of the
convergence research.
5. Neoclassical Equation for Convergence Study
The evolution of the concept of conditional convergence from its inchoate to precise stage
and the associated transition of the convergence regression from informal to formal model-based
specification are accomplished in Barro and Sala-i-Martin (1992), (henceforth BS) and Mankiw,
Romer, and Weil (1992) (henceforth MRW). In both these works, the regression specification is
derived formally from the neoclassical growth model. MRW work with the original Solow
model, while BS use the Cass-Koopmans’optimal savings version of the NCGT. Since the
appearance of these studies, the neoclassical convergence equation has occupied the center stage
of convergence research, and it is virtually impossible to review the literature without bringing
this equation into picture. It is also needed to introduce the necessary notations.45 The exercise
involves first the derivation of the law of motion around the steady state and then translation of
this motion into an estimable regression equation.
A. Deriving the Rate of Convergence
43
This enthusiasm regarding human capital was, however, not shared by all. Just about the same time as Barro’s work appeared,
Summers and DeLong (1991) produced work emphasizing the role of equipment investment in explaining growth. In addition
to showing strong positive effect on growth, they also found that this effect does not depend on ‘education infrastructure.’
Inclusion of human capital variables, which appeared in Barro’s regression, did not affect the coefficient of the equipment
investment variable. Finally, they also thought that the regression results showed presence of positive externalities arising out
of equipment investment. In Summers and DeLong (1993), they showed that the above results held for the developing
countries as well.
44
This was also evident from the fact that, in Barro’s regressions, human capital was included in the form of an indicator of
initial condition.
45
The text by Barro and Sala-i-Martin (1995) and Mankiw (1995) also provide the derivation.
18
The dynamics of capital in Solow model is given by
&
kˆ = s f (kˆ) − (n + g + δ)kˆ,
(4)
K
&
where, kˆ =
, or capital per effective labor, kˆ is the time derivative of kˆ, and f (kˆ) is the
AL
production function normalized in terms of effective labor. Also, (s, n, g, d) are the rates of
saving, population growth, technological progress and depreciation, respectively. First order
Taylor expansion of the right hand side term around the steady state gives
&
kˆ = [ s f ′
(kˆ∗ ) − (n + g + δ)] (kˆ− kˆ∗ ) .
(5)
Substituting for s using the steady state relationship, s f (kˆ∗ ) = (n + g + δ) kˆ∗ , gives
&
kˆ = ([ f ′
(kˆ∗ ) kˆ∗ / f (kˆ∗ )] − 1)(n + g + δ)(kˆ− kˆ∗ ).
(6)
Under the assumption that capital earns its marginal product, f ′
(kˆ∗ ) kˆ∗ / f (kˆ∗ ) equals to steady
state share of capital in income, a. In Cobb-Douglas case, this will also be the exponent of
capital in the production function.
Using this relationship we then get
&
kˆ = λ(kˆ∗ − kˆ) ,
(7)
(8)
where,
λ = (1 − α )(n + g + δ) .
Evidently, ? gives the speed at which the gap between the steady state level of capital and its
current level is closed and has come to be known in the literature as the rate of convergence. The
same rate holds for convergence in terms of income per effective labor. This is because
yˆ = f (kˆ), which upon expansion at kˆ∗ and differentiation with respective to time gives
(9)
&
y&ˆ = f ′
(kˆ∗ ) kˆ.
As a first order approximation, we should therefore have,
(10)
yˆ∗ − yˆ = f ′
(kˆ∗ ) (kˆ∗ − kˆ) .
19
By substitution we then get,
& = λ( yˆ∗ − yˆ) ,
yˆ
(11)
where ? is again the rate of convergence given by equation (8).
B. Deriving the Equation for Testing Convergence
Switching to the logarithms, solving this first order non-homogeneous differential equation,
and rearranging we get from (11),46
(12)
ln yˆ(t 2 ) − ln yˆ(t1 ) = (1 − e − λτ )(ln yˆ∗ (t1 ) − ln yˆ(t1 )) ,
where t1 denote the initial period, t 2 the subsequent period, and τ = (t 2 − t1 ) .47 If we now
substitute for yˆ∗ from equation (1) above, we get
α
− λτ α
− λτ
ln( s t ) − (1 − e
)
ln( nt + g + δ) − (1 − e
) ln yˆ(t1 )
(13) ln yˆ(t 2 ) − ln yˆ(t1 ) = (1 − e − λτ )
1− α
1− α
1
1
Clearly, this is again the growth-initial level equation, but now the coefficients are
formally linked with the structural parameter of the NCGT. For example, β = − (1 − e − λτ ) , and
hence it is now possible to recover the value of ? from estimate of ß. The value of ?, in
conjunction with other estimated coefficients of the equation, yields the values of other structural
parameters of the model like a. The question that now arises is how to estimate this equation.
46
47
This derivation is most detailed in Barro and Sala-i-Martin (1995, pp. 87-88)
∗
A finer point here concerns whether to make ln yˆ contingent on t1 or not. If it is assumed that the determinants of the steady
∗
state income remain constant between t1 and t 2 , it does not matter whether ln yˆ is made contingent on t1 or not.
Generally, in implementation, it is assumed that such determinants of steady state income, such as s, n, d, and a, remain the
same between t1 and t 2 . This suggests that it is not necessary to make steady state income contingent on the initial period in
considering transitional dynamics. This is certainly the case in the continuous time setting, when, theoretically, the difference
between t1 and t 2 is instantaneous. Under the assumption that s, n, d, and a, remain constant over the transition period
considered, this is also true if we are dealing with steady state income per effective labor, as in the equation (13) above.
However, when dealing with steady state income per capita (as is usually the case), this would not be true. As we can see from
gt
equation (1), the formula for steady state income per capita has the term A e , which will differ if evaluated at t 2 instead
0
of at t1 . In the logarithmic version of the equation, this will generally imply differences in terms involving g and will not
affect the basic conclusions. However, it is worth to be aware about.
20
C. Tension between Within and Across Dimensions of Convergence
Note that the above derivation of ? and equation (13) is entirely on the basis of the
accumulation process within an economy, and there is no reference to what is happening across
economies. This shows that ? essentially refers to a within-economy process and is determined
by the values of a, n, g, d of the economy concerned. It would, therefore, seem natural and
proper to estimate equation (13) and ? on the basis of time series data for a particular country.
However, as we saw, researchers have been, instead, estimating equation (13) using
cross-section data. The main reason for this is that, from the beginning, convergence arose as a
concept pertaining to across-economy growth process. The question that resides in the mind of
convergence researchers is not so much whether an individual country is closing the gap between
its own current and steady state levels of income as whether poorer countries are narrowing their
gap with the richer countries. From this conceptual point of view, cross section data is the natural
place to look for presence or absence of convergence.
But, this introduces a tension in the convergence parameter ?. While according to
equation (13), ? is the measure of speed at which an economy proceeds towards its own steady
state level, the ? estimated from a cross-section equation is generally interpreted as the speed at
which poorer economies are closing the income gap with the richer countries. This tension was
not apparent so long as the cross-country regression specifications were informal. In those
regressions, it was possible to limit to only the reduced form interpretation of ß, though this also
meant that no information regarding the structural parameters could be obtained. Thus, while
formal derivation of the neoclassical convergence equation has been a significant step forward,
this has at the same time brought to fore the within-across tension in the concept of convergence
and, correspondingly, in the parameter ?. We shall see some concrete manifestation of this
tension in the estimation results below.
6. Cross-section Approach to Convergence
A. Cross-section Estimation of the Neoclassical Convergence Equation
One of the most successful implementations of the formal cross section approach has
been MRW itself. In this seminal study, of the six elements of ?, two, namely s and n, are
assumed to differ across countries. The values of three of the rest, namely a, g, and d were
21
supposed to be the same for all countries. Differences in A0 are assumed to be part of the error
term, and this allowed estimation by OLS. MRW find that with only s and n as explanatory
variables, the regression ran well, but the implied values of a are too high.48
The accompanying result is that the implied values of the rate of convergence, ?, prove to
be too low. For NONOIL sample,49 for example, ? equals 0.00606, implying a half-life of 114
years, which is indeed very long. To overcome this problem, MRW suggest augmentation of the
Solow model by inclusion of human capital, in exactly the same way as physical capital, only
with a different exponent, say f . Regression on the basis of this augmented model produces more
desirable results. The value of a decreases to empirically plausible level (0.48 for the NONOIL
sample), and the value of ? increases to around .02 (0.0142 for the NONOIL, implying a half-life
of 49 years).
Similar results on conditional convergence across countries are also presented in BS.
These are mainly drawn from Barro (1991). However, the regressors are now clearly interpreted
as determinants of the steady state.50 And second, the structural parameter ? is now traced back
from the estimated regression coefficient. For a similar sample of 98 countries as MRW’s
NONOIL, BS report λ̂ to be 0.0184. However, although BS use the neoclassical convergence
equation to recover ?, they do not quite apply it to determine the right hand side variables of the
regression.
B. ß-Convergence across US States
48
MRW do not report the results of the restricted version of this regression, hence we do not have an unique estimate of a. Based
on the coefficient of the s variable its value would be 0.82, while that on (n + g + d), 0.68. Both of these are for the NONOIL
sample and are far greater than the share of capital in the national income in these countries, computed on the basis of the
national accounts data. Note that these values agree with the estimates produced earlier by Romer (1989a) from a growth
accounting exercise under similar assumptions.
49
This is a sample of 98 countries which included almost all the sizable countries of the of the Summer-Heston data set except
those for which extraction of oil was the dominating source of income.
50
“We interpret these variables as proxies for the steady-state value of output per effective worker and the rate of technological
progress.” (p. 246). Also, note that BS uses identical term, ‘conditional convergence,’to interpret their equation, which is
analogous to (13) above. To quote: “The theoretical relation in equation (15) predicts conditional convergence, (our italics)
that is, negative relation between log y i , t and the subsequent growth rate if we hold constant the steady state position,
0
∗
log yˆi , and the steady state growth rate.” (p. 243)
22
Convergence study is not limited to samples of independent countries. Researchers also
addressed the issue in the context of regions of the same country. In particular, whether or not
convergence holds for the states of the US has been an attractive research topic.
The initial results in this regard are again presented by BS. Since the derivation of the
convergence equation in BS proceeds from the Cass-Koopmans version of NCGT, in
implementation, they ideally need to control for the ‘deep’ behavioral parameters. However, BS
make the assumption that steady states of the US states are the same, and this relieves them from
the difficult task of getting data on the deep parameters.51 This also means that BS consider
convergence across US states to be a situation of unconditional convergence. However, in some
of their regressions BS include regional dummies and a variable proxying for output
composition. Inclusion of these variables makes it a little ambiguous whether it is unconditional
or conditional convergence that is investigated. In any case, they find significant evidence of
convergence across the US states,52 and the estimated rate of convergence turns out to be in the
neighborhood of 2 percent per year.
In contrast with BS, Holtz-Eakin (1993) emphasizes the possible differences in steady state
among the US states and, thereby, considers the situation to be one of conditional convergence.
He basically uses a human capital augmented version of the neoclassical convergence equation
and implements a variant of pooled regression. Upon inclusion of variables that either represent
or proxy for the determinants of steady state, Holtz-Eakin obtains higher estimates of the rate of
convergence.
C. Manifestation of Within vs. Across Tension in Convergence Results
The within-across tension of the convergence concept and the parameter ? can be
numerically illustrated on the basis of the results of the cross-section studies discussed above.
Equation (8) clearly shows ? to be a function of a, g, d, and n. Thus, even if a, g, and d are
assumed to be the same, different n implies different value of ?. In most convergence studies, as
we saw above, n is allowed to vary across countries. Hence, we have a situation where
51
52
∗
“... we assume that the steady state value, yˆi , and the rate of technological progress do not differ across states.” (p. 227)
This was true in terms of both per capita income and product and for different time periods considered. BS left it as an
unresolved puzzle why this rate proved to be the similar in terms of income and product. Also, in some of the regressions, BS
included regional dummies and/or a variable proxying for output-composition. Inclusion of these variables changes the nature
of convergence, to some extent, from unconditional to conditional.
23
assumption of variation runs into conflict with the assumption of commonness within the same
parameter. One way of seeing this conflict is to use the estimated common values of a, f , and the
actual values of n (g and d are given) to produce country specific values of ?. The results from
such computation show that these values vary widely across countries included in the sample.
Another way of seeing this tension is to take the converse route and compute the implied value
of a common population growth rate from the derived value of ? using equation (8). For
example, if we do this on the basis of estimated values of ?, a, and f for NONOIL sample in the
restricted version of MRW’s augmented model,53 we get n equal to -0.001, 0.0064, and 0.0028
for NONOIL, INTER and OECD samples respectively.54 These are far from the representative
values of n for any of these samples.55 This situation is not unique with MRW. The problem is
rather generic with studies that use cross section dimension of data (therefore including panel
data sets) to study convergence.
D. Other Model Based Cross Section Studies of ß-Convergence
Since the formal derivation of the convergence equation by MRW and BS, it became quite
standard for empirical work on convergence to be based on this equation. In particular, the MRW
specification, based on the original Solow model, became the point of departure for many
studies. One reason for this is its easier implementibility, because it does not involve deep
behavioral parameters. A variety of issues have been explored using MRW-specification, even
within the cross-section set-up. For example, Chua (1992) uses it to study external economies
arising from regional spillovers. His exercise shows presence of regional spillover, mainly from
human capital. However, the spillover is not strong enough to support NGT. Sala-i-Martin
(1996b) uses the neoclassical specification to study convergence across the European economies.
Shioji (1995) conducts similar analysis of convergence across the Japanese prefectures. Other
researchers have used the formal cross-section approach to explore issues of measurement error,
53
These values are 0.0142, 0.48 and 0.23 respectively.
54
These are two other samples considered in MRW. INTER is the sample of 75 countries obtained by dropping 23 countries, for
which data are less reliable, from the NONOIL sample. OECD is the sample of 22 OECD member countries.
55
The negative n for the NONOIL sample is problematic. The other issue here concerns the correspondence of the theoretical and
empirical relationships between n and ?. Theoretically, higher n would yield higher ?, which is what we got when we imposed
this theoretical relationship in our reverse calculation above. But in actuality, n is highest on average for the NONOIL sample
and lowest for OECD. Yet the estimates implied value of ? turned out to be the smallest for the former and greatest for the
latter. Other researchers have also drawn attention to these problems. See, for example, Caseli et al. (1995) and Lee et al.
(1995).
24
outliers, etc. These studies produce a wide variety of numerical results; however, the basic
conclusion regarding conditional convergence is not refuted.
E. Research on Club-Convergence
Durlauf and Johnson (1995) use the neoclassical convergence equation to investigate ‘clubconvergence.’Alluding to theoretical models of multiple equilibria, they observe that
convergence in large samples of countries (global convergence) does hold or proves weak
because, in these samples, countries belonging to different equilibria (or ‘regimes’) are lumped
together. The proper thing, according to them, is to identify groups of countries, the members of
which share the same equilibrium, and to check whether convergence holds within the groups
(local convergence). With this motivation, they conduct two sets of exercises. In the first, the
countries are classified on the basis of arbitrarily chosen cut off levels of initial income and
literacy. To the extent that such exogenous splitting may create selection bias, Durlauf and
Johnson present a second exercise in which the splitting is endogenized using the regression-tree
method. In either case, the results prove to be qualitatively similar. Estimated parameter values
differ significantly across the groups, particularly in the case of endogenous splitting. Also, the
rates of convergence within the groups, in general, prove to be higher than in the whole sample.
Durlauf and Johnson interpret the observed parameter instability as indicative of countries
belonging to different regimes.56
The difficulty here lies in distinguishing evidence of club-convergence from that of
conditional convergence. This, in turn, is related with the criteria used to group the countries in
order to demonstrate club-convergence. Clearly, steady state determinants cannot be used for this
purpose, because differences in them cause equilibrium to differ even under conditional
convergence. Use of time-varying characteristics, like (initial) level of income or literacy also
involves problems.57 Thus, Durlauf and Johnson’s finding of faster convergence within groups
than in the broader sample is also compatible with conditional convergence.
56
Since this instability pertained to groups classified according to both initial income and human capital levels, the authors
concluded that both of these variables were important in identifying the ‘regimes.’
57
All the countries, at one point of time of their history or the other, have to pass through all cut-off points (of income or
literacy). Thus, if equilibria are contingent on the cut off levels, then all the countries should end up having the same
equilibrium. It may be said that the (initial) level in combination with some time-invariant characteristic does the job. Or,
perhaps, it is the ratio of levels that matters. This is because although the countries cross all the level values sooner or latter,
25
In sum, we see that the switch from informal to formal specifications elevated the
convergence discussion from one being about broad presence or absence of convergence to one
dealing with precise values of structural parameters of growth model. The cross-section studies
agree regarding the broad result of conditional convergence, but no consensus-value of ? or a are
obtained. However, we find that, in general, the value of ? increases as more differences in the
steady state are controlled for either by inclusion of relevant variables in the regression or by
selecting more ‘homogeneous’sample.
7. Panel Approach to Convergence Study
In studying convergence, the cross-section studies focussed on the steady state
differences in preference variables, like investment rate and labor force growth rate. However, as
equation (1) shows, the steady state is also characterized by technology. It is in accounting for
differences in technology that the cross-section approached encountered important limitation.
This gave rise to the panel approach to convergence study.
A. Omitted Variable Bias Problem of the Cross-section Regression
The limitation of the cross-section approach in controlling for the technology differences
creates an omitted variable bias problem. This can be illustrated using equation (13). Although
this equation is in terms of income per effective labor, in actual implementation, researchers
invariably work with income per capita. Expressed in terms of per capita income and
rearranging, we get
(14)
α
α
ln s t1 − (1 − e λτ )
ln( nt1 + g + δ) + e − λτ ln y t1
1− α
1− α
+ (1 − e − λτ ) ln A0 + g (t 2 − e − λτ t1 ).
ln y t 2 = (1 − e − λτ )
The A0 term on the right hand side is the productivity shift term. MRW, for example, recognize
the importance of this term and observe that, “the A0 term reflects not just technology but
they may not do so observing the same proportion of the levels. However, Durlauf and Johnson found that output dominated
literacy as criterion for group/regime identification.
26
resource endowments, climate, institutions, and so on; it may therefore differ across countries.”
(p. 410-1) However, in actual estimation, they regard A0 as part of the error term and assume it
to be uncorrelated with the included variables, s and n.
But, this assumption contradicts the expansive definition of A0 that MRW themselves
provide. Given this definition, it is difficult to argue that A0 is uncorrelated with s or n. Actually,
in cross-section regression, this assumption becomes an econometric necessity for identification.
This is because there are no good measures of A0 , and, even if some proxy variables are
included, there still remains a part that is unobservable or unmeasurable and yet correlated with s
and n. On the other hand, claiming A0 to be the same for all countries is not appealing and
contradictory to its definition. Such a claim implies that the aggregate production function is
parametrically identical across countries. This kind of strict homogeneity of production function
is not realistic, and researchers have been objecting to it for quite some time.58 However,
ignoring A0 , by treating it as part of the uncorrelated error term, creates a problem of omitted
variable bias. 59
B. Panel Estimation of the Convergence Equation
Switching to the panel framework can solve this problem by allowing to control for the
unobservable and unmeasurable part of A0 in the form of individual (country) effects. Using
notations of panel data literature, equation (14) can be written as
y it = (1 + β ) y i , t − 1 + β ψ x i , t − 1 + η t + ε it ,
(15)
58
Durlauf and Johnson (1995) also noted the issue of potential differences across countries in aggregate production function.
Observing wide variation in the estimated parameter values across the groups, they observed that, “aggregate production
function differs substantially across countries.” In particular, large differences in the intercept term -- which is related with A0
of equation (14) -- led them to conclude that, “different economies have access to different aggregate technologies.” (p. 375)
Hence, they expressed the view that “... the Solow growth model should be supplemented with a theory of aggregate
production function differences in order to fully explain international growth patterns.”
59
There are other possible sources of bias for the cross-section regression as well. See Lee et al. (1995) and Evans and Karras
(1996a) on this point. Lee et al. (1995), for example, draw attention to another possible bias of the cross section growth
regression. They start with an equation similar to (13) below and assume t and t-1 to be one year apart. Since the cross-section
regression generally considers growth over a long (say, 25 year) period, they try to have the equation correspond to that by
assuming initial t=0, and iterating it forward to T. This gives rise to, among others, a composite error term
j
ξ iT = ∑ Tj=− 1
0 β ε i , T − j . Thus, possible serial correlation in e now acts as source of bias. So does possible across country
variation in
A0
and g. See Lee et al. (1995, Appendix A) for details.
27
where y it = ln y t2 , y i , t − 1 = ln y t1 , (1 + β ) = e − λτ , ψ = (− α /(1 − α )) ,
xi , t − 1 = (ln s i , t − 1 − ln( ni , t − 1 + g + δ)), µ i = (1 − e − λτ ) ln A(0) , and η t = g (t 2 − e − λτ t1 ) .60
There are many different ways to model and deal with the country effect term µ i .
However, it is clear that, in view of the correlation of A0 with s and n, random effectsspecification of µ i is not appropriate. The most appropriate choice, it seems, is Chamberlain’s
(1982, 1983) model of correlated effects and the accompanying method of Minimum Distance
(MD) estimation. 61 This was implemented in Knight et al. (1993) and Islam (1995). The
estimated values of ? for the NONOIL, INTER and OECD samples in Islam (1995) were 0.0434,
0.0417 and 0.0670 respectively. The analogous value reported in Knight et al. (1993) for
NONOIL sample was even higher, 0.0652. Also, the implied values of a (output elasticity with
respect to physical capital) were now much lower and more in conformity with its commonly
accepted empirical values. The estimated values of a in Islam (1995) for the three samples above
were 0.4397, 0.4245 and 0.2972 respectively. The value of a reported for NONOIL sample in
Knight et. al. (1993) was even lower, 0.335.
The results also indicate that the way human capital influences output is, perhaps,
different from the way physical capital does.62 In particular, it seems that human capital impacts
output largely through its influence on the overall technological level. One advantage of the
panel approach is that it produces estimates of µ i , which are indirect estimates of the aggregate
60
Recall our earlier discussion about whether the variables representing determinants of steady state should be made contingent
∗
on the initial period or not. The formulation in equation (14) is on the basis of such contingent yˆ . However under the
assumption that s and n remain constant between t and t-1, which is the natural (even necessary) assumption to make here, we
can replace x i , t − 1 in the equation above by x it .
61
Instead of hiding or sidetracking, the correlated effects model allows the correlation between µ i and x it ’s to come to fore
and play out its role in the estimation process.
62
Incorporation of human capital in the panel analysis led to ‘anomalous’ results, with the coefficient of the human capital
variable turning out to be negative and generally insignificant. This agreed with earlier results regarding human capital
obtained from pooled regressions (see for example Gregorio (1992)) and also the results obtained by Benhabib and Spiegel
(1994). To the extent that the human capital data are weak, there may be data issues related with this result too. The schooling
data used for construction of human capital variable are yet to be adjusted for quality differences. Also, many processes of
human capital formation that occur outside of formal schooling are not included in this variable. Nevertheless, these results
indicate that the channel of influence of human capital on output may be more complicated than suggested by MRW though
their proposal of multiplicative inclusion of human capital in the aggregate production function, alongside physical capital.
Benhabib and Spiegel (1994) also found support for multiple and more complex channels of influence of human capital on
output. Panel results in Islam (1995) show very strong positive correlation between measures of human capital and estimated
28
technological level. These provide the point of departure for a second level of analysis geared to
ascertaining the determinants of technological differences.
Researchers have since used the panel approach to investigate many other issues of
convergence. These are of two types. The first refers to those issues that could not be adequately
examined in the cross-section set-up. The second comprises of those new issues that are specific
to panel estimation. In the following, we discuss some issues of both these types.63
C. Issue of Endogeneity Bias
One issue of the first type is that of endogeneity bias. By itself, equation (15) does not
pose a problem of endogeneity. The variables on the right hand side are pre-determined.
However, Chamberlain’s MD estimation procedure uses both past and future values of x to
substitute out µ i (and y i 0 )64 and hence requires strict exogeneity of xit ’s for the validity of
estimation. Caseli et al. (1996) raise this issue and use Arellano’s GMM procedure to avoid this
problem. This procedure eliminates µ i by first differencing and uses lagged values of y it ’s and
xit ’s as instruments. Using this estimator, Caseli et al. find estimated value of ? and a to be 0.128
and 0.104, respectively, when the regression is based on the original Solow model. These results,
however, do not support the equality of magnitudes of coefficients for s and n, and this leads
Caseli et al. to reject Solow model. On the basis of the augmented (as per MRW) Solow model,
estimated values of ?, a, and f turn out to be 0.0679, 0.491, and –0.259 respectively. The
negative value of the human capital coefficient f now leads the authors to reject the augmented
Solow model. The authors then switch to informal, extended specifications of the growth-initial
level regression and, based on their results, suggest an estimated value of ? to be around ten
percent. They view such a value of ? to be compatible with the open economy version of the
Cass-Koopmans variant of neoclassical growth theory.
values of A0 . This provides a basis for the suggestion that the route along which human capital influences output may run
through A0 .
63
It is necessary to note that panel estimation is not the same as pooled estimation. Sometimes, researchers have labeled their
regressions as panel when, in fact, these are pooled regressions. Panel estimation presupposes explicit modeling of the
individual effect term, while in a pooled regression, this term is relegated to a single-component error. A pooled regression
may also be conducted in a SURE multi-equation framework with the equations distinguished by only the time period covered.
This involves some specification of the error covariance matrix and therefore has some semblance to the (GLS) estimation
under random-effects assumption. However, the error covariance structures, in these two cases, are not the same. Moreover, as
we have noted before, the random-effects specification is not appropriate for estimation of the convergence equation.
29
Caseli et al.’s attempt to check and correct for endogeneity-bias has been a worthy effort.
There are few issues, however. First, the authors do not work out the value of a that corresponds
to their suggested value of ?. Based on their results for the original Solow model, an ? equaling
to ten percent would imply an a equal to 0.1258, which is too low an estimate of capital’s share
in output, even if capital is defined as physical capital only. The second issue concerns their
switch from the model based formal specifications to the extended informal specifications. We
will come to this issue shortly.
D. Heterogeneity in Steady State Growth Rate
Another issue that has been explored using panel approach is the issue of further
parametric heterogeneity of equation (15). In particular, researchers have been interested in
heterogeneity of the steady state growth rate, g. The question whether or not countries have
common g is of considerable theoretical and empirical interest. Theoretically, it remains
controversial. (See Romer’s comment on Mankiw’s paper in Mankiw (1995).) Empirically,
testing of the hypothesis of common g is made difficult by the fact that data only give the actual
growth rates, which are generally a combination of steady state and transitional growth rates.
Lee et al. make a commendable effort to find out the implication of heterogeneous g for the
parameter estimates. Based on a panel for 1960-1989, they find that, with heterogeneity in g, the
rate of convergence for NONOIL, INTER, and OECD samples increase to 0.1845, 0.1521, and
0.1495, respectively.65
One consequence of extending heterogeneity to g (in addition to A0 is that this leads to a
virtual collapse of the concept of convergence, so far as its across-dimension is concerned. This
is because convergence under heterogeneity of both A0 and g implies that the economies are
converging not only to different levels of per capita income but also to different growth rates.
This interpretation of conditional convergence also makes the NCGT equilibrium situation
observationally indistinguishable from a situation characterized by NGT. One problem with Lee
et al.’s results is that the convergence rates obtained under heterogeneous g are very high. The
authors do not report the corresponding values of α, the capital share. But, when worked out, the
64
65
The procedure begins with recursive substitution for y i , t − 1 , so that at the end we just have y i 0 on the right hand side.
The correspondence between Lee et al.’s samples and MRW’s NONOIL, INTER, and OECD samples is approximate.
30
values of α are likely to be very low. This makes it somewhat unclear whether the results can be
interpreted to be supportive of heterogeneous g.
E. TFP-Convergence and the Panel Approach
One useful feature of the panel approach is that it provides a link with studies of TFPconvergence. Originally, TFP studies were based on time series data of individual countries and
were focused on computation of TFP growth rates. These studies did not consider the issue of
convergence in TFP levels. International comparison of relative TFP levels was initiated by
Jorgenson and Nishimizu (1978) and was carried forward by Christensen, Cummings, and
Jorgenson (1981). Dougherty and Jorgenson (1996, 1997) have recently resumed this line of
work. Wolff (1991) and Dollar and Wolff (1994) have also considered the issue of TFP-level
convergence along similar methodological lines. This methodology consists of two steps. The
first step consists of growth-accounting exercise using time series data in order to get the TFP
level indices. In the second step, these indices are analyzed to check for TFP-level convergence.
Dougherty and Jorgenson’s second step analysis is limited to graphical treatment. Wolff, on the
other hand, runs regression of subsequent TFP growth on initial TFP level. In both cases, the
sample consists of the G-7 countries, and the evidence generally supports TFP-convergence.
Not all researchers have adopted this two-step procedure to investigate TFP-convergence. In
an important work on this topic, Dowrick and Nguyen (1989) try to do both growth accounting
and TFP-convergence test in a cross-section regression. The specification is similar to (13).
However, they proceed from the assumption of a common capital-output ratio for all the
countries of the sample. This allows them to interpret the coefficient on the initial income
variable of the cross-section growth equation to be indicative of TFP-convergence. On the
whole, their results also support TFP-convergence in their sample of fifteen OECD countries.66
The problem that arises here is that capital-output ratio may not be the same, and labor
productivity differentials may arise from differences in both technological level and capital
intensity. This is particularly true for larger sample of countries.67
66
However, in their formulation, the initial income variable was relative to that of USA, the most advanced country of the
sample. The time period covered was between 1950 and 1985.
67
Dowrick and Nguyen try to distinguish between these two sources by including interaction term in the regression. The
interaction is between initial income variable and the average investment rate over the period. They conclude in favor of
technological diffusion. The procedure, however, involves several simplifying assumptions.
31
One difficulty in extending the time series growth accounting approach to larger sample
of countries lies in the short length of time series for most of the developing countries. It is in
this respect that the panel approach to convergence study can be of help. The estimates of µ i
produced by panel estimation can be used to recover A0 . Under the assumption that g is
common, ratios of A0 provide relative TFP levels. These TFP level indices can be used to
investigate the issue of TFP convergence in a large sample of countries. Analysis of TFPconvergence in large sample of countries can also benefit from the cross-section growth
accounting procedure suggested recently by Hall and Jones (1996, 1997). This procedure also
yields TFP indices for large sample of countries. Results of both Hall and Jones (1996) and Islam
(1995) indicate presence of huge TFP differences across countries in global samples. Whether
these differences are narrowing or widening over time is an issue that needs to be further
investigated. Such an investigation will, however, require TFP-level indices for several time
periods. This is yet to be done for large samples by either the panel regression procedure or the
cross-section growth accounting procedure.
Convergence discussion has renewed researchers’attention to TFP dynamics across and
within countries. Other researchers who have recently discussed this issue include Bernard and
Jones (1996), Parente and Prescott (1993, 1994), and Young (1995).68
C. Possibility of Small Sample Bias
The advantages of the panel approach do not come without some problems. One of these
is the possibility of small sample bias. Note that equation (15) is a dynamic panel data model.
Theoretical properties of the estimators of such models are asymptotic, and their small sample
properties are unknown. The panel estimators that have so far been used in the convergence
literature are Arellano’s GMM, Chamberlain’s MD, LSDV, and some variants of maximum
likelihood (as in Lee et al. (1995)). All these estimators are consistent,69and the only way to
know their small sample performance is through conducting Monte Carlo study. Results of
Monte Carlo studies are more useful when these studies are tailored to the equations actually
68
Maddison (1987), in his splendid growth accounting study for the developed economies, also deals with TFP catch-up, though
instead of estimating the extent of ‘catch-up,’he imputes particular value to it.
32
estimated and the data sets actually used. Monte Carlo study using Summers-Heston data set and
focusing on the convergence equation (15)70 indicates that, in general, estimators that do not use
further lagged values of the dependent variable as instruments perform better than those which
do.71 However, to the extent that Monte Carlo results tend to be data and model specific, it is
necessary that researchers using panel estimators checked into the small properties through
Monte Carlo studies designed for their respective data set and equation estimated. 72
Barro (1997) has recently drawn attention to the fact that LSDV uses only withinvariation for estimation and throws away the cross-section or between-variation. This brings us
back to the within-across tension of the convergence parameter. To the extent that the
convergence equation is based on a within-economy growth model, estimation of its parameters
on the basis of within-variation may be more desirable.73 More importantly, whether or not an
estimator uses both within and between variation, perhaps, cannot be the main criterion of an
estimator’s suitability in this case. For example, the random-effects GLS estimator uses both
within and between variation. But, as we saw, this estimator is not suitable for estimating the
convergence equation because it contradicts the correlation of the individual effect with the
included regressors.
E. Panel Analysis and Extended Cross-section Regression
69
Note that the asymptotic properties of panel estimators can be considered in the direction of N → ∞ or T → ∞ or both N
and T going to infinity. Further, N and T can go to infinity either at the same rate or different rates. Amemiya (1967, 1973) has
shown that, although LSDV is biased in the direction of N → ∞ , it is consistent in the direction of T → ∞ .
70
See Islam (1992). The dynamic panel data estimators considered in this study include: OLS, LSDV, two instrumental variable
estimators by Anderson and Hsiao (1981, 1982), two GMM estimators by Arellano and Bond (1991), 2SLS, 3SLS, generalized
3SLS and Minimum Distance estimators by Chamberlain (1982, 1983).
71
The Arellano GMM estimator, which depends heavily on lagged y’s as instruments, displayed large bias. This may, to a certain
extent, explain the very high values of λthat Caseli et al. (1996) obtain using this estimator. However, it needs to be
mentioned that the instrument set used for the Arellano estimator in the Monte Carlo study is not exactly the same that Caseli
et al. seems to have used.
72
Nerlove (1996) has argued that the change in results brought about by LSDV estimation under fixed-effects assumption in
Islam (1995) may be the result of small sample bias of this panel estimator. He conducts OLS on pooled data and GLS under
random-effects assumption, and finds that results from these procedures prove to be similar to those obtained earlier from
cross-section estimation. However, as mentioned earlier, the random-effects assumption is not appropriate for µ i in the
convergence equation. It is not surprising that GLS under random-effects specification and OLS on either cross-section or
pooled data yield similar estimates, because they all share the same assumption. They all ignore the correlation of A0 with s
and n. Also, Monte Carlo study in Islam (1992) show that, for the data and model in question, LSDV performs much better
than many other estimators. Chamberlain’s MD estimator also displayed similar superior performance.
73
Note in this context that Chamberlain’s MD estimator uses both cross-section and time-series variation in data. Also,
Chamberlain’s MD estimator is robust to presence of heteroskedasticity and autocorrelation. Note that presence of
autocorrelation induced by measurement errors has been another source of Barro’s concern regarding use of panel estimator.
33
The growth-initial level regression, which has been the mainstay of convergence
research, seems to be now entering a new phase. In the first phase, represented by works
reviewed in section-4, the specifications were informal. These regressions were basically geared
to check which variables influence growth positively and which, negatively. This led to the
emergence of a bewildering variety of explanatory variables, so much so that growth regressions
fell into some disrepute. Responding to the situation, Levine and Renelt (1992) used Leamer’s
extreme bound analysis to find out which of the suggested explanatory variables were
statistically robust and which were not. The second phase of the growth regressions started with
the switch to the formal specification of convergence equation of the type presented in section-5.
This made it possible to recover the theoretical parameters, and it also rigorously specified the
variables to be included in the regression.
It seems that growth regressions are now entering a third phase. This is manifested in the
inclusion of a large number of additional regressors in equation of the type (13). Thus, for
example, Barro (1997) presents this kind of extended cross-section regression including a host of
variables in addition to the ones that are specified by theoretical models. Caseli et al.’s last set of
regressions also falls into this category. Other examples of this type of work include Sachs and
Warner (1997), etc. Some researchers do not try to provide theoretical justification for inclusion
of the additional variables. Others, who want to keep the link with theoretical growth models, try
to do this by referring to the A0 term of equation (14). It is argued that the variables of the
expanded list are required to control for A0 in the regression. This distinguishes these
specifications from the informal specifications of the first phase, because this provides the
theoretical locus standi of the variables of the expanded list. Sala-i-Martin (1997) reports running
several million of this kind of regressions to find out the statistical properties of the estimated
coefficients of different explanatory variables. This is somewhat in the spirit of what Levine and
Renelt (1992) did earlier.
It, therefore, seems that two approaches are possible in dealing with the A0 term of the
growth-convergence equation. The first approach is to continue with the cross-section regression
and extend it to include variables that proxy for A0 . The second approach is to opt for panel
estimation and control for A0 as unobservable individual effect. The estimates of A0 are
34
analyzed in a second step. The two approaches have their respective advantages and
disadvantages. The advantage of the extended cross-section regression is that it is a one-step
procedure that addresses the issue of determinants of A0 in conjunction with estimation of other
parameters. The disadvantage is that it cannot avoid the potential problem of omitted variable
bias. The panel approach has the advantage that, in its first step, the parameters of the growth
model can be estimated corrected for omitted variable bias. The disadvantage is that the first step
of the exercise cannot reveal anything about the determinants of TFP level.
In this context it should be noted that, in equation (15), the term A0 appears with the
coefficient (1 − e − λτ ) . This implies quite a few restrictions on the coefficients of the extended
cross-section regression. First, this implies that all the variables that appear as proxies or
components of A0 should have the same coefficient. Second, this common coefficient of A0
components has to be equal to the coefficient of the investment variable and numerically equal
but opposite in sign to the coefficient of the labor force growth variable. Researchers using
extended cross-section regressions do not always recognize these restrictions. This often leads to
an eclectic approach. On the one hand, the specification is claimed to have formal link with
growth model, and that link is used to recover the rate of convergence parameter from the
coefficient of the initial income variable. On the other hand, this link is not followed through for
the coefficients of the rest of the variables of the regression.74
In sum, the panel results show that in studying convergence and estimating the
parameters pertaining to the capital deepening process, it is important to account of the
technological differences. In general, it is found that, when technology differences are allowed,
NCGT fares better in cross-country growth data. The results also indicate that capital deepening
and technological diffusion, the two processes that are supposed to yield income convergence,
may not always play symmetric roles. It appears that in small sample of developed countries, the
process of TFP convergence has aided income-convergence. But, this may not have been the
case in large, global sample where very large TFP differentials remain. Panel approach also
helped investigate other issues like endogeneity and heterogeneity of steady state growth rate.
74
Sometimes the link is not recognized to trace out the value of the capital-elasticity parameter.
35
Results from these latter investigations show the difficulty in having consensus estimates of the
structural parameters. However, the hypothesis of conditional convergence is generally upheld.
8. Time Series Approach to Convergence
The progression of convergence study from cross-section to panel and then on to time series
approach seems to be quite natural. However, in many cases time series analysts of convergence
researchers have actually taken off directly from the cross-section tradition.
A. Time Series Equation for ß-Convergence
The commonly used equation for time series analysis of convergence can be derived directly
from the equation for ß-convergence given by (15). This generally involves the assumption that
the xi , t − 1 remains unchanged over the sample period considered. Then β ψ xi , t − 1 becomes just
another time invariant term, and it can be subsumed under the term µ i . Also, note that if we
substitute t 2 = t and t1 = t − 1 in the expression for η t , we get
(16)
η t = g (t 2 − (1 + β ) t1 ) = g[t − (1 + β )(t − 1)] = (1 + β ) g − β g t .
For an individual economy, (1 + β ) g is a constant, and hence can also be subsumed under
µ i , so that η t effectively reduces to − β g t . Introducing these changes and upon rearrangement,
we get from (15), suppressing the country subscript i, the following:
(17)
y t = µ − β g t + (1 + β ) y t − 1 + εt .
Note that this is same as the Dickey-Fuller equation with a drift and linear trend. Recall
that for convergence in the usual sense, ß should be negative. In other words, we should have (1+
ß) less than one. The question then becomes whether the coefficient on y t − 1 is less than unity or
not. If we cannot reject the H 0 : (1 + β ) = 1 , then by implication we cannot reject the null ß = 0,
i.e., we cannot reject the hypothesis that there is no convergence. Hence, looked from the point
of view of individual economy’s time series, the question of convergence reduces to the standard
question of whether or not the output series is integrated.
36
Broadly, time series analyses of convergence may be categorized into two types. The first
focuses on analysis of the time series of individual economies, without reference to any other
economy. This comes closest to the study of within-convergence. The second analyzes output
series of one economy in reference to those of others. Within the second, again, several different
strands may be distinguished. We begin by looking at evidence of convergence produced by
within analysis of individual economies.
B. Time Series Analysis of Within-Convergence
As noted, convergence analysis focussing on individual economy’s time series and using
equation (17) is virtually indistinguishable from standard unit root analysis of macroeconomic
series. The difference is that while the traditional unit root analysis has been limited mainly to
output series of the developed countries, under convergence paradigm, the analysis is extended
to a larger sample of countries, including the developing ones. Thus, for example, Lee et al.
(1995) conduct an exercise along this line and find that, out of 102 countries for which the
equation is fitted, only for three the null of unit root can be rejected (at five percent level of
significance).75 To account for potential serial correlation, they also adopt the augmented
Dickey-Fuller specification with the number of lags determined by use of the SBC information
criterion. This, however, do not change the results that much.76
One well-known weakness of the Dickey-Fuller unit root tests is the formulation of the
null. The test shows when the null of unit root cannot be rejected, leaving, however, a wide range
of other non-unit root alternatives still compatible with the evidence. The test proposed by
Kiwatkowski et al. (1992) is an advance in this regard, because it takes stationarity as the null.
However, this test also has its problems; in particular, its outcome depends on the degree of
truncation. Using this test, Lee et al. find that the number of countries for which stationarity can
be rejected “fell steadily with the length of the truncation parameter.” (p. 22) When this
parameter is set at 8, the number of rejections falls to only 9.
75
Noting that standard Dickey-Fuller tests have low power, Lee et al. also use a test proposed by Im, Pesaran and Shin (1995),
referred to as the t-bar test, which is based on the average value of the DF statistics obtained across countries. The results
remain basically unchanged.
76
Based on the ADF t-statistics, the number of rejections of the unit root null ranges between 3 and 14, depending on the number
of lags and whether data were demeaned or not. Lee et al. refer to allowing for ‘common time specific effects.’(p. 21) It is
unclear what is meant because apparently the procedure works on individual time series separately. The use of an analogous tbar statistic for the ADF set-up, as proposed in Im et al., does not affect these results by that much.
37
It is important to note here that the above tests are based on the assumption of xi , t − 1 being
constant. The presence of xi , t − 1 in equation (15) is linked to the notion of conditional
convergence. The assumption that xi , t − 1 is time-invariant contradicts this notion and reduces the
analysis, in large measure, to that of unconditional convergence. Yet, elements of xi , t − 1 may
change over time even within an economy. Lee et al. themselves recognize the possibility of
‘once for all changes’taking the form of ‘shifts or take-offs,’and observe that their result of nonstationarity may be the result of not taking account of these changes.77 This has been confirmed
by research done by Ben-David, Papell, and others. They show that introduction of simple trend
breaks (either exogenous or endogenous) leads to large increase in the number of rejection of
unit root. 78
C. Time Series Analysis of Convergence across US States
The second type of time series analysis of convergence, which analyzes output series of
one economy with reference to those of others, may be termed as time series analysis of acrossconvergence.
Initial Time Series Analysis of Convergence across US States
One particular application of time series analysis of across-convergence has been to the
data of the US states. In this analysis, the output series of the individual states or regions are
studied with reference to the average for the US as a whole. For example, Carlino and Mills
(1993) analyze per capita income of eight geographic regions of the US.79 They proceed from a
definition of stochastic convergence similar to that given by equation (2) and interpret y it as y t ,
the average for the sample (i.e., for the USA a whole). Thus, they study the time series properties
77
Lee et al. observe that Solow model does not have internal explanation for such changes. However, that does not mean that the
model does not allow for the possibility of such changes
78
For details, see Ben-David and Papell (1995, 1997), Ben-David, Lumsdaine, and Papell (1997), Lumsdaine and Papell (1997),
and Zivot and Andrews (1992).
79
The regions are: New England, Mideast, Great Lakes, Plains, Southeast, Southwest, Rocky Mountains, and Far West.
38
of Dy jt = ( y t − y jt ) , where y jt are the log per capita output of the region j. The authors use an
augmented version (in the Dickey-Fuller sense) of equation (17) with y t replaced by Dy t .80
Note that, in this deviation-setup, the intercept term of the equation stands for timeinvariant differences in the determinants of the steady state across regions. This is equivalent to
allowing some components of xi , t − 1 to vary, albeit only in the direction of i. However, Carlino
and Mills also feel the necessity of allowing trend break, which can be interpreted as allowing
xi , t − 1 to vary in the direction of t, albeit in a very restricted way. These changes in the setup and
specification bring the analysis closer to that of conditional convergence, and Carlino and Mills
find that they can reject the unit root hypothesis for majority of the regions. This clearly favors
conditional convergence.
Lowey and Papell (1996) extend this line of analysis. While in Carlino and Mills, the
trend break is exogeneously set for the year 1946, Lowey and Pappel endogenize the timing of
the break. Also, they conduct the analysis at a further disaggregated level by dividing the US into
22 regions instead of 6. Again, the null of unit root can be rejected for majority of the regions.
Unit Root Analysis of Pooled Data for the US States
Although in works like Carlino and Mills and Lowey and Papell, the analysis is conducted on
the basis of data in deviation or relative form,81 the unit root tests are conducted region by
region. In contrast, in Evans and Karras’(1996b) work on convergence across US states, similar
test is conducted by pooling the deviation data. Also, Evans and Karras consider convergence at
the levels of states, instead of broader regions.82 In view of the weakness of the standard DickeyFuller test, Evans and Karras use a modified version of the unit root test proposed by Levine and
Lin (1993) designed specifically for pooled data. The results show rejection of the unit root
hypothesis even when trend breaks are not included. Note that in this setup too, the state specific
intercept term of the equation stands for (time-invariant) difference in steady state among the
individual states. Hence, this may also be interpreted as a finding of conditional convergence.
εt .
80
They also impose a time series structure on the error term
81
This distinguishes methodologically their work from that of, say, Lee et al.
82
This makes their results directly comparable with Barro and Sala-i-Martin (1992)’s analogous results obtained from crosssection approach.
39
Evans and Kerrera contrast this aspect of their result with Barro and Sala-i-Martin (1992)’s result
of unconditional convergence across US states. However, as noted earlier, Barro and Sala-iMartin, in part of their analysis, also include regional dummies and an index of composition of
output. Hence, it is a moot point whether their result is entirely of unconditional convergence.
D. Time Series Analysis of Convergence across Countries
Unit Root Analysis of Pooled Data for Countries
Evans and Karras (1996a) also conduct unit root analysis of pooled deviation data for a
sample of 56 countries. The result is similar: evidence favors rejection of unit root, i.e., favors
the hypothesis of conditional convergence. Analogous results are also obtained in Evans (1996)
from analysis of long historical data (1870-1989) for a sample of thirteen developed countries.
Co-integration Approach to Across-Convergence Study
An alternative way of analyzing output data in relative form is to use the framework of cointegration analysis. Bernard and Durlauf conduct such an analysis on the basis of a sample of 15
developed countries.83 The goal is to check whether the per capita output series of these
economies are co-integrated or not, and if they were, whether the co-integration vector is of the
form (1, -1) or (1,-a), where a is a constant. The indirect84 way of doing this is to formulate these
hypotheses in terms of conditions on the rank of the spectral density matrix at frequency zero of
∆ D Yt and ∆ Yt . Here, ∆ Yt is the first difference of Yt , which, in turn, is the vector of individual
output series, y it . Similarly, DYt is the vector of Dy it , the deviation of output of country i from
that of the reference country having index 1.85
The basic idea, from Engle and Granger (1987), is that if the number of distinct stochastic
trends in Yt is less than n (which would imply co-integration), then the spectral density matrix at
frequency zero of ∆ Yt , i.e., of f ∆Y (0) , is not of full rank. If all n countries are converging in per
83
They also consider two sub-samples: the first consisting of 11 European countries, and, the second, consisting of 6 European
countries that showed a high degree of ‘pairwise cointegration.’
84
The direct way of doing this is to do pairwise analysis. However, apart from posing severely large number of possible pairings
in any sample of respectable size, this route also faces the problem of arriving at an overall result from the pair specific results.
85
For the 15 country sample, the US is the reference country. For the European sub-samples, the reference country is France.
40
capita output, then f ∆DY (0) i = 0, ∀ i , or equivalently, the rank of f ∆DY (0) is zero. So, in
operational terms, the task is to look at the spectral density matrix at frequency zero of ∆DY and
∆Y -- the first for convergence, the second for co-integration— and check for the rank conditions
(more concretely, number of co-integrating vectors).
Bernard and Durlauf use two sets of procedures to carry out the tests: one, based on Phillips
and Ouliaris (1988), and the other, based on Johansen (1988). In either case, the conclusion is
broadly similar: there is evidence of co-integration of the form (1, -a) but not of the form (1, -1).
Bernard and Durlauf interpret this result as showing that the countries ‘shared common trends’
but did not converge. However, as we have noted before, co-integration of the form (1, -a) can
also be a manifestation of conditional convergence.86
It may be noted that, despite the apparent methodological differences, Bernard and Durlauf’s
cointegration analysis is similar to Evans and Karras’unit root analysis. In both cases, the steady
state levels of different economies are allowed to differ by only a constant proportionality factor,
though Evans and Karras, in addition, incorporate trend breaks. The results are qualitatively
similar although pertain to different samples.
In sum, time series analysis has actually been a different way of investigating ß-convergence.
Both unconditional and conditional-convergence have found their place in this analysis.
However, in time series analysis of conditional convergence, steady state variation was generally
limited to time-invariant differences and trend breaks. The broad evidence, nevertheless, favors
conditional convergence. To the extent that time series approach avoids imposition of structure,
it does not produce estimates of the structural parameters and hence does not answer questions
concerning specific values of these parameters.
9. Distribution Approach and s -Convergence
While all the approaches discussed so far -- cross-section, panel, and time-series -- have
concentrated on ß-convergence, the distribution approach has the distinction of focusing on s convergence. However, the correspondence is not so simple. Distribution approach has actually
86
Bernard and Durlauf were inclined to interpret the shared common trends as indicative of club convergence. However, we have
noted earlier the difficulty in distinguishing evidence of club convergence from that of conditional convergence in general.
41
proceeded along two lines. The first maintains relationship with ß-convergence and tries to work
out the precise relationship between ß and s . In contrast, the second line of the distribution
approach emphasizes the limitations of ß-convergence and focuses on the shape of the entire
distribution and intra-distribution dynamics.
C. Relationship between ß-convergence and s -convergence
In order to see the relationship between ß and s , we can start from the decomposition of the
cross-section variance into its constituent elements. This decomposition is already offered in BS.
They note that if all the terms other than y i , t − 1 and ε it in equation (15) are ignored, then the
evolution of σ t2 , variance of y it , under suitable assumptions on ε it , can be described by
~
σ t2 = (1 − β ) 2 σ t2− 1 + σε2 = β 2σ t2− 1 + σ ε2 ,
(18)
~
where σε2 is the variance of e, and β = (1 + β ) . Iterating backwards, this yields
 2
σ ε2
σ ε2  ~ 2t

σ =
~ + σ0 −
~2 
β .
1− β 2 
1
β
−


2
t
(19)
As t → ∞ , the above approaches the steady state value, σ ∞2 =
σ ε2
2
~ 2 . It is clear that σ ∞
1− β
increases with σε2 and decreases as β becomes more negative. What is more important is that
σ t2 can monotonically either increase or decrease to σ ∞2 depending on whether the initial
variance σ 02 is smaller or greater than the steady state variance σ ∞2 . This algebraic result again
shows that a negative β can not guarantee falling variance. However, it also shows that an
empirical finding of increasing cross-sectional variance is not incompatible with β convergence.
Similar relationships have been presented in Lee et al. Unlike BS, Lee et al. do not ignore
the terms representing variation in the steady state across countries in equation (15), and hence
they get the following more involved expression for cross-section variance:
42
~
~ 2t 2
~ 2t 2 1 − β 2t  2  1 −
σ = β σ 0 + [1 − β ]σ∗0 + 
~ 2 σε + T −
1−
1 − β 

2
t
(20)
~
β 2t  2
~ σ g ,
β2 
where σ∗20 is the cross-country variance in steady state per capita output in time 0, and σ g2 is
variance of the steady state growth rate, g. Under assumption of common g, the last term drops
out. Then the above expression is basically the same as of BS, except that it now has the
additional term involving σ∗20 . As expected, the latter term now also appears in the expression
for σ ∞2 :
σ ∞2 = σ∗20 +
(21)
σε2
~
1− β 2
Substituting for σ∗20 in equation (20) using equation (21), yields
~
σ t2 = σ 02 + [1 − β 2t ](σ ∞2 − σ 02 ) .
(22)
Note that this is the same equation as can be obtained from BS equation (19) above,
because σ∗20 gets subsumed in σ ∞2 . It also helps to see again that dispersion may either increase
or decrease towards σ ∞2 , depending on whether initial dispersion is less or greater than it.87
D. Relationship between Tests for ß- and s -convergence
The above shows how β and σ 2 are algebraically related, and value of one can be
obtained from that of the other, provided other conditions are satisfied.88 This also implies that
tests of these two concepts of convergence can be related. This is of particular importance for s convergence because, unlike β -convergence, no statistical test for s -convergence was available
and/or used. Taking up the task, Litchenberg (1994) observe that, ignoring other terms, from (15)
we can also have
(22)
87
88
σ t2
~ 2 σ ε2
=
β
+ 2
σ t2− 1
σt − 1
Drawing attention to this result, Lee et al. note that s -convergence is not an implication of the Solow model.
2
As we shall see, some researchers, indeed, produced alternative estimates of β from analysis of σ .
43
The basic information for s -convergence is the ratio σ t2 / σ t2− 1 , and one can work with this ratio
directly, as has been done by Miller (1995) and Lee et al. However, Lichtenberg shows that this
~
ratio can be estimated indirectly from β and the R 2 of the cross section regression
estimating β . Since 1 − R 2 = σε2 / σ t2 , it follows from equation (22) that
(23)
~
σ t2 / σ t2− 1 = R 2 / β 2 .
~ˆ
This shows that in order to draw inference about s -convergence, β needs to be adjusted using
R 2 to account for the distribution of the shock term. Short of this adjustment, hypothesis of
absence of s -convergence will be rejected too often.
Litchenberg suggests that the test statistic obtained from equation (23) had an Fdistribution with [n-2, n-2] degrees of freedom.89 However, Carree and Klomp (1995) point out
that Litchenberg’s conclusion regarding appropriate distribution for the test statistic implied by
(23) is not entirely correct. They draw attention to the fact that F-distribution is valid if σ t2 and
~
σ t2− 1 are independent of each other, which will not be true provided β ≠ 0 .90 However, they try
to salvage Litchenberg’s idea of using F-distribution to test for s -convergence by showing that if
~ˆ
~
~ˆ
~ˆ
sample is large, so that β can be thought ‘close’to β , then T2 = (σˆt2 / σˆt2− 1 − β 2 ) /(1 − β 2 ) will
be distributed approximately as F(n-2, n-1).91 They also suggest another variant of the statistic
~ˆ
that take into account variability in the estimate β . This is given by
~ˆ
~ˆ
T3 = (σˆt2 / σˆt2− 1 − ( β 2 − zα σˆβ~ ) 2 ) /(1 − ( β 2 − zα σˆβ~ ) 2 ) , where σˆβ~ is the standard error of
89
Note that the above relationships are based on a host of simplifying assumptions and ignoring, in particular, differences in the
steady state. Litchenberg think that similar relationship hold even when differences in steady state are allowed. There are a few
2
2
differences here. First, the relationship involves, apart from conditioning on the left hand side, both σ u and σ ε on the right
∗
∗
hand. Second, it is derived on the assumption that the steady state is not time contingent, so that yˆit is the same as yˆi , t − 1 .
Third, the relationship above is in terms of income per effective worker; it will be more complicated when transformed in
terms of income per capita. However, the basic ideas are evident from this algebra.
90
They also point out that the correct degrees of freedom is [(n-1), (n-1)] instead of [(n-2), (n-2)]
91
They denoted original Litchenberg statistic by T1 .
44
~
estimated β and zα is the adopted critical value from the standard normal distribution. The
adjustment will cause a type-I error that is lower than the significance level.92
E. Evidence regarding s -convergence
Evidence regarding s -convergence differ in different samples. For the OECD countries,
data have generally favored s -convergence. Lee et al., for example, compute variance of crosssection distribution of log of per capita income for different samples of countries for 1961
to1989 and plot them against time. Their results show that the variance for the OECD sample has
decreased over time. Miller (1995) and other researchers have produced similar results regarding
OECD sample. Note that this kind of evidence does not involve any formal statistical test.
Litchenberg, in contrast, uses his procedure to formally test the hypothesis of s convergence. He runs a simple regression of ln GDP85 on ln GPD60 for the OECD countries
and uses the result to compute the test statistic as per equation (23) above.93 Application of the
critical values from F(20, 20) distribution results in a non-rejection of the null of nonconvergence.94 But, Carree and Klomp redo the exercise and find that use of their statistics
reverses the conclusion. For the period 1960-85, all of Carree and Klomp’s three statistics report
convergence. However, when a later sample period of 1972-94 is considered, they find that
convergence does not hold any more. This, in their view, shows that while during initial years
there has been significant reduction in variance, in the more recent years σ t2 has become close to
σε2 and hence no further pronounced tendency for it to decrease is found. In other words, as the
92
One feature of these tests is that the comparison of variance is limited to that of first and last period only. In order to make use
of the information in between, Carree and Klomp, proceeding from simplified version of equation (15), iterating backwards, and
making use of the independence assumption, derive the following expression for the first difference of cross-section variance,
(24)
~ 2 (t − 1) ~ 2
2
2
2
∆σ t = β
[( β − 1)σ 0 + σε ] .
~2
2
Under the null of no convergence ( β − 1) = 0 Hence the equation above can be used to test this null by regressing ∆σ t on
~ 2( t − 1)
~
~
β
and using the t(T-2) distribution, if β is known. In practice, one has only an estimate of β and the distribution will
hold only approximately. This is their statistic, T4 .
2
2 ~ˆ2
93
He obtains a slope coefficient of 0.715 and R =0.802. This gave R / β = 1.57 which is equal to var(ln GDP85)/var(ln
GDP60), the test statistic.
94
The probability value is 0.31.
45
countries get closer to the steady states, the transition-component of the dynamics recede, and
idiosyncratic shocks take over, which then displays no systematic tendency to decrease.
Basic evidence of s -convergence has been provided for other smaller samples of
countries as well. Similar evidence of s -convergence has been reported for the US states.95 For
large, global sample of countries, however, evidence generally indicates a rise in variance. For
example, according to Lee et al.’s computation, output-variance in the sample of 102 countries
increased from 0.77 to 1.24 between 1961 and 1989. Other researchers have also produced
similar evidence.
There are different ways in which these results can be interpreted. For example, rising s
in the global sample may indicate that the steady state dispersion, σ ∞2 , itself has increased,
which, in turn, may be the result of increased dispersion of the determinants of steady state.
Alternatively, it is possible that σ ∞2 has remained unchanged, but initial variance, σ 02 , was less
than σ ∞2 , so that the variance increased from below towards the steady state variance.96 In either
case, the outcome is not incompatible with conditional ß-convergence. Similarly, we do not
know how much of the decrease in variance in small sample of developed economies is due to
negative ß and how much due to reduction in the other items of (20), including reduction in the
dispersion of the steady state determinants. All this indicates that more knowledge is needed
about the dynamics of the steady state determinants in order to understand the changes in the
cross-section variance of income.
F. Study of Evolution of Cross-sectional Distribution
The second line of the distribution approach to convergence study does not limit itself to
examination of variance. Instead, it studies the evolution of the entire shape of the distribution
and intra-distribution dynamics. Also, this line of research strives to go beyond the anonymity of
distribution, to identify the position of individual or groups of countries within the distribution,
and to see how these positions change over time. This line of research has been most vigorously,
and almost single-handedly pursued by Quah.97
95
See for example, Sala-i-Martin (1996), Miller (1996).
96
Lee et al. point out that increased variance may also be the result of increasing dispersion in g.
97
Quah has produced a series articles based on this line of research. These include Quah (1993b, 1996a, 1996b)
46
In order to follow the evolution of the entire distribution, Quah focuses on the probability
mass at different quantiles. Already, from simple plotting of the cross-section distribution of the
global sample for successive years, two features emerge: first, the cross-section distribution is
not collapsing, and, second, this distribution is becoming increasingly bi-modal. However,
because it is not known whether the plotted distributions are of steady state or not, and because
the plots of distribution cannot tell the position of individual countries, Quah performs a more
formal analysis of the distribution-dynamics. He proposes to put it in the following framework:
Ft + 1 = M Ft ,
(26)
where Ft is the cross-section distribution at time t, and Ft + 1 is the same at time t+1, and M is the
operator that maps Ft onto Ft + 1 . The goal is to know M, which determines the evolution of the
distribution. If M is assumed to be unchanged over time, then we have,
Ft + s = M s Ft ,
(27)
where s is any particular length of time (number of years, say).This framework, Quah points out,
also gives us a way of getting at the desired steady state distribution. This is obtained by taking
the limit of Ft + s with s → ∞ . Quah models M as a Markov transition matrix and calibrates it
using actual data.98
Both the calibrated transition matrices and ergodic distributions obtained on their basis
lead to similar conclusions. First is ‘persistence.’The values of the diagonal elements of the oneyear M are in the neighborhood of 0.9, implying that most of the countries continue to remain in
the same position (or range) of the distribution. Second, whatever mobility (within the
distribution) exists, it works to ‘thin out the middle,’and ‘pile up of probability mass at the two
tails.’This is Quah’s result of growing ‘twin-peakedness’or bi-modality of the distribution. The
results do not change if higher order specifications are used. In fact, these make the bi-modal
98
The variable modeled is per capita income normalized by world average. In general, Quah treats M to be time-invariant,
although he allows it to have higher (than first) order of specification. In Quah (1993b), he takes the quantiles as fixed (at .25,
.5, 1 and 2) and calibrates the transition probabilities for countries to move from one quantile to another (i.e., elements of M)
from actual data. First he obtains ‘one-step annual’transition matrix by averaging the observed one-year transitions over the
period between 1962-63 and 1984-85. He then lets it run to get the ergodic distribution on the basis of the one-year transition
matrix. He does it in two variants: one with first order and another with higher order specification. He also obtains the 23-year
transition matrix spanning the period 1962 to 1985, calibrated in analogous fashion. He then derives the ergodic distribution on
its basis of the latter matrix as well. Again, he does this in above mentioned two versions.
47
property and ‘poverty piling up’even more pronounced.99 This exercise is extended in Quah
(1993a) where he lets the quantiles to evolve (instead of being fixed).100 However, the results
remain more or less intact. In fact, now the dynamics of Q(t) further confirm these results. Thus,
altogether, Quah’s formal analysis confirms what informal plotting of distribution of successive
years already suggested.
Quah notes certain technical shortcomings of this analysis.101 However, perhaps, it is
more important to note here that conditioning variables play no role in this analysis. In fact,
Quah makes it explicit that it is his intention not to be restricted by assumptions of long term
growth. M is memory-less, and no growth theory is required for its estimation; no structure is
imposed on data. Thus, Markov-analysis of the evolution of cross-section distribution is a type of
reduced form analysis of the cross-sectional output. This contrasts with research on conditional
ß-convergence, which imposes structure and uses growth theory to decide on specification,
choice of right hand side variables, etc. The connection between reduced form, Markov analysis
and growth theory comes afterwards, when the results are confronted with the predictions of
growth theory. For example, growth theory faces the task of explaining the phenomena of
persistence, bi-modlity, etc., which this analysis has uncovered.
In sum, σ -convergence and distribution approach are not so far apart from βconvergence and the approaches that focus on β-convergence. Once it is remembered that σ convergence research generally focuses on unconditional convergence, it becomes clear that
results regarding σ -convergence largely agree with those regarding β-convergence. For
99
Quah also allows the one year transition matrix to be iterated 23 times and compares the resulting matrix (which he calls
‘stationary estimate’) with the 23-year transition matrix that is obtained from actual calibration of the data. He finds that the
long run matrix shows stronger persistence than found in the ‘stationary estimate.’
100
He fits a VAR model to forecast the quantiles, Q(t), and then takes the convolution with M raised to the appropriate power to
get the dynamic evolution of the sequence of distributions.
100
For example, the results are contingent on the arbitrary grid that is used to discretize the point in time empirical distributions.
However, as Quah notes, inappropriate discretization may destroy the Markov property of an otherwise well behaved first
order Markov process. Also, conclusions regarding piling up of probability mass are contingent on the choice of discretizing
grid and may not be robust. (See Quah 1993a, p. 437.) Finally, Quah observes that the VAR models are estimated on the basis
of only about 20 data points and hence are not that precise
101
For example, the results are contingent on the arbitrary grid that is used to discretize the empirical distributions. Quah notes
that inappropriate discretization may destroy the Markov property of an otherwise well-behaved first order Markov process.
Also, conclusions regarding piling up of probability mass is contingent on the choice of discretizing grid and may not be
robust. (Quah 1993a, p. 437) Similarly, Quah observes that the VAR models are estimated on the basis of only twenty data
points and hence are not that precise.
48
example, σ -convergence has generally been reported for the small sample of developed
economies for which there is also evidence of unconditional convergence. On the other hand, σ convergence does not hold for large, global sample of countries, for which unconditional
convergence also does not hold. Similarly, the findings of non-collapsing distribution and
increasing bi-modality are not incompatible with conditional β-convergence, once it is noted that
the determinants of steady state can not only vary cross-sectionally but also change over time.
However, fuller understanding of these results requires that growth research focused on the
dynamics of steady state determinants.
10. Conclusions
Discussion of this paper shows that convergence has, indeed, been understood and
investigated in many different ways. It began with simple bivariate regressions of growth on
initial level of income, with specifications not formally linked with growth models. The implicit
assumption of these regressions was that all countries in the sample had the same steady state
level of income. This later came to be known as unconditional convergence. Soon, it became
clear that NCGT’s convergence implication is, at best, conditional on differences in steady state.
The conceptual transition from unconditional to conditional convergence was also associated
with a switch from informal to formal specification of the growth-initial level regression. The
formal specification allowed recovery of the parameters of the growth model from the regression
coefficients. This raised the focus of discussion from broad presence or absence of convergence
to precise values of the growth model’s parameters.
Despite the differences in approach and method of investigation, some general results
have surfaced. Bulk of the evidence has favored conditional convergence. This suggests that
NCGT cannot be rejected on the basis of evidence from convergence research. This does not
mean that the dispute between NCGT and NGT has been resolved. This means that either the test
has to move to other grounds, and/or it has to be specified in other ways.
It is clear that the concept of conditional convergence has played a crucial role in
determining the outcome of the convergence debate. This concept has two other consequences.
First, it makes convergence very hollow, and it works toward making NCGT and NGT
observationally equivalent. This is particularly true when the concept of conditional convergence
49
is pushed so far as to allow countries to have their specific steady state growth rates. The second
consequence of this concept has been that it has, to some extent, diverted attention away from
important issues of growth. This is because the concept of conditional convergence abstracts
from the determinants of steady state. Yet it is these determinants on which the long run income
level of a country depend.
The current stage of research on growth started with quite a bit of concern for practical
relevance. Lucas expressed this best in his following well-known observation:
“Is there some action a government of India could take that would lead the Indian economy to
grow like Indonesia’s or Egypt’s? If so, what, exactly. If not, what is it about the “nature of India”
that makes it so? The consequences for human welfare involved in questions like these are simply
staggering: Once one starts to think about them, it is hard to think about anything else.” (Lucas
1988, p. 5)
It is in this light that it may be somewhat disturbing that more research is directed to estimation
of conditional convergence rate than to investigation of key determinants of growth, like
investment, fertility, and technology. This relative lack of focus on the steady state determinants
is also obstructing fuller understanding of such important results obtained from the distribution
approach as ‘persistence’and ‘increasing bi-modality.’It is by focusing on the dynamics of the
steady state determinants that we can grasp the underlying causes of these phenomena. Such a
change in focus may also help current growth and convergence research to have the kind of
practical relevance that Lucas expressed above so nicely. Observing the status of growth research
in the sixties and early seventies, Sen made the following remark:
“... much of modern growth theory is concerned with rather esoteric issues. Its link with public
policy is often very remote. It is as if a poor man collected his money for his food and blew it on
alcohol.” (Sen 1970, p. 9, our italic)
Romer (1989b, p. 52) noted that considerable intellectual capital accumulation was necessary
before growth research could reach this new stage. It is, therefore, important that the crurrent
stage of growth and convergence research does not head to a state similar to the one that Sen
referred.
50
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