Black market exchange rate versus the official rate in testing the PPP: an application of a non-linear test.
Bahmani-Oskooee, Mohsen ; Tanku, Altin
INTRODUCTION
The purchasing power parity theory (PPP) is perhaps one of the most
important theories in economics that has received the largest attention
in the literature (1). In its absolute form, the theory claims that in
the long run the exchange rate between two currencies will be equal to
the ratio of two corresponding national price levels. One implication of
the theory is that a country can fight inflation by managing the value
of its currency in terms of the currency of its major trading partners.
For a successful policy, the relative prices and the nominal exchange
rate should converge over time and form a stationary process. Thus, one
way of testing the PPP is to test for the mean-reverting property of the
nominal exchange rate that incorporates changes in relative prices into
its mean-reverting process, that is, testing for stationarity of the
real exchange rate.
Early studies that tested for stationarity of the real exchange
rates or other formulation of the PPP mostly rejected it. Examples
include Taylor (1988), Carbae and Ouliaris (1988, 1991), McNown and
Wallace (1989), Layton and Stark (1990), Kim (1990), Bahmani-Oskooee and
Rhee (1992), and Bahmani-Oskooee (1995). Examples of those who have
supported the PPP include Bahmani-Oskooee and Barry (1997), Lothian and
Taylor (1996), and Bahmani-Oskooee (1998). (2) Most of these studies
relied upon the standard Augmented Dickey-Fuller (ADF) test in which the
null hypothesis of nonstationarity or unit root is tested against a
linear stationarity. Recently Kapetanios et al. (2003, KSS hereafter)
have developed a new test that is based on a non-linear exponential smooth transition autoregressive (ESTAR) procedure in which the null is
still unit root but the alternative is non-linear stationarity in a
time-series variable. Application of this new test by most recent
studies reveals that the new test supports the PPP more often than the
standard ADF test implying that the mean-reverting process of real
exchange rates follows a non-linear path. Examples in this later group
of studies include Sarno (2000), Taylor et al. (2001), Liew et al.
(2004), Chortareas and Kapetanios (2004), Sarno et al. (2004), and
Bahmani-Oskooee and Gelan (2006).
One common feature of all of the studies mentioned above is that
they have all relied upon the official exchange rates in testing the
PPP. There is another part of the literature that includes studies that
have used black market exchange rates in testing the PPP in
less-developed countries (LDCs). The main argument for using the black
market or parallel exchange rates in testing the PPP is that they serve
as a proxy for floating exchange rates in LDCs. The literature reviewed
by Bahmani-Oskooee and Goswami (2005) reveals that PPP is supported more
often when the black market exchange rate is used in the testing
procedure. Studies in this group are Culbert (1975), Edwards (1989),
Phillips (1988), Bahmani-Oskooee (1993), El-Sakka and McNaab (1994),
Baghestani (1997), Sanchez-Fung (1999), Luintel (2000), Kouretas and
Zarangas (2001), Nagayasu (2000), and Bahmani-Oskooee and Goswami
(2005).
Although the above-mentioned studies validate the PPP by using the
black market more often than the official exchange rate, support for the
PPP is not unanimous across the countries. In this paper, we try to test
the PPP by using both the official as well as the black market exchange
rates. In doing so we employ both the standard ADF test and the KSS test
to determine whether the mean-reverting properties of the real black
market exchange rates are on a non-linear but stationary path. To this
end, we introduce the KSS test in the next section. The penultimate section reports the results with concluding remarks in the final
section.
THE RELATIVELY NEW KSS TEST (3)
The standard ADF test sets the stage for testing the null of
non-stationarity or unit root against the alternative of stationarity of
a time-series variable that follows a linear path. In order to see the
difference between the standard ADF test and the new KSS test, we
outline the standard ADF test first, as in equation (1) for a
time-series variable Z.
[DELTA][Z.sub.t] = [delta][Z.sub.t-1] + [n.summation over (k=1)]
[[rho].sub.k] [DELTA][Z.sub.t-1] + [[epsilon].sub.t] (1)
Kapetanios et al. (2003) built upon the standard ADF test outlined
by (1) and introduce a new test, but a relatively more powerful one, in
which the null hypothesis is still unit root but the alternative
hypothesis is a non-linear stationary smooth transition autoregressive
(STAR) process. They demonstrate that in small samples the linear ADF
test suffers from some upward size distortion whereas the size of their
new non-linear test rarely rises above the 5% level. Furthermore, the
goodness of fit is substantially better under the exponential smooth
transition autoregressive (ESTAR) specification as compared to the
simple autoregressive specification. The new test is based on the
following ESTAR specification:
[DELTA][Z.sub.t] = [lambda][Z.sub.t-1][1 -
exp(-[theta][Z.sup.2.sub.t-1]] + [[mu].sub.t] (2)
In (2) the time-series variable [Z.sub.t] could be raw data, the
de-meaned data, or the de-trended series. (4) In (2), like (1), the null
of unit root, that is, [theta]-0, is tested against the alternative of
[theta] > 0. However, Kapetanios et al. (2003) show that since
[lambda] is not identifiable under the null, (2) could be approximated
by (3) using the Taylor series.
[DELTA][Z.sub.t] = [delta][Z.sup.3.sub.t-1] + [[omega].sub.t] (3)
Furthermore, to make the residuals in (3) white noise, they include
the augmented terms as in (4):
[DELTA][Z.sub.t] = [delta][Z.sup.3.sub.t-1] + [n.summation over
(k=1)][[rho].sub.k[DELTA][Z.sub.t-k] + [[omega].sub.t] (4)
Comparing (4) to (1) we gather that in (4) non-linearity is
reflected by raising the lagged value of the time-series variable under
consideration to power three rather than to power one. Again the null of
unit root, that is, [delta] = 0 is tested against the alternative of
[delta] > 0 by the familiar t ratio obtained for [delta]. However,
the new test has a new distribution for which the critical values are
tabulated by Kapetanios et al. (2003). For selecting the lag length on
the augmented term, we follow Kapetanios et al. (2003, p. 365) and rely
upon significance of augmented terms. Since this t ratio is for a
non-linear model as outlined by (4), we shall denote it by [t.sub.NL].
THE RESULTS
In this section we apply the standard ADF test outlined by equation
(1) as well as the non-linear ADF test outlined by equation (4) to the
official and the black market real exchange rates from 24 developing
countries. The CPI data to be used in constructing the real exchange
rates and the official exchange rates come from the International
Financial Statistics of the IMF and the black market exchange rates come
from Reinhart and Rogoff (2004). (5) Study period differed from one
country to another depending on the availability of the black market
exchange rate as shown in Table 1.
The results of the unit-root tests applied to both real official
and real black market exchange rates (in log form) are reported in Table
2.
Reported in Table 2 are four statistics. The t-ratio from the
standard ADF test that includes only a constant term is denoted by ADFc.
In applying the non-linear test, since there is no constant in (4),
following Kapetanios et al. (2003) we de-mean the data and apply the
test to de-meaned data. The t-ratio for this non-linear test is reported
as [t.sub.NL-demeaned]. Following the literature, these two tests are
used to examine the mean-reverting properties of real exchange rates or
reversion to a constant as a way of testing PPP.
Let us first concentrate on the results for the real official
exchange rate (Table 2). The linear ADF test rejects the null of unit
root only in three countries, providing support for PPP (ie, Argentina,
Egypt, and Suriname). These cases are identified by a * next to their
statistics. However, when we shift to the non-linear test results, the
null of unit root is rejected in 11 countries. Therefore, like previous
research, when the official exchange rate is used, the non-linear test
supports the PPP more often than the linear or standard ADF test.
Now consider the results in Table 2 when the real black market
exchange rate is used to carry out the unit root tests. The null of unit
root is rejected in three countries by the linear ADF test. However,
when we shift to the nonlinear test, the null is rejected in 13
countries. Again, the non-linear test rejects the null much more
frequently than the linear test. Comparing these results to those of the
official exchange rates, the non-linear tests support the PPP more often
than the standard linear tests. But they support the PPP even more when
the real black market exchange rate is used compared to the real
official rate.
As an additional exercise, we tested the reversion in real exchange
rates to a trend. This amounts to including a constant and a trend term
in the standard ADF test. The statistic is denoted by [ADF.sub.t]. As
for the non-linear test, again since there is no trend term in (4),
following Kapetanios et al. (2003) we de-trend the data and apply the
test to de-trended data. The statistic for this test is denoted by
[t.sub.NL-detrend]. Table 3 reports the results.
From Table 3 we gather that the standard ADF test supports trend
stationarity of the official exchange rate only in one country (ie,
Syria) whereas, when the black market exchange rate is used, the null of
trend stationarity is supported in five countries (ie, Costa Rica,
Ethiopia, Kenya, Pakistan, and South Africa). Again, the null is
rejected more often when the black market exchange rate is used in the
testing procedure. The same is true when we consider the non-linear
tests. The null of unit root is rejected by the [t.sub.NL-detrend] for
10 countries when the official exchange rate is used. However, it is
rejected for 17 countries when the black market exchange rate is used.
Several studies have interpreted reversion to a trend as a method
of testing the productivity bias hypothesis. The hypothesis asserts that
in the long run, movement in the real exchange rate is determined by
productivity differentials. (6) Our results show that the productivity
bias hypothesis is supported more often when the black market exchange
rate rather than the official rate is used to test the hypothesis. This
supports Bahmani-Oskooee and Gelan (2006) who used regression analysis for testing the productivity bias hypothesis and showed that the
hypothesis is supported more often when the black market exchange rate
is used in the testing procedure.
SUMMARY AND CONCLUSION
The absolute PPP asserts that in the long run the exchange rate
between two currencies should be equal to the ratio of two corresponding
prices. One implication of the theory is that when the nominal exchange
rate and relative prices are combined and form a long-run cointegrating
relationship, their linear combination, which reflects the movement in
the real exchange rate, must be stationary or mean reverting to support
the PPP.
Most previous studies that tested the mean-reverting properties of
the real exchange rate relied upon the standard ADF test, which sets the
stage for testing the null hypothesis of non-stationarity or unit root
against the alternative of linear stationarity. Since introduction of a
new test by Kapetanios et al. (2003) the emphasis has shifted to
determine whether the mean-reverting property of real exchange rates are
on a non-linear path. Almost all studies that have used the relatively
new non-linear test have shown that the new test validates the PPP more
often than the standard ADF test. All these studies, however, have used
the real official exchange rates to support their assertion.
There is yet another group of studies that has emphasised the use
of black market exchange rates in testing the PPP on the grounds that
such rates are proxies for the floating exchange rates. The floating
exchange rate system lubricates the adjustment path between exchange
rates and relative prices, thus supporting the PPP more often than the
fixed exchange rate system. Although almost all studies have shown that
using the black market exchange rate supports the PPP more often than
using the official exchange rate, this conclusion is not unanimous and
is based on the application of standard and linear ADF tests.
In this paper, we test the PPP by using the standard ADF as well as
the new non-linear ADF test. The main purpose here is to determine
whether the non-linear test validates the PPP more often than the
standard ADF test when we employ the black market exchange rates. To
this end, we use data from 24 developing countries. Two main conclusions
stand out. First, the non-linear test provides support for the PPP more
often than the standard ADF test when either the official exchange rate
or the black market rate is used. Second, concentrating on the
non-linear tests PPP receives somewhat more support when the black
market exchange rate is used in testing the PPP as compared to using the
official exchange rate. The same is true when we test reversion to a
trend as a means of testing the productivity bias hypothesis. This
hypothesis receives relatively more support when the non-linear test is
applied to black market exchange rates as compared to official rates.
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(1) Valuable comments of two anonymous referees and Richard Perlman
are greatly appreciated. Remaining errors, however, are ours.
(2) For a review article, see Rogoff (1996). A few studies have
supported the PPP, for example, Bleaney (1991), Beach et al. (1993),
Moosa (1994), Bahmani-Oskooee and Barry (1997), Lothian and Taylor
(1996), and Bahmani-Oskooee (1998).
(3) This section closely follows Bahmani-Oskooee et al. (2007a, b).
(4) [mu] is an error term with usual properties.
(5) The data are actually on Reinhart's website at
http://www.puaf.umd.edu/faculty/papers/ reinhart/reinhart.htm.
(6) See for example Bahmani-Oskooee et al. (2007b).
MOHSEN BAHMANI-OSKOOEE & ALTIN TANKU
The Center for Research on International Economics and The
Department of Economics, The University of Wisconsin-Milwaukee,
Milwaukee, WI 53201, USA. E-mail: bahmani@uwm.edu
Table 1: Study period for each country
Country Starting date Ending date
Algeria 1974Q1 1998Q4
Argentina 1968Q4 1998Q4
Chile 1973Q1 1998Q4
Costa Rica 1957Q1 1998Q4
Egypt 1957Q1 1998Q4
Ethiopia 1970Q3 1998Q4
Indonesia 1968Q1 1998Q4
India 1957Q1 1998Q4
Jordan 1976Q1 1998Q4
Kenya 1966Q4 1998Q4
Malaysia 1957Q1 1998Q4
Myanmar 1970Q1 1998Q4
Mexico 1957Q1 1998Q4
Morocco 1959Q1 1998Q4
Nigeria 1970Q3 1998Q4
Pakistani 1957Q1 1998Q4
Paraguay 1957Q1 1998Q4
Philippine 1957Q1 1998Q4
South Africa 1957Q1 1998Q4
Sri Lanka 1957Q1 1998Q4
Suriname 1970Q3 1998Q4
Syria 1970Q3 1998Q4
Thailand 1965Q1 1998Q4
Turkey 1980Q1 1998Q4
Table 2: The results for mean reversion of the real official and
the black market exchange rate
Official exchange rate
Country [ADF.sub.c] [t.sub.NL-demeaned]
Algeria -0.48730 -0.82218
Argentina -2.7161 * -4.0398 *
Chile -1.0987 -1.3953
Costa Rica -1.1423 -4.8530 *
Egypt -2.6673 * -2.3525
Ethiopia -2.1103 -1.5793
India -0.036137 -1.6294
Indonesia 0.60163 -8.0719 *
Jordan -2.0881 -1.6661
Kenya -1.4377 -3.0284 *
Malaysia -1.0527 -3.5438 *
Myanmar -0.70773 -0.96901
Mexico -2.4185 -1.8229
Morocco -2.0961 -3.2377 *
Nigeria -1.6962 -2.5996
Pakistani 0.14959 -0.023946
Paraguay -1.5300 -2.1968
Philippine -2.1561 -0.88850
South Africa -1.6778 -5.4370 *
Sri Lanka -0.86966 -0.86564
Suriname -2.9486 * -3.4762 *
Syria -1.4530 -3.2904 *
Thailand -1.2475 -6.9223 *
Turkey -1.8554 -3.8390 *
10% Critical value -2.57 -2.66
Black market exchange rate
Country [ADF.sub.c] [t.sub.NL-demeaned]
Algeria -1.7331 -1.4106
Argentina -2.3959 -6.9858 *
Chile -1.7609 -1.3149
Costa Rica -2.8778 * -6.0357 *
Egypt -1.8641 -3.4307 *
Ethiopia -1.7763 -0.89291
India -1.4264 -2.0218
Indonesia 0.20028 -7.6739 *
Jordan -1.5256 -1.5419
Kenya -3.7860 * -3.1587 *
Malaysia 0.059040 -4.1983 *
Myanmar -2.1250 -2.9965 *
Mexico -2.2236 -3.9350 *
Morocco -1.7957 -1.7174
Nigeria -1.7088 -3.1691 *
Pakistani -1.0578 -1.4942
Paraguay -1.4322 -1.9730
Philippine -2.1277 -5.2079 *
South Africa -3.2575 * -3.3577 *
Sri Lanka -2.1161 -1.4978
Suriname -1.8716 -3.0207 *
Syria -1.4784 -1.9948
Thailand -0.97783 0.25155
Turkey -2.2565 -3.3951 *
10% Critical value -2.57 -2.66
Note: Critical values come from Kapetanios et at. (2003, p. 364).
Table 3: The for trend-stationarity of the real official and the black
market exchange rate
Official exchange rate
Country [ADF.sub.t] [t.sub.NL-detrended]
Algeria -2.2443 -1.4841
Argentina -2.7548 -3.9591 *
Chile -0.83752 -1.1328
Costa Rica -1.9987 -3.6371 *
Egypt -2.9633 -2.8591
Ethiopia -2.3088 -2.0881
India -2.0691 -2.7175
Indonesia -1.0974 -8.9847 *
Jordan -2.5144 -3.1149
Kenya -2.1501 -3.1605 *
Malaysia -2.5051 -4.5851 *
Myanmar -1.3843 -1.7461
Mexico -2.5391 -1.7750
Morocco -2.8014 -2.9015
Nigeria -1.7084 -2.8037
Pakistani -2.7075 -4.4616 *
Paraguay -2.5966 -2.6481
Philippine -2.9038 -2.7762
South Africa -2.6334 -5.4858 *
Sri Lanka -1.1876 -1.6419
Suriname -2.9267 -3.4349 *
Syria -3.3356 * -3.9554 *
Thailand -2.0011 -8.1923 *
Turkey -2.5308 -2.3250
10% Critical value -3.12 -3.13
Black market exchange rate
Country [ADF.sub.t] [t.sub.NL-detrended]
Algeria -2.0589 -3.5951 *
Argentina -2.6656 -6.7001 *
Chile -1.8507 -0.73845
Costa Rica -3.6053 * -6.6801 *
Egypt -2.3080 -3.4316 *
Ethiopia -3.2922 * -4.0022 *
India -2.2356 -3.1398 *
Indonesia -1.4868 -8.3244 *
Jordan -1.6160 -6.1116 *
Kenya -3.8344 * -3.4926 *
Malaysia -1.4677 -5.7319 *
Myanmar -1.9652 -2.9212
Mexico -2.3406 -4.0544 *
Morocco -2.1817 -3.1416 *
Nigeria -2.4589 -3.5022 *
Pakistani -3.2837 * -4.7848 *
Paraguay -2.0312 -2.3147
Philippine -2.9599 -6.0179 *
South Africa -4.0747 * -3.6068 *
Sri Lanka -2.2071 -4.9789 *
Suriname -1.8404 -3.1131
Syria -1.7789 -2.3754
Thailand -1.9121 0.26016
Turkey -2.8861 -2.1510
10% Critical value -3.12 -3.13
Note: Critical values come from Kapetanios et al. (2003, p. 364).
