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Economics Letters 56 (1997) 143–147
Time-series based tests of the convergence hypothesis: Some
positive results
a
b,
David Greasley , Les Oxley *
a
Department of Economic History, University of Edinburgh, Edinburgh, UK
Department of Economics, University of Waikato, Hamilton, New Zealand
b
Received 15 May 1996; received in revised form 13 May 1997; accepted 18 May 1997
Abstract
Using time-series based tests, proposed by Bernard and Durlauf (1996), and annual time series data, 1900–1987, on
several OECD countries, the paper identifies bivariate convergence between: Belgium and The Netherlands; France and
Italy; Australia and the United Kingdom; and Sweden and Denmark.  1997 Elsevier Science S.A.
Keywords: Economic growth; Convergence; Discontinuities
JEL classification: C22; C52
1. Introduction
Tests of the convergence hypothesis, or the tendency for per capita income levels to equalize over
time, have attracted considerable attention, see for example, Barro and Sala i Martin (1994); Dowrick
and Nguyen (1989); Quah (1993). For an excellent overview of the area including economic
underpinnings and econometric evidence see Barro and Sala i Martin (1995). In a series of recent
developments Bernard and Durlauf have extended the traditional testing methodology by arguing in
favour of time series rather than cross-sectional based tests, see Bernard and Durlauf, hereafter BD
(Bernard and Durlauf, 1995, 1996) and Durlauf and Johnson (1995). Although these developments
are both innovative and appealing, they have generally found little evidence of convergence in the
comparisons investigated.
In this paper we will demonstrate that convergence can be identified for several pairs of countries;
in particular: Sweden and Denmark; France and Italy; Belgium and The Netherlands; and Australia
and the United Kingdom. This will be demonstrated via a series of bi-variate, (two country
differences), unit root tests based upon both traditional ADF approaches and those of Perron (1989);
Zivot and Andrews (1992). The latter methods prove useful given the well known propensity for
discontinuities to bias test results in favour of non-rejection of the unit root null hypothesis. Section 2
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D. Greasley, L. Oxley / Economics Letters 56 (1997) 143 – 147
144
outlines the basis of the BD time-series tests of convergence, with Section 3 presenting the results.
Section 4 concludes.
2. Unit root-based tests of convergence
Bernard and Durlauf use time-series tests to consider convergence in an explicitly time series
setting based upon differences between countries GDP per capita. In particular consider the
following: define y i as the log real GDP per capita in country i and likewise y j for country j. Define
the differences in real GDP per capita in countries i and j, y i 2 y j . Define It as the information set
available at period t. Bernard and Durlauf (1996), p. 165, defines (Definition 2):
Convergence as equality of long term forecasts at a fixed time. Countries i and j converge if the
long term forecasts of ( log) per capita output for both countries are equal at a fixed time t.
lim E( y i,t1k 2 y j,t 1k uIt ) 5 0
k→`
(1)
In a time series testing framework, testing such a definition relies upon the time-series properties of
the output per capita series. In particular, Bernard and Durlauf (1996), p. 170, demonstrate via
Proposition 5 that: If y i 2y j contains either a non zero mean or a unit root, then Definition 2 (stated
above), of convergence is violated.
3. The data and univariate unit root tests
The data used relates to real GDP per capita, 1900–1987, and is taken from BD (1995), which is
based upon Maddison’s (Maddison, 1982, 1989) GDP and population estimates. Fig. 1 reproduces log
per capita GDP for: Sweden and Denmark; Belgium and The Netherlands; France and Italy; and
Australia and the United Kingdom.
Table 1 below presents the results of ADF tests of the bi-variate differences in real output per
capita.
On this basis of the ADF tests, the unit root null is rejected in the case of: France and Italy;
Belgium and The Netherlands; and Australia and the United Kingdom, where both intercept and trend
terms are insignificantly different from zero. However, for Sweden and Denmark the unit root null is
not rejected, implying violation of BD’s Proposition 5, and, apparently, non-convergence. It is well
known, however, that discontinuities in the bivariate differences may bias the test statistic in favour of
the unit root hypothesis and Table 2 presents the results of applying the Perron (1989) approach.
The discontinuity most likely to affect the Nordic economies comparative performance is Swedish
neutrality and the occupation of Denmark during World War Two. When a 1939 ‘crash’ is modelled
according to the Perron (1989) approach, there is strong evidence in favour of convergence between
Sweden and Denmark, with the null hypothesis of a unit root strongly rejected, see Table 2a. Using
the Zivot and Andrews (1992) (ZA) approach confirms the optimality of a 1939 break. However,
although for the full sample, an insignificant time trend is estimated indicating common trends, a
significant intercept is identified—this is what the 1939 ‘crash’ is identifying; see Fig. 2.
Table 2b, which presents results for the period 1940–87 without discontinuities, establishes
D. Greasley, L. Oxley / Economics Letters 56 (1997) 143 – 147
145
Fig. 1. Real GDP per capita, 1900–1987. X axis, time is measured (1900–1987); Y axis, numbers refer to logs. Countries identified are:
Belgium, Netherlands, France, Italy, Australia, UK, Denmark and Sweden.
Table 1
Unit root tests (without discontinuities) (log) differences, GDP per capita, 1900–1987
Countries
k
ADF
p value on
intercept
p value on
time trend
LM(SC)
( p value)
France–Italy
Belgium–The Netherlands
Australia–UK
Sweden–Denmark
3
3
2
2
24.196 *
23.469 *
23.710 *
21.709
0.759
0.367
0.261
0.199
0.829
0.328
0.722
0.165
0.633
0.178
0.582
0.296
* Denotes significant at the 5% level based upon MacKinnon (1991); k is the degree of augmentation; p value is the
probability value; LM(SC) is the LM statistic on the test of AR1 errors in the ADF equation; k is chosen on the basis of AIC
and BIC measures, conditional upon the removal of serial correlation in the errors of the ADF equation. These criteria are
used in all the results.
Table 2
Unit root tests
Countries
(a)
Sweden–Denmark
(b)
Sweden–Denmark
Period
Crash
k
p value on
intercept
p value on
time trend
LM(SC)
( p value)
1900–87
25.364 *
1
0.000
0.536
0.432
1940–87
27.816 *
1
0.237
0.470
0.242
Crash denotes ADF(k) statistic based upon the Perron (1989) ‘crash’ model.
* Significant at the 5% level based upon Perron (1989) or MacKinnon (1991), as appropriate.
For the (a) section, 1939 was year of discontinuity.
D. Greasley, L. Oxley / Economics Letters 56 (1997) 143 – 147
146
Fig. 2. Log per capita GDP (Sweden and Denmark). X axis, time is measured; Y axis, measures logs.
convergence via a rejection of the unit-root null and the insignificance of the time trend and the
intercept term.
4. Conclusions
Using bivariate comparisons and a careful use of unit root tests allowing (when necessary), for
discontinuities, we have identified bivariate convergence between: France and Italy; Belgium and The
Netherlands; Australia and the United Kingdom; and Sweden and Denmark. Other bi-variate
comparisons drawn from the BD (1995), data set, lead to a rejection of the convergence hypothesis.
Convergence in the strict sense of BD appears to be a difficult hypothesis to confirm, however,
examples can be identified and further work in this area will clearly be desirable.
References
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