Testing for Unit Root Processes for South
American Countries
Stefan Emmoth, Nicklas Nordfors
Abstract
In this thesis, we aim to test the Purchasing Power Parity (PPP) hypothesis
on data from ten different South American countries, using different statistical
methods. We begin by testing the hypothesis using the standard linear tests
augmented Dickey-Fuller and Phillips-Perron. We then continue to test in case
of nonlinear processes with methods developed to increase the power of the
tests; exponential smooth transition autoregressive (ESTAR) and assymetric
ESTAR (AESTAR). Finally, the panel unit root Choi test is applied. We use
monthly real effective exchange rate data from 10 different South American
countries, provided by Bruegel. Our results show that the p-values generally
increase when comparing the linear Dickey-Fuller and Phillips-Perron test to
the ESTAR and AESTAR tests. We reject the null hypothesis of a unit root for
one country using the ESTAR test and for three countries using the AESTAR.
We have ambiguous changes in the p-values when comparing the AESTAR to
the ESTAR. For the Choi test, we reject the null hypothesis of a unit root
for all tests but the ESTAR test. Rejecting the null hypothesis of a unit root
indicates that the PPP hypothesis holds. Hence, new tests, e.g. nonlinear and
panel data tests that have a higher power compared to the standard linear
tests, may lead to the vindication of the PPP hypothesis.
Supervisor: Prof Kristofer Månsson
Bachelor Thesis in Statistics
Department of Economics
University of Gothenburg
Contents
1 Introduction 3
2 Literature review 4
3 Method 7
3.1 Linear testing for unit root processes . . . . . . . . . . . . . . . . . . 8
3.2 Nonlinear testing of a unit root process . . . . . . . . . . . . . . . . . 9
3.3 Panel data test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 Model diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.5 Structural change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Data 16
5 Results 17
5.1 Test for the presence of structural change . . . . . . . . . . . . . . . . 17
5.2 Test for the presence of a unit root process . . . . . . . . . . . . . . . 17
6 Conclusion 22
References 27
A Appendix 31
A.1 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
A.2 Bank of International Settlements Data . . . . . . . . . . . . . . . . . 32
1 Introduction
Our aim with this thesis is to apply a range of different tests to test for a unit root
process and evaluate their performances. The tests are applied to time series data on
the real effective exchange rates (REER) of ten different South American countries.
The rationale of choosing to test for unit root processes in REER data is the fact that
it is connected to the purchasing power parity (PPP) hypothesis. Crownover et al.
(1996) defines the purchasing power parity as "an equilibrium condition equating
the nominal exchange rate between two countries with the price ratio of an identical
bundle of goods in each country". When able to reject the null hypothesis of a unit
root, it is an indication that the PPP hypothesis holds. Therefore, researchers have
been trying to determine whether there is a unit root process in REER data.
In some cases when testing for relative PPP, researchers are trying to determine
whether a unit root exists between nominal exchange rate and a ratio of price indices.
If the researchers fail to reject the null hypothesis of a unit root it suggests that
arbitrage does not eliminate the difference between the nominal exchange rate and
the ratio of price indices which would be evidence that relative PPP would not hold
(Crownover et al., 1996).
The first attempts to validate the PPP hypothesis used the augmented Dickey-
Fuller (ADF) unit root test and failed to reject the null hypothesis, i.e. implicitly
rejecting the PPP hypothesis. This outcome was attributed to the low power of the
ADF-test. Since then, the literature has gone separate ways; nonlinear tests have
been introduced and some researchers have instead turned to panel unit root tests
(Bahmani-Oskooee et al., 2008).
For this thesis, we will apply and evaluate a total of five different tests to deter-
mine whether the time series contains a unit root process. Two of these tests are
the often-used augmented Dickey-Fuller test by Dickey and Fuller (1979) and the
Phillips-Perron test, originated by Phillips and Perron (1988). We will also apply
two newer, non-linear tests: a test for an exponential smoothing transitional autore-
gressive (ESTAR) process (Kapetanios et al., 2003) and the asymmetric exponential
smoothing transitional autoregressive (Sollis, 2009). Finally, we will also apply a
panel test for a unit root developed by Choi (2001).
These tests are then applied on real effective exchange rate data from bruegel.org,
from which we have chosen ten different South American countries. We chose all
South American countries for which we could retrieve sufficient data, and these are:
Argentina, Bolivia, Brazil, Chile, Colombia, Ecuador, Paraguay, Peru, Uruguay and
Venezuela. As far as we know, this is the first thesis to make use of data from South
American countries, which are all developing countries as well, and apply the ESTAR,
3
AESTAR and Choi tests. We have 541 monthly observations for all countries with the
exception of Brazil, Paraguay and Peru. Brazil has 422 observations and Paraguay
and Peru both have 289 observations each. Countries with 541 observations have
data from January 1970, Brazil has data from December 1979 while Paraguay and
Peru has data from January 1991. The data continues until January 2015.
We find that with the Dickey-Fuller test, we can reject the null hypothesis of a
unit root in 2 out of 10 countries. There is only a slight change when applying the
Phillips-Perron test, where it is possible to reject the null hypothesis in 3 out of 10
countries. With the standard tests for a unit root process, we do not manage to
reject the hypothesis of a unit root process for 7 out of 10 countries.
The results for the ESTAR test shows a significant difference: it is now possible
to reject the hypothesis of a unit root process for 1 out of 10 countries. This shows a
decreased power of the ESTAR test compared to the previous tests. Results for the
AESTAR shows an increase in the power of the test and we can reject the null for
2 out of 10 countries. These results differ somewhat from those found by Bahmani-
Oskooee et al. (2008). There does not seem to be nonlinearities for South American
countries. It should also be noted that we take GARCH effects into account which
Bahmani-Oskooee et al. (2008) does not.
Finally, the results of the Choi test show that we can reject the null hypothesis in
all cases when using the p-values derived from the tests with no corrected standard
errors. After having corrected for heteroskedasticity and serial correlation, the null
hypothesis is rejected for all tests but the ESTAR.
2 Literature review
One of the first observations of the purchasing power parity was made by Cassel
(1918). In the article, Cassel argues that the exchange rate between two countries is
determined foremost by "the quotient between the internal purchasing power against
goods of the money of each country". He also proposes the name purchasing power
parity for this parity (Cassel, 1918). Cassel goes on to state that in order for the
exchange rate not to deviate (to a large degree) from the purchasing power parity, two
conditions must be met: free movement of merchandise as well as some sort of trade
volume between the two countries. The condition of free movement of merchandise
is not violated unless there is a disturbance in trade that affects one country more
than the other (Cassel, 1918).
There are two measures of PPP; absolute and relative. Absolute PPP indicates
that the nominal exchange rate is solely determined by the ratio of price levels while
relative PPP states that the change in the nominal exchange rate is determined by
4
the change in the countries respective price levels, i.e the inflation. Sometimes PPP
is also referred to as the law of one price (Crownover et al., 1996).
The absolute purchasing power parity can be formally expressed as follows:
Pt =
P ∗t
St
(1)
where P t denotes the price level of in one country, Pt∗ denotes the price level in
a foreign country and St denotes the nominal exchange rate between the currencies
of the two countries, all at time t. Price level is usually represented by a Consumer
Price Index (CPI).
Relative purchasing power parity on the other hand can be expressed as follows:
∆P ∗t
P ∗t−1
=
∆St
St−1
+
∆P t
Pt−1
(2)
where the change in the nominal exchange rate and the change in domestic price
level equals the change in the foreign price level.
Testing whether the PPP hypothesis holds in the long-run have commonly been
performed through analyzing real effective exchange rates (REER), e.g. through
testing the null hypothesis of a random walk process in REER data. Later, testing
for the presence of a unit root, using e.g. augmented Dickey-Fuller, was used since
the earlier tests failed to reject the null hypothesis (Taylor, 2003). The Dickey-Fuller
test was developed by Dickey and Fuller (1979) and tests for the presence of a unit
root process for an autoregressive model. If the null hypothesis of a unit root cannot
be rejected, it also implies that the time series is non-stationary.
A common problem with these tests is that they have suffered from a lack of
power. Using Monte-Carlo simulations, Lothian and Taylor (1997) find that these
two conventional unit-root tests have a low degree of power with regards to sample
sizes of both 20 and 50 as well as two different cases of first order autocorrelations.
When increasing the sample size to 100 observations, the power of the test also
increases, but stil remains very low for higher values of autocorrelation; around 50
percent, i.e. a fifty percent chance of rejecting the null hypothesis of a unit root.
Only at a sample size of 200 does the tests demonstrate a high power for both cases
of first order autocorrelation (Lothian and Taylor, 1997).
Possible solutions to the problem of the lack of power when using the aforemen-
tioned tests have included using longer spans of data, which bears the problem that
there is a higher chance of having structural changes in the data. There is also the
possibility that the exchange rate regime could have changed when the time span is
longer. Another attempt to solve the problem has been to use panel data to increase
the number of REERs to test (Taylor, 2003).
5
More recently, studies on purchasing power parity and real exchange rates have
indicated that real exchange rates can exhibit nonlinearity rather than linearity.
There are at least three different sources of nonlinearity in real exchange rates: the
first is due to nonlinearities in international goods arbitrage, causing the goods of
markets that are geographically separated to differentiate. The second is the inter-
action of heterogeneous agents in foreign exchange markets, i.e. agents exhibiting
different opinions regarding exchange rates. The third source of nonlinearity could
be political interventions in the foreign exchange markets (Taylor, 2003).
An exchange rate is the price of one currency expressed in another currency. This
is called the nominal exchange rate, and it differs from the real exchange rate. The
real exchange rate is measure of the relative price of two countries' goods; it shows
how much of a (or several) good that is needed in country A to purchase the same
good in another country B. In order to construct a real exchange rate that reflects
the weight each trading partner has with a country. This can be done by creating a
weighted average measure of the price of goods for the foreign countries relative to
the prices domestic country. A trade weight is applied to the bilateral exchange rate
of every country to create a measure of the real effective exchange rate (Feenstra and
Taylor, 2008).
In order to deal with nonlinearity in real exchange rates, researchers have de-
veloped and proposed new statistical tests. One test proposed to deal with nonlin-
earities is the test developed by Kapetanios et al. (2003). The test is developed to
detect nonstationarity against an alternative hypothesis of nonlinear and globally
stationary exponential smoothing transition autoregressive processes. The test is
most commonly referred to as the ESTAR but is also referred to as the KSS-test.
Through Monte-Carlo simulations, Kapetanios et al. (2003) find that their test has
more power around the null hypothesis when compared to the standard Dickey-Fuller
test.
Apart from applications on simulated data, the ESTAR test has also been put to
test by Bahmani-Oskooee et al. (2008), who applies both the ESTAR test and the
conventional augmented Dickey-Fuller test to test the validity of the PPP hypothesis
for 88 different countries, using monthly real effective exchange rate data. His sample
consists of developing countries, which is argued to fill a gap in the literature, since
most of the previous studies have been done on data from industrialized countries.
Bahmani-Oskooee et al. (2008) find support for the PPP hypothesis for 12 out of 88
countries using the ADF test. Using the ESTAR test, support is found for another
19 countries, i.e. a total of 31 countries.
Bahmani-Oskooee et al. (2008) also suggest that for developing countries, there is
a higher degree of nonlinear adjustment towards PPP, due to more foreign exchange
6
interventions as well as more barriers to trade in developing countries.
A further development of the ESTAR test has been provided by Sollis (2009).
While some tests test a unit root hypothesis against the alternative of an ESTAR
process is "implicitly symmetric", the author proposes an asymmetric exponential
smooth transition autoregressive (AESTAR) test in order to test against the alter-
native of a symmetric or asymmetric process. Using Monte-Carlo simulations, the
author finds that the AESTAR test has more power than the test developed by
Kapetanios et al. (2003), provided the data generating process (DGP) is an asym-
metric ESTAR model. Furthermore, applying the test to real exchange rates from
Nordic countries, Sollis (2009) finds that while neither the Dickey-Fuller test nor the
ESTAR is able to reject the null hypothesis of a unit root. AESTAR on the other
hand is able to reject the null hypothesis in two countries (Sollis, 2009).
As previously mentioned, panel data tests were considered one possible solution
for the problem of low power (Taylor, 2003). The weakness of these multivariate
tests is that since the null hypothesis is, normally, that all of the time series contain
a unit root process. Therefore, a rejection of the null hypothesis will not determine
how many of the time series that are stationary. Taylor and Sarno (1998) showed
that in a sample of four series, one stationary process along with three unit root
processes, the single stationary process lead to a rejection of the null hypothesis at
a 5 % significance level.
Unlike many other panel data unit root tests, Choi (2001) devised a test aimed
to work under more general assumptions. For this test, the number of groups can
be either finite or infinite (as opposed to only infinite) and the time span for the
groups is assumed to be different. In Choi (2001), the p-values of a unit root test,
that has been applied to each group, are combined and tested. Choi also apply the
test to real exchange rate data, and find evidence in favor of the PPP hypothesis,
suggesting that the test could have improved power in small samples.
3 Method
In this section, the methods used to test for PPP that were introduced in the pre-
vious section are more thoroughly discussed. We start by introducing what a unit
root process is and continue with the Dickey-Fuller and Phillips-Perron tests before
moving on to present the non-linear ESTAR and AESTAR tests. We finish this
section by describing the Choi test and the model diagnostics.
7
3.1 Linear testing for unit root processes
In time series analysis one is concerned with whether a process changes over time
or not. A process is said to be stationary if the time series exhibits mean reversion,
has a finite variance and has a theoretical correlogram that diminishes as lag length
increases (Enders, 2010).
A nonstationary series has no long-run mean, the variance is dependent on time
and goes to infinity when time increases towards infinity as well as theoretical auto-
correlations in finite samples dies out slowly. A unit root process is non-stationary
(Enders, 2010).
The Dickey-Fuller test was developed by Dickey and Fuller (1979) and it is widely
used in order to find the prevalence of unit root processes in time series data. This
test was further developed by Elliott et al. (1992), resulting in the Augmented Dickey-
Fuller test.
In Dickey and Fuller (1979) considered three equations for testing the prevalence
of a unit root;
∆yt = γyt−1 + εt (3)
∆yt = a0 + γyt−1 + εt (4)
∆yt = a0 + γyt−1 + a2t+ εt (5)
The difference between the three equations concerns the presence of the deter-
ministic elements of a0 and a2t. The first equation is a pure random walk model,
the second one includes an intercept or drift term while the third equation includes
both a drift and a linear trend. When testing the PPP hypothesis we are testing the
null hypothesis γ = 0 or not in Equation 4 (Enders, 2010). If the null is not rejected
the time series has a unit root process. When testing for a unit root one needs to be
aware that the critical t-values depend on whether an intercept is included and/or
if a time trend is included. The t-values in a Dickey-Fuller has its own distribution
because a non-stationary process has infinite variance compared to the regular t-
distribution which has a finite variance. For a sample size of 100 we have that the 5
percent critical value is -2.89 and the 1 percent critical value is -3.51 (Enders, 2010).
There is also an augmented version of the Dickey-Fuller test, which controls for
potential higher-order autocorrelations. It is formalized as
∆yt = a0 + γyt−1 +
p∑
i=2
βi∆yt−1+1 + εt (6)
8
and thus allows for autoregressive processes up until the pth order. (Enders, 2010)
The second linear test we will apply in this thesis is the Phillips-Perron test for
a unit root process. The test, developed by Phillips and Perron (1988), tests for
a unit root process in univariate time series against the alternatives of stationarity
or a deterministic trend. The reasoning to include the Phillips-Perron test is that,
compared to the Augmented Dickey-Fuller, it uses Newey-West standard errors and
is thus robust against serial correlation and heteroskedasticity. (StataCorp, 2013)
Newey-West standard errors are heteroskedasticity and autocorrelation consistent
(HAC). The standard errors must be calculated conditional on a choice of maximum
lag length. The autocovariances of the standard errors are computed at the chosen
maximum lag length. Usually the lag length is chosen to exceed the periodicity of
the data, and for monthly data then at least 12 lags are sufficient. (Wooldridge,
2009)
Let us consider two least-squares regression equations
yt = µˆ+ αˆyt−1 + uˆt, (7)
yt = µ˜+ β˜(t−
1/2T ) + α˜yt−1 + u˜t, (8)
where (µˆ, αˆ) and (µ˜, β˜, α˜) are the conventional least-squares regression coeffi-
cients.
The null hypothesis of the coefficients in the regression models is thus α = 1 and
µ = β = 0. Introducing a non-zero drift µ 6= 0 in the generating process does not
effect α˜ and its t-statistic. Following, we can rewrite Equation 8 above as
yt = µ+ αyt−1 + ut (9)
where the distributions of the above-mentioned statistics are unchanged. Thus,
we use Equation 9 to test for a unit root in our time series.
3.2 Nonlinear testing of a unit root process
Linear tests, such as the standard Dickey-Fuller test, persistently fails to reject the
null of a unit root when applied to economic data, which has led some researchers
to reconsider the economic theories tested, while some researchers have turned to
developing alternative methodologies (Kapetanios et al., 2003). An example of a
new test is the ESTAR test; Kapetanios et al. (2003) have developed a nonlinear test
for a unit root against the alternative of a nonlinear exponential smooth transition
autoregressive process (ESTAR) which is globally stationary. Through Monte Carlo
9
simulations the authors find that the power of the ESTAR test is better than the
Dickey-Fuller around the null hypothesis.
In order to formalize the ESTAR model we first need to introduce a STAR(1)
model,
∆yt = α + βyt−1 + γyt−1ϑ(θ; yt−d) + εt (10)
where β and γ are unknown parameters and the error term is independently and
identically distributed with mean 0 and variance σ2.
The transition function adopted in this case is of the exponential form, and is
defined as follows
ϑ(θ; yt−d) = 1− exp(−θy
2
t−d) (11)
Where we assume θ ≥ 0, and d ≥ 1 is the delay parameter. Also, θ : R ∈ (0, 1).
By joining the two previous equations we then get the exponential STAR, i.e.
ESTAR, model,
yt = α + βyt−1 + γyt−1[1− exp(−θy
2
t−d)] + εt (12)
The above equation can be converted into:
∆yt = γyt−1[1− exp(−θy
2
t−d)] + εt (13)
where θ = β−1. If we add an intercept and impose θ = 0 and d = 1, our ESTAR
will look as follows,
∆yt = α + γyt−1[1− exp(−θy
2
t−d)] + εt. (14)
The reason for these impositions is that the Augmented Dickey-Fuller test is not
very powerful in the case where the process is stationary but nonlinear. This develops
a powerful framework (Kapetanios et al., 2003).
If θ is positive it determines the speed of the mean reversion. Many economic
models predict that the reversion towards equilibrium is faster when being far away
from it, but a rather unstable reversion the closer it is to equilibrium.
In Equation 14 above, we would test whether θ = 0 against the alternative θ > 0.
However, it is not feasible since we have another unobserved parameter, γ, in the
equation. Kapetanios et al. (2003) derived a first order Taylor series approximation
to test for unit root under the ESTAR framework with the null being δ = 0 against
the alternative δ < 0
10
∆yt = α + δy
3
t−1 + εt (15)
As we can see this is much easier to test and to do so we need a t-statistic which
is obtained as follows:
tNL =
δˆ
s.e.(δˆ)
(16)
where δˆ is the OLS estimate of δ and s.e(δˆ) is the standard error of δˆ. Unlike
when testing linearity against nonlinearity for the stationary process, the tNL test
does not have an asymptotic standard normal distribution. Since the tNL does not
have an asymptotic standard normal distribution, the tNL distribution has to be
simulated in order to allow for valid conclusions from the ESTAR tests.
However, a more general case can be presented where the errors in Equation are
serially correlated. If the serially correlated errors enter in a linear fashion, we can
use the corrections established by Dickey and Fuller (1979) and to extend Equation
into
∆yt = α +
p∑
j=1
ρi∆yt−i + γyt−1[1− exp(−θy
2
t−1)] + εt (17)
where εt ∼ iid(0, σ2). The tNL statistic for testing θ = 0 for this equation is found
in the same manner as in Equation 16 where γˆ is the OLS estimate of γ and s.e(γˆ)
is the standard error of γˆ obtained from the following auxiliary regression with the
p augmentations:
∆yt = α +
p∑
i=1
ρj∆yt−j + δy
3
t−1 + ut. (18)
The null hypothesis is that ∆ = 0 and if this is rejected, we conclude that the
time series contains a unit root. This equation is used to determine whether our time
series contains a unit root process.
There are reasons to believe that real exchange rates will have an asymmet-
ric adjustment if policymakers behave asymmetrically in the international exchange
markets. To test whether this is true or not we will make use of the asymmetrical
ESTAR, AESTAR. The AESTAR model employs both an exponential function and
a logistic function. (Sollis, 2009)
∆yt = Gt(θ1, yt−1)[(St(θ2, yt−1)ρ1 + (1− St(θ2, yt−1))ρ2]yt−1 + εt (19)
11
Gt(θ1, yt−1) = 1− exp(−θ1(y
2
t−1)), where θ1 ≥ 0 (20)
St(θ2, yt−1) = [1 + exp(−θ2yt−1)]
−1, where θ2 ≥ 0 (21)
εt ∼ iid(0, σ
2) (22)
To test this for a unit root we need an auxiliary model by taking a Taylor expan-
sion of the exponential function in the original model around γ = 0. This is done by
Sollis (2009). The derivations are outside the scope of this thesis but can be found in
the aforementioned article. In essence, the author conducts a few Taylor expansions
and end up with the following Taylor approximation,
∆yt = α + φ1y
3
t−1 + φ2y
4
t−1 +
k∑
i=1
Ki∆yt−i + ηt (23)
where α is the intercept, φ1 = ρ∗2γ1, φ2 = a(ρ
∗
2 − ρ
∗
1), and ηt = ε + Rt, where Rt
is what is left after the Taylor approximation.
When calculating the F-value for the AESTAR test in Equation 23 is the unre-
stricted model. The restricted model tests wether φ1 = φ2 = 0. Thus, it follows
that the restricted model drops φ1 and φ2. Also, for the F-statistic to be valid, φ1
in Equation 23 needs to be smaller than zero. An asymptotic F-distribution has
been simulated to provide correct p-values since the AESTAR does not follow a
conventional F-distribution.
The lag lengths chosen when applying both the linear and nonlinear unit root
tests are determined using the the Schwarz Bayesian information criterion (SBIC)
(Schwarz et al., 1978), which we can be formally defined as follows:
SBIC = ln(σˆ2) +
k
T
ln(T ) (24)
The SBIC is considered to be the most conservative of the information criteria: in
comparison to Akaike information criteria(AIC), it yields an asymptotically correct
model while the AIC will suggest too large of a model. The SBIC delivers a more
conservative model because it has a larger penalty term. The SBIC is also strongly
consistent, meaning that it asymptotically delivers the correct model order, while
the AIC is more efficient. AIC will on average deliver too large models, even with
infinite amount of data (Brooks, 2008).
12
3.3 Panel data test
As previously mentioned, the direction of research to improve upon unit root tests
seems to have gone in two ways: either focusing on panel data tests or to the devel-
opment of nonlinear tests (Bahmani-Oskooee et al., 2008). While the focus in this
thesis is on alternative and non linear tests, one panel unit root test is applied as
well: the Choi panel unit root test developed by Choi (2001). Although the Choi test
has been applied to the p-values of more standard tests of unit root processes, e.g.
ADF, there are to our knowledge no examples where it has been applied to p-values
resulting from nonlinear testing. Hence, we are interested in seeing if the results of
the test are different when applied to the linear and nonlinear tests.
Choi's test posits the null hypothesis that all the time series are unit root non
stationary against the alternative hypothesis that at least one time series is non
stationary while the others are not, in the case of finite N , i.e. number of groups
Choi (2001). In the case where N is finite, the following test statistics are appropriate
Choi (2001) (following the notation of Choi (2001)):
• P is called the inverse chi-square test, developed by Fisher in 1932:
P = −2
N∑
i=1
ln(pi) (25)
• Z is called the inverse normal test, developed by Stouffer et al. (1949):
Z =
1
√
N
N∑
i=1
Φ−1(pi) (26)
where Φ() is the standard normal cumulative distribution function, and Φ−1() is
its inverse.
• L is the logit test developed by George (1977):
L =
N∑
i=1
ln(
pi
1− pi
) (27)
The pi for all test statistics are the p-values derived from a unit root test, e.g.
augmented Dickey-Fuller or Phillips-Perron. In our case, we have also used the
p-values derived from the ESTAR and AESTAR test.
13
The rejection rules are as follows, where ciα represents the critical values at sig-
nificance level α:
P > cpα (28)
cpα is the value of the upper tail of chi-square distribution, and the degrees of
freedom is 2N
Z < czα (29)
czα is the value of the lower tail of the normal distribution,
L∗ < clα (30)
clα is the value of the lower tail of the t-distribution, and the degrees of freedom
is determined by 5N + 4.
3.4 Model diagnostics
There are three central assumptions concerning the residuals of time series regres-
sions:
• the error terms are homoskedastic,
• the errors are not serially correlated,
• the errors are normally distributed.
A standard approach to test for the presence of correlation in errors for two time
periods (serial correlation) is the Durban-Watson test Durbin and Watson (1971) and
the Breusch-Godfrey test developed by Breusch (1978) and Godfrey (1978). In this
thesis we have employed the Breusch-Godfrey test for serial correlation seeing as we
have dynamic models and the Durbin-Watson is biased for such models. (Wooldridge,
2009)
Heteroskedasticity can be tested using the Breusch-Pagan test developed by
Breusch and Pagan (1979). Serial correlation and heteroskedasticity can be cor-
rected with the Newey-West method, developed by Newey and West (1986). After
concluding heteroskedasticity or serial correlation in our ESTAR models, we have
corrected the residuals by using Newey-West standard errors. Also, GARCH effects
can be corrected with Newey-West standard errors. This was shown by Kim and
Schmidt (1993).
14
However, we are unable to use Newey-West standard errors for our AESTAR
since the correction does not provide sum of squares and therefore we cannot com-
pute a F-test. Instead, we have used sandwich covariance matrices to correct for
heteroskedasticity as well as autocorrelation. The sandwich covariance matrix was
introduced by Huber (1967), Eicker (1967) and White (1980). White developed the
commonly known White test for heteroskedasticity. The sandwich covariance matrix
is used in the Phillips-Perron test, it corrects for both heteroskedasticity as well as
autocorrelation. (StataCorp, 2013).
Some time series displays autoregressive conditional heteroskedasticity (ARCH)
and this needs to be tested for; ARCH was developed by Robert Engle (1982),
and it was later developed to a generalized ARCH (GARCH) by Bollerslev (1986).
ARCH processes are stochastic processes which have a mean of zero and are serially
uncorrelated processes. ARCH processes also have non constant variances conditional
on the past as well as constant unconditional variances. Since the ARCH model
allows for the forecast variance to change over time and since the variance is predicted
by past forecast errors, it is considered useful Engle (1982).
3.5 Structural change
A structural change or structural break is a change in the parameters of a time
series model. While an assumption of stationarity in time series data signifies that
parameters are constant over time, a structural break happens when one or more
of the parameters changes at a certain date, called the break date. The impact of
a change in a parameter will of course depend on the model in question and the
parameter that has changed and it is important to account for the possibility of a
structural change in time series since it can be harmful to inference, forecasting as
well as policy recommendations (Hansen, 2001).
One of the earliest tests for a structural break, where the break date is modelled as
unknown, was introduced by Chow (1960) and is commonly referred to as the Chow
test. The Chow test was later developed by Quandt (1960). The most relevant aspect
for this thesis is the possible effect of structural changes on unit root tests: under
the presence of a structural break, e.g. the Dickey-Fuller test is biased towards not
rejecting the null hypothesis of a unit root process (Enders, 2010).
In order to control for structural changes, the time series plots of all countries
are inspected visually. We also apply a formal test: the supF-test of Andrews (1993)
for multiple structural changes. The null hypothesis for the supF-test is that there
is no structural change in the time series.
In order to adjust for these structural changes, the breakpoint, i.e. the observation
15
where the change occurred, is needed. The breakpoint is calculated using Zeileis et al.
(2001), which implements the methodology of Bai and Perron (2003). Zeileis et al.
(2001) computes the breakpoints through minimizing the residual sum of squares of
the equation:
yt = xt
>β + ut (31)
4 Data
The data used is real effective exchange rates (REER) for ten South American coun-
tries: Argentina, Bolivia, Brazil, Chile, Colombia, Ecuador, Paraguay, Peru, Uruguay
and Venezuela. The data set is provided by Bruegel, and contains monthly observa-
tions, computed using annual CPI's. There are two versions of the data: the broad
index, which is calculated using 172 trading partners, and the narrow index which
is calculated using 67 trading partners. The broad index is available from 1960 and
forward, and the narrow index is available from 1995 (Darvas, 2012).
Table 1: Number of observations and the selected lag length.
Countries Observations Lag length Time span
Argentina 541 2 Jan 1970 - Jan 2015
Bolivia 541 6 Jan 1970 - Jan 2015
Brazil 422 1 Dec 1979 - Jan 2015
Chile 541 3 Jan 1970 - Jan 2015
Colombia 541 1 Jan 1970 - Jan 2015
Ecuador 541 1 Jan 1970 - Jan 2015
Paraguay 289 1 Jan 1991 - Jan 2015
Peru 289 5 Jan 1991 - Jan 2015
Uruguay 541 0 Jan 1970 - Jan 2015
Venezuela 541 1 Jan 1970 - Jan 2015
Table 1 shows the countries included in the analysis, along with the number of
available monthly observations, as well as the lag length chosen for each country. We
have 541 REER observations for most countries. However, Brazil has 422 observa-
tions and there are 289 observations for Peru and Paraguay.
The time period of the data spans between January 1970 and January 2015 for
all countries with 541 observations, whereas the time period for Brazil starts in
16
December 1979, and Chile and Colombia have observations from January 1991. The
time series plots are shown in Figure 1 through 10.
We easily detect sudden changes in the data. For Bolivia, there is an increase
from one month to another; from a value of 144 in the end of 1984 to a value of over
1500 towards the end of 1985. This rather drastic change is caused by hyperinflation.
There is also a similar situation for Chile, where there is a large increase in value in
the fall months of 1973 before the value plummets again: from 1024 in October of
1973 to 204 in November.
5 Results
5.1 Test for the presence of structural change
The presence of structural change in time series can lead to erroneous inference and
cause unit root tests, e.g. the Dickey-Fuller test, to be biased. Therefore, we are
interested in discovering and testing for any possible structural changes in our data.
The first step is the visual inspection of each country's REER development over
time in a time series plot, as can be seen in Figure 1 through 10. Through visual
inspection of the graphs it is easy to spot sudden changes in the trend of some of the
data. The time-series graphs show that the REER has been quite volatile over time
in all countries. For most, if not all, countries, it is also easy to draw the conclusion
that we have structural breaks in the data set. We also test for the presence of
structural change using the supF-test Andrews (1993).
Below, Table 2 is presented, containing the results of the supF-test; the supF-
statistic is presented along with the p-values. As can be seen in the table, the p-
values of all time series are 0, which tells us that the null hypothesis of no structural
change can be rejected for all time series. Therefore, the conclusion is that there is
a structural change in the time series, which must be accounted for.
When we have calculated the breakpoints or break dates for each country, we can
adjust the Augmented Dickey-Fuller test to take the structural change into account.
This adjustment is done as shown in (Enders, 2010, Chapter 4), and the results of
this adjustment are presented in Table 4.
5.2 Test for the presence of a unit root process
The following tables contains the p-values from the four tests, i.e. the Augmented
Dickey-Fuller, Phillips-Perron, ESTAR and AESTAR test, for the presence of a unit
root process. Table 3 presents the p-values without any corrections made to the
17
Table 2: supF-test of structural change: supF-statistic and p-value.
Country supF-statistic P-value
Argentina 277.1307 0
Bolivia 111.7408 0
Brazil 206.8212 0
Chile 725.9274 0
Colombia 569.4104 0
Ecuador 1901.053 0
Paraguay 390.7971 0
Peru 169.3872 0
Uruguay 402.5082 0
Venezuela 196.5973 0
residuals. The test have been applied to all ten of the South American countries.
When applicable, the number of lags used is shown in Table 1. The p-values dis-
played in Table 3 suffer from heteroskedastic and in some cases also autocorrelated
residuals. We decided to present this Table to illustrate the large differences between
uncorrected residuals and corrected residuals. P-values that are correct, in the sense
that the residuals have been corrected for heteroskedasticity and autocorrelation, are
displayed in Table 4. The p-values for the tests of heteroskedasticity and serial cor-
relation can be found in Table 6 in the Appendix. Only the ESTAR and AESTAR
estimates are different.
In order to calculate the p-values, we have generated a t-distribution for the
ESTAR test and a F-distribution for the AESTAR tests.This has been done with a
Monte Carlo simulation in which we have simulated 10 000 t-values to obtain the
correct t-distribution and F-distribution.
Table 3 displays the results from the tests of unit root. Both of the linear tests,
the Dickey-Fuller and the Phillips-Perron, display similar results. Using these tests,
we can reject the null hypothesis of a unit root for only two countries with the
Dickey-Fuller, Bolivia and Chile, and for Peru when using the Phillips-Perron test.
Argentina is significant at the ten percent level in the Dickey-Fuller test but not in the
Phillips-Perron. Moving on to our non-linear tests, the ESTAR and the AESTAR, we
have very different results. With the ESTAR test we can reject the null hypothesis
of a unit root for four countries; Argentina, Bolivia, Brazil, and Chile. Using the
AESTAR test, rejection of the null hypothesis is made for seven countries at the
5 percent level. Generally, the AESTAR test increased the power for all countries
18
Table 3: P-values from unit root tests without correcting the residual.
Countries Dickey-Fuller Phillips-Perron ESTAR AESTAR
Argentina 0.0700 0.1144 0.0057 0.0100
Bolivia 0.0000 0.0000 0.0000 0.0000
Brazil 0.0843 0.2101 0.0464 0.0360
Chile 0.0131 0.0002 0.0000 0.0000
Colombia 0.2953 0.4220 0.4095 0.2893
Ecuador 0.1388 0.1623 0.0514 0.0032
Paraguay 0.8540 0.9360 0.8246 0.8465
Peru 0.1558 0.0315 0.1171 0.0267
Uruguay 0.2457 0.2457 0.2474 0.3267
Venezuela 0.9803 0.9933 1.0000 0.0018
except for Paraguay and Uruguay.
In Table 4, we display the p-values where we have corrected the residuals for
heteroskedasticity and serial correlation. The Dickey-Fuller values has been corrected
for structural breaks and the t-statistics has been compared to the ones found in
Perron (1989). Note that we do not use p-values since we could more easily compare
our t-statistics to the ones found in Perron (1989) without simulating the appropriate
distribution.
We can reject the null for Bolivia and Colombia, while all other countries are
insignificant at the 10 percent level, while there are very few significant countries.
We can reject the null hypothesis of a unit root for only one country using the
ESTAR test, namely Bolivia. Chile has a p-value of 0.0611 and can thus be rejected
at the ten percent level. Regarding the AESTAR test, we can reject the null in three
countries; Bolivia, Peru and Venezuela. After correcting the residuals we can see that
the p-values are generally very high and evidence for PPP is thus very weak. Also,
we can see that the ESTAR and the AESTAR tests have decreased power compared
to the linear tests for some countries, which was not expected. It should be noted
that nonlinear tests are robust to structural change.
Comparing Table 3 and Table 4 there is a drastic change. For example, Argentina
had very low p-values using both the ESTAR and the AESTAR but when correcting
the residual, the p-value increased tenfold and this pattern is clear for all countries.
We have also run the analysis on data supplied by the BIS and found similar
results. Unfortunately, the data from BIS only has observations for six South Amer-
ican countries and for a shorter time span. The results were similar to the ones
19
Table 4: P-values from unit root tests with corrected residuals.
Countries Dickey-Fuller (t-statistic) Phillips-Perron ESTAR AESTAR
Argentina -2.8634 0.1144 0.5615 0.6040
Bolivia -10.6707* 0.0000 0.0000 0.0000
Brazil -2.1151 0.2101 0.1691 0.1668
Chile -1.5831 0.0002 0.0611 0.0000
Colombia -6.8406* 0.4220 0.2445 0.1119
Ecuador -2.2116 0.1623 0.6420 0.7675
Paraguay -.7046 0.9360 0.8602 0.9135
Peru -2.9350 0.0315 0.1869 0.4350
Uruguay -2.7927 0.2457 0.6597 0.7144
Venezuela 1.2374 0.9933 1.0000 0.0392
* significant at the 1% level.
presented above and can be found on the last page of the Appendix.
Below, we find a table over the observed test statistics (P , Z, L) for the Choi test
and their respective critical values at α = 0.01. The null hypothesis that all time
series are unit root non stationary will be evaluated according to the rejection rules
given in Choi (2001), i.e.: P > cpα, Z < czα and L < clα.
For the p-values derived from tests without correcting for heteroskedasticity and
serial correlation, the decision is to reject the null hypothesis for all tests at the 1
% significance level. For the p-values where standard errors are corrected, the null
hypothesis can be rejected in 3 out of 4 cases, only for the ESTAR p-values is it not
possible to reject the null hypothesis.
20
Table 5: Choi test statistics and critical values.
Test P Z L
Augmented Dickey-Fuller 52.0530 -2.9033 -19.0506
Phillips-Perron 59.5689 -2.9027 -20.6331
ESTAR 71.3608 -2.9060 -14.4774
AESTAR 91.9802 -6.1566 -43.3001
Test, corrected standard errors.
Augmented Dickey-Fuller 52.0530 -2.9033 -19.0506
Phillips-Perron 59.5689 -2.9027 -20.6331
ESTAR 38.3376 -0.3843 0.2794
AESTAR 58.1911 -2.8381 -22.0974
Critical values 37.5662 -2.3974 -2.3263
21
6 Conclusion
The results presented in the previous section, are somewhat consistent with many
other studies on the subject. Linear tests, in this case the Dickey-Fuller and Phillips-
Perron test for a unit root process, fails to reject the hypothesis in most cases. There
is a difference in P-values for both of the tests, and the p-values are generally higher
using the Phillips-Perron test due to the Newey-West corrected errors. For some
countries, the P-values decrease, e.g. Peru and Chile.
When correcting for heteroskedasticity and autocorrelation in the ESTAR test,
the P-values are higher for many countries when compared to the two linear tests.
Colombia is the only country to display a large decrease in p-value and only one
country is significant.
The ESTAR and the AESTAR generally display similar results, however we can
see that the p-value for Peru increases and that the p-value for Venezuela dramatically
decreases. Notably, we can only reject the null hypothesis for three countries with
the AESTAR test.
Without correcting the residuals for the nonlinear tests the results are quite differ-
ent when compared to (Bahmani-Oskooee et al., 2008). After correcting the residuals
for serial correlation and heteroskedasticity, our results remain rather different from
the results obtained in the previously mentioned study. In Bahmani-Oskooee et al.
(2008), the null hypothesis is rejected for two countries in South America: Chile and
Ecuador.
In contrast, we are able to reject the null hypothesis for only one country with
the ESTAR test, Bolivia. However, we find that Ecuador has a p-value of 0.6420,
which means that we can no longer reject the null hypothesis of a unit root processs.
Chile has a p-value of 0.06 which is fairly close to being significant. When residuals
are corrected, we can only reject the null hypothesis of a unit root process for one
country; which indicates that these nonlinear tests are not able to confirm the PPP
hypothesis when correcting the residuals.
The Choi test is also applied on the p-values derived from the above mentioned
tests. The result, at a 1 % significance level, is that we find evidence to reject the
null hypothesis in for all tests except for the ESTAR test. The conclusion we can
draw from this is that by applying this panel unit root test, we do find evidence in
support of the PPP hypothesis when applying ADF, PP and the AESTAR test.
When rejecting the hypothesis of a unit root process, we implicitly find evidence
in favor of the power purchasing parity hypothesis. Using the other tests included
in this thesis, we have not found evidence for PPP, due to lack of power. Another
possible explanation for not being able to reject the hypothesis for the Dickey-Fuller
22
test could be the presence of structural breaks. It should also be noted that the
failure to vindicate the PPP hypothesis could be the simple explanation that the
nations in question violate the assumptions of the hypothesis through barriers to
trade or similar interventions.
Possible weaknesses in this thesis is the choice of data: while we have tried to
corroborate our results using data from the Bank of International Settlements, we still
have the problem of possible exchange regime changes in the period of the chosen
countries. This could affect our results, as mentioned by Taylor (2003). Another
weakness lies in the application of the Choi panel data test: when we have p-values
that are practically zero, it influences the test strongly towards rejecting the null
hypothesis.
23
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Figure 10: Venezuela
26
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A Appendix
A.1 Tables
Table 6: Tests of autocorrelation, heteroskedasticity, and ARCH
Country Test Autocorrelation Heteroskedasticity ARCH
Argentina ESTAR 0.7203 0.0000 0.3016
AESTAR 0.1697 0.0000 0.0293
Bolivia ESTAR 0.0000 0.0000 0.0000
AESTAR 0.0243 0.0000 0.2427
Brazil ESTAR 0.1691 0.0000 0.0000
AESTAR 0.1484 0.0000 0.0000
Chile ESTAR 0.0000 0.0000 0.0000
AESTAR 0.0001 0.0000 0.6882
Colombia ESTAR 0.3296 0.0008 0.2281
AESTAR 0.2998 0.0001 0.2732
Ecuador ESTAR 0.8439 0.0000 0.4644
AESTAR 0.3789 0.0000 0.3465
Paraguay ESTAR 0.1356 0.0000 0.0000
AESTAR 0.1357 0.0000 0.0000
Peru ESTAR 0.0020 0.0000 0.0033
AESTAR 0.0018 0.0000 0.0002
Uruguay ESTAR 0.7016 0.0000 0.7300
AESTAR 0.7227 0.0000 0.7313
Venezuela ESTAR 0.2399 0.0000 0.5713
AESTAR 0.3313 0.0000 0.6185
31
A.2 Bank of International Settlements Data
Table 7: Number of observations and the selected lag length.
Countries Observations Lag length Time span
Argentina 252 1 Jan 1994 - Mar 2015
Brazil 252 2 Jan 1994 - Mar 2015
Chile 252 1 Jan 1994 - Mar 2015
Colombia 252 1 Jan 1994 - Mar 2015
Peru 252 1 Jan 1994 - Mar 2015
Venezuela 252 1 Jan 1994 - Mar 2015
Table 8: P-values from unit root tests without correcting the residual.
Countries Dickey-Fuller Phillips-Perron ESTAR AESTAR
Argentina 0.5885 0.6527 0.3917 0.0495
Brazil 0.4565 0.4534 0.4181 0.5094
Chile 0.0654 0.1260 0.0726 0.1001
Colombia 0.0889 0.1566 0.071 0.0869
Peru 0.1401 0.2494 0.1326 0.1941
Venezuela 0.9934 0.9968 1.0000 0.0097
32
Table 9: P-values from unit root tests with corrected residuals.
Countries Dickey-Fuller Phillips-Perron ESTAR AESTAR
Argentina 0.5885 0.6527 0.7686 0.5440
Brazil 0.4565 0.4534 0.3573 0.4641
Chile 0.0654 0.1260 0.0726 0.1001
Colombia 0.0889 0.1566 0.071 0.0461
Peru 0.1401 0.2494 0.1326 0.2145
Venezuela 0.9934 0.9968 1.0000 0.0051
33
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Figure 11: Argentina
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400
Ch
ile
1995m1 2000m1 2005m1 2010m1 2015m1time
Source: BIS.org
Figure 13: Brazil.
0
100
200
300
400
Co
lom
bia
1995m1 2000m1 2005m1 2010m1 2015m1time
Source: BIS.org
Figure 14: Colombia
34
0100
200
300
400
Pe
ru
1995m1 2000m1 2005m1 2010m1 2015m1time
Source: BIS.org
Figure 15: Chile.
0
100
200
300
400
Ve
nez
uel
a
1995m1 2000m1 2005m1 2010m1 2015m1time
Source: BIS.org
Figure 16: Ecuador
35