Download Convergence in GDP per capita and real wages: Some results for

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MATHEMATICS
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COMPUTERS
IN SIMULATION
Mathematics and Computers in Simulation 43 (1997) 4 2 9 4 3 6
EI~qEVIER
Convergence in GDP per capita and real wages: Some results
for Australia and the UK
a •
Les Oxley • , David Greasley b
a Department of Economics, Universi~ of Waikato, Private Bag 3105, Hamilton, New Zealand
b Department of Economic History, University of Edinburgh, Edinburgh, Scotland
1. Introduction
Most tests of the convergence hypothesis utilise cross-sectional data and report convergence for
industrial economies (see, for example, Refs. [2,7]). Outside the industrial world convergence club,
there appears less tendency for per capita income differences to narrow (see Refs. [6,11]). Although
diminishing returns provide a simple economic underpinning for the convergence hypothesis, Barro and
Sala i Martin [1] and Mankiw et al. [13] argue that investment in human capital might reduce the
tendency for returns to diminish. Alternatively doubts have grown around the ability of cross-sectional
tests to distinguish convergence. In particular, Bernard and Durlauf [4] identify inconsistencies between
cross-sectional and time-series tests, favouring time-series methods for pure tests of the convergence
hypothesis. Using such tests, Bernard and Durlauf [3,5] reject convergence, even among industrial
economies.
In addition, tests of the convergence hypothesis have considered only convergence in GDP per capita.
However, the mechanism of convergence should relate equally to factor shares, for example, real
wages. In this paper, we will consider to what extent convergence exists using both data for GDP per
capita and real wages for Australia and the UK. Given the close trade, immigration and educational
links between these two countries, it is interesting to test whether any evidence of convergence exists.
This paper deploys time-series unit root-based tests to consider the convergence in GDP per capita
and real wages between Australia and the UK, during the period 1870-1992. Investigating pairwise
GDP per capita convergence between Australia and the UK should shed light on Britain-Australia
economic relations, and on the convergence hypothesis more generally.
Bernard and Durlauf [4] produce a time-series based test definition of convergence. Here, we extend
their work by giving attention to the possibility of structural discontinuities in the convergence process.
* Corresponding author.
0378-4754/97/$17.00 ~ 1997 Elsevier Science B.V. All rights reserved
PH S 0 3 7 8 - 4 7 5 4 ( 9 7 ) 0 0 0 2 8 1
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L. Oxley, D. Greasley /Mathematics and Computers in Simulation 43 (1997) 429-436
Thereafter, we define two convergence hypotheses, report the pairwise time-series test results, and
consider their implications for Australian and British macroeconomic history.
2. Econometric testing of the convergence hypothesis
The economic underpinnings of the convergence hypothesis arise naturally within the standard or
augmented Solow neoclassical growth model. Differences in initial endowments are seen to have no
long-term effects on growth with deficient countries able to catch-up to the leaders that suffer from
diminishing returns. In contrast, Rebelo-type models [18] imply leadership can be maintained, with
non-convergence the likely outcome. As such, not only are tests of convergence interesting in their own
right, but they emerge as one natural testable implication of alternative models of growth. However,
convergence is but one implication of such models and does not in itself represent a full test of the
competing approaches. In order to test for convergence, some form of clear definition and some
appropriate form of data are required. Currently, three basic estimation and related testing approaches
are used.
(i) Barro-type time series o f cross sections: Here, the cross-section correlation between the initial per
capita output levels and subsequent growth rates for a group of countries is examined. Evidence of
convergence is implied if a negative correlation exists, i.e. countries with low per capita initial incomes,
on average, grow faster than those with higher initial per capita incomes. We define the average growth
rate for each of I economies as gi,T
T t (Yi,v - Yi,0). As such, tests of convergence typically involve
estimation of Eq. (1):
gi,T = Ct -k- /3Yio -q- Ei,T
( 1)
where T is a fixed horizon, and e a random disturbance term with mean zero. Support for the hypothesis is inferred if /3 < 0, with the alternative hypothesis, /3 > 0. Some formulations would add
extra control variables to Eq. (1), see for example Ref. [13], where savings and population growth rates
are included. In such cases, /3 < 0 implies that convergence holds conditionally on some set of
exogenous factors.
The main technical problem with such tests is that they cannot identify groupings of countries which
are converging. Furthermore, they are tests of catching-up, not convergence as the /3 < 0 property
represents the process o f converging (catching-up), not convergence itself. Tests of this cross-sectional
type will not be utilised in this study.
(ii) Bernard-Durlauf-type time series [4]: In this approach, tests of convergence examine the longrun behaviour of differences in per capita output across countries. The main feature of this class of tests
is that convergence implies such differences will always be transitory in the sense that long-run
forecasts of the difference between any pair o f countries converge to zero as the forecast horizon grows.
The main testing feature of this approach is that output differences between two economies contain a
unit-root or time trends if the countries have converged. However, the approach also distinguishes
convergence from the tendency to catch-up, where again a strong implication is the absence of a unit
root, although significant time-trend effects can exist. In particular, Durlauf [8], and Bernard and
Durlauf [5], utilise the Dickey-Fuller unit root testing procedure as a time-series based test of
convergence. Here, convergence implies output innovations in one economy should be transmitted
L. Oxley, D. Greasley /Mathematics and Computers in Simulation 43 (1997) 429-436
431
internationally. Utilising slightly different definitions to Bernard and Durlauf [4], this can be illustrated
via the concepts of catching-up and long-run convergence.
Definition.
Catching-up: Consider two countries i and j, and denote their log per capita real output as Yi and yj.
Catching-up implies the absence of a unit root in their difference Yi - Yj. Non-stationarity in Yi - Yj
must violate the proposition, although the occurrence of a non-zero time trend in the deterministic
process in-itself would not.
L o n g - r u n convergence: Consider two countries i and j, and denote their log per capita real output as Yi
and yj.
L o n g - r u n c o n v e r g e n c e implies the absence of a unit root in their difference Yi - Yj a n d the absence of
a time trend in the deterministic process.
Catching-up differs from long-run convergence in that the latter relates to some particular period T
equated with long-run equilibrium. In the former case, the existence of a time trend in the stationary
Yi - Yj series would imply a narrowing of the (log per capita output) gap, or simply that the countries
though catching-up had not y e t converged. This catching-up could be oscillatory, but must imply nondivergence of output differences. Conversely, the absence of a time trend in the stationary series implies
that catching-up has been completed.
As defined in the foregoing, tests of catching-up and long-run convergence hinge, therefore, on the
time-series properties of Y i - Yj. The natural route for such tests involves Dickey-Fuller-type tests
based on the bi-variate difference in log per capita output between pairs of countries, i and j, i.e.
Yit -- Yj, = # + a(Yi,t 1 -- Yj,t 1) -~- /3l -'['- ~
6kA(yi,t k -- Yj,t k) + Ct
(2)
k 1
where y indicates the logarithm of per capita output. If the difference between the output series contains
a unit root, a = 1, output per capita in the two economies will diverge. The absence of a unit root,
c~ < 1, indicates either catching-up, if/3 ¢ 0, or long-run convergence, if/3 = 0.
A testing strategy: Testing for c o n v e r g e n c e within a Bernard-Durlauf-type framework, involves a
two-stage process. Firstly, check for the existence of a unit root in the difference b e t w e e n p e r capita
i n c o m e in the two countries concerned. The non-rejection of a unit root implies non-stationarity and
rejection of the time-series property implications of the convergence hypothesis. Contingent on
rejection of the unit root hypothesis is the convergence criteria requirement that significant time-trend
effects are absent. Furthermore, strong convergence would imply the insignificance of the constant
term.
Testing for catching-up involves the first two stages of testing for convergence, i.e. rejection of a unit
root and checks on the significance of the time trend, but becomes an issue only if the time trend is
significant. Notice that the concept does not require a zero constant. However, catching-up as a longrun property of the model would be inconsistent with a constant, non-time varying, time trend as,
asymptotically, a constant time-trend effect would always imply d i v e r g e n c e as t ~ e~. Therefore,
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L. Oxley, D. Greasley /Mathematics and Computers in Simulation 43 (1997) 429~436
catching-up characterised by stationary output differences and a constant time trend is relevant only for
a particular finite T, and only on the basis that the countries have not already converged. Hence,
checking for convergence would always be the first stage, which requires stationary output differences,
followed by tests for catching-up. Note, stationary output differences would imply either convergence,
catching-up on 'parallelism', i.e. identical growth rates for all time periods.
The main reservation surrounding the robustness of unit-root tests in general, and therefore their
application to tests of convergence in particular, concerns the possibility that structural discontinuities
in the series may lead to erroneous acceptance of the unit-root hypothesis. Perron [14] and Rappoport
and Reichlin [17], consider the importance of incorporating the effects of discontinuities when
investigating the statistical properties of long-run historical series. Alternatively, Zivot and Andrews
[19], contend that discontinuities have been too readily accepted and that macroeconomic time series
usually contain a unit root.
Applying Perron's unit-root testing strategy requires the prior specification of breakpoint years. Zivot
and Andrews [19] favour an alternative approach based upon recursive searching for endogenous
discontinuities at every year within the sample. They deploy dummies similar to Perron's to incorporate
crash and trend breaks, but dispense with the single-year dummy. Critical values reported by Zivot and
Andrews [19] for testing the significance of a with breaks at any year much greater in absolute value
than Perron [14], raising a stringent barrier to the rejection of a unit root.
(iii) Haldane and Hall [9] time varying parameter models. One of the main problems with the
Bernard-Durlauf approach relates to the testing implications of the catching-up proposition. Without
careful application, that is, treating tests of convergence as a pre-test, tests of catching-up inevitably
imply asymptotic divergence, although after a period of catching-up and convergence at some T.
Furthermore, a fixed time trend does not permit study of the process of converging~catching-up.
Haldane and Hall [9] utilise the properties of the Kalman filter, time-varying parameter, approach to
permit and, therefore, study the evolution of the trend in, in their case, bilateral exchange rates. The
approach can be straightaway applied to the model of per capita output differences giving the advantage
of providing a time series of time-trend coefficients.
3. Time-series test results for convergence
This section reports tests for long-run convergence, and catching-up. On the basis of the results in
Table 1 based on Eq. (1), for the 1870-1992 period, neither version of the convergence hypothesis
receives support, since a unit root cannot be rejected in the cross-country differences in GDP per capita.
However, the likelihood of structural continuities in the British and Australian growth records
suggests their impact on the convergence process warrants investigation.
Table l
Unit root tests. Differences in G D P per capita (without discontinuities)
Countries
Sample
ADF a
LM (SC)
Q6
U K-Australia
UK-Australia
1870-1992
1892-1992
- 3.250
-5.531 *
0.792
0.309
4.029
7.969
* Significant at the 5% level, based on MacKinnon [12].
a A D F denotes ADF(4).
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L. Oxley, D. Greasley/Mathematics and Computers in Simulation 43 (1997) 429-436
Table 2
Unit root tests: differences in GDP per capita [19] approach (results based upon ADF(4))
Country
Year
Crash
Trend
UK-Australia +
1870-1992
-5.418 *
[1981]
-4.192
[1935]
a
Crash and trend
a
-5.440 *
[1891]
a
* Significant at the 5% level, based upon Zivot and Andrews [19].
a [] denotes the year of the maximum absolute value of the ADF.
.063425
.011288
J
/
-. 040849
/
I'
I
"
0929861870
.............
1901 . . . . . . . . . . . . . . . . .
1932 ............
1"963 . . . . . . . . . . . . .
199~
Fig. 1. Time-varying parameter methods. Differences in GDP per capita, UK-Australia coefficient on time-varying time trend
(plus 2 s.e. bands.).
Table 2 reports the results of applying the Zivot and Andrews [19] searching methods to the
G D P per capita output differences between Australia and the UK. Here, a crash, or crash-andtrend change, dated at 1891, overturns the unit-root hypothesis implying either catching-up or
convergence.
However, the lack o f a significant time-trend effect and, therefore, the importance of the crash
effect alone favours the stronger effect o f convergence. The second set of entries in Table 1
reinforce these results, showing that for the period 1 8 9 2 - 1 9 9 2 the unit-root hypothesis is
rejected.
Application o f the Haldane and Hall [9] time-varying parameter approach supports such conclusions
(see Fig. 1), which plots the coefficient on the time-varying time trend.
The results therefore show how the omission of significant discontinuities can lead to incorrect
inferences being drawn regarding convergence and, what is more important, the possible causes of
economic growth. In contrast with previous time-series studies, the results show that Australia and the
U K appear to have attained long-run convergence. The results lend support to exogenous, Solow-type
L. Oxley, D. Greasley/Mathematics and Computers in Simulation 43 (1997) 429~136
434
Table 3
Unit root tests: differences in real wages (without discontinuities)
Countries
Sample
ADF a
LM (SC)
Q6
UK-Australia
1855-1988
-4.105 *
2.353
1.030
* Significant at the 5% level, based on MacKinnon 112].
a ADF denotes ADF(2).
• 16594
.095979
• 026022
/
J
/
-. 043935
r854
.................
.......................
.................
i996
Fig. 2. Time-varying parameter methods. Differences in real wages, UK-Australia coefficient on time-varying time trend
(plus 2 s.e. bands).
growth modelling strategies, while no case supports one important implication of the Rebelo model,
namely long-term non-convergence. However, as stated earlier, such implications do not constitute full
tests of the growth models.
Turning to results for differences in real wages between Australia and the UK for the longer period,
1855-1988, Table 3 shows that the unit hypothesis is rejected without recourse to Zivot and Andrews
methods. However, the significant time-trend effect now favours the weaker notion of catching-up,
rather than convergence.
Again, the Haldane and Hall [9] methods, presented as Fig. 2, which plot the coefficient on the timevarying trend, support such a conclusion.
As a final set of tests, consider whether British and Australian wages are cointegrated. Table 4 shows
that British and Australian real wages are individually integrated, of order 1, I(1).
However, Table 5, which reports the results of implementing the Johansen [10], cointegration
approach, shows that they are cointegrated, strengthening the notion that real wages will eventually
converge, once the process of catching-up has been completed.
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L. Oxley, D. Greasley /Mathernatics and Computers in Simulation 43 (1997) 429-436
Table 4
Unit root tests: real wages
Countries
Sample
ADF a
LM (SC)
Q6
UK
Australia
1855-1988
1855-1988
-1.555
-2.029
1.161
2.987
1.236
5.471
ADF denotes ADF(2).
Table 5
Testing for cointegration, Johansen [10] method, real wages
Variable
Maximum eigenvalue Trace statistic
statistic
H0
H1
/3
VAR
Real wages
25.431 *
0.899
r= 0
r < 1
r= 1
r : 2
1.408
1
26.330 *
0.899
* Significant at the 95% level, r = 0 represents no cointegrating vector;/3 the coefficient value from vector 1 normalised on
the UK; VAR defines the lag in the vector auto-regression chosen on the basis of economic interpretation and the properties of
the cointegrating vector residuals.
4. C o n c l u d i n g
remarks
D o u b t s h a v e g r o w n in r e c e n t y e a r s a b o u t the v a l u e o f the c o n v e r g e n c e h y p o t h e s i s . Partly, the
s c e p t i c i s m c o n c e r n s the i n a b i l i t y o f c r o s s - s e c t i o n a l s t u d i e s to d i s c e r n c o n v e r g e n c e o t h e r than b e t w e e n a
n a r r o w r a n g e o f i n d u s t r i a l c o u n t r i e s , b u t m o r e f u n d a m e n t a l l y the b a s i c u t i l i t y o f the c r o s s - s e c t i o n a l
a p p r o a c h h a s b e e n q u e s t i o n e d (see, f o r e x a m p l e , R e f s . [3,4,15,16]). M o r e o v e r , t h e r e c e n t d e v e l o p m e n t
o f t i m e - s e r i e s c o n v e r g e n c e tests h a s l e d to r e s u l t s w h i c h h i t h e r t o h a v e b e e n w h o l l y u n f a v o u r a b l e to the
c o n v e r g e n c e h y p o t h e s i s , e v e n f o r i n d u s t r i a l e c o n o m i e s . T h e n o v e l t y o f this p a p e r ' s s t a t i s t i c a l f i n d i n g s
lies in t h e s u p p o r t f o u n d f o r the c o n v e r g e n c e h y p o t h e s i s v i a a t i m e - s e r i e s p e r s p e c t i v e . I n c o r p o r a t i n g
d i s c o n t i n u i t i e s l e d to the r e j e c t i o n o f a u n i t r o o t f o r U K - A u s t r a l i a n G D P p e r c a p i t a a n d U K - A u s t r a l i a n
r e a l - w a g e s series.
In t h e m s e l v e s t h e s e r e s u l t s o f f e r s u p p o r t f o r the c o n v e r g e n c e h y p o t h e s i s , w h e n a p p l i e d to G D P p e r
c a p i t a a n d c a t c h i n g - u p in t h e c a s e o f r e a l w a g e s .
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