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. Sarno Economics Letters 66 2000 127 –136
respectively. While the real exchange rate may be subject to short-run variation, a necessary condition for PPP to hold in the long run is that the real exchange rate be covariance stationary, not driven by
permanent shocks since nonstationarity of a real exchange rate implies theoretically infinite divergence of purchasing power across the countries considered, expressed in the same currency.
The enormous relevant empirical literature provides mixed results on the validity of long-run PPP for full surveys see Froot and Rogoff, 1995; Rogoff, 1996; for a brief excursus, see the introduction
in Taylor and Sarno, 1998. Nevertheless, an interesting emerging line of research stresses the importance of allowing for nonlinearity in the adjustment of the real exchange rate toward its long-run
equilibrium, illustrating how linear nonstationarity tests may not detect mean reversion if the true data generating process DGP of the real exchange rate is a stationary nonlinear process e.g. inter alios,
Michael et al., 1997. Also, the fact that PPP appears to hold more closely when there are large variations in relative prices e.g. Frenkel, 1976; Taylor and McMahon, 1988 suggests that the speed
of adjustment toward long-run PPP may vary more than proportionately with the size of the shock to the real exchange rate, and the degree of mean reversion in the real exchange rate may vary
nonlinearly with the level of the real exchange rate; this fact is particularly relevant in the present context, given the high inflation rates experienced by some of the countries examined, especially in
the 1980s. Theoretically, moreover, nonlinear, smooth, symmetric real exchange rate adjustment to a stable equilibrium level is predicted by a number of recent contributions on real exchange rate
determination in presence of transactions costs or transport costs, mainly following Dumas 1992.
In this letter, using nonlinear econometric modeling techniques, strong evidence is provided suggesting that deviations from PPP revert to a constant equilibrium level in a nonlinear fashion
during the recent float for each of the Middle Eastern countries considered, also offering a potential explanation to the mixed results of Bahmani-Oskooee 1998. The estimated models display local
instability and global stability, predicting that the real exchange rate is a unit root process in the neighborhood of its long-run equilibrium, whilst adjusting faster with the absolute size of the
deviation from equilibrium. The model proposed may be viewed as a nonlinear error correction model ECM in the form of a smooth transition regression STR which, after imposing a number of
restrictions, becomes the smooth transition autoregressive STAR process used by some early empirical literature using nonlinear models in this context. Testing these restrictions in the STR
reveals, however, that assuming a priori a STAR model for the real exchange rate often leads to misspecification, as the restrictions only hold for a limited number of countries.
2. Nonlinear real exchange rate adjustment
A nonlinear model considered by researchers modeling real exchange rates is the exponential ¨
¨ STAR ESTAR model Granger and Terasvirta, 1993; Terasvirta, 1994, where adjustment takes
place in every period but the speed of adjustment varies with the extent of the deviation from equilibrium. The following general point can be made, however, against the practice of assuming a
priori an ESTAR process for the real exchange rate. Each of s , p and p is assumed and typically
t t
t
found to be an integrated process of order one, so that q is stationary if s , p and p contain the
t t
t t
same stochastic trend i.e. cointegrate and an ECM characterizes the dynamic relationship between
L . Sarno Economics Letters 66 2000 127 –136
129
them: this then implies that an ESTAR model for q implicitly imposes some restrictions, namely
t 1,2
restrictions on the dynamic relationship between s , p and p .
t t
t
¨ In this letter, I consider an exponential STR ESTR reparametrized in the form Terasvirta, 1998:
n n
n
Ds1p2p 5a 1rs1p2p 1a Dp 2a Dp 1
O
a Ds 2
O
a Dp 1
O
a Dp
t t 21
1 t
2 t
3i t 2i
4i t 2i
5i t 2i
i 51 i 51
i 51
1
n n
n
1 a 1rs1p2p
1 a Dp 2a Dp 1
O
a Ds 2
O
a Dp 1
O
a Dp F? 1u
F G
t 21 1
t 2
t 3i
t 2i 4i
t 2i 5i
t 2i t
i 51 i 51
i 51
where Dy 5 y 2 y ; y, u is white noise and F ? is the exponential function [1 2 exp
h 2 uz 2
t t
t 21 t
t 2
c j] with u . 0 determining the speed of transition and z denoting the transition variable.
1E t
21
Alternatively, one may consider the logistic function [1 2 exp h 2 uz 2 c j]
and the resulting
t 1L
model would be the logistic STR LSTR. The transition function of the LSTR is monotonically increasing in z and yields asymmetric adjustment toward equilibrium in the model, whereas the
t
transition function of the ESTR, which is bounded between zero and unity, is symmetric about c ,
1E
although the tendency to move back to equilibrium is stronger the larger the deviation from equilibrium. Clearly, the real exchange rate behavior suggested by the discussion in Section 1 is
symmetric and consistent with an ESTR, not a LSTR. Specifically, I consider an ESTR with z given
t
by lagged values of the deviation from PPP, q d . 0.
t 2d
The ESTR 1 may be seen as a nonlinear ECM which becomes linear if the a values are all zero
or if u 5 0 in F ? . While in a linear ECM the error correction coefficient must be significantly
negative for the model to be stable, the prediction of equilibrium models in presence of transactions costs of arbitrage that the larger the deviation from PPP the stronger is the tendency to return to
equilibrium implies that r 0 is admissible in Eq. 1, but one must have r , 0 and r 1 r , 0. Thus,
for small deviations from equilibrium, q may adjust very slowly or not adjust at all, but for large
t
deviations q adjusts rapidly to its equilibrium level, being therefore globally stable.
t
The ESTR 1 clearly nests the ESTAR model considered by the recent literature in this context e.g. inter alios, Michael et al., 1997, written in first difference as:
n n
2
Dq 5 b 1 r9q 1
O
b Dq 1
b 1 p9q 1
O
b Dq [1 2 exp
h 2 u [q 2 c ]
j 1 e
S D
t t 21
j t 2j
t 21 j
t 2j t 2d
2E t
j 51 j 51
2
1
This issue is the nonlinear analogue of the issue discussed in the context of a linear framework by Kremers et al. 1992, who show that imposing invalid restrictions in the dynamic relationship between cointegrating variables yields a misspecified
univariate model, also reducing the power of conventional nonstationarity tests on the variable constructed by imposing the restrictions.
2
Unlike Bahmani-Oskooee 1998, both my discussion and empirical analysis focus on real bilateral exchange rates rather than real effective exchange rates. Since real effective exchange rates are computed as weighted averages of real bilateral
rates, however, the intuition and the results of this letter may be extended to real effective rates. Also note that, while the frequency of the data is the same as Bahmani-Oskooee 1998, in the empirical analysis I prefer to use 1973Q1 as the
starting date rather than 1971Q1, consistent with the majority of empirical studies on PPP during the recent float; however, all the empirical results reported below were qualitatively identical when using 1971Q1 as the starting date.
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. Sarno Economics Letters 66 2000 127 –136
and it is then straightforward to derive the restrictions under which Eq. 2 is a correctly specified model, if the true DGP for q is the ESTR 1. If a 5 a 5
a 5 a 5 0, a 5 a 5 a for i 5 3, 4, 5,
t 1
2 1
2 i 1
i 2 i 3
b 5 b 5 b for i 5 3, 4, 5, then the ESTR 1 becomes the ESTAR 2; if these restrictions do no
i 1 i 2
i 3 3
hold, however, Eq. 2 is misspecified. Nevertheless, as a preliminary to model specification and estimation, it is reasonable to execute
¨ linearity tests using the auxiliary ordinary least squares regression Granger and Terasvirta, 1993:
ECM 2
3
9 9
9 9
3 v 5
g W 1 g W z 1
g W z 1
g W z 1 innovations
t t
1 t t 2d
2 t t 2d
3 t t 2d
ECM
where v denotes the residuals from the linear ECM for Ds 1 p 2 p as a function of W , which is
t t
t
the vector comprising the explanatory variables Ds 1 p 2 p , Dp
, Dp and Ds
for
t 21 t 2i
t 2i t 2j
i 5 0, . . . , n and j 5 1, . . . , n; g , g and g are vectors of parameters. A general test for linearity
1 2
3
against STR is then the F-test of the null hypothesis H : g 5 g 5 g 5 0 for d [ h1,2, . . . , Dj, where
0L 1
2 3
0 is a null vector, while the choice between LSTR and ESTR may be based on a sequence of nested tests within H . Since for any ESTR
g 5 0 and a priori the LSTR may be ruled out as implausible
0L 3
for modeling real exchange rates, however, it is convenient to test for linearity against ESTR by testing the null hypothesis H :
g 5 g 5 0 ug 5 0 using an F-test say F . If linearity is rejected for
1 1
2 3
1
more than one value of d [ h1,2, . . . , Dj, d is set equal to the value which minimizes the p-value of
4
the linearity test.
3. Data and empirical analysis
