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working paper
department
of economics
Convergence in International Output
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
Convergence in International Output
Andrew
Steven
No.
93-7
B.
N.
Bernard
Durlauf
May 1993
AUG
1
2 1993'
Convergence
Andrew
in International
B. Bernard
Steven N. Durlauf
Department of Economics
Department
MA
of
Economics
Stanford University
M.I.T.
Cambridge,
Output 1
02139
Stanford,
CA
94305
May, 1993
*We thank Suzanne Cooper, Chad Jones, and Paul Romer for useful discussions. The authors
thank participants at the NBER Summer Workshop on Common Elements in Trends and Fluc-
also
and LSE, and 3 anonymous referees for helpful comments.
Economic Policy Research provided financial support. Bernard gratefully acknowledges dissertation support from the Alfred P. Sloan Foundation. All errors are ours.
tuations, seminars at Cambridge, Oxford,
The Center
for
Summary
This paper proposes and tests new definitions of convergence and
We
capita output.
define convergence for a group of countries to
common
mean
has identical long-run trends, either stochastic or deterministic, while
for proportionality of the stochastic elements.
These
reject
trends allow
definitions lead naturally to the use of
convergence but find substantial evidence for
European countries
that each country
common
cointegration techniques in testing. Using century-long time series for 15
we
trends for per
common
also reject convergence but are driven
OECD
economies,
trends. Smaller samples of
by a lower number of common
stochastic trends.
1
Introduction
One
of the most striking features of the neoclassical growth
model
is
its
implication for
cross-country convergence. In standard formulations of the infinite- horizon optimal growth
problem, various turnpike theorems show that steady-state per capita output
dent of
output
initial
levels.
is
indepen-
Further, differences in microeconomic parameters will gener-
ate stationary differences in per capita output and will not imply different growth rates.
when one observes
Consequently,
differences in per capita output
growth across countries,
one must either assume that these countries have dramatically different microeconomic
characteristics, such as different production functions or discount rates, or regard these
discrepancies as transitory.
Launched primarily by the theoretical work of Romer (1986) and Lucas (1988), much
attention has been focused on the predictions of dynamic equilibrium models for long-term
behavior when various Arrow-Debreu assumptions are relaxed.
shown that divergence
scale associated with
in
Lucas and Romer have
long-term growth can be generated by social increasing returns to
both physical and human
capital.
An
empirical literature exploring
convergence has developed in parallel to the new growth theory. Prominent
contributions
is
the work of
Baumol
(1986),
DeLong
among
(1988), Barro (1991) and
these
Mankiw,
Romer, and Weil (1992). This research has interpreted a finding of a negative cross-section
correlation between initial income
The use
and growth rates as evidence
in favor of convergence.
of cross-section results to infer the long-run behavior of national output ignores
valuable information in the time series themselves, which can lead to a spurious finding of
convergence. First,
it is
possible for a set of countries which are diverging to exhibit the sort
of negative correlation described by
is
diminishing.
Baumol
et al. so long as the
marginal product of capital
As shown by Bernard and Durlauf (1992), a diminishing marginal product
1
of capital
means that short-run
be mixed up
will
dynamics and long-run, steady-state behavior
transitional
in cross-section regressions.
Second, the cross-section procedures work
with the null hypothesis that no countries are converging and the alternative hypothesis
that
all
countries are, which leaves out a host of intermediate cases.
In this paper
Our research
we propose a new
and
definition
set of tests of the
from most previous empirical work
differs
framework.
explicitly stochastic
If
same across
countries.
we
test
convergence
in
an
long-run technological progress contains a stochastic
trend, or unit root, then convergence implies that the
the
in that
convergence hypothesis.
The theory
permanent components
output are
in
of cointegration provides a natural setting for testing
cross-country relationships in permanent output movements.
Our
examines annual log
analysis, which
real
output per capita for 15
OECD
economies
from 1900 to 1987, leads to two basic conclusions about international output fluctuations. 1
First,
we
find very little evidence of convergence across the economies.
deviations do not appear to disappear systematically over time. Second,
is
common
strong evidence of
countries.
As a
result,
Per capita output
we
find that there
stochastic elements in long-run economic fluctuations across
economic growth cannot be explained exclusively by idiosyncratic,
country-specific factors.
A
relatively small set of
common
long-run factors interacts with
individual country characteristics to determine growth rates.
Our work
Quah
(1990)
is
related to studies by
who have
Campbell and Mankiw (1989), Cogley (1990), and
explored patterns of persistence in international output.
quarterly post- 1957 data, Campbell and
exhibit both persistence
and divergence
Mankiw demonstrate
in output. Cogley,
examining
using a similar data set to the one here, concludes that persistence
countries; yet at the
same time he argues that common
OECD
9 OECD
that 7
is
Using
economies
economies
substantial for
many
factors generating persistence imply
Quah
finds
a lack of convergence for a wide range of countries on the basis of post- 1950 data.
Our
that "long run dynamics prevent output levels from diverging by too much."
analysis differs from this previous
work
relationship between cointegration,
to distinguish
between
common
in three respects.
common
factors,
First,
we
directly formulate the
and convergence, which permits one
sources of growth and convergence. Second,
we attempt
to
determine whether there are subgroups of converging countries and thereby move beyond
the
all
or nothing approach of previous authors. Third,
we employ
different
econometric
techniques and data sets which seem especially appropriate for the analysis of long-term
'The countries are: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany,
Japan, the Netherlands, Norway, Sweden, the United Kingdom, and the United States.
Italy,
growth behavior.
The
spirit of
our analysis has much
in
common
(1993) and Lee, Pesaran, and Pierse (1992),
the
US and UK
respectively.
with work by Pesaran, Pierse and Lee
who study
multisector output persistence for
These papers derive methods to measure the long run
of a shock originating in one sector on
all
effects
sectors in the economy. While the focus of that
research has been on measuring persistence rather than convergence, a useful extension
of the current paper would be the application of the multivariate persistence measures to
international data to both provide additional tests of convergence as well as to provide a
framework
for
measuring the sources of growth.
The plan
common
of the paper
is
as follows: Section 2 provides definitions of convergence
trends using a cointegration framework. Section 3 outlines the test statistics
Section 4 describes the data. Section 5 contains the empirical results.
use.
and
we
The evidence
from the cross-country analysis argues against the notion of convergence for the whole
common
stochastic
work come from employing stochastic
definitions
sample. Alternatively there do appear to be groups of countries with
elements.
Convergence
2
The organizing
for
in stochastic
principles of our empirical
environments
both long-term economic fluctuations and convergence. These definitions
rely
on the
notions of unit roots and cointegration in time series.
We
model the individual output
series as satisfying
a(L)Yi<t =
where a(L) has one root on the unit
circle
m4
and
Equation
2.1:
(2.1)
£i,i
£ 1){
is
a
mean
zero stationary process.
This formulation allows for both linear deterministic and stochastic trends in output. The
interactions of both types of trends across countries can be formalized into general definitions
of convergence and
common
trends.
Definition 2.1. Convergence
in
per capita output
Log per capita outputs in countries
1.
Yi
2.
/x,
3.
Yi
t
t,
•
=
t
t,
.
converge
if
-,YPt t are cointegrated with a cointegrating matrix,
is
a
p—\
Definition 2.2.
Long-run
1.
.,p
Vi,j,
0'
where Ip -\
.
-lYpj satisfy Equation 2.1,
fij
.
1,.
log
identity matrix
Common
=
[/„_!,
and
e v -\
/3',
such that
-e p _i]
wop-lxl
(2-2)
vector of ones.
treads in per capita output
per capita outputs in countries l,...,p are determined by
the individual output series, Yi it ,
.
.
-lYpj, satisfy Equation 2.1,
common
trends if
=
2.
m
3.
yj.t,
The
the
V
fij
. .
,
.
first
i,j,
Yp j
are cointegrated.
definition gives us a formal definition of convergence. If countries are to attain
same long-run growth
rates with output levels separated only by a stationary difference,
then they must satisfy Definition
2.1.
Each
series
must contain the same time trend and be
cointegrated with every other series with the cointegrating vector (1,-1).
However,
if
a group of output series does not satisfy Definition 2.1, but instead
satisfies
the weaker Definition 2.2, then output levels will be cointegrated but the stochastic trends
for the
group
across countries.
sibility
not be equal.
will
This
is
It will
remain true that permanent shocks
the natural definition to employ
that there are a small
number of
we
if
be related
will
are interested in the pos-
stochastic trends affecting output which differ in
magnitude across countries.
The
role of linear deterministic trends in our analysis
is
straightforward.
outputs contain linear trends, then long-run levels and growth rates
those trends are identical across
group require that
output for
Our
al.
all
countries have the
countries in our sample
definition of convergence
who have
between
all
countries.
all
is
Thus both convergence and common
same
well
mean
countries'
be equal only
will
linear trend.
In practice
we
Our
if
any
that
modeled by a stochastic trend. 2
that there
is
Baumol
et
a negative cross-section correlation
income and growth, thereby inferring long-run output behavior from
section behavior.
in
will find
substantially different from that employed by
defined convergence to
initial
is
If
cross-
analysis studies convergence by directly examining the time series
properties of various output series, which places the convergence hypothesis in an explicitly
dynamic and stochastic environment.
One
potential difficulty with the use of unit root tests to identify convergence
presence of a transitional component in the aggregate output of various countries.
series tests
moments
assume that the data are generated by an invariant measure,
of the data are interpretable as population
moments
the
Time
the sample
for the underlying stochastic
process. If the countries in our sample start at different initial conditions
to,
i.e.
is
and are converging
but are not yet at a steady state output distribution, then the available data
may be
generated by a transitional law of motion rather than by an invariant stochastic process.
Consequently, unit root tests
may
erroneously accept a no convergence null. Simulations in
Bernard and Durlauf (1992) suggest that the
Results are available
size distortions are unlikely to
upon request fiom the authors.
be
significant.
Output relationships across countries
3
Econometric methodology
3.1
In order to test for convergence
and common trends, we employ multivariate techniques
developed by Phillips and Ouliaris (1988) and Johansen(1988)
Let
j/i tt
denote the log per capita output
output in country
i
from output
of output deviations,
The
Z?j/,,t,
1, i.e.
AY as the first
and ADY the first
of the individual output levels,
t
t
work
starting point for the empirical
-
j/i jt
y,- >t
.
Wold
t
is
r cointegrating vectors then
Phillips
A
t
then natural to write a
It is
C(l)
is
(3.1)
.
t
the p output series are cointegrated in levels with
if
of rank
p—r and
number of linearly-independent
there
is
a vector
ARMA representation.
stochastic trends has been developed by
and Ouliaris (1988) who analyze the spectral density matrix
second test
vector
representation of output as
As shown by Engle and Granger (1987),
the
,
vector
the finding that the individual elements
t
first test for
Z?y, if the deviation of
differences of the deviations.
AY = n + C(L)e
A
and
t
of the per capita output vector are integrated of order one.
multivariate
i
Y is defined as the n x 1
difference of Y DY as the (n-l)xl
country
in
country
level of
at the zero frequency.
due to Johansen (1988, 1989) who estimates the rank of the cointegrating
is
matrix.
For a vector of output
series,
convergence and
common
trends impose different restric-
AY ,f^y(0)-
on the zero frequency of the spectral density matrix of
tions
requires that the persistent parts be equal;
common
t
Convergence
trends require that the persistent parts
of individual output series be proportional. In a multivariate framework, proportionality
and equality of the persistent parts corresponds to
as a condition
if
not of full rank.
the
number of distinct
If all
Vt, or equivalently, the
have
common
which
is
formalized
on the rank of the zero-frequency spectral density matrix. From Engle and
Granger (1987),
is
linear dependence,
stochastic trends in
n countries are converging
rank of /ady(0)
is
0.
On
Y
t
in per capita
is
less
=
output, then /^dk(0),,,
the other hand,
persistent parts, the output deviations from a
than n, then /Ay(0)
if
several output series
benchmark country must
all
have zero- valued persistent components.
Spectral-based procedures devised by Phillips and Ouliaris permit a test for complete
convergence as well as the determination of the number of common trends for the 15 output
series.
The
tests
make
use of the fact that the spectral density matrix of
at the zero frequency will be of rank q
< n where
q
is
the
first differences
number of linearly-independent
stochastic trends in the data and n
rank
If
the zero frequency matrix
eigenvalues will also be q
test that
We
the
number
of series in the sample. This reduction
captured in the eigenvalues of the zero frequency of the spectral density matrix.
in
is
is
<
The
n.
examines the smallest
than
less
is
m
full
rank, q
particular Phillips-Ouliaris test
= n—q
eigenvalues to determine
use two critical values for the bounds test, C\
m
values assess the average of the
the eigenvalues.
critical value,
The
first critical
the
number
we employ
if
of positive
is
a bounds
they are close to zero.
= 0.10^ and Ci =
These
0.05.
critical
smallest eigenvalues in comparison to the average of
value, Cj,
5%
C?, corresponds to
< n then
is
m
all
x 10% of the average root. The second
of the total variance.
For the Johansen tests we impose some additional structure on the output
assume that a finite- vector autoregressive representation
exists
We
series.
and rewrite the output vector
process as,
£Y =
t
T(L)&Yt + nYt-i+ii +
et
(3.2)
where
= -(A t+1 + ...-A k ),
T,
(t
=!,...,*-!),
and
ll
II
= -(I-A l -...-A k ).
represents the long-run relationship of the individual output series, while T(L) traces
out the short-run impact of shocks to the system.
relationships,
matrix,
II,
and thus
all
must be stationary
different choice of
=
a
in
Equation
a(3'
p.
3.2.
/?
(3.3)
is
t
II is
<
If
the rank of
r
<
II
is
not uniquely determined; a
II
II is
equals p, then Yt
still
is
related to the
number of
a stationary process.
If
the
p, there are r cointegrating vectors for the individual series
and hence the group of time
rank of
/?
satisfying Equation 3.3 will produce a different cointegrating matrix.
cointegrating vectors.
the rank of
the matrix of cointegrating vectors,
However,
Regardless of the normalization chosen, the rank of
Y
come from the
which can be written as
with q and 0, p x r matrices of rank r <
in
are interested only in the long-run
the tests and estimates of cointegrating vectors
II
as /3'Yt-k
We
series is
being driven by
p—r common
shocks.
If
the
equals zero, there are p stochastic trends and the long-run output levels are not
related across countries. In particular, from Definition 2.1, for the individual output series
to converge there
long-run trend.
must be p— 1 cointegrating vectors of the form
(1,-1) or
one
common
Two
test statistics
proposed by Johansen to
are derived from the eigenvalues of the
MLE
rank of the cointegrating matrix
test the
estimate of
II.
If
n
of
is
full
rank, p, then
will
have no eigenvalues equal to zero.
will
have p—r zero eigenvalues. Looking at the smallest p-r eigenvalues the
however,
If,
it is
of less than
p
trace
full
rank, r
<
p,
it
then
it
statistics are
p
= T ]T
A,
;
«
-2ln(Q;r,p)
^
= -T
«=r+l
/n(l
-
A,)
(3.4)
i=r+l
and
maximum
The
eigenvalue
= TA r+1
ss
— 2ln(Q;r, r+
1)
= — Tln{\ —
A r +i)
(3.5)
trace statistic tests the null hypothesis that the rank of the cointegrating matrix
against the alternative that the rank
hypothesis that the rank
is
is
p.
The maximum eigenvalue
r against the alternative that the rank
is
r
is
statistic tests the null
r+1. Critical values
for
the asymptotic distributions of both statistics are tabulated in Osterwald-Lenum (1992).
Data
4
The data used
in the empirical exercise are
ternational dollars.
GDP
The
series
annual log real
GDP
per capita in 1980
in-
run from 1900-1987 for 15 industrialized countries with the
data drawn from Maddison (1989) and the population data from Maddison (1982).
Population for 1980-1987 comes from IFS yearbooks.
The population data
as published in
current national borders, while the
GDP
Maddison (1982) are not adjusted to conform
data are adjusted. Failure to account for border
changes can lead to large one time income per capita movements as population
lost.
For example,
loss of the
GDP
per capita in the
UK
jumps
in
is
gained or
1920 without a correction for the
population of Ireland in that year. To avoid these discrete jumps we adjust the
population to reflect modern borders. 3
The
to
year-to-year
The
GDP
data set also has a few minor problems.
movements during the two world wars
Austria are constructed from
GDP
for
Belgium and during
WWI
for
estimates of neighboring countries.
Empirical results on convergence and cointegration
5
In testing for convergence
and common trends, we use three separate groupings of countries:
15 countries together, the 11 European countries and finally a subset of 6 European
all
This type of gain or loss affects Belgium, Canada, Denmark, Fiance, Italy, Japan, and the UK at
If territory, and thus population, are lost by country X in year Ti, we adjust earlier years by
extrapolating backward from T\ using the year-to- year population changes of country X.
3
least once.
countries which exhibit a large degree of pairwise cointegration. 4 Results from the PhillipsOuliaris procedures
on convergence and common trends are
Results using the Johansen methods are in Table
We
convergence
initially test for
Ouliaris
bounds
off the
US
output
levels
tests
output.
If
on the
first
indicates the presence of
of the
number
of
roots. If the first
common
p < n
is
n
to find
is
tests for the
for the
95%
more than one
we can cannot
between these extremes,
lies
As an
If
the lower
the lower
If
reject the
95%
than
less
we can
bound
is
in
of eigenvalues that account for
sum
bound on the
no convergence
95%
this
of the
the upper
p
same
distinct roots.
more of the
We
is
is
often not significant for any
are presented for null hypotheses on
The evidence on convergence
is
samples the convergence hypothesis
number
common
For
quite striking.
fails.
no-convergence null for both
The
less
than
p or more
also look for the
maximum
number
eigenvalue statistics on
number of cointegrating
of
different
then we cannot
VAR
lag length of
maximum
statistics give different estimates of the cointegrating vectors; the
eigenvalue test
Ad-
total variance.
multivariate results from the Johansen trace and
The two
null.
two
bound
critical value
largest
we conclude
reject the null hypothesis that there are
greater than the
or
if
convergence and cointegration are presented in Tables 3a and 3b for a
vectors. Test results
trends ranging from
all
test statistics
1
and
to 15.
in all three
direct convergence test in Table
critical levels as
the largest eigenvalue
*The 6 European countries are Austria, Belgium, Denmark, France,
Italy
5
Cogley (1990) uses a similar measure.
K, the size of the Daniell window was chosen to be T 6 or 27 for our sample.
7
Reducing the lag length increased the number of trends somewhat.
'
,
is
1
cannot
statistically
and the Netherlands. Bernard
(1991) finds cointegration in 10 of 15 pairs.
6
n
alternative measure
of the total variance
each group. First,
reject the hypothesis that there are at least
reject the
idiosyncratic
distinct roots for
stochastic trend for the group. Table 2 presents
number of common trends
distinct roots.
7
If
more of the sum, then we conclude
or
groups mentioned above. 6
acounts for
the critical value for a given p,
2.
distinct root in
stochastic trends for the block. 5
common
ditionally, if the largest root
The
1
at the cumulative percentage of the
largest roots contribute
greater than the critical level
that there
to find
presents the Phillips- Ouliaris bounds tests for convergence and the cumulative
1
sums of the eigenvalues
root
of significant roots
we look
respectively.
having subtracted
t ,
from a benchmark country.
trends in international output.
trends,
that there are p important
Table
number
common
ADY
we would expect
we would expect
trends dominate for every country, then
and 2
1
3.
difference of output deviations,
in the deviations
countries in the levels. If the
Tables
country group by performing the Phillips-
in the 15
countries converge, then
and no roots
in
for all three groupings. Additionally,
from zero
different
maximum
both the trace and the
eigenvalue statistics reject convergence in every group in Table 3a.
Having
failed to find evidence for convergence, or
the test for the
number
country sample and
more
distinct roots
of
common
the alternative critical value of
more
or
is
distinct roots but
a large
common
96.7% of the
C\ (= ^)
critical value
and cannot
The
trends.
a single long-run trend, we turn to
Phillips-Ouliaris statistics for the fifteen
reject the null hypothesis that there are 7 or
reject the null that there are at least 4 distinct roots.
5%
of the
now cannot
sum
total, coinciding
with the results from the test
the largest root accounts for barely
50% and
the
5%
level
The Johansen
and 13 or fewer
six largest roots
statistics.
On
common
account for
the other hand,
the largest two roots for about
variance, which argues against the existence of just a single
for convergence.
reject for 7
reject for at least 5. This leads us to posit that there
component over the sample. The
stochastic
we again
of the eigen values, Ci,
75%
factor, as
is
of total
required
trace statistic rejects 12 or fewer cointegrating vectors at
10%
at the
level for the entire fifteen
country sample. This
implies that there are only two or three long-run shocking forces for the entire group.
maximum
eigenvalue statistic does not reject for any
Taking
all
there are 6 or
statistic
11 of the
more trends and cannot
statistic rejects 5 or
for
European countries
and that there are
more
at least 3 trends with the
trends, while the
Turning to the results for the
six
distinct trends with
rejects 2 or
Using
maximum
more trends with the
CI
statistic.
The Johansen
C2
trace
eigenvalue test again cannot reject
results suggest that there are
on the order
European countries.
both the C\ and Ci
MLE
trends.
reject the null hypothesis that
European countries, we
null that there are at least 3, again with
three largest eigenvalues.
we
The
reject that there are at least 4 trends with the
of 4-5 long-run processes driving output in the
more
number of
as a group,
any number of cointegrating vectors. These
are 4 or
With
reject the null that there are
critical values
and cannot
reject the
both values. 97.8% of the sum comes from the
statistics,
the smaller six European country group
trace statistic and 5 or
more with the maximum eigenvalue
test.
6
Conclusions
This paper attempts to answer empirically the question of whether there
output per capita across countries.
We first
based on the theory of integrated time
countries can
fail
to converge only
if
is
convergence
in
construct a stochastic definition of convergence
series.
Time
series for per capita
output of different
the persistent parts of the time series are distinct.
10
Our
analysis of the relationship
little
among long-term output movements
evidence of convergence. Virtually
hypothesis of no convergence.
cointegration across
OECD
On
all
of our hypothesis tests cannot reject the null
the other hand,
we
find evidence that there
is
some
set of
is
substantial
economies. The number of integrated processes driving the 15
countries' output series appears to be on the order of 3 to 6.
that there
across countries reveals
common
Our
results therefore imply
factors which jointly determines international output
growth.
11
References
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13
Review of Income and
Table
la.
Bounds Tests
Bounds Tests**
6 European Countries
Trends Lower Upper
Phillips-Ouliaris
All Countries
Trends
<1
Upper
1.60+
3.09
<1
0.68+
1.32
Convergence
11 European Countries
Trends
<1
Lower
Upper
0.29+
0.46
bound is below the critical value for the largest root, reject null of no convergence.
bound is above the critical value for the largest root, cannot reject null of no convergence.
If
the upper
If
the lower
+
Lower
for
Significant for critical value of 0.05.
**
These
statistics are calculated
countries, the
11
European
US
is
subtracted
countries, France
off.
is
on the vector of first differences of GDPi — GDPk. For all
For the 6 European countries, France is subtracted off. For the
subtracted
off.
Table lb. Cumulative Percentage from p Largest Eigenvalues
6 European Countries
All Countries
11 European Countries
Trends Cumulated % Trends Cumulated % Trends Cumulated %
Largest
Smallest
1
0.74
1
0.69
1
0.69
2
0.88
2
0.89
2
0.89
3
0.93
3
0.98
3
0.95
4
0.96
4
1.00
4
0.97
5
0.97
5
1.00
5
0.98
6
0.98
6
0.99
7
0.99
7
1.00
8
0.99
8
1.00
9
1.00
9
1.00
10
1.00
10
1.00
11
1.00
12
1.00
13
1.00
14
1.00
14
L
Table
Bounds Tests for Cointegration
European Countries
11 European Countries
Trends Lower Upper Trends Lower Upper
Phillips- Ouliaris
2a.
All Countries
6
Trends
Lower
Upper
15
0.00
0.00
14
0.00
13
0.00
6
0.00
0.00
11
0.00
5
0.00
4
0.01
0.01
0.02
0.03*+
0.00
0.00
10
0.00
0.00
9
0.00
0.01
0.11
8
0.01
0.01
0.40
7
0.01
0.02
6
0.02
0.03" +
12
0.00
0.00
3
0.06*+
11
0.00
0.01
2
0.23
10
0.01
0.01
9
0.01
0.01
5
0.03
0.06
8
0.01
0.02
4
0.06+
0.10
7
0.03
0.04*+
3
0.12*
0.19
6
0.04
0.07
2
0.30
0.46
5
0.07+
0.10
4
0.11*
0.17
3
0.19
0.29
2
0.39
0.57
Cumulative Percentage from p Largest Eigenvalues
All Countries
6 European Countries
11 European Countries
Trends Cumulated % Trends Cumulated % Trends Cumulated %
Table 2b.
Largest
the upper
bound
is
0.52
1
0.69
1
0.63
2
0.76
2
0.91
2
0.84
3
0.87
3
0.98
3
0.92
4
0.92
4
0.99
4
0.95
5
0.95
5
1.00
5
0.98
6
0.97
6
1.00
6
0.99
7
0.98
7
0.99
8
0.99
8
1.00
9
0.99
9
1.00
10
1.00
10
1.00
11
1.00
11
1.00
12
1.00
13
1.00
14
1.00
15
1.00
Smallest
If
1
bound
above the
is
below the
critical value,
• Significant at O.lOm/n, n
+
Significant at
critical value, reject null
5%
of the
is
P
cannot reject null of at least
the
sum
of
number of
countries,
of the roots.
15
m
is
more
or
P
the
distinct roots. If the lower
distinct roots.
number of roots
=
0.
Table
Multivariate Tests for Convergence and Cointegration
3.
(VAR
lag length
Table 3a.
> 14
>13
> 12
> 11
> 10
553.99
>9
>8
>7
>6
>5
>4
>3
>2
European 6
62.00*
53.84*
31.89*
Eig
Trends
>
71.00
51.06
147.60
38.29
292.99
48.89
244.10
41.07
156.82
34.66
162.59
36.03
126.56
32.61
93.95
31.23
>
62.71
24.52
38.19*
17.22
9.08
1.38
Rejects at 5%.
t
Distributed x 2 (p
—
1)
where p
is
the
79
62.41
23.27
17.55
21.59
14.02
34.80J
>
1
74.51
22.49
>0
0.57
52.02
18.07
33.95
15.95
18.00
9.16
1
8.84
7.55
>0
1.28
1.28
number of
countries.
Q
Eig
39.14*
7.57
109.31*
16
7'5
40.54*
53.06
59.38
1.38
102.95
198.66
352.37
>0
Trace
251.72
>9
>8
>7
>6
>5
>4
>3
>2
Max
Trends
>5
>4
>3
>2
68.62
10.51
European 6
Eig
60.46
62.00
10.46
Max
312.18
414.37
20.97
Trace
10
482.99
*
Cointegration
European
Max
1
>
Convergence*
European 6
Table 3b.
Trace
2)
All
All
Trends
=
7.00
0.57
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