Int. Fin. Markets, Inst. and Money 17 (2007) 125–139
A re-examination of international inflation
convergence over the modern float
William J. Crowder a,∗ , Chanwit Phengpis b,1
b
a Department of Economics, University of Texas at Arlington, Arlington, TX 76019, USA
Department of Finance, California State University, Long Beach, Long Beach, CA 90840, USA
Received 25 July 2005; accepted 10 September 2005
Available online 16 November 2005
Abstract
Crowder [Crowder, W.J., 1996. The international convergence of inflation rates during fixed and floating
exchange rate regimes. Journal of International Money and Finance 15, 551–576] provided evidence that
inflation rates among the seven largest industrialized economies shared one common stochastic trend in the
post-war era, over both the fixed and floating exchange rate regimes. The convergence of inflation rates over
the floating exchange rate period implies less insulation for the domestic economy from idiosyncratic shocks
in the rest of the world, thereby reducing the attractiveness of flexible exchange rates. Several subsequent
studies have found much less convergence than suggested in Crowder’s original results, suggesting a higher
degree of insulation of flexible exchange rates. In this study, we revisit the question of the degree of inflation
convergence among the G-7 nations over the modern float. Employing a host of diagnostic methods, we
conclude that there is in fact one common trend underlying the inflation rates of the G-7 nations. Consistent
with Crowder [Crowder, W.J., 1996. The international convergence of inflation rates during fixed and floating
exchange rate regimes. Journal of International Money and Finance 15, 551–576], we cannot attribute the
source of the underlying common trend to any one particular country.
© 2006 Elsevier B.V. All rights reserved.
JEL classification: C32; E31
Keywords: Inflation convergence; G-7; Floating exchange rate regime
∗
1
Corresponding author. Tel.: +1 817 272 3147; fax: +1 817 272 3145.
E-mail addresses: crowder@uta.edu (W.J. Crowder), pchanwit@yahoo.com (C. Phengpis).
Tel.: +1 562 985 1581.
1042-4431/$ – see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.intfin.2005.09.002
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1. Introduction
One of the central advantages of a floating exchange rate system is that it should provide
a degree of insulation from idiosyncratic shocks in the rest of the world. But as economies
become increasingly integrated around the globe, the ability of flexible exchange rates to absorb
perturbations becomes suspect. It is well understood that under a fixed exchange rate system,
inflation rates in the participating countries must be equal in steady state equilibrium. This implies
a dynamic relationship in a stochastic environment in which inflation rates cannot diverge over
the long-run. But under a floating regime, it is not necessary that inflation rates move together
since diverging rates of inflation will simply be reflected in ever depreciating (or appreciating)
exchange rates. But it is a well documented stylized fact that over the modern floating exchange
rate regime, exchange rate depreciations are not permanent. Nominal exchange rate changes
are stationary. This implies that inflation rates should not diverge across countries forever, but
should in fact move together in a manner consistent with a constant rate of depreciation or
appreciation.
The period of floating exchange rate regimes for major developed nations began after the
collapse of the Bretton Woods System in the early 1970s when these countries abandoned the
fixed parities of their currencies vis-à-vis the US dollar. A few exceptions include, for instance,
European Union (EU) or European Monetary System (EMS) nations. These countries allowed
their currencies to float against non-EMS currencies, but to fluctuate only within pre-specified
margins relative to other EMS currencies via the Exchange Rate Mechanism (ERM) during the
period from 1979 to 1998.2
In theory, the exchange rate regime influences the degree of interrelatedness among inflation
rates in various nations. Crowder (1996) detects six cointegrating vectors (henceforth, CIVs) or
long-run equilibria among inflation rates in seven major industrialized countries or the G-7 nations
over the period from the beginning of the floating era in 1973 until 1993. This finding implies that
the G-7 inflation rates completely converge or are driven by a single common stochastic trend
in the long-run.3 Evidence of one common trend, rather than multiple common trends or even
non-convergence, is quite contradictory to the monetarist view. The monetarist framework argues
that while domestic inflation should converge on the inflation rate in the reserve currency country
(i.e., the US) under the fixed regime, it should be insulated from the rest of the world under the
floating regime.
Accordingly, Crowder (1996) details several potential contributing factors to international
transmission of inflation rates which lead to similar inflation experiences faced by various nations.
These factors include: (1) policy coordination among central banks in different countries; (2)
influences of currency substitution which makes domestic monetary policies partially contingent on those in other nations; (3) adjustments towards the relative purchasing power parity
(PPP) which implies that if nominal exchange rate changes have a constant mean, inflation
rates will converge but differ only by this constant magnitude in the long-run; (4) adjustments of domestic output to equalize domestic inflation with world inflation through the trade
2 Further, individual currencies of the EMS nations which later joined the European Economic and Monetary Union
(EMU) were irrevocably fixed to a common currency, euro, upon the official inception of the EMU in January 1999. These
currencies were subsequently withdrawn from circulation and completely replaced by the euro in 2002.
3 In the system of p cointegrated variables with r CIVs, there must be (p − r) common trends. Based on Hafer and Kutan
(1994), complete convergence is inferred if there are (p − 1) CIVs and thus one common trend, while partial convergence
is inferred if there are at least one but fewer than (p − 1) CIVs and hence multiple common trends.
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127
multiplier and (5) common stochastic shocks including rising oil prices because the G-7 inflation rates are found to be cointegrated not only among themselves but also with the oil price
inflation.
Several other studies however provide relatively tenuous evidence of complete international
convergence of inflation rates over the modern float. It is found that inflation rates do not converge
unless the subject countries are part of formal exchange rate arrangements, especially the ERM for
EMS nations. Moreover, unlike those in other countries, the EMS inflation rates can completely
converge due to intra-EMS policy mandates, such as the Maastricht Treaty (1992). This treaty
stipulates economic convergence criteria which explicitly require EMS nations to converge their
key economic variables including exchange rates and inflation rates prior to becoming EMU
members in 1999. Westbrook (1998) detects a single common trend among inflation rates in five
EMS countries during the period from March 1979 to December 1992. With monthly data covering
the period May 1986 to December 1990, Caporale and Pittis (1993) report non-convergence of
inflation differentials (vis-à-vis Germany) in three non-EMS countries including Switzerland, the
UK and US, but strong convergence of the differentials in six EMS nations (among which the
number of common trends varies between one to three depending on deterministic specifications
and lag lengths in the VAR). They accordingly suggest that participation in the EMS has counterinflationary benefits from the leading role of Germany’s Bundesbank in pursuing strict monetary
policy and hence price stability.
Nonetheless, some other studies find that formal exchange rate arrangements or explicit policy
mandates do not necessarily result in complete convergence of inflation rates among the subject
countries. This observation is partially corroborated by inconclusive evidence of a single common
trend by Caporale and Pittis (1993) discussed above. Holmes (1998) also finds that inflation rates
based on output prices in the manufacturing sectors as well as those in the service sectors in six EU
nations over the period from January 1980 to May 1995 converge only partially because multiple
common trends are detected in both cases. Further, Trivez (2001) employs monthly data from
January 1980 to November 1999 and reports the absence of bivariate cointegrating relations for
several pairs of inflation rates in eleven EMU countries. This result thus implies the presence of
multiple common trends driving inflation rates within the EMU zone even though EMU member
nations were part of the ERM for a few decades, have been subject to Maastricht Treaty (1992)
for several years and have been under the same monetary policy by the European Central Bank
(ECB) via the euro since January 1999.
Given conflicting results in the existing literature, this study re-examines international convergence of inflation rates in G-7 countries over the period of modern float from March 1973
until February 2003 via the conventional Johansen cointegration methodology augmented with
several diagnostic techniques to ensure the robustness of test results. This investigation expectedly
provides at least two contributions to the existing literature.
First, the updated sample period enables an improved investigation into inflation convergence
among G-7 countries. Given the long history of ERM and the Maastricht Treaty (1992) and the
approximately 4-year period since France, Germany and Italy began sharing a common currency
and monetary policy in January 1999, their inflation rates should be driven by a single common
trend according to the monetarist view. Canada, Japan, the UK and US do not have formal exchange
rate arrangements with other countries. Hence, if the monetarist framework is the only contributing
factor to international inflation convergence, inflation rates in these four countries should be driven
uniquely by their own domestic situations, thereby bringing the total common trends in the system
to five. The smaller number of common trends (i.e., less than five) is entirely possible depending
on the explanatory power of additional converging factors including policy cooperation among
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central banks, currency substitution, the relative PPP, domestic output adjustments and common
stochastic shocks such as the oil price inflation, among others. The main research question is
whether or not these factors contribute so strongly to convergence that a single common trend in
the G-7 inflation rates can be claimed.
Second, because an inference regarding the number of common trends is the crux of the
investigation, several diagnostic techniques that have not been all inclusive in prior empirical work
are employed to ensure the validity of test results from conventional Johansen cointegration tests.
This study incorporates the recursive tests of the stability of cointegrating parameters (Hansen
and Johansen, 1999), tests of alpha restrictions across equations in the VAR (Horvath and Watson,
1995) and unit root tests of CIV and common trend estimates (Dickey and Fuller, 1979, 1981;
Ng and Perron, 2001). It also includes the rolling cointegration tests (e.g., Rangvid and Sorensen,
2002; Pascual, 2003) which hold the number of observations and thus the test power constant
as the estimation window rolls forward and therefore enable inferences concerning convergence
over the moving but fixed time intervals within the full sample period.
This study finds that there exist two common trends among the G-7 inflation rates based solely
on conventional Johansen cointegration tests. However, additional diagnostic tests strongly reveal
the presence of only one common trend and complete convergence of inflation rates among G-7
countries, thereby reiterating the results in Crowder (1996). Further, neither one of the included
inflation rates can be omitted from long-run equilibria or can be considered weakly exogenous or
the source of the underlying common trend.
The rest of this study is organized as follows. Section 2 describes the data and delineates the
econometric methodology employed, with estimation results and related findings presented in
Section 3. Section 4 delivers conclusions and implications.
2. Data and methodology
2.1. Data
Monthly data for non-seasonally-adjusted consumer price indices (CPIs) in Canada, France,
Germany, Italy, Japan, the UK and US are obtained from the International Financial Statistics
(IFS) databank compiled by the International Monetary Fund (IMF). The data covers the period
from February 1973 to February 2003. Accordingly, inflation rates from March 1973 to February
2003 are computed from the first differences of the natural log of the corresponding CPIs. The
March 1973 starting month is also designated as the beginning of the modern float era in Lastrapes
and Koray (1990) and Crowder (1996).
2.2. Methodology
2.2.1. Univariate analysis
Cointegration presupposes that variables in the system are non-stationary and integrated of
the same order. The Augmented Dickey–Fuller (ADF) unit root tests (Dickey and Fuller, 1979,
1981) are employed to investigate univariate properties of the G-7 inflation rates under the null
hypothesis that the series is non-stationary and integrated of order one. Further, it is well known
that the ADF tests have low power and lead to large size distortion in the presence of large
and negative moving average (MA) terms in the data generating process (e.g., Schwert, 1989).
Therefore, the Ng–Perron unit root tests (Ng and Perron, 2001) are also conducted to confirm
the ADF test results. Based on the local GLS trending method, the Ng–Perron tests produce two
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129
test statistics, MZ␣ and MZt , and are found empirically to have greater power and much less size
distortion than the ADF tests.
2.2.2. Multivariate analysis
The Johansen cointegration procedure (Johansen, 1988, 1991, 1992a,b, 1994, 1995) is implemented by constructing a VAR process as in (1).
Xt = Φ1 Xt−1 + · · · + Φk Xt−k + μ + δt + εt
(1)
where Xt is a p-dimensional vector of non-stationary variables (i.e., inflation rates under investigation); Φj the coefficient matrices; μ a vector of constants; δ a vector of coefficients on linear trend
terms; εt the white noise error vector with non-diagonal covariance matrix and k is the minimum
lag length that reduces serial correlation in residuals to zero statistically in each equation in the
VAR based on the Ljung–Box (L–B) Q-statistics.
The VAR in (1) is then transformed into its error correction model (VECM) as in (2).
Xt = Γ1 Xt−1 + · · · + Γk−1 Xt−k+1 + ΠXt−1 + μ + δt + εt
(2)
The long-run multiplier matrix Π = Φ(1) − I can be decomposed into two (p × r) matrices such
that ␣␤ = Π. The ␤ matrix contains parameters for r CIVs or long-run equilibria which imply
the presence of (p − r) common stochastic trends underlying the system of included variables,
while the ␣ matrix contains error correction coefficients which measure the extent to which each
variable responds to deviations from the long-run equilibria.
The test statistic for the null hypothesis of at most r against the alternative of p CIVs is the λtrace
statistic given in (3) where λi is an eigenvalue obtained from maximum likelihood estimation of
(2) via reduced rank regression.
p
λtrace (r) = −T
ln(1 − λi )
(3)
i=r+1
The distribution of the λtrace statistic is non-standard and predicated upon, among other things,
the specification of deterministic components μ and δ in the VAR in (1). Hence, inferences
regarding the number of CIVs and subsequent hypothesis tests may be invalid if deterministic
components are improperly specified (e.g., Hansen and Juselius, 1995).
Johansen (1994) details the tests for different deterministic specifications in the VAR. Conditional on r CIVs, the test of the null hypothesis that linear trends in the levels of data are eliminated
by cointegrating relations (such that linear trend terms can be excluded from the cointegration
space) against the alternative hypothesis that the linear trends are not eliminated by cointegrating
relations (such that linear trend terms must be present in the cointegration space) can be achieved
by computing the G(r) statistic in (4).4
r
(1 − λ∗j )
G(r) = −T
ln
(4)
∼ χ2 (r)
(1 − λj )
j=1
λ∗j
where
and λj are the eigenvalues from the VARs under the alternative and null hypotheses,
respectively.
4 Quadratic trends in the levels of data are irrelevant for inflation rates and possibly for other economic variables as
well. The quadratic trends would imply that changes in variables occur at an ever increasing or ever decreasing rate.
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Non-rejection of the above null hypothesis enables the test of the more restrictive null hypothesis that there are no trends in the levels of data (but constant terms in the cointegration space)
against the alternative hypothesis that the linear trends are eliminated by cointegrating relations
(i.e., the specification under the null in (4)). This test can be accomplished by computing the G(r)
statistic in (5).
p
(1 − λ∗j )
G(r) = −T
ln
(5)
∼ χ2 (p − r)
(1 − λj )
j=r+1
where λ∗j and λj are the eigenvalues from the VARs under the alternative and null hypotheses,
respectively.
Because inferences regarding the number of CIVs and common trends are the centerpiece of this
study, additional diagnostic techniques to ensure the validity of test results from the conventional
Johansen procedure are crucial. First, the recursive stability tests of cointegrating parameters
(Hansen and Johansen, 1999) are conducted since the parameters should be stable if the model is
to be useful. These tests can be achieved by: (1) holding short-run dynamics constant at their full
sample estimates but allowing long-run relations to change over time; (2) estimating the VECM
over the base period and (3) keeping initial observations in the base period fixed and increasing
one additional observation at each iteration to re-estimate the VECM such that the last recursive
estimation period is equal to the full sample. Contingent on r CIVs over the full sample, the test of
the null hypothesis that the beta estimates derived from one recursive iteration do not statistically
differ from the full sample estimates of beta yields the likelihood ratio test statistic which is
distributed as χ2 with (p − r)r degrees of freedom.
Second, the CIV and common trend estimates from conventional Johansen tests are subject to
the unit root tests. Gonzalo and Granger (1995) show that in the cointegrated system of p variables
with r CIVs and thus (p − r) common trends, the vector of variables Xt can be decomposed into
stationary and permanent/non-stationary components as in (6).
Xt = Stationary component + permanent component
= ␣(␤ ␣)
−1 −1 ␣⊥ Xt
␤ Xt + ␤⊥ (␣⊥ ␤⊥ )
(6)
where ␣⊥ and ␤⊥ are the (p × (p − r)) matrices that satisfy ␣ ␣⊥ = 0 and ␤ ␤⊥ = 0, and the CIVs
and common trends are defined as ␤ Xt and ␣⊥ Xt , respectively. Using this decomposition, the
estimates of ␤ Xt and ␣⊥ Xt are expected to exhibit sharply contrasting properties in that the first
should be stationary while the latter should be non-stationary.
Third, the Wald tests of restrictions on alphas across VECM equations are conducted. Following
Horvath and Watson (1995), the VECM is estimated conditional on r CIVs. Evidence of r CIVs
must be corroborated by the finding that the inflation rate in at least one nation has a statistically
significant alpha coefficient and hence responds to divergence from each one of those r long-run
equilibria.5
Fourth, the rolling cointegration tests (e.g., Rangvid and Sorensen, 2002; Pascual, 2003) are
performed. The length of the rolling estimation window is chosen to be fixed at a 10-year interval
because cointegration is a long-run property and reasonably long time spans of data may be needed
to detect the existing CIVs (Hakkio and Rush, 1991). As the estimation window rolls forward, an
5 In other words, H : α = α = . . . = α = 0 for CIV (where j = 1, 2, . . ., or r) across all seven VECM equations for
0
1,j
2,j
7,j
j
the G-7 inflation rates. The resultant likelihood ratio test statistic is distributed as χ2 (7).
W.J. Crowder, C. Phengpis / Int. Fin. Markets, Inst. and Money 17 (2007) 125–139
131
additional set of λtrace statistics is computed. The plots of the rolling λtrace statistics through time
show the strength of cointegrating relations (and thus the degree of convergence) over adjacent
fixed intervals within the full sample period.6
Further, even when a single common trend is detected, an immediate conclusion that the
G-7 inflation rates completely converge may be misleading because the estimates of beta coefficients may be statistically insignificant or have incorrect signs. Such convergence requires further
hypothesis testing that the normalized beta coefficient for each of the included inflation rates
is statistically significant and negative with respect to the inflation rate in the country arbitrarily chosen as a numeraire.7 Finally, the weak exogeneity tests (Johansen, 1992b) or the tests of
restrictions on alphas across CIVs in each VECM equation are performed.8 The null hypothesis
is that the inflation rate in country i is weakly exogenous such that it is irresponsive to deviations
from long-run equilibria and thus one of the (p − r) underlying common trends in the system.
3. Summary of results
Table 1 presents the results from the ADF and Ng–Perron Unit Root Tests. The ADFc test
statistics indicate non-rejection of the null hypothesis of a unit root at the 5% significance level
for all inflation rates, except for the inflation rate in Japan. Similarly, ADFct statistics suggest nonrejection of the null hypothesis of a unit root at the 5% significance level for all inflation rates,
except for those in Japan and the UK. However, it is found from fitting the ARIMA(1,0,1) model
for each inflation rate that the MA(1) coefficient is large and negative. The coefficient varies
between −0.94 for the inflation rate in Japan and −0.69 for the inflation rate in Italy. Hence,
inferences based solely on the ADF test results may be inappropriate because the ADF tests are
found in prior empirical work to have large size distortion in the presence of large and negative
MA errors. In fact, the MZ␣ and MZt statistics from the Ng–Perron Tests result in non-rejection
of the unit root null hypothesis at the 5% level for each inflation rate irrespective of whether the
data are demeaned or detrended by the GLS procedure. Thus, based on the Ng–Perron tests which
have greater power and less size distortion in the presence of large and negative MA errors than
the ADF tests, all G-7 inflation rates are non-stationary.
Table 2 shows the results from the conventional Johansen cointegration procedure. The VAR
is first estimated based on the deterministic specification that there exist linear trends in the levels
of data which are not eliminated by cointegrating relations. By varying the lag length k in the
VAR from 1 to 11, serial correlation in residuals cannot be reduced to zero statistically based
on the L–B Q-statistics. For example, at k equal to 11, the L–B Q-statistics are still statistically
significant at the 5% level in the equations for Japan and the UK (Column 2, Panel A, Table 2).
6 Pascual (2003) suggests that the rolling λ
trace statistics (from the rolling tests) are preferred to the recursive λtrace
statistics (from the recursive tests). The test power remains constant for each rolling iteration due to a fixed number of
observations, while the test power rises with each recursive iteration due to increasing numbers of observations. Hence,
larger values of the recursive λtrace statistics towards the end of the full sample period may not necessarily imply increasing
convergence but rather reflect the fact that they converge on their long-run values because of increases in the test power.
7 For example, if one common trend is detected, each of the six CIVs can be expressed as π
US + βi π i via a matrix
normalization of beta, where πUS is the US inflation rate used as a numeraire and πi is the inflation rate in country i for
i = the US. Complete convergence implies that βi are statistically significant and negative so that an increase (decrease)
in the US inflation rate is associated with an increase (decrease) in the inflation rate in country i to retain cointegrating
relations in the long-run.
8 In other words, H : α = α = . . . = α = 0 for a VECM equation for the inflation rate in country i. The resultant
0
i,1
i,2
i,r
likelihood ratio test statistic is distributed as χ2 (r).
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Table 1
Unit root tests of inflation rates
Country
Canada
France
Germany
Italy
Japan
UK
US
ADF testsa
MA(1)b
ADFc
ADFct
−1.91
−1.46
−2.80
−1.62
−4.26*
−2.12
−2.21
−2.60
−1.83
−3.06
−3.31
−4.14*
−3.48*
−2.53
−0.88
−0.78
−0.92
−0.69
−0.94
−0.86
−0.79
Ng–Perron tests–demeanedc
Ng–Perron tests–detrendedd
MZ␣
MZt
MZ␣
MZt
−2.33
−1.83
−0.54
−3.42
−6.31
−2.17
−2.18
−1.00
−0.93
−0.46
−1.21
−1.68
−1.04
−1.03
−2.38
−1.92
−1.40
−9.03
−3.30
−5.17
−2.88
−0.97
−0.86
−0.67
−2.12
−1.26
−1.60
−0.92
a Lag length in the ADF test equation is chosen based on the Akaieke Information Criterion (AIC). ADF and ADF
c
ct
test statistics are from the ADF equation that includes a constant but no time trend and the ADF equation that includes
both constant and time trend, respectively. Critical values for ADFc and ADFct obtained from MacKinnon (1996) are
−2.8613 and −3.4098, respectively, at the 5% level.
b The MA(1) coefficient estimates from the ARIMA(1,0,1) model.
c Data are demeaned by the GLS procedure for the Ng–Perron tests. Critical values for the corresponding MZ and
␣
MZt statistics obtained from Ng and Perron (2001) are −8.10 and −1.98, respectively, at the 5% level.
d Data are detrended by the GLS procedure for the Ng–Perron tests. Critical values for the corresponding MZ and MZ
␣
t
statistics obtained from Ng and Perron (2001) are −17.30 and −2.91, respectively, at the 5% level.
* Denotes statistical significance at the 5% level.
Conversely, when k is increased to 12, serial correlation is reduced to zero statistically. The L–B
Q-statistic is not statistically significant in any one of the equations in the VAR (Column 3, Panel
A, Table 2).
With 12 lags in the VAR, the first set of G(r) statistics is computed under the null hypothesis
that linear trends in the levels of data are eliminated by cointegrating relations (Column 2, Panel
B, Table 2). This null hypothesis cannot be rejected at the 5% level across all possible r’s, except
for r equal to 2 or 3 where the associated G(r) statistic of 7.46 or 8.38 is just marginally significant
at the 5% level. Further, the second set of G(r) statistics is calculated under the null hypothesis
that there are no trends in the levels of data but constant terms in the cointegration space (Column
4, Panel B, Table 2). This null hypothesis cannot be rejected at the 5% level across all possible r’s,
thereby implying that the restriction imposed by the null is unquestionably not binding. Thus, it
appears that the most appropriate deterministic specification in the VAR is the one with no trends
in the levels of data but constant terms in the cointegration space. Conditional on this deterministic
specification, the λtrace statistics lead to non-rejection of the null hypothesis of r ≤ 5 (Panel C,
Table 2), thereby implying five CIVs and resultantly two common trends in the system of G-7
inflation rates.9
However, the results from additional diagnostic tests support the presence of six CIVs and
hence only one common trend among the G-7 inflation rates. Fig. 1 shows the recursive tests
of the stability of cointegrating parameters. The parameters for five CIVs and for six CIVs are
stable with the passage of time through 1986 and through 1989, respectively, as indicated by the
plots of the normalized recursive likelihood ratio statistics below the critical value line of one.
These findings therefore suggest that the VECM based on six CIVs does not significantly alter
the stability of cointegration space in relation to the VECM based on five CIVs.
9 The finding of five CIVs retrospectively confirms that the deterministic specification chosen for the VAR is appropriate.
This is because the rejection of the null hypothesis for the first set of G(r) statistics occurs only when r is equal to 2 or 3.
W.J. Crowder, C. Phengpis / Int. Fin. Markets, Inst. and Money 17 (2007) 125–139
133
Table 2
Conventional Johansen cointegration tests of G-7 inflation rates
Panel A: lag length (k) selectiona
Equation
k = 11
k = 12
Canada
France
Germany
Italy
Japan
UK
US
21.21
28.32
33.64
43.25
61.57*
65.84*
36.31
25.53
37.16
45.74
46.90
32.95
45.49
40.45
Panel B: deterministic specification
r
G(r) statisticsb
p−r
G(r) statisticsc
1
2
3
4
5
6
0.34
7.46*
8.38*
9.31
10.20
12.16
6
5
4
3
2
1
1.77
1.69
1.52
1.31
0.62
0.15
Panel C: cointegration resultsd
H0
λtrace e
r=0
r≤1
r≤2
r≤3
r≤4
r≤5
r≤6
229.95*
143.86*
98.36*
61.82*
33.84**
14.65
3.34
a The VAR is estimated under the deterministic specification that linear trends in the levels of data are not eliminated by
cointegrating relations. The numbers shown above are the L–B Q-statistics distributed as χ2 (36) under the null hypothesis
of no serial correlation in residuals.
b Asymptotically distributed as χ2 (r) under the null hypothesis that linear trends in the levels of data are eliminated by
cointegrating relations against the alternative hypothesis that the linear trends are not eliminated by cointegrating relations.
c Asymptotically distributed as χ2 (p − r) under the null hypothesis of no trends in the levels of data against the alternative
hypothesis that there exist linear trends in the levels of data that are eliminated by cointegrating relations.
d Based on the VAR assuming no trends in the levels of data but constant terms in the cointegration space.
e Critical values from Table B.2 in Johansen (1995).
* Denotes statistical significance at the 5% level.
** Denotes statistical significance at the 10% level.
Further, Table 3 show the results from unit root tests of the estimates of six CIVs and one
common trend according to the Gonzalo and Granger (1995)’s decomposition. It is apparent that
all six CIV estimates and the common trend estimate conform to their theoretical property in
that the first are stationary while the latter is non-stationary. The unit root null hypothesis can
be rejected at the 5% level in both ADF and Ng–Perron tests for each CIV estimate, while the
opposite is true for the common trend estimate. The results from the likelihood ratio tests of
restrictions on alpha coefficients across all seven VECM equations shown in Table 4 also reiterate
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Fig. 1. Recursive stability tests. Note: Conditional on five CIVs or on six CIVs, the plots show the recursive likelihood
ratio statistics scaled by 5% critical values against the passage of time. The 1973:03–1983:07 period is the base or the
first estimation period which is increased recursively until the last estimation period covering the full sample. The values
below one indicate non-rejection of the null hypothesis that the ␤ estimates for the corresponding recursive estimation
period are not statistically different from the ones derived from the full sample at the 5% significance level.
the presence of six CIVs in the system. The likelihood ratio test statistics lead to rejection of the
null hypothesis that none of the G-7 inflation rates responds to deviations from CIVj for j = 1, 2,
. . ., or 6 at the 5% level. Hence, all of the six CIVs are economically meaningful in that there are
dynamic adjustments of the inflation rates in the system to retain these equilibria in the long-run.
Table 3
Unit root tests of CIV and common trend estimates
Estimates
ADF test statisticsa
CIV1
CIV2
CIV3
CIV4
CIV5
CIV6
Common trendd
−5.28*
−4.96*
−6.84*
−5.79*
−3.92*
−5.37*
−1.85
MA(1)b
−0.81
−0.81
−0.26
−0.09
0.32
−0.02
−0.75
Ng–Perron test statisticsc
MZ␣
MZt
−24.58*
−3.50*
−3.40*
−3.19*
−3.71*
−2.90*
−3.23*
−1.49
−23.14*
−20.61*
−28.66*
−17.68*
−21.00*
−4.76
a The ADF test equation includes a constant (but no time trend) and six augmented lags. The critical value obtained
from MacKinnon (1996) is −3.3361 at the 5% level.
b The MA(1) coefficient estimates from the ARIMA(1,0,1) model.
c Data are demeaned by the GLS procedure for the Ng–Perron tests. Critical values for MZ and MZ obtained from
␣
t
Ng and Perron (2001) are −8.10 and −1.98, respectively, at the 5% level.
d The estimate of a common trend is derived from the Gonzalo and Granger (1995)’s decomposition.
* Denotes statistical significance at the 5% level.
W.J. Crowder, C. Phengpis / Int. Fin. Markets, Inst. and Money 17 (2007) 125–139
135
Table 4
Horvath–Watson cointegration tests
CIVs
Test statistics
CIV1
CIV2
CIV3
CIV4
CIV5
CIV6
41.04*
47.50*
28.64*
44.28*
38.89*
30.22*
Conditional on six CIVs, the tests are the Wald tests [based on Horvath and Watson (1995)] under the null hypothesis
that none of the G-7 inflation rates responds to deviations from CIVj for j = 1, 2, . . ., or 6. Test statistics are distributed as
χ2 (7).
* Denotes statistical significance at the 5% level.
Additionally, Fig. 2 graphs the 10-year rolling trace statistics normalized by the 10% critical values. Evidence of six CIVs begins to emerge approximately in 1991 (i.e., for the 1981:02–1991:01
rolling window). Further, it is increasingly discernible towards the end of the sample period (i.e.,
towards the 1993:03–2003:02 rolling window) that there are six CIVs; only the bottommost line
which represents the normalized rolling λtrace statistics for H0 : r ≤ 6 is consistently below the
critical value line of one. Hence, evidence of five CIVs inferred from conventional Johansen tests
over the full sample may be induced by relatively unstable and tenuous cointegrating relations
(according to the recursive stability tests as well as the rolling tests) during the beginning portion
of the floating era, even though the presence of six CIVs and thus one common trend in the system
of G-7 inflation rates is more probable.
Fig. 2. Rolling trace tests. Note: The plots show the rolling trace statistics scaled by the 10% critical values. The rolling
estimation window equals to the 10-year period. Lag length of 5 is chosen; it is the minimum necessary to eliminate
serial correlation in residuals based on L–B Q-statistics which are distributed as χ2 (36). The first estimation window is
from 1973:08 to 1983:07, while the last estimation window is from 1993:03 to 2003:02. The number of lines above one
indicates the number of CIVs at the 10% significance level.
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Table 5
Beta coefficients normalized by the US inflation rate
CIVs
Country
Beta coefficientsa
Test statistics for H0 : βi = 0b
CIV1
CIV2
CIV3
CIV4
CIV5
CIV6
Canada
France
Germany
Italy
Japan
UK
US
−0.82
−0.60
−2.38
−0.47
−1.87
−0.87
1.00
−13.23*
−12.77*
−7.83*
−12.05*
−11.20*
−9.46*
N/A
Test statistic for H0 : βi = −1 for all i = USc
33.66*
a
The US inflation rate is chosen as a numeraire. If inflation rates completely converge, the beta coefficient must be
negative with respective to the normalized numeraire.
b Asymptotically valid t-statistics under the null hypothesis that inflation rate in country i does not have long-run
relations with the US inflation rate.
c Distributed as χ2 (6) under the null hypothesis that each of the CIVs is a one-to-one relationship between the US
inflation rate and the inflation rate in country i for all i = US.
* Denotes statistical significance at the 5% level.
To ascertain that a single common trend truly implies complete convergence, Table 5 shows the
normalized beta coefficients for the six CIVs with respect to the US inflation rate as a numeraire.
This normalization allows each CIV to represent the bivariate relationship between the US and
non-US inflation rates. It is found that the coefficients are negative (Column 3, Table 5) and
statistically significant at the 5% level (Column 4, Table 5) for every CIVs. These findings thus
support complete convergence of G-7 inflation rates in that an increase (decrease) in the US
inflation rate is associated with increases (decreases) in other G-7 inflation rates in the long-run.
Nonetheless, the test of a more restrictive null hypothesis that each of the six CIVs is a one-to-one
relationship between the US and non-US inflation rates yields the χ2 statistic of 33.66 which can
be clearly rejected at the 5% level (last row, Table 5). This result implies that the convergence takes
a weak form rather than a strong form in that the G-7 inflation rates completely converge, but may
not differ from one another in the long-run by only a constant mean according to the relative PPP.
Further, because the EMS (or currently the EMU) countries, unlike other nations, are subject
to cross-country policy mandates, cointegration tests within the EMU group are performed.
The results are presented in Table 6. The null hypothesis of r ≤ 2 cannot be rejected at any
conventional significance level (Panel A, Table 6). Therefore, in contrast to the complete set of
G-7 inflation rates, the two CIVs from conventional Johansen test indicates one common trend
among the three EMU inflation rates without having to further perform supplementary diagnostic
tests. The normalized beta coefficients for the French and Italian inflation rates with respect to
the German inflation rate as a numeraire are negative and thus have a correct sign for complete
convergence (Panel B, Table 6). Nonetheless, the coefficients differ considerably from −1 and
the null hypothesis that each of the two inflation rates has a one-to-one relationship with the
German inflation rate can be clearly rejected at the 5% level (last row, Panel B, Table 6). This
finding further implies that while the Bundesbank (in the past) and the ECB (at the present time)
may play an important contributing role in EMU inflation convergence based on the monetarist
view, their influences are not so robust that a one-to-one relationship between inflation rates in
Germany and another EMU country can result. The complete convergence of EMU inflation
rates exhibits a weak-form as does the convergence in the entire group of G-7 inflation rates.
W.J. Crowder, C. Phengpis / Int. Fin. Markets, Inst. and Money 17 (2007) 125–139
137
Table 6
Cointegration test within the EMS group
Panel A: cointegration resultsa
H0
λtrace b
r=0
r≤1
r≤2
54.71*
18.44**
4.01
Panel B: beta coefficients normalized by the German inflation rate
CIVs
Country
Beta coefficientsc
Test statistics for H0 : βi = 0d
CIV1
CIV2
France
Italy
Germany
−0.29
−0.22
1.00
−6.84*
−7.27*
N/A
Test statistic for H0 : βi = −1 for all i = Germanye
18.84*
a
Based on the 12-lag VAR assuming no trends in the levels of data but constant terms in the cointegration space.
Compared against critical values from Table B.2 in Johansen (1995).
c The German inflation rate is chosen as a numeraire. If inflation rates completely converge, the beta coefficient must
be negative with respective to the normalized numeraire.
d Asymptotically valid t-statistics under the null hypothesis that inflation rate in country i does not have long-run
relations with the German inflation rate.
e Distributed as χ2 (2) under the null hypothesis that each of the CIVs is a one-to-one relationship between the German
inflation rate and the inflation rate in country i for all i = Germany.
* Denotes statistical significance at the 5% level.
** Denotes statistical significance at the 10% level.
b
Table 7
Weak exogeneity tests
Country
Test statistics
Canada
France
Germany
Italy
Japan
UK
US
21.83*
16.33*
18.92*
27.50*
72.20*
21.79*
18.39*
The tests are conditional on the presence of six CIVs under the null hypothesis that the inflation rate in country i is weakly
exogenous. Test statistics are distributed as χ2 (6).
* Denotes statistical significance at the 5% level.
Finally, Table 7 sets forth results from the weak exogeneity tests. The null hypothesis that the
inflation rate in one country is weakly exogenous thus the source of the common stochastic trend
can be rejected at the 5% level for each inflation rate. In other words, all G-7 inflation rates are
endogenous in that their linear combinations form a single common trend driving the system of
G-7 inflation rates over extended time horizons.
4. Conclusions and implications
Prior empirical work provides mixed evidence concerning whether inflation rates in various
countries converge, and if so, whether complete or partial convergence is present. This study
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intends to resolve this issue by employing the updated set of data from March 1973 to February
2003 for the G-7 countries and by incorporating several diagnostic techniques to ensure the
robustness of test results from conventional Johansen tests. The main research question is whether
or not various possible converging factors including the monetarist view, central bank policy
coordination, currency substitution, the relative PPP, domestic output adjustments and common
stochastic shocks contribute so collectively and strongly to international inflation convergence
that complete convergence of the G-7 inflation rates over the modern float can be inferred.
This study detects complete convergence of EMU inflation rates based on conventional
Johansen tests and more importantly complete convergence G-7 inflation rates based on several
diagnostic tests in addition to conventional Johansen tests. In either case, complete convergence
exhibits a weak form rather a strong form (or a one-to-one relationship with respect to numeraire
implied by the relative PPP). Further, this study finds that all G-7 inflation rates are endogenous
in that none is the sole source of the underlying common trend.
These results have useful implications. First, even a careful and thorough implementation of
the conventional Johansen cointegration methodology may not necessarily result in indisputable
inferences concerning the number of CIVs and common trends. Supplementary tests may be
needed in obtaining correct and sound results. Second, complete convergence of EMU inflation
rates may mislead researchers to conclude that the monetarist view is a single explanatory factor
for such convergence. However, other aforementioned factors are potentially important as well in
converging inflation rates internationally even though the subject countries do not share a common
currency and monetary policy nor formally collaborate with one another in their economic policies.
This inference is supported by the findings that the EMU convergence takes a weak-form as does
the convergence in the entire group of the G-7 inflation rates, that the EMU and non-EMU inflation
rates alike cannot be excluded from long-run equilibria (with respect to the US inflation rate as a
numeraire) and that all G-7 inflation rates are jointly driven by only one common trend. Third, the
endogeneity of each G-7 inflation rate implies that no major currency (e.g., the US dollar, the EMU
euro or the Japanese yen) can be considered a dominant reserve currency and that no individual
country exerts leading influences on economic policies of other G-7 nations in the long-run.
Acknowledgement
This paper has benefited substantially from the comments of an anonymous referee.
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