On Testing Conditional Sigma – Convergence*
| Author | Michael Pfaffermayr,Peter Egger |
| Date | 01 August 2009 |
| Published date | 01 August 2009 |
| DOI | http://doi.org/10.1111/j.1468-0084.2007.00467.x |
453
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OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 71, 4 (2009) 0305-9049
doi: 10.1111/j.1468-0084.2007.00467.x
On Testing Conditional Sigma – Convergence*
Peter Egger† and Michael Pfaffermayr‡
†Ifo Institute, University of Munich and CESifo, Munich, Germany (e-mail: egger@ifo.de)
‡Department of Economics, University of Innsbruck, Innsbruck, Austria
(e-mail: michael.pfaffermayr@uibk.ac.at)
Abstract
In a cross-section where the initial distribution of observations differs from the steady-
state distribution and initial values matter, convergence is best measured in terms of
-convergence over a fixed time period. For this setting, we propose a new simple
Wald test for conditional -convergence.According to our Monte Carlo simulations,
this test performs well and its power is comparable with the available tests of uncon-
ditional convergence. We apply two versions of the test to conditional convergence
in the size of European manufacturing firms. The null hypothesis of no convergence
is rejected for all country groups, most single economies, and for younger firms of
our sample of 49,646 firms.
I. Introduction
Cross-sectional studies on convergence typically rely on the concept of -conver-
gence, which refers to the negative correlation of initial values and growth rates. The
Cross-sectional approach to convergence is warranted, if starting values matter and
if we observe data over a fixed period of time in transition towards their steady-state
limiting distribution. Examples are data on firm size or firm-level output per worker.
Firm size or output per worker may be way off the steady-state level, if the distribution
of the starting values differs markedly from the limiting distribution.
In a stationary setting, convergence defined as the decline in the dispersion of a
distribution over time requires reversion to the long-run mean after the occurrence
of a random shock. In addition, it is necessary that the process starts out at a higher
*We are grateful to JonathanTemple, Janette Walde, Rainer Winkelmann, Ludger W¨ossmann, the partici-
pants of the University of Zurich research seminar and an anonymous referee for constructive and helpful
comments.
JEL Classification numbers: C21, L11, O47.
454 Bulletin
dispersion than that of the steady-state distribution. As a consequence of mean rever-
sion, the impact of the initial distribution fades away over time and the variance
of firm size or output per worker falls in spite of constant parameters of the cor-
responding first-order autoregressive process. Hence, in this setting convergence is
interpreted as the dissipation of the initial differences over time. It is well known that
-convergence is a necessary, however, not a sufficient condition for convergence
in this sense. Hart (1995) among others argues that one may simultaneously infer
-convergence and a constant or even increasing variance between two points in
time. To mistakenly infer a decline in variance from evidence of -convergence
(mean reversion) is known as committing Galton’s fallacy. Therefore, Carree and
Klomp (1997), Friedman (1992), Hart (1995, 2000) and Lichtenberg (1994) empha-
size that tests for convergence should investigate whether the variance indeed
decreases between two points in time, i.e. whether there is -convergence (Barro
and Sala-i-Martin, 1995).
This paper proposes a simple Wald test for conditional -convergence in a cross-
sectional with a fixed time interval between the first observed and the final values. The
Wald test is suited to test either unconditional or, more importantly, conditional con-
vergence in a cross-sectional setting and it is very easy to implement. It only requires
the estimation results of the respective -convergence regression and the variance
of the values at some earlier point in time. We abstract from the analysis of twin-
peaks as emphasized by Quah (1993). Rather, we follow the literature on conditional
-convergence by assuming that exogenous variables account for group-specificor
individual-specific steady states (see Barro and Sala-i-Martin, 1995). The Wald test
allows for heterogeneous, even individual-specific steady states, which cannot be
tackled by the available tests for -convergence of Carree and Klomp (1997).
Our Monte Carlo simulations indicate that the proposed Wald test performs well.
It is properly sized and reveals sufficient power in the samples usually used in conver-
gence studies. Under unconditional convergence the power of the Waldtest is similar
to the available tests. Specifically, the Monte Carlo simulations demonstrate that the
power of the Waldtest rises with the extent of the variance reduction in the considered
time interval. It is higher the closer the initial observation is to the birth date of the
process. The power decreases with the length of time interval under consideration.
These findings suggest that the average growth rates used as the dependent variable
in the convergence equation should be based on short time intervals starting not too
much later than the birth of the process.
The Wald test is applied to conditional convergence in the size of 49,646 Euro-
pean manufacturing firms in 23 European countries. We reject the null hypothesis of
no convergence for all considered country groups, for most of the single countries,
and for the younger firms of our sample of western European and central and eastern
European firms. To a large extent, our estimation results confirm the rejection of
Gibrat’s law (i.e. that firm growth is independent of initial size) of many studies based
on -convergence regressions. For eight small countries and, with a few exceptions,
for firms of age 40 or higher the Wald test does not reject. The risk of committing
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford 2007
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