Half Life of Real Exchange Rate:
Evidence from Nonlinear Approach
Chin-Ping King*
Abstract
The famous “purchasing power parity puzzle”, which indicate the question in
reconciling the high short-term volatility of real exchange rates with high persistence
of real exchange rates, has attract numerous research interest. However absence of
perfect international arbitrage in the presence of market frictions can result in
nonlinear adjustment of real exchange rates. Recently, the more evidences show the
real exchange rate reveals the non-linearity. We choose the prominent nonlinear model,
threshold autoregressive model (TAR model), to conduct our analysis. The empirical
results about OECD countries show that the real exchange rates of U.S. dollar-Spain
peseta and U.S. dollar-Korea won can fit the two-regime TAR model. The half-life
estimates for real exchange rate can be computed through the application of
generalized impulse response analysis on estimated TAR model. The half-life
estimates which are smaller than or equal to one year in our study is evidently shorter
than 3-5 years, which are found in previous studies. These empirical evidences can
resolve the purchasing power parity puzzle, which has been a controversial issue over
the years.
Keywords: Purchasing Power Parity Puzzle, Half-life, Market Frictions, Threshold
Autoregressive Unit Root Test, Threshold Autoregressive Model,
Generalized Impulse Response Function
JEL Classification: C15, C22, F31, G15
*Assistant Professor, Department of Finance, Providence University
1
1. Introduction.
The famous “purchasing power parity puzzle”, which indicate the question in
reconciling the high short-term volatility of the real exchange rates with their high
persistence, has attract numerous research interest. It seems that there are no receiving
model can simultaneously explain both the inappropriate slow convergence to PPP
and the immense short-term volatility in the real exchange rate. However, the speed of
parity reversion among the present prevailing empirical studies has reported relatively
longer, 3-5 years, half-life estimates. Such high persistence of the real exchange rate
has resulted in tremendous exploration from various aspects. As well discussed, the
cause of short-term nominal exchange rate volatility is often due to the nominal shock
under price rigidity. Rogoff (1996) propose if nominal rigidity is responsible for
short-term volatility, substantial convergence to PPP should be one to two years.
Several recent relevant papers try to resolve the puzzle. Cheung and Lai (2000)
use impulse response analysis based on ARMA models to calculate half-lives of PPP
deviation, and also compute its confidence intervals. They show the lower bounds of
confidence interval for half-life estimate are low enough to resolve the puzzle. Murry
and Papell (2002) use DF and ADF regression separately to conduct analysis. They
apply the least square and median-unbiased estimate of lagged term in the regression
and impulse response function to compute the half-life measure and its associated
confidence interval. They also take serial correlation and small sample bias into
account to compute half-life estimates and their confidence interval. Calculating
confidence intervals as well as point estimates for long-horizon post-1973 data, they
find that most of the point estimates lie within the 3-5 years range. Taylor and Peel
(2000) relate the exchange rate to economic fundamentals such as the monetary model
while they focus long-run exchange rate behavior. However they find the relationship
between the exchange rate and the economic fundamentals is nonlinear. Their
empirical study provides the degree of exchange rate overvaluation and
undervaluation of the dollar relative to sterling (or to mark) over the recent float.
Taylor, Peel and Sarno (2001) seek to resolve the purchasing power parity puzzle
through an analysis of nonlinearity in the real exchange rate adjustment toward
long-run equilibrium. They find the speed of adjustment to the real exchange rate
shock occurred in nonlinear model is faster than that in linear model.
Past literatures show while we apply traditional ADF test to investigate the real
2
exchange rates, we often cannot reject unit root null hypothesis. But traditional ADF
test has very low power of test while time series is near to unit root. So even the real
exchange rates do not have unit root, we often cannot reject false null hypothesis.
However if the real exchange rates are not stationary, then deviation of the real
exchange rates from equilibrium will not revert back to original equilibrium, i.e. PPP
will not hold. Past studies use market frictions (like transaction cost, tariff, trade
barriers and so on) to explain the random walk behavior of the real exchange rates.
These market frictions will impede international arbitrage. Based on this incomplete
arbitrage, the real exchange rates will not have mean-reverting property.
However some studies try to resolve above problem. Frankel and Rose (1996),
Lothian (1997) and Taylor, Peel and Sarno (2001) apply panel data to test several real
exchange rates jointly. Panel unit root test can increase the power of test. And in many
of these studies, the unit root null hypothesis is rejected for such groups of the real
exchange rates. But while we apply panel unit root test to investigate jointly the real
exchange rates of several countries, the rejection of null hypothesis only imply at least
one of these several countries is stationary. We can’t guarantee all countries under
consideration are stationary. Cheung and Lai (1993) and Lothian and Taylor (1996)
increase the length of the sample period to enhance the power of test of unit root tests.
This approach can reject unit root null hypothesis. But these long run data are
extracted from different historical periods and from different nominal exchange rate
regimes. So we should not use these data to evaluate the post-1973 floating exchange
rates.
From another aspect, absence of perfect international arbitrage in the presence of
market frictions can result in nonlinear adjustment of the real exchange rates.
Recently, the more evidences show the real exchange rate reveals the nonlinearity.
Hence the application of linear model cannot account for the adjustment mechanism
of real exchange rate completely and precisely. Instead of applying linear model
specification, we need to test and estimate nonlinear model. As known, all nonlinear
autoregressive models can be distinguished as the TAR model (threshold
autoregressive model) and the STAR model (smooth transition autoregressive model).
The TAR model represents time series can suddenly and abruptly switch between
different regimes. The STAR model represents the switching between different
regimes is slow, continuous and transitional. So in some degree, we can take the
STAR model as slightly nonlinear model, and take the TAR models as heavily
nonlinear models. From another aspect, heavy market frictions will result in serious
lack of perfect international arbitrage. This circumstance should be appropriate to use
3
the TAR models to delineate behavior of real exchange rates. Slight market frictions
will result in a little lack of perfect international arbitrage. Then this circumstance
should be appropriate to use the STAR models to delineate behavior of the real
exchange rates. However market frictions should be interpreted as resulting not only
from transportation costs and trade barriers, but also from sunk costs of international
arbitrage and the resulting tendency of traders to respond only to sufficiently large
arbitrage opportunities. However real world economies own large degree of market
frictions. Especially, less highly developed countries and median industrial countries
often suffer heavier market frictions. While less highly developed countries and
median industrial countries engage in international business, these countries evidently
have the problems of heavier market frictions. Recent investment theory suggests that
if international arbitrage involves such degree of market frictions, then unhedgeable
uncertainty can be levered up into relative large and sharp adjustment of real
exchange rates. So we can appropriately use threshold to characterize these degree of
market frictions. Hence we apply the threshold autoregressive model (TAR model) to
implement our study.
We choose the prominent nonlinear model, threshold autoregressive model (TAR
model), to conduct our analysis. We apply the TAR model to fit nonlinearity for the
following reasons. First, for many economic phenomena, the discrete adjustment may
often prevail. While there are fixed cost of adjustment, economic agents don’t adjust
continuously. For instance, in financial markets, the presence of transaction costs may
create the bands in which asset price can be free to fluctuate (i.e. there are arbitrage
possibilities). If the deviation from the equilibrium exceeds the bands, agent will act
to move the economy back towards the equilibrium. Second, recently the statistical
framework for the TAR model are well developed, the application of the TAR model
don’t counter the problem of statistical inference. Third, all nonlinear autoregressive
models can be distinguished as the TAR model and the STAR model. In some degree,
we can take the STAR model as slightly nonlinear model, and take the TAR models as
heavily nonlinear models. From another aspect, we can use the TAR models to
characterize heavier market frictions, which often occur in less highly developed
countries. Heavier market frictions can impede smooth operation of international
arbitrage. So we can’t use the STAR models to deal with countries, which have
heavier market frictions. We can apply the TAR models to analyze behavior of
financial asset price of such kind countries, which suffer heavier market frictions.
Fourth, through serious and formal statistical testing we can certain that the real
exchange rates, which we choose to fit the TAR models, are indeed threshold
autoregressive processes.
4
In the present study, we apply the TAR model to fit the real exchange rate. We
conduct the hypothesis testing for whether the real exchange rates follow the TAR
process. We focus on the real exchange rates about U.S. dollar vs. currencies of all
other 29 OECD countries. The empirical results about OECD countries show that the
real exchange rates of U.S. dollar-Spain peseta and U.S. dollar-Korea won can fit the
two-regime TAR model. It’s not surprising that no major countries of OECD (like
U.K., German, France, Japan and so on) real exchange rates can fit the TAR models.
Because market frictions will impede perfect international arbitrage, the incomplete
arbitrage will result in the nonlinear adjustment of real exchange rates. More market
frictions make real exchange rates to be more nonlinear. In some degree, we can take
the STAR model as slightly nonlinear model, and take the TAR models as heavily
nonlinear models. From another aspect, heavy market frictions will result in serious
lack of perfect international arbitrage. This circumstance should be appropriate to use
the TAR models to delineate behavior of real exchange rates. Slight market frictions
will result in a little lack of perfect international arbitrage. Then this circumstance
should be appropriate to use the STAR models to delineate behavior of the real
exchange rates. The market frictions of major countries of OECD seem to be slight.
So it is reasonable that the real exchange rates for major countries of OECD cannot fit
the TAR models. However the market frictions of median countries of OECD (like
Spain and Korea) seem to be heavier. So both U.S. dollar-Spain peseta and U.S.
dollar-Korea won real exchange rates can fit the TAR model.
Due to the non-linearity, the traditional impulse response function analysis
cannot be applied appropriately. Then for the sake of quantifying the speed of parity
reversion, we perform the generalized impulse response function analysis, which is
developed by Koop, Pesaran and Potter (1996). The half-life estimate for the real
exchange rate can be computed through the application of generalized impulse
response analysis on estimated TAR model. The empirical results show that when the
shock is 100%, the half-life for U.S. dollar-Spain peseta real exchange rate is only
0.67 years and the half-life for U.S. dollar-Korea won is 0.42 years. When the size of
shocks decreases, the half-life estimates increase. This empirical evidence can resolve
the purchasing power parity puzzle, which has been a controversial issue over the
years.
The rest of the present paper can be structured as the following. Section 2 we
discuss how to model the non-linearity and how to apply generalized impulse
response function analysis for nonlinear model to compute half-life estimate. Section
5
3 we report our empirical results. We state conclusion in Section 4.
2. Measuring the Speed of Nonlinear PPP Reversion
Past papers show when we apply traditional ADF test to test the real exchange
rates, we cannot reject the unit root null hypothesis. If the real exchange rates are not
stationary, then deviation of the real exchange rates from equilibrium will not revert
back to original equilibrium, i.e. PPP will not hold. Several studies try to clarify this
issue. Frankel and Rose (1996), Lothian (1997) and Taylor, Peel and Sarno (2001)
apply panel data to test several real exchange rates jointly. But when we apply panel
unit root test to investigate jointly the real exchange rates of several countries, the
rejection of null hypothesis only imply at least one of these several countries is
stationary. We cannot guarantee all countries under consideration are stationary.
Cheung and Lai (1993) and Lothian and Taylor (1996) extend the length of the sample
period to increase the power of test of unit root tests. But these long-term data are
acquired from different nominal exchange rate regimes. So we cannot appropriately
use these data to evaluate the post-1973 floating exchange rates.
As known, lack of perfect international arbitrage in the presence of market
frictions can result in nonlinear adjustment of real exchange rates. Past studies assume
the real exchange rates are linear and use traditional ADF test to investigate
stationarity of the real exchange rates. O’Connell (1998) has demonstrated that if true
real exchange rates are threshold autoregressive model (TAR model, one important
type of nonlinear models), then application of traditional ADF test will result in more
decline in power of test. Hence we should apply nonlinear model, especially TAR
model, to delineate the behavior of the real exchange rates.
All nonlinear autoregressive models can be distinguished as the TAR model and
the STAR model. In some degree, we can take the STAR model as slightly nonlinear
model, and take TAR models as heavily nonlinear models. From another aspect,
heavy market frictions will result in serious lack of perfect international arbitrage.
This circumstance should be appropriate to use the TAR models to delineate behavior
of real exchange rates. Slight market frictions will result in a little lack of perfect
international arbitrage. Then this circumstance should be appropriate to use the STAR
6
models to delineate behavior of real exchange rates. However market frictions should
be interpreted as resulting not only from transportation costs and trade barriers, but
also from sunk costs of international arbitrage and the resulting tendency of traders to
respond only to sufficiently large arbitrage opportunities. Real world economies
should own large degree of market frictions. O’Connell (1998) thinks a variety of
market frictions can impede smooth operation of international arbitrage. Recent
investment theory suggests that if international arbitrage involves such degree of
market frictions, then unhedgeable uncertainty can be levered up into relative large
and sharp adjustment of real exchange rates. So we can appropriately use threshold to
characterize these degree of market frictions. Hence we apply the threshold
autoregressive model (TAR model) to implement our study.
Moreover, to compare with the past linear adjustment, the nonlinear specification
can provide more elaborate exploration for the adjustment mechanism of the real
exchange rates. Previous studies apply linear model specification to the issue of
persistence, which can be quantified by half-live measure. The half-life indicates the
time needed to dissipate the half amount while the certain amount shock occurs. From
impulse response function analysis we can formally compute the half-life. But we
cannot directly use traditional impulse response function to analyze nonlinear models.
So we will apply appropriate method, generalized impulse response function, to
analyze dynamic adjustment of nonlinear models.
2.1. Non-linearity of the real exchange rates
At the beginning, we conduct linearity test for the real exchange rates. Brown,
Durbin and Evans (1975) develop a famous linearity test, CUSUM test. CUSUM test
investigates whether the time series has structural changes. Kramer, Ploberger and Alt
(1988) extend above original CUSUM test, which can only tests time series models of
non-stochastic regressors, to autoregressive models. So the null hypothesis of this
kind of extended CUSUM test is that real exchange rates are linear AR model. The
alternative hypothesis of this kind of extended CUSUM test is there are structural
changes in real exchange rates. We can use this extended CUSUM test to conduct
linearity test for real exchange rates.
2.2 Threshold type non-linearity and stationary
Next if real exchange rates are nonlinear under investigations of above extended
CUSUM test, then which type of non-linearity the real exchange rates have, and
7
whether the TAR models can fit these nonlinear real exchange rates well. Besides, we
should test whether real exchange rates are stationary. We discuss non-linearity of real
exchange rates firstly. The above CUSUM test can only test whether time series is
nonlinear. But we cannot distinguish which type of non-linearity the time series has.
However the test of threshold non-linearity can help us to certain whether nonlinear
time series is threshold type of non-linearity.
Successively we discuss stationary of real exchange rates. Past literatures almost
show real exchange rates cannot reject unit root hypothesis through traditional
Augmented Dickey-Fuller test. Taylor, Peel and Sarno(2001) think the power of
univariate ADF test is too low. So statistical inference made from univariate ADF test
is not so credible. They suggest applying multivariate ADF test, which has higher
power of test, to investigate whether real exchange rates have unit root (i.e.
non-stationary). However traditional ADF tests (include univariate and multivariate)
assume real exchange rates are linear under null. But if real exchange rates are
genuinely nonlinear, then traditional ADF tests are not proper choices to test
stationary of real exchange rates. Because we apply traditional ADF test to test a TAR
process will result in decline in the power of test.
Hence Caner and Hansen (2001) develop threshold autoregressive unit root test
to solve this problem. Threshold autoregressive unit root test combine the test of
threshold non-linearity and the test of stationary. That is to say, this is a joint test of
threshold non-linearity and stationary. We can apply threshold autoregressive unit root
tests to test threshold non-linearity and stationary simultaneously.
We deflate the nominal exchange rate by the ratio of domestic price level to
foreign price level to define the real exchange rate. To take logarithm, the above
definition can be represented as: qt=st+pt*-pt=ln(StPt*/Pt). qt is the real exchange rate.
st is logarithm of the nominal exchange rate, i.e. the amount of domestic currency
exchanged by per foreign currency. pt and pt* are logarithm of domestic price level
and logarithm of foreign price level.
To apply threshold autoregressive unit root test, firstly we can use threshold type
of augmented Dickey-Fuller regression to model real exchange rates as the following:
μ1+ρ1qt-1 +α11Δqt-1+α12Δqt-2+ … +α1pΔqt-p + et
Δqt-1  λ
Δqt =
(1)
μ2+ρ2qt-1 +α21Δqt-1+α22Δqt-2+ … +α2pΔqt-p + et
8
Δqt-1>λ
If we wish to test whether real exchange rates have threshold effect, the hypothesis
testing of threshold test can be stated as the following:
H0: μ1=μ2 ,ρ1=ρ2 , α11=α21 , α12=α22 , … and α1p=α2p
(2)
H1:
otherwise
The null hypothesis of threshold test part of this threshold autoregressive unit root test
is real exchange rates do not have threshold effect. The alternative hypothesis of
threshold test part of this threshold autoregressive unit root test is real exchange rates
have threshold effect.
We apply the Sup-Wald statistics to test the above hypothesis testing. The
standard Wald statistics WT is:

WT(γ1, … ,γ1n) =T(( 
0
2

/  2(γ1, … , γ1n) )–1)

Where  2(γ1, … , γ1n) is defined as the residual variance estimate from nonlinear
alternative which dependent on threshold values, γi , i=1,…,n, are the threshold values

and 
0
2
is the residual variance estimate from OLS estimation of the null linear


model. We choose the optimal threshold values  1 , … , 
1n
to achieve the supreme of
Wald statistics WT . Hence we have the Sup-Wald statistics is the following:

WT(  ) = sup WT(γ)
(3)


Whereγis a (1  n) vector, γ=(γ1, … ,γ1n) and  is the optimal threshold value of


γ,  = ( 

1
, … ,  1n).
Due to the hypothesis testing for linear null versus TAR alternative, the test
statistics sup WT(γ) under null hypothesis has nuisance parameter, i.e. threshold

9
valuesγ. The nuisance parameter is just the threshold values γthat don not appear
in the null hypothesis. Because theoretical limiting distribution of test statistics
sup WT(γ) contains nuisance parameters (threshold values here) γ, this nuisance

parameter (threshold values here)γis unknown parameter for null hypothesis. So we
cannot use mathematical formula of theoretical limiting distribution of sup WT(γ) to

calculate the asymptotic critical values of theoretical limiting distribution of
sup WT(γ). That is to say, asymptotic critical values cannot be tabulated. So we also

cannot conduct statistical inference. For resolving this problem in order to conduct
inference, we use bootstrap method to replicate the sampling distribution of the test
statistics. Then we can base on bootstrap distribution to make appropriate statistical
inference. Following Hansen(1996), We apply bootstrap method to repeat random
sampling from the residuals of estimated null model and to calculate sup WT(γ)

statistics from replications produced by this estimated null model. We can state this
bootstrap procedure precisely. While we once randomly sample from the residuals of
estimated null model, we can acquire new time series data. Then we can use these
new data to separately estimate linear null model and TAR alternative model.
Evidently, we can get estimate of threshold value in TAR alternative model. So we
can indeed to calculate one value of sup WT(γ). To repeat the above procedure many

times, bootstrap method can generate many values of sup WT(γ). Then these many

values of sup WT(γ) can form the sampling distribution of the sup WT(γ) statistics.


Hence we can base on the sampling distribution of the sup WT(γ) statistics to

conduct hypothesis testing. Hansen’s Monte Carlo experiment indicates the testing
procedures stated above have enough power of test.
Because the parameters ρ1=ρ2 control the stationary of the real exchange rate
process qt . So if we wish to test whether real exchange rates have unit root, the null
hypothesis of unit root test can be stated as the following:
H0: ρ1=ρ2 =0
(4)
10
The null hypothesis of unit root test part of this threshold autoregressive unit root test
is real exchange rates have unit root. That is to say, if we cannot reject null hypothesis,
then the real exchange rates are non-stationary. The alternative hypothesis of unit root
test part of this threshold autoregressive unit root test is real exchange rates do not
have unit root in at least one regime. So we have two possible alternative hypotheses.
One alternative hypothesis is the real exchange rates do not have unit root in each
regime. This alternative hypothesis is:
H1: ρ1 < 0 and ρ2 < 0
(5)
This first alternative hypothesis means the real exchange rates in both regimes do
not have unit root. Another alternative hypothesis is the real exchange rates do not
have unit root in only one regime. This alternative hypothesis is:
H2:
ρ1 < 0 and ρ2 = 0
or

(6)
ρ1 = 0 and ρ2 < 0
This second alternative hypothesis means the real exchange rate has partial unit
root. That is to say, the real exchange rate in one regime do not have unit root and the
real exchange rate in another regime have unit root. If H2 holds, then the real
exchange rate process qt will behave like a stationary process in one regime, but will
behave like a unit root process in another regime.
Caner and Hansen (2001) suggest test statistic to conduct unit root test part of
threshold autoregressive unit root test. The test statistic is


RT = t12 1{  1<0} + t22 1{  2<0}

Where 
(7)

1

and 
2
are estimates in model (1) mentioned above. t1 and t2 are the t

ratios for  1 and  2 from the estimation of model (1). 1{.} is the indicator function,
which is 1 while event in the bracket occurs and is 0 while event in the bracket
doesn’t occur. This test statistic is testing the null of unit root test H0: ρ1=ρ2 =0
against the alternative ρ1<0 or ρ2<0 (alternative hypothesis include H1 and H2).
11
If we reject the null, it cannot reveal which alternative will be appropriate. So we
need another statistics to help us to distinguish between H1 and H2. The above

individual t statistics t1 and t2 which are the t ratios of estimates 

1
and 
2
can
provide us proper test statistics to solve this problem. If both of t1 and t2 are all
statistically significant, then real exchange rates in both regimes are consistent with
alternative H1. However if only one of t1 and t2 are all statistically significant, then
real exchange rates is consistent with partial unit root alternative H2.
We can compare threshold type of augmented Dickey-Fuller regression (1) with
linear augmented Dickey-Fuller regression that is used to conduct conventional
augmented Dickey-Fuller test. The linear augmented Dickey-Fuller regression is the
following:
Δqt
= μ+ρqt-1 +α1Δqt-1+α2Δqt-2+…+αpΔqt-p + et
Assume that the true process for real exchange rate qt is threshold type of
augmented Dickey-Fuller regression as model (1), but we make a misspecification to
use linear augmented Dickey-Fuller regression to model real exchange rate qt. The
estimates of parameter ρ in above linear augmented Dickey-Fuller regression model
will locate between ρ1 and ρ2 . If we use the data in regime ofΔqt-1  λ, we can get
estimate of parameterρ1 . If we use the data in regime ofΔqt-1>λ, we can get estimate
of parameterρ2 . So if we mix above data in both regimes, the estimate of parameterρ
must locate between estimate of parameterρ1 and estimate of parameterρ2.
Even though the alternative hypothesis of unit root test part of threshold
autoregressive unit root test, H1: ρ1 < 0 and ρ2 < 0, does not hold, the partial unit
root alternative hypothesis of unit root test part of threshold autoregressive unit root
test , H2 , still provide good explanation about mean-reverting property of the real
exchange rates. In the following, we will indicate partial unit root process is
consistent with results of conventional unit root test on the real exchange rates. And
we will show how partial unit root process can explains the mean-reverting behavior
of the real exchange rates.
Firstly, while true process of the real exchange rate is threshold type of
augmented Dickey-Fuller regression as model (1). And by unit root test part of
threshold autoregressive unit root test, we know partial unit root alternative holds. In
the partial unit root case, either the case is: ρ1 < 0 and ρ2 = 0 or the case is: ρ1 =
12
0 and ρ2 < 0. This means the real exchange rate will behave like a stationary
process in one regime, but will behave like a unit root process in another regime. If
we use linear augmented Dickey-Fuller regression to conduct conventional linear
augmented Dickey-Fuller test, then we may not reject conventional unit root null
hypothesis ρ= 0. Because partial unit root case is consistent with conventional unit
root null hypothesis ρ= 0. Although past literatures show real exchange rates do not
reject conventional unit root null hypothesis ρ= 0, but real exchange rates may still be
partial unit root process.
Next, the effect of transaction costs suggests that the larger deviation from PPP,
the stronger tendency to move back to equilibrium. That is to say, small deviations of
real exchange rates from equilibrium may be characterized by unit root behavior, but for
large deviations of real exchange rates from equilibrium the real exchange rates process
is mean reverting. So we will explain how the partial unit root process can
characterize the above mean-reverting phenomena.
We can use indicator function to represent model (1) as the following:
Δqt =(μi+ρiqt-1 +αi1Δqt-1+αi2Δqt-2+ … +αipΔqt-p + et )*(1-1{Δqt-1>λ})
(μj+ρjqt-1 +αj1Δqt-1+αj2Δqt-2+ … +αjpΔqt-p + et )*1{Δqt-1>λ}
(8)
Where i=1,2, j=1,2, and i  j. 1{.} is the indicator function, which is 1 while event in
the bracket occurs and is 0 while event in the bracket does not occur.
The above TAR model assumes that the border between two regimes is given by a
specific value λ of the threshold variableΔqt-1. We can obtain more gradual
transition between different regimes by replacing the indicator function 1{Δqt-1>λ}
by a continuous functionΦ[Δqt-1;θ,λ], which changes smoothly from 0 to 1 as Δqt-1
increase. Then above the TAR model will transform to the STAR model.
Δqt =(μi+ρiqt-1 +αi1Δqt-1+αi2Δqt-2+ … +αipΔqt-p + et )*(1- Φ[Δqt-1;θ,λ])
(μj +ρjqt-1 +αj1Δqt-1+αj2Δqt-2+ … +αjpΔqt-p + et )*Φ[Δqt-1;θ,λ] (9)
Where i=1,2, j=1,2, and i  j.
The above two STAR regressions can be rewritten as the following general
STAR regression:
13
Δqt =μi+ρiqt-1 +αi1Δqt-1+αi2Δqt-2+ … +αipΔqt-p + [(μj-μi)+(ρj-ρi)jqt-1
+(αj1-αi1)Δqt-1+(αj2-αi2)Δqt-2+ … +(αjp-αip)Δqt-p]*Φ[Δqt-1;θ,λ] + et
(10)
Where i=1,2, j=1,2, and i  j. Φ[Δqt-1;θ,λ] is transition function of the STAR model.
Taylor, Peel and Sarno(2001) set this transition function is exponential function :Φ
[Δqt-1;θ,λ]=1-exp[-θ(Δqt-1-λ)2]. The exponential transition function is bounded
between 0 and 1, Φ:R  [0,1], has the properties Φ[Δqt-1;θ=0,λ]=0 and lim Φ[Δ
  
qt-1;θ,λ]=1. The parameterθ,θ  0, determines the smoothness of the transition from
one regime to the other. From another aspects, the transition functionΦ[Δqt-1;θ=0,λ]
determines the degree of mean-reversion and is itself governed by the transition parameter
θ, which effectively determines the speed of mean-reversion, and the parameter λ is the
equilibrium level ofΔqt. While small deviation of the real exchange rates from
equilibrium occur, this means transition functionΦ[Δqt-1;θ,λ] in STAR model is
approach zero. From aspect of the TAR model, real exchange rates now locate in the i
regime. If ρi <0, then small deviation of real exchange rates from equilibrium reveal real
exchange rates which is near around the equilibrium are unit root process. However while
large deviation of real exchange rates from equilibrium occur, this means transition
functionΦ[Δqt-1;θ,λ] in the STAR model is approach one. From aspect of the TAR
model, real exchange rates now locate in the j regime. If ρi +(ρj-ρi) <0 (this is equal
to ρj <0), then large deviation of real exchange rates from equilibrium reveal real
exchange rates which is far away from the equilibrium are stationary process. So real
exchange rates deviate largely from the equilibrium will finally revert back to the original
equilibrium.
The effect of transaction costs suggests that the larger deviation from PPP, the
stronger tendency to move back to equilibrium. This implies that while If ρi = 0 is
admissible, we must have ρi+(ρj-ρi) < 0, i.e. ρj< 0. That is for small deviationsΔqt
may be characterized by unit root behavior, but for large deviations the process is
mean reverting. Because the estimate of parameterρmust locate between estimate of
parameterρ1 and estimate of parameterρ2. And by unit root test part of threshold
autoregressive unit root test, we know partial unit root alternative holds. In the partial
unit root case, either the case is: ρ1 < 0 and ρ2 = 0 or the case is: ρ1 = 0 and
ρ2 < 0. Hence while the partial unit root alternative of unit root test part of threshold
autoregressive unit root test holds, the conventional unit root null hypothesis ρ= 0 may
not be rejected, but the true nonlinear process is globally stable withρi+(ρj-ρi) < 0, i.e.
14
ρj< 0. That is to say, we can explain partial unit root case by the following economic
reasoning: small deviations of real exchange rates may be characterized by unit root
behavior, but for large deviations of real exchange rates the process is mean reverting.
Balke and Fomby(1997) have defined stationary of TAR models. If time series has
unit root in one regime and time series is stationary in another regime, then the whole
TAR model can still be globally stationary. While time series locate in stationary
regime, time series will tend to revert to equilibrium. While time series locate in unit
root regime, so long as the intercept parameters (drift parameters) act to push time
series back towards threshold value. Then time series may move across threshold
value and enter into the stationary regime. The time series is still taken as stationary
globally. Hence if real exchange rates are partial unit root cases, we can still take the
real exchange rates as stationary globally and fit these real exchange rates as TAR
modes.
2.3 The determination of specific threshold autoregressive model
Then we test and estimate the TAR model. The TAR model is when real
exchange rate is greater (or smaller) some critical value, the real exchange rate will
follow different time series (i.e. the real exchange rate will be located in different
regime). We therefore model the real exchange rate as the following n-regime TAR
model:
qt =

α10 +α11 qt-1+…+α1p qt-p+εt
qt-d  γ1
α20 +α21 qt-1+…+α2p qt-p+εt

γ1 < qt-d  γ2
α(n-1)0 +α(n-1)1 qt-1+…+α(n-1)p qt-p+εt
γ1n-1< qt-d  γn
αn)0 +αn1 qt-1+…+αnp qt-p+εt
γ1n< qt-d 2
(11)
n is the numbers of regimes, qt is the real exchange rate. p is the lag order which
series of each regime owns in TAR model. qt-d is threshold variable, 1  d  p. γ is the
threshold value, i.e. the band of regime.
We can use the following sequential testing procedure to determine the number
15
of regimes. For testing whether the real exchange rate can be fitted by threshold
autoregressive model and how many regimes threshold autoregressive model the real
exchange rate can be fitted, we can conduct the following sequential hypothesis
testing.
The followings are sequential testing procedure. The preceding hypothesis test is
the linear autoregressive null versus the nonlinear n-regimes TAR model alternative:
H0: qt =α0 +α1 qt-1+…+αp qt-p+εt
(12)
H1: the n-regimes TAR model stated as in the equation (11).
The successive hypothesis test is the nonlinear n-regimes TAR model null versus
the nonlinear (n+1)-regimes TAR model alternative:
H0: the n-regimes TAR model stated as in the equation (11).
(13)
H1: the (n+1)-regimes TAR model
We also apply the Sup-Wald statistics sup WT(γ) to test the above hypothesis

testing.
If we reject null hypothesis of the above first hypothesis testing (12) and cannot
reject null hypothesis of the above second hypothesis testing (13), then the TAR
model should have n regimes. But if we reject null hypothesis of the above first
hypothesis testing (12) and also reject null hypothesis of the above second hypothesis
testing (13), then we should continue to repeat this sequential hypothesis testing until
we reject null hypothesis of linear null vs. (n+k)-regimes alternative and cannot reject
null hypothesis oh (n+k)-regimes vs. (n+k+1)-regimes (k=1,…,m). Then we can
finally conclude that the TAR model should have (n+k) regimes.
We can state more concretely about the above sequential testing procedure. If the
null hypothesis of testing for linear autoregressive null versus the two-regime TAR
alternative and the null hypothesis of testing for linear autoregressive null versus the
three-regime TAR alternative are all rejected, then this represent there are
non-linearity in real exchange rates. For determining the number of regimes, we
should go on investigating the testing for the testing for the two-regime TAR null
16
versus the three-regime TAR alternative. If the null hypothesis of testing for the
two-regime TAR null versus the three-regime TAR alternative cannot be rejected, then
we can conclude real exchange rates can fit the two-regime TAR model. However if
the null hypothesis of testing for the two-regime TAR null versus the three-regime
TAR alternative is also rejected, then we should go on investigating the testing for the
testing for linear autoregressive null versus the four-regime TAR alternative and the
testing for the three-regime TAR null versus the four-regime TAR alternative. If one
of the above two hypothesis testing cannot be rejected, then we can conclude real
exchange rates can fit the three-regime TAR model. If both of the above two
hypothesis testing is rejected, then we should go on conducting this type sequential
testing procedure until we can determine the exact number of regimes in the TAR
model.
Tong (1990) proposes modification of Akaike Information Criterion (AIC) to
determine the lag order of independent variables, which are the lagged dependent
variable, in threshold autoregressive model. This modified AIC is the following:
n
AIC (p) =


Ni ln 
i
2
+ 2n(p+1)
(14)
i 1
Where n is the number of regimes in TAR model, Ni (i=1,…,n) is the number of

observations in the ith regime, and  i2, i=1,…,n., is the variance of the residuals in
the ith regime. p is the lag order of independent variables in each regime of TAR
model. We can choose an optimal lag order p to minimize this modified AIC.
Besides, we also should determine the lag order d of threshold variable qt-d.
Threshold variable qt-d is chosen from the lagged term of independent variable qt-i
(i=1,…,p) in TAR model. We can determine the lag order d of threshold variable qt-d
through estimation procedure of TAR model. For model (11), the n-regime TAR
model, there are three types of parameters to be estimated. The first type of
parameters is coefficientsαij (i=1,…,n and j=0,…,p) in each regime. The second type
of parameters is threshold values γi (i=1,…,n). The third type of parameters is the lag
order d of threshold variable qt-d. From model (11), the n-regime TAR model, we can
calculate sum of square error which is the function of coefficients αij in each regime,
threshold values γi and the lag order d of threshold variable. So we can use OLS
method to chooseαij , γi and d to minimize the sum of square error. But because TAR
model is nonlinear and discontinuous, we can apply sequential conditional least
17
square method to estimate the TAR model more efficiently. The following is
sequential conditional least square method in estimating the TAR models. Firstly, we
assumeγi and d are given. So we can use OLS method to estimate αij. Based on

estimates 
ij
, we can calculate residuals from estimated the TAR model. These
residuals are function of γi and d. Then we can calculate sum of square residuals which
is still the function of γi and d. Hence we can choose optimal threshold values γi and
optimal lag order d of threshold variable to minimize the above sum of square
residuals. Through above estimation procedure we can estimate coefficients αij in
each regime, threshold values γi ,and lag order d of threshold variable in the TAR
models.
Hansen (2000) developed the distribution theory for estimate of the TAR model.
The distribution for threshold value estimate is nonstandard which is composed of
complex Brownian motions, but the distribution for intercept parameters and slope
parameters estimate in each regime is just normal.
2.4 Generalized impulse response function
For the sake of half-life measure, we perform impulse response function analysis
to calculate the half-life of real exchange rate. The half-life is defined as the expected
number of years for a PPP deviation to decay by 50% amount. If model is linear, the
traditional impulse response can be implemented to compute half-life. Cheung and
Lai (2000) and Murray and Papell (2002) who all fit the linear model follow this
approach. For nonlinear model, the traditional impulse response function will depend
on initial conditions (history dependence) and the size and sign of the innovation
(shock dependence). Traditional impulse response function can be stated as the
following:
IX(n,δ,ωt-1)=E[Xt+n| vt=δ, vt+1=0,…, vt+n=0, ωt-1]
-E[Xt+n| vt=0, vt+1=0,…, vt+n=0, ωt-1]
(15)
where vt+i is a shock occurs at t+i period( i=0,…,n), ωt-1 is information set at t-1
period. Above traditional impulse response function is the effect of the shock which
has size of δ and occurs at t period on the variable at period n.
18
For linear models, traditional impulse response function has the following good
properties: (i) Positive and negative shocks have symmetric effects. (ii) For different
sizes of shocks, there are proportional effects on the variable. (iii) The impulse
responses are independent from the particular past histories. That is to say, The
impulse responses are independent from the particular information set ωt-1. However
for nonlinear models traditional impulse response function does not own above good
properties again.
Gallant, Rossi and Tauchen(1933) have ever proposed a method to calculate
impulse response function of nonlinear model. The procedure of calculation of the
baseline forecast used to compute their impulse response function is similar to the
traditional impulse response function. So their method is also based on a given history
and a given shock to calculate impulse response function of nonlinear models. The
change of histories and shocks will affect the results of their impulse response
function. These are not good properties for well-constructed impulse response
function. So we must have another well-defined dynamic structure for nonlinear
model to effectively solve above problem.
Hence Koop, Pesaran and Potter (1996) develop generalized impulse response
function to solve the problem that is how to apply impulse response function on
nonlinear models. The generalized impulse response functions treat histories and
shocks as random variables. So histories and shocks on which computation of
generalized impulse response function is based are drawn from distributions. That is
to say, the generalized impulse response function does not depend on particular
histories and shocks. Hence we have well-constructed dynamic structure for nonlinear
model. The generalized impulse response function are simulated realizations obtained
by iterating the time series model, randomly drawing from the Gaussian distribution,
and then averaging over the number of random draws. The generalized impulse
response function can be expressed as the difference between two conditional
first-moment profiles:
GIX(h, Vt , Ωt-1)=E[Xt+h | Vt, Ωt-1]-E[[Xt+h |Ωt-1]
(16)
Where GIX is the generalized impulse response function of a variable X, h is the
forecasting horizon, vt is the perturbation to the process at time t, vt is randomly
drawn from Vt. ωt-1 is the conditioning information set at time t-1 (reflecting the
history or initial conditions of the variable) and ωt-1 is also randomly drawn from Ωt-1.
E[.] is the expectations operator. The above expression provides a way of measuring
19
the effect of the shock on the conditional mean of the process.
Next we discuss the general impulse response function and computation of the
confidence interval of generalized impulse response function. We randomly draw initial
valueωt-1 from information set Ωt-1. If the size of shock occurs at t period is δ, i.e.
vt=δ, then we randomly draw vt+1, vt+2,…, vt+n from standard normal distribution.
Hence we can separately compute X1t+i (vt=δ,ωt-1) ,i=1,…,n and X1t+i (ωt-1) ,i=1,…,n,
from time series model. To repeat the above steps R times, we can separately compute
R
sample mean X
R,t+i
(vt=δ,ωt-1)= (1/R)  Xjt+i (vt=δ, ωt-1), i=1,…,n and X
R,t+i
(ωt-1)
j 1
R
=(1/R)  Xjt+i (ωt-1) ,i=1,…,n. By the Law of Large Numbers, the above Monte
j 1
Carlo replications will separately converge to conditional expectation E[Xt+n| vt=δ,
ωt-1] and E[Xt+n| ωt-1]. We take the difference between E[Xt+n| vt=δ, ωt-1] and
E[Xt+n| ωt-1], then we can get the generalized impulse response function.
3. Empirical Results
3.1 Data
We extract U.S.A., Spain and Korea monthly data from the International Monetary
Fund’s International Financial Statistics database. The data include the U.S.A., Spain
and Korea consumer price indices and the U.S. dollar-Spain peseta and U.S.
dollar-Korea won nominal exchange rate. All data cover the sample period from
January 1973 through December 1998. The real exchange rates of Spain and Korea
vis-à-vis United State are under investigation. All the series of real exchange rate are
expressed in logarithm.
3.2 Testing for non-linearity
Firstly, we conduct linearity test for real exchange rates of U.S. dollar-Spain
peseta and U.S. dollar-Korea won. Kramer, Ploberger and Alt (1988) proposed a
special CUSUM test that can be applied to autoregressive model. The null hypothesis
of this kind of CUSUM test is linear AR series. The alternative hypothesis of this kind
20
of CUSUM test is there are structural changes in time series. We use this kind of
CUSUM test to conduct linearity test for real exchange rates of U.S. dollar-Spain
peseta and U.S. dollar-Korea won. We apply Akaike Information Criterion (AIC) to
determine the lag order in AR model for real exchange rates of U.S. dollar-Spain
peseta and U.S. dollar-Korea won. While we fit both real exchange rates into linear
AR model, the lag orders which minimize AIC for both real exchange rates are all
equal to 1. Hence we set the null hypothesis of CUSUM test is AR (1) model for real
exchange rates of U.S. dollar-Spain peseta and U.S. dollar-Korea won. The alternative
hypothesis of CUSUM test is there are structural changes for both real exchange rates.
Based on Kramer, Ploberger and Alt (1988), the critical value of this kind of
CUSUM test statistics under linear autoregressive null are separately 0.85 at
significance level of 10%, 0.984 at significance level of 5% and 1.143 at significance
level of 1%. In the Table 1, we calculate the CUSUM statistics of null linear AR
model for U.S. dollar-Spain peseta real exchange rate to be 15.98. And we also
calculate the CUSUM statistics of null linear AR model for U.S. dollar-Korea won
real exchange rate is 8.30. Both real exchange rates all reject the linear autoregressive
null. These mean that U.S. dollar-Spain peseta real exchange rate and U.S.
dollar-Korea won real exchange rate all have structural changes. Hence both real
exchange rates of U.S. dollar-Spain peseta and U.S. dollar-Korea won indeed reveal
nonlinear phenomena.
Table.1 CUSUM test for non-linearity
Real exchange rates
CUSUM test statistics
U.S. dollar-Spain peseta
15.98***
U.S. dollar-Korea won
8.30***
*** means we can reject null hypothesis at significance level of 1%.
In the figure 1, we plot U.S. dollar-Spain peseta and U.S. dollar-Korea won real
exchange rates. The sample periods of both real exchange rates data are from January
1973 to December 1998. Heckscher (1916) has suggested there may be significant
deviations from PPP for the sake of international transaction costs between spatially
separated markets. So there may be nonlinearities in real exchange rates. Market
frictions, which include transaction costs, transportation costs, tariff, non-tariff
barriers, trade restrictions, sunk cost of international arbitrage, lag in information and
so on, will impede perfect international arbitrage. These lacks of perfect international
21
arbitrage will affect adjustment of PPP. These market frictions are the source of
nonlinear in the real exchange rates. Recently many studies have developed models of
nonlinear real exchange rate adjustment arising from market frictions in international
arbitrage. Williams and Wright(1991), Dumas(1992), Sercu, Uppal and Hulle(1995),
Coleman(1995), Obstfeld and Taylor(1997), O’Connell(1998), Taylor and Peel(2000)
and Taylor, Peel and Sarno(2001) apply nonlinear models to characterize the behavior
of real exchange rates and to analyze dynamic adjustment of real exchange rates.
Figure.1 U.S. dollar-Spain peseta real exchange rate and U.S. dollar-Korea won real
exchange rate
Fig. 1 The sample periods of both real exchange rates data are from January 1973 to December 1998.
3.3 Testing for threshold effect and testing for unit roots
In this subsection, we firstly discuss the stationary issue about real exchange rates.
We should investigate whether U.S. dollar-Spain peseta and U.S. dollar-Korea won
real exchange rates are stationary. Because some degree of stationary in real exchange
rates will guarantee that the deviation of real exchange rates from equilibrium will
finally return back to original equilibrium, i.e. mean reversion property of PPP.
Firstly we use traditional augmented Dickey-Fuller test to primarily test the
stationary of U.S. dollar-Spain peseta and U.S. dollar-Korea won real exchange rates.
We list results of the ADF test, which has intercept, but do not have time trend to
investigate whether real exchange rates are stationary. Other types of ADF tests also
have similar results. To use ADF test, we should take the lagged terms of dependent
22
variable, which is the differencing variable as independent variables. So we have to
determine the lagged order of these independent variables in applying ADF test. The
criterion of choice is we should include enough lagged order of these independent
variables to let that the error term in ADF test is white noise. We can use Lujing-Box
Q test to investigate whether the error term in ADF test is white noise. The null
hypothesis of Lujing-Box Q test is the time series under investigation is white noise.
The empirical results indicate while we choose to include one order lag differencing
term as independent variable into the ADF model of U.S. dollar-Spain peseta and U.S.
dollar-Korea won real exchange rates, the Lujing-box Q statistics for error term in
such ADF model of U.S. dollar-Spain peseta is 1.0179 and the Lujing-box Q statistics
for error term in such ADF model of U.S. dollar-Korea won is 1.7688. However the
critical values at significance level of 5% are separately 7.37 for the right tail of null
distribution and 0.05 for the left tail of null distribution. So we cannot reject null
hypothesis, the error terms in above ADF test of U.S. dollar-Spain peseta and U.S.
dollar-Korea won real exchange rates are white noise. Hence we include one lag order
differencing term, which is taken as independent variable into ADF regression of U.S.
dollar-Spain peseta and U.S. dollar-Korea won real exchange rates. The ADF test
statistics for U.S. dollar-Spain peseta is –1.71 and the ADF test statistics for U.S.
dollar- Korea won is –2.20. The critical value of ADF test is –2.84 at significance
level of 5%. So we cannot reject null hypothesis for both real exchange rates. Both of
U.S. dollar-Spain peseta and U.S. dollar-Korea won real exchange rates all have unit
root through traditional ADF test.
In fact, majority of past literatures almost show real exchange rates cannot reject
unit root null hypothesis through traditional Augmented Dickey-Fuller test. As known,
the power of traditional ADF test (also includes other types of unit root tests) is too
low to result in no definite conclusions about the stationarity of real exchange rates.
However traditional ADF tests assume real exchange rates are linear under null. But if
real exchange rates are truly nonlinear, then traditional ADF tests are not appropriate
to investigate stationary of real exchange rates. Because we apply traditional ADF test
to test a nonlinear process will result in more largely decline in the power of test.
Because we have applied the CUSUM test mentioned above to confirm both of U.S.
dollar-Spain peseta and U.S. dollar-Korea won real exchange rates are nonlinear. So
the empirical results of traditional ADF test are evidently not the good basis from
which we can make further inference.
However after the confirmation of non-linearity in both U.S. dollar-Spain peseta
and U.S. dollar-Korea won real exchange rates through the above CUSUM test, we
23
should also further investigate whether non-linearity of U.S. dollar-Spain peseta and
U.S. dollar-Korea won real exchange rates are threshold type non-linearity. Hence we
apply threshold autoregressive unit root test developed by Caner and Hansen (2001)
to conduct a joint test of threshold non-linearity and stationary. That is to say, we can
test threshold non-linearity and stationary simultaneously. The null hypothesis of
threshold test part of this threshold autoregressive unit root test is real exchange rates
do not have threshold effect. The alternative hypothesis of threshold test part of this
threshold autoregressive unit root test is real exchange rates have threshold effect.
We distinguish two types of threshold autoregressive ADF regression. One
threshold autoregressive ADF regression contains intercept, but does not contain time
trend. Another threshold autoregressive ADF regression contains intercept and time
trend. The empirical results of threshold autoregressive ADF regression with intercept
but without time trend are listed in Table 2. The empirical results of threshold
autoregressive ADF regression with intercept and time trend are listed in Table 3. The
empirical results in Table 2 show U.S. dollar-Spain peseta real exchange rates reject
null hypothesis at significance of 1% and U.S. dollar-Korea won real exchange rates
reject null hypothesis at significance of 10%. The empirical results in Table 3 show
U.S. dollar-Spain peseta real exchange rates reject null hypothesis at significance of
1% and U.S. dollar-Korea won real exchange rates reject null hypothesis at
significance of 5%. Hence we can conclude that both U.S. dollar-Spain peseta and
U.S. dollar-Korea won real exchange rates have threshold type of non-linearity.
The null hypothesis H1 of unit root test part of this threshold autoregressive unit
root test is real exchange rates in each regime have unit root. The alternative
hypothesis of unit root test part of this threshold autoregressive unit root test is real
exchange rates do not have unit root in at least one regime. So there are two possible
alternative hypotheses (H1 and H2). One alternative hypothesis H1 is real exchange
rates do not have unit root in each regime. Another alternative hypothesis H2 is real
exchange rates do not have unit root in only one regime. This second alternative
hypothesis H2 means real exchange rate has partial unit root. That is to say, real
exchange rate in one regime do not have unit root and real exchange rate in another
regime have unit root. If H2 holds, then the real exchange rate process qt will behave
like a stationary process in one regime, but will behave like a unit root process in
another regime.
As mentioned above, some degree of stationary in real exchange rates can
guarantee that the deviation of real exchange rates from equilibrium will return back
24
to the original equilibrium again. This is a solid basis from which we can analyze
reversion speed of real exchange rates under shocks. However if real exchange rates
reveal partial unit root phenomena, then this means some degree of stationary still
exist in real exchange rates. That is to say, the real exchange rate is stationary in at
least one regime, although real exchange rates have unit root in another regime. While
real exchange rates deviate slightly from equilibrium, the behavior of real exchange
rates may like unit root process, which can be represented by unit root process in one
regime of the TAR model. While real exchange rates deviate largely far away from
equilibrium, real exchange rates in another regime of the TAR model reveal stationary
property, which guarantees real exchange rates will revert back to the original
equilibrium. So stationary exist in only one regime can provide us enough basis to
conduct the analysis for reversion speed of the real exchange rates.
In Table 2, the unit root part of threshold autoregressive unit root test includes
three test statistics, RT, t1 and t2. The RT test statistic is testing the null of unit root test
H0: ρ1=ρ2 =0 against the alternative ρ1<0 or ρ2<0 (alternative hypothesis include
H1 and H2).
If we reject the null, it cannot reveal which alternative will be appropriate. So we
need another statistics to help us to distinguish between H1 and H2. The individual t


statistics t1 and t2 which are the t ratios of estimates  1 and  2 in each regime of
threshold autoregressive ADF regression can provide us such test statistics. If both of
t1 and t2 are all statistically significant, then real exchange rates in both regimes are
consistent with alternative H1. However if only one of t1 and t2 are statistically
significant, then real exchange rates is consistent with partial unit root alternative H2.
Because in traditional linear ADF regression, we have investigated which lag
order of differencing term, which is taken as independent variable, is appropriate. As
found above, one lag order of differencing term Δqt-1 as independent variable will let
the error term of ADF regression to be white noise. So we take one lag of differencing
term Δqt-1 as independent variable in threshold autoregressive ADF regression.
In Table 2, the unit root part of threshold autoregressive unit root test without
time trend indicates RT statistics for both U.S. dollar-Spain peseta and U.S.
dollar-Korea won real exchange rates are significant at significance level of 5%. That
is to say, RT statistics for both U.S. dollar-Spain peseta and U.S. dollar-Korea won
real exchange rates all reject null hypothesis H0 of unit root test at significance level
25
Table.2 Threshold autoregressive unit root test without time trend
Threshold Test
supWT
Real exchange
rates
U.S.dollarSpain peseta
Bootstrap
p-value
0.001***
Unit Root Test
RT
Bootstrap
p-value
0.032**
t1
t2
Bootstrap
p-value
Bootstrap
p-value
0.658
0.013**
U.S.dollarKorea won
0.0535*
0.048**
0.559
0.022**
*** means we can reject null hypothesis at significance level of 1%. ** means we can reject null
hypothesis at significance level of 5%. * means we can reject null hypothesis at significance level of
10%. The estimated bootstrap p-value is the percentage of simulated test statistics, which exceeds the
actual test statistics.
of 5%. So both U.S. dollar-Spain peseta and U.S. dollar-Korea won real exchange
rates do not have unit root in at lest one regime of threshold autoregressive ADF
regression. We also wish to distinguish which alternative is appropriate. The t1
statistics for both U.S. dollar-Spain peseta and U.S. dollar-Korea won real exchange
rates are all insignificant at significance level of 10%. We can’t reject ρ1=0 at
significance level of 10%. The t2 statistics for both U.S. dollar-Spain peseta and U.S.
dollar-Korea won real exchange rates are all significant at significance level of 5%.
We can reject ρ2=0 in favor of ρ2< 0. That is to say, the empirical results reveal
that alternative hypothesis H2, partial unit root case, holds for both U.S. dollar-Spain
peseta and U.S. dollar-Korea won real exchange rates. In Table 3, the unit root part of
threshold autoregressive unit root test with time trend has similar empirical results.
The empirical results of Table 3 show that RT statistics for U.S. dollar-Spain peseta
real exchange rate is significant at significance level of 10% and RT statistics for U.S.
dollar-Korea won real exchange rate is significant at significance level of 5%. The t1
statistics for both U.S. dollar-Spain peseta and U.S. dollar-Korea won real exchange
rates are all insignificant at significance level of 10%. The t2 statistics for U.S.
dollar-Spain peseta real exchange rate is significant at significance level of 5% and
the t2 statistics for U.S. dollar-Korea won real exchange rate is significant at
significance level of 1%. If we include time trend into threshold autoregressive ADF
26
regression, then empirical results reveal that alternative hypothesis H2, partial unit
root case, still holds for both U.S. dollar-Spain peseta and U.S. dollar-Korea won real
exchange rates.
Engel (1996) propose that real exchange rates may contain nonstationary as well
as stationary components. So both U.S. dollar-Spain peseta and U.S. dollar-Korea
won real exchange rates are partial unit root case, which is stationary in one regime
and has unit root in another regime seem to be reasonable. Balke and Fomby(1997)
indicate if time series has unit root in one regime and time series is stationary in
another regime, then the whole TAR model can still be globally stationary. When time
series locate in stationary regime, time series will tend to revert to equilibrium. When
time series locate in unit root regime, the intercept parameters (drift parameters) may
act to push time series back towards threshold value. So time series may move across
threshold value and enter into the stationary regime. The time series can be taken as
stationary globally. Hence if real exchange rates belong to partial unit root cases, we
can still take the real exchange rates as stationary globally and fit these real exchange
rates as the TAR models.
Table.3 Threshold autoregressive unit root test with time trend
Threshold Test
supWT
Real exchange
rates
Bootstrap
p-value
Unit Root Test
RT
Bootstrap
p-value
t1
Bootstrap
p-value
t2
Bootstrap
p-value
U.S.dollarSpain peseta
0.002***
0.0525*
0.791
0.019**
U.S.dollarKorea won
0.010**
0.010**
0.722
0.004***
*** means we can reject null hypothesis at significance level of 1%. ** means we can reject null
hypothesis at significance level of 5%. * means we can reject null hypothesis at significance level of
10%. The estimated bootstrap p-value is the percentage of simulated test statistics, which exceeds the
actual test statistics.
27
3.4 Estimation and testing for threshold autoregressive model
Before conducting estimation of threshold autoregressive models, we firstly
implement the hypothesis testing which can be represented as the linear
autoregressive null versus the threshold autoregressive model alternative, or the
threshold autoregressive model with one certain number of regimes null versus the
threshold autoregressive model with another number of regimes alternative, to
determine whether the threshold autoregressive models of real exchange rates have
certain numbers of regimes. Table 4 reports the bootstrap p-value of the Sup-Wald
statistics for these sequential hypothesis testing.
Table 4 shows the results for the hypothesis testing of linear autoregressive null
versus two regime TAR model alternative3, the p-value of both U.S. dollar-Spain
peseta and U.S. dollar-Korea won real exchange rates do not exceed 5%. So at
significance level of 5%, both U.S. dollar-Spain peseta and U.S. dollar-Korea won
real exchange rates all reject null hypothesis. For hypothesis testing of linear
autoregressive null versus three regimes TAR model alternative, the p-value of both
U.S. dollar-Spain peseta and U.S. dollar-Korea won real exchange rates do not exceed
10%. So at significance level of 10%, both U.S. dollar-Spain peseta and U.S.
dollar-Korea won real exchange rates all reject null hypothesis. For hypothesis testing
of two-regime TAR model null versus three-regime TAR model alternative, the
p-values of both real exchange rate series exceed 10%. At significance level of 10%,
U.S. dollar-Spain peseta and U.S. dollar-Korea won real exchange rate cannot reject
null hypothesis of two-regime TAR model. To synthesize the above results, U.S.
dollar-Spain peseta and U.S. dollar-Korea won real exchange rate should be
two-regime TAR model. Thus we can base on the results of Table 4 to infer that both
of the U.S. dollar-Spain peseta and U.S. dollar-Korea won real exchange rate series
can fit the two-regime TAR model.
Besides, we also apply the modified Akaike Information Criterion (AIC)
proposed by Tong (1990) to determine the lag order of independent variables in TAR
model for real exchange rates of U.S. dollar-Spain peseta and U.S. dollar-Korea won.
Table 5 reports the relevant results. For TAR model of U.S. dollar-Spain peseta real
exchange rate, the lag order 2 minimize the modified AIC. So we should set time
series in each regime of TAR model for U.S. dollar-Spain peseta real exchange rate
are AR (2). For TAR model of U.S. dollar-Korea won real exchange rate, the lag order
1 minimize the modified AIC. So we should set time series in each regime of the TAR
28
Table.4 P-value of Sup-Wald test
U.S. dollar-Spain peseta
U.S. dollar-Korea won
Bootstrap p-value
H0：linear model
H1：two-regime TAR model
0.0396**
0.0068**
0.094*
0.0768*
0.485
0.813
H0：linear model
H1：three-regime TAR model
H0：two-regime TAR model
H1：three-regime TAR model
** means we can reject null hypothesis at significance level of 5%. * means we can reject null
hypothesis at significance level of 10%. The estimated bootstrap p-value is the percentage of simulated
Wald statistic which exceeds the actual Wald statistic.
model for U.S. dollar-Korea won real exchange rate are AR (1).
Moreover we can estimate TAR model for both real exchange rates to determine
their each lag order d of threshold variable. Form results of Table 6, we get d=2 for
U.S. dollar-Spain peseta real exchange rate. So the threshold variable of U.S.
dollar-Spain peseta TAR model is qt-2. However form results of Table 6, we get d=1
for U.S. dollar- Korea won real exchange rate. So the threshold variable of U.S.
dollar- Korea won TAR model is qt-1.
Results of testing support two-regime TAR model. So we can estimate the
two-regime TAR models for the U.S. dollar-Spain peseta and U.S. dollar-Korea won
real exchange rate series. The estimates of slope parameters in each regime, the
associated standard deviation of slope parameters in each regime, the estimates of
threshold variable and the corresponding interval of threshold variable are obtained
through estimation procedure introduced before.
29
Table.5 AIC for TAR models
TAR model
of real exchange
rates
U.S.dollarSpain peseta
U.S.dollarKorea won
P
1
-2142.87
-2205.42*
2
3
-2145.48* -2135.12
-2108.18
-2114.39
4
5
6
-2130.05
-2121.54
-2116.19
-2107.07
-2093.97
-2088.10
* means the minimum of AIC
Table 6 below reports the estimated lag order of threshold variable, the estimates
of intercept parameters and slope parameters in each regime and their associated
standard deviation, and also reports the estimates of threshold value and their
associated confidence interval for both U.S. dollar-Spain peseta and U.S. dollar-Korea
won real exchange rates. We firstly analyze the estimated results of TAR model for
U.S. dollar-Spain peseta real exchange rates. The estimated lag order of threshold
variable is 2. So the threshold variable of U.S. dollar-Spain peseta TAR model is qt-2.
The estimate of threshold value γ in TAR model of U.S. dollar-Spain peseta real
exchange rate is 4.8279. By modified AIC we know the lag order of independent
variable in TAR model of U.S. dollar-Spain peseta real exchange rates is 2. So there
are three coefficient parameters to be estimated in each regime of TAR model. These
include α0, α1, α2 in first regime (threshold variable qt-2 is smaller than or equal to
threshold value 4.8279) and β0, β1, β2 in second regime (threshold variable qt-2 is
larger than threshold value 4.8279). The intercept parameter (α0) of first regime for
U.S. dollar-Spain peseta real exchange rate is significantly different from zero at
significant level of 1%. The slope parameter (α1) of first regime for U.S. dollar-Spain
peseta real exchange rate series is significantly different from zero at significant level
of 1%. Another slope parameter (α2) of first regime for U.S. dollar-Spain peseta real
exchange rate is also significantly different from zero at significant level of 1%. The
intercept parameter (β0) of second regime for U.S. dollar-Spain peseta real exchange
rate is significantly different from zero at significant level of 10%. The slope
parameter (β1) of second regime for U.S. dollar-Spain peseta real exchange rate is
significantly different from zero at significant level of 1%. Another slope parameter
(β 2) of second regime for U.S. dollar-Spain peseta real exchange rate is not
30
significantly different from zero at significant level of 10%.
We successively analyze the estimated results of TAR model for U.S. dollarKorea won real exchange rates. The estimated lag order of threshold variable is 1. So
the threshold variable of U.S. dollar-Korea won TAR model is qt-1. The estimate of
threshold value γ in TAR model of U.S. dollar-Korea won real exchange rate is
7.0969. By modified AIC we know the lag order of independent variable in TAR
model of U.S. dollar- Korea won real exchange rates is 1. There are two coefficient
parameters to be estimated in each regime of TAR model. These include α0, α1 in first
regime (threshold variable qt-1 is smaller than or equal to threshold value 7.0969) and
β0, β1 in second regime (threshold variable qt-1 is larger than threshold value
7.0969). The intercept parameter (α0) of first regime for U.S. dollar-Korea won real
exchange rate is not significantly different from zero at significant level of 10%. The
slope parameter (α1) of first regime for U.S. dollar-Korea won real exchange rate is
significantly different from zero at significant level of 1%. The intercept parameter
(β 0) of second regime for U.S. dollar-Korea won real exchange rate series is
significantly different from zero at significant level of 1%. The slope parameter (β1)
of second regime for U.S. dollar-Korea won real exchange rate is significantly
different from zero at significant level of 1%.
Table 6 also reports the associated confidence interval of threshold estimates (γ) in
two-regime TAR model for U.S. dollar-Spain peseta and U.S. dollar-Korea won real
exchange rate series. According to Hansen (2000), the confidence interval of
threshold estimates can be computed by inverting likelihood ratio test statistics, which
is constructed to test threshold estimates γ. Hansen demonstrate the asymptotic
distribution of threshold values γ is highly relative to the asymptotic distribution of
likelihood ratio test statistics. The confidence interval of asymptotic distribution of
likelihood ratio test statistics is easily computed. Due to the application of likelihood
ratio test for testing threshold estimate γ, likelihood ratio test statistics is a function of
threshold estimate γ. So we can use inverse function of likelihood ratio test statistics
and the asymptotic distribution of likelihood ratio test statistics to compute the
confidence interval of threshold estimate γ.
Although we have found both U.S. dollar-Spain peseta and U.S. dollar-Korea won
real exchange rates are partial unit root cases. But we can still analyze the stationarity
of each regime in the TAR models. We apply threshold autoregressive ADF regression
to investigate the whole stationarity of threshold autoregressive real exchange rate
Table.6 Estimates of two-regime TAR model
31
Table.6 Estimates of two-regime TAR model
U.S. dollar-Spain peseta
U.S. dollar-Korea won
Parameters of Model
Lag Order of
Threshold Variable
d=2
d=1
First Regime
(qt-d  γ)
α0
α1
α2
1.4969***
0.1122
(0.4971)
(0.1063)
1.0627***
0.9837***
(0.1067)
(0.0155)
-0.3775***
(0.1441)
Second Regime
(qt-d>γ)
β0
0.1007*
(0.0591)
2.5700***
(0.8694)
β1
0.9864***
(0.0694)
0.6415***
(0.1211)
β2
-0.0065
(0.0695)
Threshold Value γ
4.8279
[4.6091,5.4788]
7.0969
[6.6899,7.4761]
qt-1 is threshold variable. γ is threshold value. The number in parentheses reports the standard errors
for the corresponding model coefficient estimates. [.,.] represents 95% confidence interval. The number
in right portion of confidence interval is the upper bound of threshold value under confidence
coefficient 0.95 and the number in left portion of confidence interval is the lower bound of threshold
value under confidence coefficient 0.95. Statistical significance is indicated by a double asterisk ** for
the 5% level and a triple asterisk *** for the 1% level.
32
processes. Even threshold autoregressive ADF regression estimated in above
subsection show real exchange rates are partial unit root cases. We can still investigate
the stationarity of each regime of the TAR models. The TAR models estimated in this
subsection are formal and true threshold autoregressive structures of real exchange
rates. That is to say, threshold autoregressive ADF regression estimated in above
subsection and true TAR models estimated in this subsection are two different models.
We apply threshold autoregressive ADF regression to investigate whether real
exchange rate processes are stationary. But we fit real exchange rates by true TAR
models. So the estimates of threshold autoregressive ADF regressions are certainly
different from the estimates of true TAR models. So even both real exchange rates are
partial unit root cases, we can still analyze the stationarity of each regime of true TAR
models estimated here.
For U.S. dollar-Spain peseta real exchange rate, each of two regimes in TAR
model is AR (2) model. For U.S. dollar- Korea won real exchange rate, each of two
regimes in TAR model is AR (1) model. Greene (1997) defines stationary of AR (1)
model. For AR (1) model: qt =α0 +α1qt-1+ εt, if |α1|<1, then qt is stationary. Greene
(1997) also defines stationary of AR (2) model. For AR (2) model: qt =α0 + α1qt-1+
α2qt-2 + εt, if |α2|<1, α1+ α2<1 and α2-α1<1. Because Hansen (1997) has demonstrated
the asymptotic distributions of intercept parameters and slope parameters in each
regime of TAR model are normal distributions. So we can apply t statistics to test
stationary conditions in each regime of U.S. dollar- Korea won TAR model. In the
first regime of U.S. dollar- Korea won TAR model, the estimate of slope parameter is
0.9837 and the standard deviation of estimate of slope parameter is 0.0155. If we wish
test stationary condition of this regime, we can form two hypothesis testing: one has
α1  1 null vs. α1<1 alternative which is a left test, another has α1  1 null vs. α1>1
alternative which is a right test. If we all reject null hypothesis about above two
hypotheses, then we can infer that |α1|<1, i.e. qt is stationary in the first regime of U.S.
dollar- Korea won TAR model. The t statistics of first hypothesis testing is –18.5752.
From t distribution, we can reject null hypothesis at significance level of 1%. The t
statistics of second hypothesis testing is 2260.5877. From t distribution, we can reject
null hypothesis at significance level of 1%. So we can conclude that qt is stationary in
the first regime of U.S. dollar- Korea won TAR model.
In the second regime of U.S. dollar- Korea won TAR model, the estimate of
slope parameter is 0.6415 and the standard deviation of estimate of slope parameter is
01211. If we wish test stationary condition of this regime, we can form two
hypothesis testing: one hasβ1  1 null vs. β1<1 alternative which is a left test,
33
another has β1  1 null vs. β1>1 alternative which is a right test. If we all reject null
hypothesis about above two hypotheses, then we can infer that |β1|<1,i.e. qt is
stationary in the first regime of U.S. dollar- Korea won TAR model. However if we
wish test stationary condition of this regime, we can form two hypothesis testing: one
has β1  1 null vs. β1<1 alternative which is a left test, another has β1  1 null vs.
β1>1 alternative which is a right test. If we all reject null hypothesis about above two
hypotheses, then we can infer that |β1|<1,i.e. qt is stationary in the second regime of
U.S. dollar- Korea won TAR model. The t statistics of first hypothesis testing
is –52.2904. From t distribution, we can reject null hypothesis at significance level of
1%. The t statistics of second hypothesis testing is 239.4275. From t distribution, we
can reject null hypothesis at significance level of 1%. So we can conclude that qt is
stationary in the second regime of U.S. dollar- Korea won TAR model.
Hansen (1997) has substantiated the asymptotic distributions of intercept
parameters and slope parameters in each regime of TAR model are normal
distribution. So we can apply likelihood ratio test to investigate whether above
stationary conditions in each regime of U.S. dollar-Spain peseta TAR model hold.
This likelihood ratio test statistics has F distribution. Applying this likelihood ratio
test, we find that each AR (2) process of two regimes in U.S. dollar-Spain peseta TAR
model satisfy above stationary conditions. These above results are consistent with the
results of generalized impulse response function analysis about each regime of TAR
models for both real exchange rates.
3.5 Generalized impulse response analysis and half-life estimates
The estimated models presented in Table 6 provide a basis for the estimation of
the real exchange rate half-lives. The way to obtain the speed of parity reversion of
the estimate models is through the impulse response function. The traditional impulse
response function of linear model does not be affected by the change in the history, in
the sign of shock and in the size of shock. However the traditional impulse response
function of nonlinear model cannot hold these properties. Koop, Pesaran and Poter
(1996) develop the generalized impulse response function to analyze the dynamic
structures of nonlinear model. The generalized impulse response function is varying
with respect to the history, the sign of shock and the size of shock.
We apply generalized impulse response function to analyze adjustment dynamics
in each regime of the TAR models for U.S. dollar-Spain peseta and U.S. dollar-Korea
won real exchange rate. Dynamic structures in each regime of the TAR models for
34
U.S. dollar-Spain peseta real exchange rate at 100% shock are graphed in Figure 2.
Dynamic structures in each regime of the TAR models for U.S. dollar- Korea won real
exchange rate at 100% shock are graphed in Figure 3.
Figure.2 The generalized impulse response function in each regime of U.S.
dollar-Spain peseta TAR model
Fig. 2 Adjustment dynamics of real exchange rate in each regime of U.S. dollar-Spain peseta TAR
model for 100% shock.
We represent z percent shock as ln(1+(z/100)),z=100,10,1 here. By definition, we
can represent real exchange rates qt as qt=ln(St )+ln(Pt*)-ln(Pt)=ln(StPt*/Pt). We add
this shock term to real exchange rates qt. Then occurrence of z percent shock is
ln(1+(z/100)) + qt. Through derivation, we can get that the occurrence of z percent
shock is ln((StPt*/Pt)*(1+(z/100)). This just takes natural log function of
(StPt*/Pt)*(1+(z/100)).
If a 100% shock occurs in regime 1 of U.S. dollar-Spain peseta TAR model, then
through generalized impulse response function, then the half-life estimate of regime 1
is 0.38 year. If a 100% shock occurs in regime 2 of U.S. dollar-Spain peseta TAR
model, then through generalized impulse response function, then the half-life estimate
of regime 2 is 2.83 years. Shocks occur in regime 1 will decay very soon. But shocks
occur in regime 2 will decay relatively very slow.
35
If a 100% shock occurs in regime 1 of U.S. dollar-Korea won TAR model, then
through generalized impulse response function, then the half-life estimate of regime 1
is 3.50 years. If a 100% shock occurs in regime 2 of U.S. dollar-Korea won TAR
model, then through generalized impulse response function, then the half-life estimate
of regime 2 is 0.25 year. Shocks occur in regime 1 will decay relatively very slow. But
shocks occur in regime 1 will decay very soon.
Figure.3 The generalized impulse response function in each regime of U.S.
dollar-Korea won TAR model
Fig. 3 Adjustment dynamics of real exchange rate in each regime of U.S. dollar- Korea won TAR
model for 100% shock.
In the following paragraph, we figure out the generalized impulse response
function for U.S. dollar-Spain peseta and U.S. dollar-Korea won real exchange rate.
Firstly we compare the three different shocks that are 100%, 10%, and 1% shocks
separately. The evidence show the half-life will be shorter while the larger shock
occurs, vice versa. The graphs of the 48 months generalized impulse responses for the
above three shocks are displayed in Figure 4.
36
Figure.4 The generalized impulse response functions for three different size shocks
(100%, 10% and 1%)
Fig. 4 Adjustment dynamics of real exchange rates for 100%, 10% and 1% shock separately. S100%,
S10% and S1 %are the generalized impulse response functions of 100%, 10% and 1% shock for U.S.
dollar-Spain peseta separately. K100%, K10% and K1% are the generalized impulse response functions
of 100%, 10% and 1% shock for U.S. dollar–Korea won separately.
Through the generalized impulse response function analysis based on estimated
nonlinear model, we can compute the speed of PPP reversion of real exchange rate
after real exchange rate suffers shock. The half-life is often defined as the number of
years for a certain size shock to decay by 50%. In other words, shorter half-life
estimate means speed of convergence of real exchange rate to PPP will be higher.
Longer half-life estimate means speed of convergence of real exchange rate to PPP
will be slower. Moreover we also compare the half-life estimates that are
corresponding to the different size shocks.
The half-life estimates for the three different size shocks are reported in Table 7.
While the shock is 100%, the half-life estimate of U.S. dollar-Spain peseta real
exchange rate is 0.67 years and the half-life estimate of U.S. dollar-Korea won real
exchange rate is 0.42 years. While the size of shocks decrease, the value of half-life
estimate increase. Moreover we can compare the half-life estimate obtained in the
present paper to those of previous literatures.
We can compare our results with those of linear models. Several past literatures,
37
Abuaf and Jorion (1990), Lothian and Taylor (1996), Frankel and Rose (1996), Wei
and Parsley (1995), Cheung and Lai (2000) and Murry and Papell (2002) apply linear
model to analyze the speed of mean-reversion of real exchange rates. Firstly, these
linear models evidently do not take market frictions into considerations. Market
frictions will hamper perfect international arbitrage. Linear models imply perfect
arbitrage. We cannot use linear models to characterize these incomplete arbitrages.
However nonlinear models are an appropriate choice to analyze the behavior of real
exchange rates under circumstance of market frictions. Secondly, at the presence of
market frictions real exchange rates have great possibility to be nonlinear. If we use
linear models to investigate the stationarity of real exchange rates, we often cannot
reject unit root null hypothesis. This problem will impede study about mean-reversion
of real exchange rates. Thirdly, several previous papers fit real exchange rates as
linear models and analyze the speed of mean-reversion of real exchange rates.
Abuaf and Jorion (1990) reported an average half-life of 3.3 years for eight series of
real exchange rates. Lothian and Taylor (1996) estimated the half-life for the
dollar-pound rates to be 4.7 years. In examining pooled data on real rates of a group
of currencies, Frankel and Rose (1996) found the half-life to be roughly 2.5 years,
where Wei and Parsley (1995) obtained half-life estimates around 4.5 years. Cheung
and Lai (2000) report the average of all half-life estimates is approximately 3.3 years.
Recently, Murry and Papell (2002) analyze post-1973 data and estimate half-life to be
3-5 years. Based on the TAR model, our results get half-life estimates to be under 1
Table.7 Half-life persistence estimate
Speed of PPP reversion
U.S. dollar-Spain peseta
U.S. dollar-Korea won
100% shock
0.67
0.42
0.75
0.50
0.83
0.63
10% shock
1% shock
The point estimates of the half-life persistence measured in years
38
year. So our results about half-life estimates are consistent with the high short-term
volatility of real exchange rates
Taylor, Peel and Sarno (2001) apply the STAR model to fit the real exchange rate
series and estimate half-lives through alternative nonlinear dynamic structure (cf.
Gallant, Rossi and Tauchen (1993)). Their empirical results state the average of
half-live estimate is 1 year under larger shock and the average of half-life estimates is
3 years under smaller shock. If market frictions are serious, then TAR model will be
more appropriate than the STAR model. Heavier market frictions will impede perfect
international arbitrage more seriously. Then real exchange rates tend to adjustment
sharply and abruptly. Behavior of real exchange rates will very like threshold
autoregressive process. Our empirical results about half-life estimates are shorter than
1 year for different size of shocks. So our results indicate real exchange rates with
more heavy market frictions can more effectively resolve purchasing power parity
puzzle.
When we take the threshold autoregressive model of real exchange rates into
account, both of the persistence in the real exchange rate and high short-term
volatility can be explained consistently. Notably if short-term nominal exchange rate
volatility is caused by monetary shocks under rigid price, adjustment of price level
should complete in one year. When adjustment of price level has completed, the real
exchange rate will revert back to original equilibrium. So the reasonable convergence
to PPP should be 1-2 years. For the sake of above nominal rigidity the half-life
estimate seems to be under one year or slightly over one year. It is evident that our
half-life estimates are appropriate and reasonable.
The half-life estimates, which are less than 1 year in our paper, are evidently
shorter than 3-5 years, which are found in previous papers. It seems that our TAR
modeling can more effectively resolve the purchasing power parity puzzle. Therefore
our results about the half-life estimates can reconcile the persistence of the real
exchange rate (slow adjustment of real exchange rate after occurrence of shock) with
their high short-term volatility.
4. Conclusion
39
Absence of perfect international arbitrage in the presence of market frictions
can result in nonlinear adjustment of real exchange rates. Recent investment theory
suggests that if international arbitrage involves heavier degree of market frictions,
then unhedgeable uncertainty can be levered up into relative large and sharp
adjustment of real exchange rates. So we can appropriately use threshold to
characterize such degree of market frictions. Hence we apply the threshold
autoregressive model (TAR model) to implement our study. Our empirical results
indicate that the both real exchange rates in the present paper are well fitted by
threshold type nonlinear mean reverting processes over the floating rate period. Then
we estimated the generalized impulse response functions for our estimated threshold
type nonlinear real exchange rate models. The adjustment dynamics of real exchange
rate in response to shocks can be characterized using generalized impulse response
analysis. The half-life estimates of present nonlinear model found to be smaller than 1
year are evidently shorter than those of past linear model. Simultaneously while the
size of shocks decrease, the half-life estimates increase.
We focus on the real exchange rates about U.S. dollar vs. currencies of all other
29 OECD countries. The empirical results about OECD countries show that U.S.
dollar-Spain peseta and U.S. dollar-Korea won real exchange rate own heavier
nonlinear phenomena. Hence we can properly apply threshold autoregressive models
to fit these both real exchange rates. The empirical results about U.S. dollar-Spain
peseta and U.S. dollar-Korea won real exchange rate presented here provide us a
meaningful resolution to the previous prevailing issue. Rogoff has ever indicated the
appropriate half-life estimate that can explain consistently both of the persistence and
the high short-term volatility for real exchange rate should to be approximately under
one year or over one year. Our empirical evidences undoubtedly confirm that the
nonlinearity for real exchange rate is crucial to resolve the purchasing power parity
puzzle.
40
Reference
1. Abuaf, N. and P. Jorion (1990) “Purchasing power parity in the long run”. Journal
of Finance,45,157-174.
2. Balke, N. S. and T. B. Fomby (1997) “Threshold cointegration”. International
Economic Review, 38, No.3, 627-645.
3. Brown,R. , J. Durbin and J.M. Evans(1975)” Techniques for testing the constancy
of regression relationships over time” Journal of the Royal Statistical
Society,Series B, Vol.37, No.2,149-192
4. Caner, M. and B. E. Hansen (2001) “Threshold autoregression with a unit root”
Econometrica, Vol.69, No.6, 1555-1596
5. Chan, K. S.(1991), “Percentage points of likelihood ratio tests for threshold
autoregression,” Journal of the Royal Statistical Society, Series B, 53, 691-696
6. Cheung Y. W. and K.S. Lai (2000) “On the purchasing power parity puzzle”.
Journal of International Economics, 52, 321-330.
7. Coleman, A. M. G. (1995) “Arbitrage, shortage and the low of one price:new
theory for the time series analysis of an old problem” Unpublished, Princeton
university
8. Dumas, B. (1992) “Dynamic equilibrium and the real exchange rate in a spatially
separated world” Review of Financial Studies, 5, 153-180
9. Engel, C. (1996) “Long-run PPP may not hold after all” Working paper series,
No.6546, NBER
10. Frankel, J.A. and A.K. Rose (1996) “A panel project on purchasing power
parity：Mean reversion within and between countries” Journal of International
Economics,40, 209-224.
11. Gallant, A.R. , P.E. Rossi, and G. Tauchen(1993) “ Nonlinear Dynamic Structures
41
” Econometrica, 61, 871-908.
12. Greene, W. H. (1997) “Econometric Analysis” Prentice-Hall International, Inc.
13. Hansen, E. B. (1996) “Inference when parameter is not identified under the null
hypothesis” Econometrica, Vol.64, No.2, p.413-430
14. Hansen, B. (1997) “Inference in TAR Models” Studies in Nonlinear Dynamics
and Econometrics,Vol.2, No.1,1-14.
15. Hansen, B.(2000) “Sample Splitting and Threshold Estimation ”. Econometrica,
Vol.68, No.3, 575-603.
16. Hansen, B. and B. Seo (2002) “Testing for two-regime threshold cointegration in
vector error-correction models” Journal of Econometrics, 110, 293-318.
17. Heckscher, E. F. (1916) “Vaxelkursens grundval vid pappersmyntfot” Ekonomisk
Tidskrift, 18, 309-312
18. Kilian, L. (1998) “Small-sample confidence intervals for impulse response
functions”.The Review of Economics and Statistics, 218-230.
19. Koop, G., M.H. Pesaran and S.M. Potter (1996) “Impulse response analysis in
nonlinear multivariate models”. Journal of Econometrics,74, 119-147.
20. Kramer, W., W. Ploberger and R. Alt (1988)”Testing for structural change in
dynamic models” Econometrica, Vol.56, No.6,1355-1369
21. Lothian, J.R. and M.P. Taylor (1996) “Real exchange rate behavior: The recent
float from the perspective of the past two centuries” Journal of Political
Economy.104,488-509.
22. Lothian, J.R. (1997) “Multi-country evidence on the behavior of purchasing
power patrity under the current float” Journal of international money and
finance, 16,19-35
23. Meese, R. and A. K. Rose (1990) “Nonlinear, nonparametric, nonessential
exchange rate estimation” American Economic Review Papers and
42
Proceedings, 80:192-196.
24. Meese, R. and A. K. Rose (1991) “An empirical assessment of nonlinearities in
models of exchange rate determination” Review of Economic Studies,
80:608-619.
25. Murray, C. J. and D.H. Papell (2002) “The purchasing power parity persistence
paradigm” Journal of International Economics,56,1-19.
26. Obstfeld, M. and A.M. Taylor (1997) “Nonlinear aspects of good-market
arbitrage and adjustment: Heckscher’s commodity points revisited
”.Journal of the Japanese and international Economies,11,441-479.
27. O’Connell, P.G.J.(1998) “Market frictions and real exchange rates ”. Jouranl of
International Money and Finance,17,71-95.
28. Rogoff, K. (1996) “ The purchasing power parity puzzle” Journal of Economic
Literature, Vol. 34, No.2 , 647-668
29. Runkle, D.E. (1987) “Vector autoregressions and reality”. Journal of Business &
Economic Statistics, Vol.5, No.4,437-442.
30. Sercu, P., R. Uppal and C. V. Hulle (1995) “The exchange rate in the presence of
transaction costs: implications for tests of purchasing power parity” Journal
of Finance, 50, 1309-1319
31. Taylor, M.P. and D.A. Peel (2000) “Nonlinear adjustment, long-run equilibrium
and exchange rate fundamentals ”.Journal of International Money and
Finance,19,33-53.
32. Taylor, M.P. , D.A. Peel and L. Sarno(2001) “ Nonlinear mean-reversion in real
exchange rates: toward a solution to the purchasing power parity puzzles
”.International Economic Review,Vol.42, No.4,1015-1042.
33. Tong, H. (1990) “Non-linear time series: a dynamic systems approach” Oxford:
Oxford university press
34. Wei, S. J. and D. C. Parsley (1995) “Purchasing power dis-parity during thr
43
floating rate period” working paper 5032, NBER.
35. Williams, J. C. and B. D. Wright (1991) “Storage and commodity
markets”,Cambridge university press, Cambridge, MA.
44