Very Preliminary Draft
Nonlinear Adjustment and a Long Memory Model in
Real Exchange Rate
By
Sang-Kuck Chung
School of Economics and International Trade
INJE University
Obang-dong 607, Kimhae city 621-749
Korea
E-mail: [email protected]
Office: +82-55-320-3124
Fax: +82-55-337-2902
And
Bong-Han Kim
Department of International Trade
Kongju National University
Shinkwan-dong 182, Kongju city 314-701
Korea
E-mail: [email protected]
Office: +82-41-850-8391
Fax: +82-41-850-8390
May 2004
Abstract
In this paper we considered new time series model which can describe long memory and
nonlinearity simultaneously, and which can be used to assess the relative importance of
these attributes in empirical time series. Upon fitting it to the monthly deviations from
PPP for six countries, we found that a parsimonious version of the model captures the
salient features of the data rather well. When we compared the model with various
competitive models, we found that a linear fractionally integrated model could certainly
be improved upon by including nonlinear features. Indeed, once we added these, there
remained no evidence of nonlinearity and time-varying parameters. However, the
introduction of long memory into a STAR model did not lead to an improved fit. The
key distinction between these two models lies in their implied long-run properties. We
highlighted these differences using impulse response functions and related measures of
the persistence of shocks.
Keywords: ESTAR; real exchange rate; PPP; Fractional Integration; Generalized
Impulse Response Function
JEL classification: C53; F31
Acknowledgement
This work was supported by the Brain Korea 21 Project in 2004. The results reported in
this paper were generated using GAUSS 5.0.
2
I. INTRODUCTION
Two dominant stylized facts have characterized the evolution of real exchange rates
during the last decade. Firstly, real exchange rates have shown a sustained trending
appreciation and depreciation. Up until recently, such an appreciation was viewed as
primarily due to a Balassa-Samuelson effect affecting the sector of tradables, see for
instance, De Broeck and Torsten (2001), Bostgen and Coricelli (2001), Arratibel,
Rodriguez-Palenzuela and Thimann (2002), Egert (2002a, 2002b), Strahilov (2002).
However, whether the Balassa-Samuelson has been strong is today a subject of
controversy. Indeed, other factors have been at play and may also have their importance
in explaining part of the observed real appreciation. This first feature is commonly
called long memory, which have been successfully implemented for exchange rates
(Diebold et al., 1991; Cheung, 1993; Baillie and Bollerslev, 1994), inflation rates
(Hassler and Wolters, 1995; Baillie et al., 1996), and unemployment (Diebold and
Rudebusch, 1989; Tschernig and Zimmermann, 1992; Koustas and Veloce, 1996; Crato
and Rothman, 1996), see Baillie (1996) for a broad survey. The time series models that
are used in these studies to describe this feature belong to the class of fractionally
integrated models, introduced by Granger and Joyeux (1980) and Hosking (1981).
The second feature is commonly called nonlinearity. There are lots of recent
studies on nonlinear models of real exchange rate behavior in the empirical international
finance literature. They provide strong evidence that nonlinear model specifications of
real exchange rate behavior are well motivated by theoretical models incorporating
transaction costs such as transportation costs, tariffs and nontariff barriers, as well as
any other costs that incur in international trade (Obstfeld and Rogoff, 2000). Intuitively,
transaction costs create a band for the real exchange rate within which arbitrage is not
profitable, so that deviations from purchasing power parity (real exchange rate
fluctuations) are not adjusted. However, outside of the band, arbitrage works and the
real exchange rate is mean-reverting. The time series models which are considered most
often for describing and forecasting the nonlinear properties of real exchange rates are
either of the Markov-switching type, see Hamilton (1989), or of the threshold
autoregressive (TAR) type, see Tong (1990), or of the smooth transition autoregressive
(STAR) type, see Granger and Terasvirta (1993), Terasvirta (1994, 1998) and van Dijk
et al. (2002)1.
Interestingly, at least as far as we know, there have been few attempts to
1
These three model classes assume the presence of two or more regimes, within which the time series
requires different (linear) models for description and forecasting.
3
discriminate between the long memory and nonlinear properties of real exchange rates,
and we attempt to describe these two dominant features simultaneously with a single
time series model. Such attempts would be useful though, as several studies document
that one may be tempted to fit a long-memory model to nonlinear data, and vice versa.
For example, Granger and Terasvirta (1999) and Davidson (2000) demonstrate that
nonlinear models may generate time series to which one may want to fit linear longmemory models. On the other hand, data generated from long-memory models may
appear to have nonlinear properties, see Andersson et al. (1999). As occasional
structural breaks amount to a rather stylized nonlinear model, one may expect that
neglecting structural breaks also spuriously suggests the presence of long memory2. The
most recently work of Dufrenot et al. (2003) suggests long-memory regime-switching
models to capture the adjustment towards the long run equilibrium, in which we take
into account two types of persistence: a permanent component due to the influence of
real factors and a nonlinear component where persistence is associated with timedependent effects.
The outline of our paper is as follows. In Section II, we discuss the
representation of the model, which we will briefly introduce the fractionally integrated
smooth transition autoregressive (FI-STAR) model in subsection 1, and tests for
nonlinearity in fractionally integrated autoregressive models in subsection 2, and
generalized impulse response function considering the dynamic behavior of an
estimated FI-STAR model in subsection 3. In Section III, we specify a FI-STAR model
for a time series of monthly real exchange rates and provide the empirical results. In
Section IV, we conclude the paper with a range of possible topics for further research.
II. METHDOLOGY
2.1 The FI-ESTAR Model
A model that allows for long memory in an observed time series {rt } basically builds
upon
(1)
(1  L) d rt  u t ,
2
see Koop and Potter (2000) and Lundbergh et al. (2002) for discussions on structural change and
nonlinearity and see Bos et al. (1999), Granger and Hyung (1999) and Diebold and Inoue (2001) for their
applications to a long memory.
4
where u t is a covariance stationary process, e.g., an autoregressive moving average
(ARMA) process, and the parameter d is possibly noninteger, in which case the time
series {rt } is called fractionally integrated, see Granger and Joyeux (1980) and
Hosking (1981). The series {rt } is covariance stationary if d  0.5 and invertible
if d  0.5 . The autocorrelation function of {rt } does not decline at an exponential rate,
which is typical for covariance stationary ARMA processes, but rather at a (much)
slower hyperbolic rate. For 0  d  0.5 , {rt } possesses long memory, in the sense that
the autocorrelations are not absolutely summable. Finally, for 0.5  d  1 , {rt } is
nonstationary, but the limiting value of the impulse response function is equal to 0, such
that shocks do not have permanent effects.
To capture nonlinear features in a time series {rt } , one can choose from a wide
variety of nonlinear models, see Franses and van Dijk (2000) for a recent survey. A
model which enjoys a fair amount of popularity, mainly due to its empirical tractability,
is the smooth transition autoregressive (STAR) model, given by
(2)

rt   1,0 


p







1, j rt  j 1  G ( s t ;  , c)   2,0 
j 1
p

j 1

 2, j rt  j G ( s t ;  , c)  et ,


rt  is a stationary and ergodic process,  t ~ iid(0,  2 ) and  , c R   R, where R
denotes the real line  ,   and R  the positive real line 0,   . The transition
function G(st ;  , c) determines the degree of mean reversion and is bounded between ‘0’
and ‘1’. The parameter  effectively determines the speed of mean reversion, and the
parameter c may be seen as the equilibrium level of rt . A simple transition function
suggested by Granger and Terasvirta (1993) and Terasvirta (1994), which is particularly
attractive in the present context, is the exponential function:
(3)

Gs t ;  , c   1  exp   s t  c 2
,
in which case equation (2) would be termed an exponential STAR or ESTAR model.
The exponential transition function, G : R  0,1 , has the properties G0  0 for s t  c
and lim x Gx  1 , and is symmetrically inverse-bell shaped around zero 3 . These
3
In the STAR model as discussed in Terasvirta (1994), the transition variable s t is assumed to be a
lagged endogenous variable, that is, s t  rt l for certain integer d > 0. However, the transition variable
5
properties of the ESTAR model are attractive in the present modeling context because
they allow a smooth transition between regimes and symmetric adjustment of the real
exchange rate for deviations above and below the equilibrium level. The transition
parameter  determines the speed of transition between the two extreme regimes, with
lower absolute values of  implying slower transition4.
It is also instructive to reparameterize the STAR model (2) as follows:
(4)

rt   1,0  rt 1 


p 1



1, j rt  j 1  G ( s t ;  , c)    2,0   * rt 1 
j 1




p 1

2, j rt  j
j 1

G( s ;  , c)  e ,
t
t


where rt  j  rt  j  rt  j 1 . In this first difference version of STAR model, the parameters
 and  * are crucial. The discussion of the effect of transactions costs in the previous
section suggests that the larger the deviation from PPP the stronger will be the tendency
to move back to equilibrium5.
In this paper we combine the two representations in (1) and (2) into the
following new time series model introduced by Dijk el al. (2002):
(5)
(1  L) d rt  xt

x t  1,0 


p

j 1






1, j x t  j 1  G( s t ;  , c)   2,0 
p

j 1

 2, j xt  j G( s t ;  , c)   t


 1' wt 1  G(st ;  , c)  2' wt G(st ;  , c)   t
 1' wt   2  1 ' wt G(st ;  , c)   t .
can also be any exogenous variable ( s t  z t ), or a possibly nonlinear function of lagged endogenous
variables, s  h~
x ;   for some function h. A lagged endogenous variable is simply used in this paper.
t
t
4
Another plausible transition function for some applications is the logistic function suggested by
Granger and Terasvirta (1993) and Terasvirta (1994), resulting in a logistic STAR or LSTAR model. To
model the behavior of real exchange rate, the LSTAR model may not be appropriate because the LSTAR
model implies asymmetric behavior of rt  according to whether it is above or below the equilibrium
level. That is to say, it is not straightforward to think of economic reasons why the speed of adjustment of
the real exchange rate should vary according to whether the specific currency is appreciated or
depreciated from the equilibrium value.
5


This implies that while   0 is admissible, one must have  *  0 and    *  0 . That is, for
small deviations rt  may be characterized by unit root or even explosive behavior, but for large
deviations the process is mean reverting.
6
where wt  (1, xt 1 ,...,xt  p ) , i  (i,0 , i,1 ,...,i, p )' for i = 1, 2, and G(st ;  , c) is the
exponential function given in (2). We assume that  t is a martingale difference
sequence with E ( t |  t 1 )  0 for the history of the time series up to time t-1, which is
denoted as t 1  {rt 1 , rt 2 ,...,r1( p1) , r1 p } . For simplicity, we also assume that the
conditional variance of  t is constant, E( t2 | t 1 )   2 , although this assumption can
be relaxed if necessary.
This FI-ESTAR model restricts the long-run properties of the time series rt 
to be constant, as these are determined by the fractional differencing parameter d.
However, the short-run dynamics, which to a large extent are determined by the AR
coefficients, are allowed to vary as the AR coefficients can switch between 1 ' and
 2 ' , depending on the value of the exponential function G(s t ;  , c) . This makes the
model potentially useful for capturing both nonlinear and long-memory features of the
time series rt .
The above formulation is equivalent to assume that the rt  series directly
follows an infinite order of fractionally integrated ESTAR model:

rt   1,0 


(6)
p

j 1






1, j (1  L) d x t  j 1  G( s t ;  , c)   2,0 

where L is the lag operator and (1  L) d  
j 0
O( j
d 1
p

j 1

 2, j (1  L) d x t  j G( s t ;  , c)   t .


 d( j  d )
. The lag weights are of
(1  d )( j  1)
) , and hence decline to zero when d<1, but are nonsummable when d>0,
although square summable when d<1/2.
2.2 Linearity Tests
Testing linearity against STAR may be complicated by the presence of unidentified
nuisance parameters under the null hypothesis because the STAR model contains
parameters which are not restricted by the null hypothesis. Davies (1977, 1987) first
considers the problem of unidentified nuisance parameters under the null hypothesis.6
The main consequence of the presence of such nuisance parameters is that the
6
See subsequent studies from Andrews and Ploberger (1994), Hansen (1996) and Stinchcombe and
White (1998) for recent general accounts.
7
conventional statistical theory is not available for obtaining the asymptotic null
distribution of the test statistics. To overcome this problem, Luukkonen, Saikkonen and
Terasvirta (1988) propose the solution, which is to replace the transition function
Gs t ;  , c  by a suitable Taylor series approximation. In the reparametrized equation, the
identification problem is no longer present, and linearity can be tested by means of a
Lagrange Multiplier [LM] statistic with a standard asymptotic F-distribution under the
null hypothesis.
In order to derive a linearity test against (2), we follow the procedure suggested
by Luukkonen et al. (1988) and Saikkonen and Luukkonen (1988). If the delay
parameter d is fixed, the linearity test against ESTAR consists of estimating by
ordinary least squares the auxiliary regression:
(7)
rt   0' xt  1' xt st   2' xt st2   3' xt st3  et ,
where  i
   i,0 ,  i,1 ,...,  i, p '


for
i  0,1,2,3
and
et   t   2  1 ' xt R3 s t ;  , c 
with
R3 s t ;  , c  the remainder term from the Taylor expansion. The expressions for  i show
that the restriction   0 corresponds with the null hypothesis H 0 : 1   2   3  0
against the general alternative that H 0 is not valid in equation (7). The Lagrange
multiplier test statistic ( LM E ) which tests this null hypothesis has an asymptotic Fdistribution with 3 p  1 and T  4( p  1) degrees of freedom. Luukkonen et al. (1988)
also suggested that a parsimonious, or economy, version of the LM E statistic can be
obtained by modifying the auxiliary regression (8) with regressors st2 and s t3 :
(8)
rt   0' xt  1' xt st   2,0 st2   3,0 st3  et ,
and testing the null hypothesis H 0' : 1  0 and H 0' :  2,0   3,0  0 . The resultant test
statistic, denoted
LM E' ,
has an asymptotic F-distribution with
T  ( p  4) degrees of freedom. The advantage of the
LM E'
 p  3 and
statistic over the
7
LM E statistic is that it requires considerably less degrees of freedom .
The LM-type tests in fact assume constant conditional variance. Neglected
heteroskedasticity has similar effects on tests for nonlinearity as residual autocorrelation,
in the sense that it may lead to spurious rejection of the null hypothesis. Davidson and
As usual, the F-version of the test statistic is preferable to the  2 -version in small samples because its
size and power properties are better.
7
8
MacKinnon (1985) and Wooldridge (1990, 1991) develop specification tests which can
be used in the presence of heteroskedasticity, without the need to specify the form the
heteroskedasticity (which often is unknown) explicitly. Their procedures may be readily
applied to robustify the linearity tests against STAR, see also Granger and Terasvirta
(1993, pp. 69-70). The heteroskedasticity robust variants of the LM E and LM E'
statistics based upon (8) and (9) can be straightforwardly computed, in which case LMtype statistics would be denoted LM H and LM H' .
To determine the delay parameter l, Tsay (1989) suggest the procedure in his
specification of TAR models. This involves repeating the linearity test for a set of
plausible values of l and simply choosing the delay parameter that minimize the p value of the linearity test.
Assuming that linearity is rejected, we can proceed to estimate and evaluate the
nonlinear model. Estimation of the parameters in the FI-ESTAR model (5) and (6) are a
relatively straightforward application of nonlinear least squares (NLS). Under the
additional assumption that the errors are normally distributed, NLS is equivalent to
maximum likelihood. Otherwise, the NLS estimates can be interpreted as quasi
maximum likelihood estimates. Under certain regularity conditions, which are discussed
in Wooldridge (1994) and Potscher and Prucha (1997), among others, the NLS
estimates are consistent and asymptotically normal.
2.3 Generalized Impulse Response Function
Another useful way of considering the dynamic behaviour of estimated FI-ESTAR and
ESTAR models is to examine the effects of the shocks  t on the future patterns of the
general time series {rt } . Impulse response functions are a convenient tool to carry out
such an analysis, as they provide a measure of the response of {rt  h } , h = 1, 2,… to a
shock or impulse  at time t.
In nonlinear models, the impact of a shock depends on the history of the
process, as well as on the sign and the size of the shock. Furthermore, if the effect of a
shock on the time series h > 1 periods ahead is to be analyzed, the assumption that no
shocks occur in intermediate periods may give a misleading picture of the propagation
mechanism of the model. The Generalized Impulse Response Function (GIRF),
introduced by Koop et al. (1996) offers a useful generalization of the concept of impulse
response to nonlinear models. The GIRF for a specific shock  t   and history  t 1
is defined as
9
(9)
GIRF r (h,  , t 1 )  E[rt  h |  t   , t 1 ]  E[rt  h | t 1 ]
for h = 0, 1, 2,…. In the GIRF, the expectation of {rt  h } given that a shock  occurs
at time t is conditioned only on the history and this shock. Put differently, the problem
of dealing with shocks occurring in intermediate time periods is handled by averaging
them out. Given this choice, the natural benchmark profile for the impulse response is
the expectation of {rt  h } conditional only on the history of the process  t 1 . Thus, in
the benchmark profile the current shock is averaged out as well. It is easily seen that for
linear models the GIRF is equivalent to the TIRF.
The GIRF is a function of  and  t 1 that are realizations of the random
variables  t and  t 1 . Koop et al. (1996) point out that GIRF r (h,  , t 1 ) itself is a
realization of a random variable, defined as
(10)
GIRF r (h,  t ,  t 1 )  E[rt  h |  t ,  t 1 ]  E[rt  h |  t 1 ]
Definition (10) allows a number of conditional versions of potential interest. For
example, one might consider only a particular history  t 1 and treat the GIRF as a
random variable in terms of  t , that is,
(11)
GIRF r (h,  t ,  t 1 )  E[rt  h |  t , t 1 ]  E[rt  h | t 1 ]
It is equally possibly to reverse the roles of the shock and the history by fixing the shock
at  t   and defining the GIRF to be a random variable with respect to the history
 t 1 . In general, one might consider the GIRF to be random conditional on particular
subsets A and B of shocks and histories respectively, that is, GIRF r (h, A, B) . For
example, one might condition on all histories in a particular regime and consider only
negative shocks8.
III. EMPIRICAL ANALYSIS
3.1 Data Analysis
8
In case of the STAR model, analytic expressions for the conditional expectations involved in the GIRF
are not available for h > 1. Stochastic simulation has to be used to obtain estimates of the impulse
response measures. See Koop et al. (1996) for a detailed description of the relevant techniques.
10
The series we consider represent the real exchange rate at the monthly frequency
covering the period January 1975 until December 1998 (288 observations) for Germany,
France, and Italy and covering the period January 1975 until November 2002 (335
observations) for UK, Japan, and Switzerland. The series rt  may be expressed in
logarithmic form as rt  st  pt  pt* , where s t is the logarithm of the nominal exchange
rate (domestic price of foreign currency), pt and p t* denote the logarithms of the
domestic and foreign consumer price levels, which are seasonally adjusted respectively.
The real exchange rate may thus be interpreted as a measure of the deviation from PPP.
Table 1(a) gives some summary statistics for the deviation from PPP, nominal
exchange rate, and domestic and foreign price levels. Mean returns are on average
positive in all markets. From the skewness and kurtosis of the series, m3 and m 4 , it is
evident that the m3 is symmetric, whilst the dispersion of a large number of observed
values is very small, which implies a leptokurtic frequency curve. This means that
returns do not follow a normal distribution, but present a sharp peak and fat tail
distribution. This is confirmed by the Jaque-Bera test for normality. In addition, the
Ljung and Box (1976) Q statistics, which evaluates the independence between the
series in conditional mean, denotes that all return series investigated are suffering from
long-run dependencies. Table 1(a) also reports Ljung and Box (1976) Q 2 portmanteau
statistics for the tenth autocorrelations of the squared real exchange rates, which is clear
that there is significant, nonlinear temporal dependence in the squared adjusted returns
series, suggesting that the volatility of adjusted returns follows an ARCH-type model.
Figure 1 plots the lag 1 through 100 sample autocorrelations of real exchange rates. The
autocorrelations do clearly exhibit a pattern of slow decay and persistence, in which
they do not appear in the two 95% Bartlett (1946) confidence bands for no serial
dependence. From figure 1, we roughly know that the fractionally integrated models
may be applied to real exchange rates.
As a preliminary exercise, we test for unit root behavior of each of S t , Pt and
Pt* by calculating standard augmented Dickey-Fuller (ADF) test statistics, reported in
Panel (b) of Table 1. In each case, the number of lags is chosen such that no residual
autocorrelation was evident in the auxiliary regressions.9 In keeping with the very large
number of studies of unit root behavior for these time series and conventional finance
theory, we are in each country unable to reject the unit root null hypothesis applied to
9
Moreover, using non-augmented Dickey-Fuller tests or augmented Dickey-Fuller tests with any number
of lags in the range from 1 to 20 yielded results qualitatively identical to those reported in Panel (b) of
Table 1, also regardless of whether a constant or a deterministic trend was included in the regression.
11
each of the nominal exchange rate for all countries at conventional nominal levels of
significance. On the other hand, the results in domestic price series are mixed, where we
can strongly reject the unit root hypothesis for Italy, UK, and Japan, but not for the
other countries.
Most of standard tests for stationarity involve a null hypothesis containing a
unit root. Classical statistical hypothesis testing generally ensures that a null hypothesis
is only rejected in the face of very strong evidence against it. If an investigator wishes to
test stationarity as a null and has strong priors in its favour, then it is not clear that the
conventional Dickey-Fuller parameterization is very useful. Kwiatkowski, Phillips,
Schmidt, and Shin (1992) (henceforth KPSS) have developed an alternative approach of
testing for unit roots and they impose stationarity under the null. Table 2 presents the
results of applying the ADF and KPSS tests to the six real exchange rates. Although it is
possible to strongly reject I(0) for all countries, there is not clear evidence that real
exchange rates may not be generated by an I(0) or I(1) process and is not indicative of
fractional integration. However, the results do not show strongly a rejection the unit root
null hypothesis applied to rt , suggesting non-stationarity of the deviation from PPP
and possibly the absence of a cointegrating relationship between the nominal exchange
rate, and domestic and foreign price indices.
In order to examine whether real exchange rates are fractionally cointegrated
we consider Geweke and Porter-Hudak (1983)’s test and its results are reported in Table
3. The results vary little across the different values of  under consideration. Table 3
shows that all of the estimates of d lie between 0 and 1, suggesting possible fractional
integration behavior. The results of the formal hypothesis testing indicate significant
evidence of d<1 in three countries of Germany, France, and Italy, though there is no
clear evidence for non-EMU countries of UK, Japan, and Switzerland. Moreover, in all
cases the estimates of d are significantly greater than zero. The results thus indicate the
presence of cointegration and possibly fractional cointegration for EMU countries and
not for non-EMU countries10.
3.2 Linearity Test Results
We start by specifying a fractionally integrated autoregressive (ARFI) model. We allow
10
Especially, we should be careful to determine whether the real exchange rate is cointegrated or
fractionally cointegrated. As shown in Chung (1997), the GPH test for cointegation appears to have
slightly better statistical power against autoregressive alternatives than the ADF-t test. However, the PP
tests have a more statistical power against autoregressive alternatives, and fractional alternatives than the
GPH test when autoregressive parameter () and d are less than ½, respectively.
12
for a maximum lag of 18 first differences. Both AIC and BIC indicate that ARFI(p,d)
models with 1 lagged level are appropriate. The ARFI models that form the basis of our
linearity tests are shown in Table 4. In particular, the estimates of d range from 0.264
for France to 0.983 for Italy, suggesting that real exchange rates are stationary for
Germany and France and nonstationary but mean-reverting for the other countries. The
model appears adequate in that the errors seem serially uncorrelated from the Q
statistic, whereas the excess kurtosis and apparent heteroskedasticity are caused entirely
by large positive residuals from the Q 2 statistic. The skewness of the errors is a more
serious problem for Japan, as it does not appear to be due to only a few aberrant
residuals.
The next stage is to test linearity against STAR nonlinearity using the LM-type
statistics discussed in Section II. We set the maximum value of the delay parameter
d max equal to 12 for the transition variable s t  rt  d for d  1,2,..., d max . Table 5 contains
p -values of the standard ( LM E and LM E' ) and heteroskedasticity robust ( LM H and
LM H' ) tests with rt d , d  1,2,...,12 as transition variable. The tests are based on an
ARFI model with 15 lagged levels under the null hypothesis. The p -values of the
standard tests indicate that linearity can be rejected most strongly when d=1 for GER
and UK, d=2 or 3 for FRN, d=3 for ITA, d=4 for JPN, and d=10 for SWI. The results
of the robust tests suggest that the evidence for nonlinearity might perhaps be due to
neglected heteroskedasticity, in which the test statistics of LM H and LM H' correspond
with those of LM E and LM E' , respectively. From the results in Table 5, we note that
there is serious evidence of heteroskedaticity for all countries and thus
heteroskedasticity robust test statistics are appropriate for our purpose. From the p values of the heteroskedasticity robust, linearity can be rejected most strongly when
d=7 for GER and FRN, d=4 for ITA, d=6 for UK, d=12 for JPN, and d=10 for SWI11.
Furthermore, an ESTAR model is appropriate for selected optimal transition variables
from p -values of the LM-type statistics which test the sub-hypotheses in the
specification procedures of Terasvirta (1994), Escribano and Jorda (1999).12
11
From a linear AR model parameterized in first differences, including a single level term at the first lag,
we allow for a maximum lag of 18 first differences. Both AIC and BIC indicate that AR(p) models with 1
lagged first differences are appropriate. From the p -values of the heteroskedasticity robust, linearity can
be rejected most strongly when l=6 for GER, l =3 for FRN and ITA, l =6 for UK and l =5 for JPN, and l
=4 for SWI. The results for this are not reported here for reasons of spaces.
12
For the transition function selection, an ESTAR model is not most appropriate for all candidate
transition variables and even according to whether the specification procedures of Terasvirta (1994) or
Escribano and Jorda (1999) is used. The results for this are available from authors on request but are not
reported here for reasons of spaces.
13
3.3 Estimated ESTAR and FI-ESTAR Models
The results reported and discussed in the previous section led to the choice of ESTAR
and FI-ESTAR models for each of the real exchange rates examined, with the various
autoregressive lag lengths and delay parameters. Starting with an unrestricted AR 15
lagged first differences and ARFI models with 15 lagged levels in both regimes, we
sequentially remove the lagged variables with the lowest t-statistic (in absolute value)
until all parameters of the remaining lagged first differences have t-statistics exceeding
1 in absolute value. The final model is estimated and Table 6 and 7 present parameter
estimates and residual diagnostics of ESTAR and FI-STAR models for six countries’
CPI-based real exchange rates, respectively.
In the short-run, the deviation from PPP is dominated by nonlinear patterns.
Table 6 provides the empirical results of ESTAR models. The  estimates vary widely
across countries, with the speeds of adjustment for some real exchange rates being much
higher than others. Their values generally support the ESTAR model’s adequacy, with
for most series being clearly distinguishable from zero. None of countries satisfy the
condition ˆ1  0 , indicating explosive behavior in the middle regime. However, the
stability condition that 1   2  0 is satisfied. For all series, ̂1 cannot be
distinguished from zero (unit-root behavior in the middle regime), and the model is
essentially defined by ˆ . It must be noted that evidence of nonlinear dynamic
adjustment to PPP as well as the magnitude of the speed-of-adjustment coefficient 
by country can vary. Residual diagnostics indicate that the model generates nonautocorrelated errors at the five per cent level for every series studied. ARCH effects are
present in the residual series for all countries from the squared Ljung-Box statistics. In
summary, the ESTAR models appear to provide a clearly acceptable representation for
the adjustment process toward PPP for the CPI-based real exchange rates. Models of
linear adjustment may be rejected for the majority of series originally considered, and
for many of those series, the ESTAR model provides a superior alternative.
Based on the linearity test results, we proceed by estimating a FI-ESTAR model
with the chosen transition variable and the chosen unbalanced autoregressive order.
From Table 7, the point estimates of ˆ range from 3.670 for UK to 24.818 for Japan,
implying that the transition between the regimes associated with the lower regime and
upper regime in the FI-ESTAR model occurs quite fast. The estimates of the fractional
differencing parameter d are all significantly different from both 0 and 1. To further
examine whether the FI-STAR model improves upon ESTAR models, we estimate FIESTAR models with d=1 imposed a priori. The residual variances of the FI-ESTAR
14
model are clearly smaller than those of the ESTAR model except France and Italy. Both
the skewness and excess kurtosis are reduced in the FI-ESTAR model, although
normality of the errors is still rejected. However, according to both AIC and BIC the FIESTAR model is not preferred over the ESTAR model. The tests against ARCH do not
reject the null hypothesis any longer. Finally, results of the diagnostic tests suggest that
the model is adequate as there is no evidence for remaining residual autocorrelation,
time-variation in the parameters or remaining nonlinearity13.
Figure 2 graphs the estimated transition functions for time series of the CPIbased deviations from PPP. The figure confirms the nonlinearity of the series and the
appropriateness of the exponential transition function. The range of the transition
function values indicates that convergence to long-run PPP is very quickly. It is
interesting that the limiting case of G(.)  1 is actually attained over the range of
observed PPP deviations for all countries.
3.4 Generalized Impulse Response Functions
To gain further insight in the dynamic properties of the estimated FI-ESTAR model, we
assess the propagation of shocks by computing generalized impulse response functions.
We compute history and shock specific GIRFs as defined in (11) for all observations
and values of the normalized initial shock equal to  / ˆ   3,2,1 , where ˆ  denotes
the estimated standard deviation of the residuals from the FI-ESTAR model. For each
combination of history and initial shock, we compute GIRF r (h,  , t 1 ) for horizons h =
0, 1, …, N with N selected arbitrarily to compare with ESTAR models. To generate
future sample paths of {rt } from the FI-ESTAR model, we use the infinite order
ESTAR representation truncated at 120 lags. The conditional expectations in (11) are
estimated as the means over 500 realizations of {rt  h } , with and without using the
selected initial shock to obtain {rt } and using randomly sampled residuals of the
estimated FI-ESTAR model elsewhere. We follow the same procedure to compute
GIRFs for the ESTAR model in first differences with a lagged level term. All GIRFs are
normalized such that they equal  / ˆ  at h = 0.
The estimated generalized impulse response functions in both FI-ESTAR and
ESTAR models are graphed in Figure 3 for each of the real exchange rates, conditional
on average initial history of the real exchange rates over the sample period. It appears
clearly that two interesting points in the generalized impulse responses exist. First, these
graphs illustrate very clearly the nonlinear nature of the adjustment, with the
13
Detailed results of these misspecification tests are available on request.
15
generalized impulse response functions for larger shocks decaying much faster than
those for smaller shocks. Second, a notable difference between the GIRFs of the
ESTAR and FI-ESTAR models is that the impulse responses for the FI-ESTAR model
are very persistent, but also the GIRFs in the ESTAR model decay much faster than the
GIRFs in the FI-ESTAR model. In particular, the GIRFs in the FI-ESTAR model tend
to be hyperbolically decayed during the first 20 months for Germany, after which their
effect gradually declines towards zero. It is also noticeable that the GIRFs from the
ESTAR model exhibit a relatively rapid exponential decay, which is marked contrast to
those of the FI-ESTAR model.
IV. CONCLUSIONS
In this paper we considered new time series model which can describe long memory and
nonlinearity simultaneously, and which can be used to assess the relative importance of
these attributes in empirical time series. Upon fitting it to the monthly deviations from
PPP for six countries, we found that a parsimonious version of the model captures the
salient features of the data rather well. When we compared the model with various
competitive models, we found that a linear fractionally integrated model could certainly
be improved upon by including nonlinear features. Indeed, once we added these, there
remained no evidence of nonlinearity and time-varying parameters. However, the
introduction of long memory into a STAR model did not lead to an improved fit. The
key distinction between these two models lies in their implied long-run properties. We
highlighted these differences using impulse response functions and related measures of
the persistence of shocks.
16
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21
Table 1: Preliminary data analysis
Countries
Germany
France
Italy
UK
Japan
Swiss
Panel A: Summary Statistics
Mean
0.5490
1.7465
7.4135
-0.4217
4.9324
0.4609
Skewness
0.9847
1.0845
0.3197
0.4501
0.1213
0.4093
Kurtosis
3.5862
3.7765
2.9366
3.6596
2.1078
2.8995
Q(10)
2302.55
2263.19
2223.88
2383.72
2767.99
2565.26
Q2(10)
2315.89
2284.66
2228.54
2197.50
2768.20
2558.31
50.66
63.69
4.95
17.38
11.93
9.49
JB
Panel B: Unit Root Tests
s t(c)
-2.0695
-2.1919
-2.3331
-2.5371
-1.7351
-2.5379
st(c,t )
-2.5319
-1.9416
-2.1104
-2.4900
-2.1637
-2.8408
p t(c )
-1.6790
-2.8013
-4.1246
-3.4320
-3.2633
-1.4842
pt(c,t )
-1.8830
-2.0048
-1.2767
-2.4787
-1.7936
-0.9076
p t*(c )
0.8290
-1.0011
pt*(c,t )
-3.1085
-3.3985
Note: The full Sample is 288 monthly observations from 1975:1 to 1998.12 for Germany, France, and
Italy and 335 monthly observations from 1975:1 to 2002.11 for UK, Japan, and Switzerland. The real
exchange rates are calculated from the consumer price, which is seasonally adjusted. In panel (a), Q(10)
and Q2(10) are the Ljung-Box statistics for tenth-order serial correlation in the residuals and squared
residuals, respectively. The critical values at the 0.05 significance level is 18.31 for 10 degrees of
freedom. The standard errors for skewness and kurtosis are (6/T)0.5 =0.162 and (24/T)0.5 =0.324,
respectively for GER, FRN and ITA (0.147 and 0.295 for the other countries), where T is the number of
observations. In Panel (b), statistics are augmented Dickey-Fuller test statistics for the null hypothesis of a
unit root process; the (c) and (c, t ) superscript indicates that a constant (a constant and a linear trend)
was (were) included in the augmented Dickey-Fuller regression. The critical value at the five percent
level of significance is -1.95 to two decimal places if neither a constant nor a time trend is included in the
regression, -2.86 if a constant only is included, and -3.41 if both a constant and a linear trend are included
(Fuller, 1976; MacKinnon, 1991).
22
Table 2: Tests for order of integration of different countries’ real exchange rates
Country
H0: I(1)
H0: I(0)
Z (t * )
Z (t̂ )
ˆ 
̂
Germany
France
Italy
UK
Japan
-2.271
-2.343
-2.350
-2.602
-2.257
-2.304
-2.229
-2.441
-2.680
-2.178
0.565
0.496
0.343
0.243
0.545
0.582
0.497
0.959
0.676
3.257
Switzerland
-2.649
-2.609
0.488
0.752
Note: Z (t * ) and Z (t̂ ) are the augmented Dickey-Fuller (ADF) t-statistics of the lagged dependent
variable in a regression with intercept only, and intercept and time trend included, respectively. The
critical value at the five and one percent level of significance is -2.86 and -3.43 if a constant only is
included, and -3.41 and -3.96 if both a constant and a linear trend are included, respectively. ˆ  and ̂
are the KPSS test statistics based on residuals from regressions with an intercept and an intercept and time
trend, respectively. The 5% critical values for ˆ  and ̂ are 0.463 and 0.146, respectively; while the
1% critical values are 0.739 and 0.216, respectively.
23
Table 3: Results of the GPH Test for Cointegration
 =.55
 =.575
 =.60
Germany
France
Italy
UK
Japan
Switzerland
d
0.342
0.398
0.318
0.097
0.198
0.259
H0:d=1
1.415
1.620
2.142
0.364
1.124
0.814
H0:d=0
5.548
5.691
8.867
4.107
6.802
3.959
d
0.295
0.309
0.196
0.233
0.185
0.129
H0:d=1
1.333
1.424
1.423
0.958
1.035
0.473
H0:d=0
5.857
6.034
8.686
5.075
6.624
4.138
d
0.177
0.218
0.142
0.110
0.021
0.026
H0:d=1
0.898
1.154
1.156
0.495
0.125
0.111
H0:d=0
5.976
6.449
9.312
4.998
6.146
4.430
Note: The sample size for the GPH test is given by n=T . There exist two alternatives; One is that the
hypothesis H0: d=1 is tested against the one-sided alternative of d<1. The other one is that the hypothesis
H0: d=0 is tested against the two-sided alternative of d0. Critical values at the 10%, 5% and 1% level are
-1.39, -1.84, -2.67 for  =.55, -1.35, -1.82, -2.61 for  =.575, and -1.34, -1.75, -2.52 for  =.60,
respectively. Critical values are based on simulated values given in Chung (1997).
24
Table 4: Fractionally Integrated Autoregressive Models: Equation (1)
Countries
GER
FRN
ITA
UK
JPN
SWI
0.274
0.264
0.983
0.872
0.947
0.905
(0.086)
(0.085)
(0.137)
(0.131)
(0.129)
(0.131)
0.0001
0.0002
-0.0003
-0.001
-0.001
-0.0005
(0.001)
(0.001)
(0.001)
(0.001)
(0.001)
(0.002)
0.945
0.946
0.363
0.419
0.386
0.404
(0.035)
(0.034)
(0.160)
(0.150)
(0.149)
(0.152)
Parameter estimates
d
0
1
Residual diagnostics
Tˆ 2
5.04
4.47
3.77
8.98
10.80
13.83
SK
-0.078
0.188
0.261
-0.061
-0.514
0.014
KURT
2.976
3.855
4.061
3.771
3.731
3.227
JB
0.221
7.645
12.230
6.529
17.050
0.558
Q 2 (10)
246.5
158.9
158.9
249.1
203.7
262.9
Q (15)
370.5
233.6
226.4
325.2
277.5
384.2
Q(10 )
4.480
6.948
6.461
6.444
11.110
6.582
Q(15)
5.949
9.297
7.088
15.550
16.880
6.774
2
Note: The table presents parameter estimates and diagnostic tests for the estimated ARFI model for the
level of real exchange rates over the sample period January 1975 – December 1998 for Germany, France
and Italy, and over the sample period January 1975 – November 2002 from real exchange rates of UK,
Japan, and Switzerland. Standard errors are given in parentheses. ˆ 2 denotes the residual variance,
SK is skewness, KURT is the kurtosis, JB the Jarque-Bera test of normality of the residuals, Q(q)
and Q2(q) are the Ljung-Box statistics for tenth-order serial correlation in the residuals and their squared
up to and including order q.
25
Table 5: LM-type tests for STAR nonlinearity after estimation of Model (1)
rt l
l=1
l =2
l =3
l =4
l =5
l =6
l =7
l =8
l =9
l =10
l =11
l =12
GER LM E
0.313
0.094
0.127
0.375
0.803
0.349
0.119
0.514
0.216
0.196
0.180
0.066
LM E'
0.019
0.054
0.097
0.376
0.236
0.040
0.054
0.391
0.592
0.653
0.879
0.652
LM H
0.417
0.284
0.501
0.406
0.944
0.249
0.516
0.718
0.627
0.475
0.323
0.429
LM H'
0.077
0.200
0.333
0.565
0.251
0.059
0.089
0.472
0.595
0.685
0.883
0.604
FRN LM E
0.090
0.020
0.024
0.100
0.228
0.178
0.011
0.095
0.048
0.001
0.001
0.009
LM E'
0.006
0.002
0.002
0.013
0.015
0.005
0.007
0.084
0.346
0.274
0.298
0.084
LM H
0.334
0.132
0.356
0.394
0.722
0.202
0.345
0.466
0.283
0.162
0.104
0.052
LM H'
0.028
0.026
0.018
0.094
0.116
0.036
0.066
0.129
0.194
0.359
0.489
0.197
ITA LM E
0.166
0.019
0.002
0.033
0.669
0.062
0.186
0.512
0.780
0.629
0.629
0.629
LM E'
0.443
0.001
0.001
0.132
0.963
0.146
0.082
0.945
0.816
0.503
0.639
0.366
LM H
0.184
0.644
0.128
0.092
0.794
0.183
0.302
0.204
0.800
0.556
0.485
0.485
LM H'
0.206
0.267
0.015
0.045
0.897
0.471
0.241
0.669
0.751
0.133
0.666
0.265
UK LM E
0.025
0.412
0.125
0.604
0.286
0.215
0.656
0.664
0.585
0.233
0.710
0.952
LM E'
0.170
0.120
0.367
0.947
0.125
0.362
0.492
0.206
0.470
0.577
0.934
0.787
LM H
0.473
0.790
0.845
0.930
0.302
0.105
0.677
0.759
0.587
0.646
0.728
0.933
LM H'
0.206
0.428
0.615
0.926
0.155
0.061
0.389
0.163
0.549
0.467
0.896
0.665
JPN LM E
0.754
0.583
0.459
0.053
0.196
0.227
0.795
0.149
0.768
0.972
0.631
0.718
LM E'
0.862
0.320
0.131
0.876
0.056
0.531
0.544
0.648
0.470
0.987
0.566
0.331
LM H
0.579
0.684
0.606
0.543
0.484
0.379
0.806
0.157
0.700
0.895
0.624
0.658
LM H'
0.586
0.229
0.475
0.975
0.152
0.733
0.449
0.609
0.573
0.971
0.465
0.116
SWI LM E
0.833
0.355
0.055
0.747
0.772
0.181
0.804
0.128
0.874
0.025
0.375
0.458
LM E'
0.889
0.124
0.764
0.645
0.328
0.080
0.272
0.511
0.430
0.017
0.068
0.072
LM H
0.136
0.824
0.094
0.523
0.836
0.372
0.769
0.537
0.892
0.088
0.673
0.402
LM H'
0.723
0.477
0.819
0.656
0.431
0.434
0.358
0.811
0.358
0.057
0.215
0.095
Note: p-values of F variants of the LM-type tests for STAR nonlinearity of the monthly deviation from
PPP. The tests are applied in an ARFI(1) model. The LM E and LM E' statistics are based on the
auxiliary regression models given in (19) and (20), respectively. The LM H and LM H' statistics
correspond with LM E and LM E' with considering heteroskedasticity in the residual.
26
Table 6: Estimated ESTAR Models: Equation (4)
Countries
GER
FRN
ITA
UK
JPN
SWI
Parameter estimates
Tr  rt l
l=6
l =3
l =3
l =6
l =5
l =4
ˆ
103.052
364.387
15.292
31.177
14.515
425.172
(40.425)
(111.226)
(6.418)
(12.757)
(7.021)
(120.445)
-0.006
0.023
0.036
-0.015
0.017
0.036
(0.002)
(0.001)
(0.005)
(0.002)
(0.004)
(0.001)
0.023
0.021
0.002
-0.002
0.001
0.032
(0.024)
(0.036)
(0.018)
(0.018)
(0.010)
(0.040)
-0.042
-0.023
-0.027
-0.080
-0.057
-0.035
(0.013)
(0.020)
(0.016)
(0.021)
(0.022)
(0.010)
ĉ
1
2
Residual diagnostics
Tˆ 2
5.17
3.66
3.17
7.87
11.44
11.65
SK
-0.121
0.355
0.202
-0.019
-0.607
0.081
KURT
3.010
3.600
3.982
3.558
4.147
3.562
JB
0.661
9.680
12.630
4.114
36.72
4.510
AIC
-7.318
-7.473
-7.497
-7.373
-7.168
-7.161
BIC
-7.091
-7.205
-7.070
-7.136
-6.894
-6.899
2
Q (10)
356.9
197.3
185.9
368.2
185.7
267.6
Q 2 (15)
530.5
316.1
278.3
478.7
257.6
411.1
Q(10 )
4.639
7.890
3.120
1.902
5.983
2.740
Q(15)
6.170
8.752
5.170
3.033
6.315
4.041
Note: The table presents parameter estimates and diagnostic tests for the estimated ESTAR model for the
first difference with lagged level term over the sample period January 1975 – December 1998 from real
exchange rates of Germany, France and Italy, and over the sample period January 1975 – November 2002
from real exchange rates of UK, Japan and Switzerland. The others see the notes to table 4.
27
Table 7: Estimated FI-ESTAR Models: Equation (5)
Countries
GER
FRN
ITA
UK
JPN
SWI
Parameter estimates
Tr
rt 7
d̂
ˆ
1,1
1, 2
rt 7
rt  4
rt 6
rt 10
0.231
0.152
0.236
0.231
0.229
0.085
(0.095)
(0.058)
(0.057)
(0.118)
(0.116)
(0.036)
10.502
7.713
14.233
3.670
24.818
13.487
(9.457)
(7.549)
(11.910)
(2.153)
(9.255)
(4.915)
0.998
0.998
0.985
0.940
1.090
1.061
(0.219)
(0.060)
(0.016)
(0.160)
(0.206)
(0.175)
0.066
0.008
-0.671
-0.059
(0.225)
(0.176)
(0.281)
(0.257)
-0.298
0.573
-0.324
(0.190)
(0.290)
(0.276)
1,3
 2,1
rt 12
0.891
0.935
0.963
1.173
1.107
1.212
(0.056)
(0.030)
(0.008)
(0.145)
(0.136)
(0.080)
-0.575
-0.236
-0.427
(0.186)
(0.131)
(0.115)
0.631
0.121
0.287
(0.199)
(0.112)
(0.115)
 2, 2
 2,3
Residual diagnostics
Tˆ 2
4.59
4.29
3.60
6.74
7.43
9.77
SK
-0.045
0.163
0.049
-0.069
-0.636
0.121
KURT
2.977
3.770
3.771
3.665
4.410
3.312
JB
0.077
6.113
5.289
4.946
38.65
1.673
AIC
-7.175
-7.224
-7.306
-7.264
-7.130
-6.991
BIC
-7.047
-7.113
-7.194
-7.085
-6.992
-6.895
Q 2 (10)
254.5
167.4
182.7
240.2
157.8
238.9
Q (15)
384.8
251.9
264.4
336.9
212.1
351.3
Q(10 )
5.80
8.658
10.68
7.79
0.788
0.443
Q(15)
7.57
10.910
11.44
20.46
1.696
1.148
2
Note: see the notes to table 4.
28
Figure 1: Autocorrelation functions for deviations from PPP
Germany, France, and Italy
Note: The figure plots the lag 1 through 100 sample autocorrelations of real exchange rates, with which
the upper and lower 95% Bartlett (1946) confidence bands appear over the sample period January 1975 –
December 1998 from real exchange rates of Germany, France and Italy.
UK, Japan, and Swiss
Note: The figure plots the lag 1 through 100 sample autocorrelations of real exchange rates, with which
the upper and lower 95% Bartlett (1946) confidence bands appear over the sample period January 1975 –
November 2002 from real exchange rates of UK, Japan and Switzerland.
29
Figure 2: Transition function for FI-ESTAR Model
Germany
France
Italy
UK
Japan
Swiss
Note: Transition functions in FI-ESTAR model for monthly real exchange rates against the chosen
transition variable.
30
Figure 3: Generalized Impulse response function for FI-ESTAR and ESTAR Models.
Germany
France
Italy
Note: Mean over all histories in the set
 t 1
of the
GIRF r (h,  , t 1 ) in the FI-ESTAR model (left hand) for the level and
ESTAR model (right hand) for first differences with lagged level term estimated for the monthly real exchange rate.
31
(Continued)
UK
Japan
Switzerland
32