Studies using the dynamic panel regression approach have found the speed of income convergence among the world and regional economies to be high. For example, Lee et al. (1997, 1998) report the income convergence speed to be 30% per annum. This note argues that their estimates may be seriously overstated. Using a factor model, we show that the coefficient of the lagged income in their specification may not be the long-run convergence speed, but the adjustment speed of the short-run deviation from the long-run equilibrium path. We give an example of an empirical analysis, where the short-run adjustment speed is about 40%.Convergence speed, Dynamic panel regression, Factor model

Masahiko Shibamoto

Yoshiro Tsutsui

English

Research Papers in Economics

 
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*OTPNFDBTFTBXSJUUFODPOTFOUPGUIFBVUIPSNBZCFSFRVJSFE Note on the Interpretation of Convergence Speed in the
Dynamic Panel Model
MASAHIKO SHIBAMOTOa∗AND YOSHIRO TSUTSUIb
a Research Institute for Economics and Business Administration, Kobe University
b Graduate School of Economics, Osaka University
January 20, 2011
Abstract
Studies using the dynamic panel regression approach have found the speed of income
convergence among the world and regional economies to be high. For example, Lee
et al. (1997, 1998) report the income convergence speed to be 30% per annum. This
note argues that their estimates may be seriously overstated. Using a factor model,
we show that the coeﬃcient of the lagged income in their speciﬁcation may not be the
long-run convergence speed, but the adjustment speed of the short-run deviation from
the long-run equilibrium path. We give an example of an empirical analysis, where
the short-run adjustment speed is about 40%.
JEL Classiﬁcation: O40;
Keywords: convergence speed, dynamic panel regression, factor model
∗Corresponding author: Research Institute for Economics and Business Administration, Kobe Univer-
sity, 2-1 Rokkodai, Nada, Kobe 657–8501, Japan. E-mail: shibamoto@rieb.kobe-u.ac.jp.
11 Introduction
This note focuses on the interpretation of the speed of convergence with which the log-
arithm of per-capital real income tends to its steady-state value in the dynamic panel
regression model. Researchers using the dynamic panel regression approach, e.g., Islam
(1995, 1998), Lee et al. (1997, 1998), and Shioji (2001), argue that economies converge at
a relatively high β-convergence speed as compared to the speed obtained in the analyses
of conventional cross-sectional regression.
In this note, we argue that their estimates of the speed of β-convergence may be
seriously overstated. Our point is that their calculations of the speed of β-convergence
do not take into account the dynamics of a common component of the per capita real
income, which may be highly persistent. Consequently, their estimates are not of the
convergence speed in the long run, but the adjustment speed of the short-run deviation
from the long-run equilibrium path.
2 Controversies on the convergence speed obtained using
dynamic panel regression
We denote the logarithm of the per capita real income of the economy i at time t as yit.
Let us start from the most basic model of dynamic panel regression:
yit = μ + byit−1 + γ
 
Θt + ζit, for all i =1 ,···,N, (1)
where ζit is an independent disturbance. μ is an economy-wide intercept that reﬂects the
initial resource endowments. Θt is a vector of some time-speciﬁc eﬀects (e.g., a linear time
trend, an economy-wide average of real income, and a shock common to all economies)
that reﬂects the technological progress across economies.1
This regression is a panel analog to the Barro regression and it is natural to interpret
the coeﬃcient of the lagged real income term as a function of the speed of convergence
β: namely, b = exp(−β). Many cross-sectional analyses employing the Barro regressions,
1For simplicity, we exclude some exogenous variables as the determinants of economic growth such
as the savings rate, measures of investment in physical and human capital, and so on to focus on the
interpretation of the convergence speed using the panel data approach.
2e.g., Barro (1991) and Barro and Sala-i-Martin (1992a,b), report that the estimate of β is
around 2% per annum.
Islam (1995) points out that allowing for intercept heterogeneity, i.e., μi instead of μ in
equation (1), increases the estimate of the speed of convergence to 5% per annum. Indeed,
if there is a positive correlation between these intercepts and the lagged real incomes, the
conventional cross-section estimate of b will be biased upward.
While Islam (1995) estimates the model where only the steady-state level of income is
allowed to diﬀer across economies and where the assumption that economies shared the
same steady-state growth rate, given as γi = γ for all i =1 ,···,N is made. Lee et al.
(1998) criticize Islam (1995) for failing to allow for diﬀerent common trends across the
countries.2 Their model is given as
yit = μi + biyit−1 + γ
 
iΘt + ζit, for all i =1 ,···,N, (2)
which allows for the existence of non-parallel balanced growth paths. Lee et al. (1997,
1998) argue that homogeneity in the growth eﬀects of convergence forces the estimate of b
toward 1, because it renders the panel estimator inconsistent.3 Based on this model, they
report that the speed of β-convergence increases to around 30% per annum.
We support the generalization by Lee et al. (1997, 1998), since such a model is ﬂexible
to describe various economies. However, we cast doubt on the interpretation that the
coeﬃcient of the lagged dependent variable, bi, represents the convergence speed in the
long run, which is discussed in the next section.
2In response, Islam (1998) argues that Lee et al. (1997, 1998) are assessing an economically uninteresting
form of convergence when they allow for trend diﬀerences. Durlauf et al. (2005) comment that “this debate
is an excellent example of the issues of interpretation that are raised in moving between speciﬁc economic
hypotheses and more general statistical models.”
3We can understand that Lee et al. (1997, 1998)’s speciﬁcation is appropriate by looking at the recent
literature on the panel unit root test, e.g., Phillips and Sul (2003). Phillips and Sul (2003) argue that if
the cress-sectional correlation for the error term is not accounted for, the estimates of the autoregressive
coeﬃcients will be biased. In order to control the cross-sectional correlation, the error term is given by a
three-component model that contains a ﬁxed eﬀect (μi), a common factor (Θt), and a purely idiosyncratic
factor (ζt). Then, it is standard practice to use a similar speciﬁcation for equation (2) in the panel unit
root test.
33 Interpretation of the key coeﬃcient in dynamic panel re-
gression in terms of a factor model
The model described as equation (2) is similar to the common factor model by previous
studies, e.g., Bai and Ng (2004), in that it allows for heterogeneous (non-parallel) growth
paths. Based on the factor model, we will show that the interpretation of bi as the long-run
convergence speed is inappropriate.
Bai and Ng (2004) develop the “Panel Analysis of Nonstationarity in Idiosyncratic and
Common Components (PANIC).” The idea of PANIC is to decompose the observed panel
data into the common and idiosyncratic components, and then test for stationarity in each
component separately. Bai and Ng (2004) consider the following factor model:
yit = vi + λ
 
iFt + eit, (3)
where Ft is the common component consisting of r factors followed by the economies, λi is
a parameter vector that represents the diﬀerent weights assigned to the common factors,
and eit for i =1 ,···,N is the idiosyncratic component with a zero mean and is orthogonal
to the common factors Ft and to itself.
Let us assume that each component, eit,F t, follows the autoregressive regression model:
eit = bieit−1 + ζit, for all i =1 ,···,N, (4)
Ft = φFt−1 + vt, (5)
where ζit for all i =1 ,···,N and vt are the error terms with a zero mean. An ζit shock
propagates through the autocorrelation structure of equation (4).
From equations (3) and (4), we can derive
yit = vi + λ
 
iFt + bieit−1 + ζit
= vi(1 − bi)+biyit−1 + λ
 
iFt − biλ
 
iFt−1 + ζit, for all i =1 ,···,N, (6)
which is observationally equivalent to the equation (2) with μi = vi(1−bi), γi =[ λ
 
i −biλ
 
i]
 
,
and Θt =[ F
 
t F
 
t−1] . As the result shows, the coeﬃcient of the lagged real income b in
4equation (6), which is the coeﬃcient of the lagged income in equation (2), is nothing but
the adjustment speed of the idiosyncratic component eit, as equation (4) shows.
If the idiosyncratic component is stationary, bi represents the adjustment speed of
short-run deviation. Even if the idiosyncratic component is non-stationary, bi is biased
as the long-run convergence speed, as long as the common factor is non-stationary. In
other words, the interpretation of bi as the long-run convergence speed is valid only if the
common factors Ft do not exist or they are stationary processes, which seems improbable
in most cases.4
4 Example
Let us give an example of the adjustment speed of the short-run deviation. The example
is an analysis using the per capita real income data for 46 Japanese prefectures, for the
period 1955–1999. The estimation of the Bai and Ng (2004) model reveals that there
exists one common component Ft, which is nonstationary, and idiosyncratic components
eit for i =1 ,···,46, which are all stationary (see Shibamoto et al., 2011). Thus, in this
example, the estimate of bi in a type of equation (2) is nothing but the adjustment speed
of the short-run deviation from the long-run path. In this case, the per capita incomes of
the economies cointegrate with the common component with diﬀerent long-run weights,
which is characterized by vi + λiFt.5 This implies that the economies have heterogeneous
balanced growth paths. On the other hand, the idiosyncratic component, eit, which has
only a transitory eﬀect on the per capita real income, reﬂects the short-run deviation from
the long-run path for the economy i.
Then, we estimate equation (4) to get the estimate of bi by ordinary least squares.6
The mean value of the estimated bi across prefectures was 0.67, implying that the speed
is about 40% (= −log(0.67)) per annum. This estimate is very large if we regard this as
the long-run convergence speed, but it is a reasonable value for the adjustment speed of
the short-run deviation, because short-run variations in economies, i.e., business cycles,
4Although we explain the case wherein eit and Ft follow the ﬁrst-order autoregressive model in equations
(4) and (5) for simplicity, the argument applies for any order of the autoregressive model.
5Bernard and Durlauf (1995)’s Deﬁnition 2.2 of “common trends in output” embodies this idea.
6We add as explanatory variables four lags of the ﬁrst diﬀerence of ˆ eit in equation (4) to remove the
autocorrelation of the residuals.
5normally return to their former levels in several years. This ﬁgure is comparable to 30% of
the “speed of β-convergence” reported by Lee et al. (1997, 1998), which suggests that the
ﬁgure reported by them is not really the long-run convergence speed, but the adjustment
speed in the short-run.7
References
Bai, Jushan, Serena Ng. 2004 “A PANIC Attack on Unit Roots and Cointegration”, Econo-
metrica, 72(4): 1127–1177.
Barro, Robert J., Xavier Sala-i-Martin. 1992a “Convergence”, Journal of Political Econ-
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Barro, Robert J., Xavier Sala-i-Martin. 1992b “Regional Growth and Migration: A Japan-
United States Comparison”, Journal of the Japanese and International Economies, 6(4):
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Barro, Robert J.. 1991 “Economic Growth in a Cross Section of Countries”, Quarterly
Journal of Economics, 106(2): 407–443.
Bernard, Andrew B., Steven N. Durlauf. 1995 “Convergence in International Output”,
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Durlauf, Steven N., Paul A. Johnson, Jonathan R.W. Temple. 2005 “Growth Economet-
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7They admit that the interpretation of this value as the β-convergence speed is highly questionable.
6Lee, Kevin, M. Hashem Pesaran, Ronald Smith. 1998 “Growth Empirics: A Panel Data
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Note on the Interpretation of Convergence Speed in the Dynamic Panel Model

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Note on the Interpretation of Convergence Speed in the Dynamic Panel Model

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