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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
L .
Sarno Economics
Letters 66
2000
127 –
136
131 Table 1
a
Linearity tests results: F statistics
1
d 5 1 d 5 2
d 5 3 d 5 4
d 5 5 d 5 6
d 5 7 d 5 8
Bahrain 17.1601 0.0000 3.3988 0.0001 7.3413 0.0000 6.6726 0.0000 1.4458 0.1373 3.8830 0.0000 6.3912 0.0000 2.7182 0.0018
Egypt 0.8544 0.6522 3.1866 0.0003 1.4445 0.1358 2.1871 0.0102 0.6296 0.8859 3.0760 0.0004 0.7941 0.7213 1.1390 0.3435
Iran 7.4904 0.0000 1.6822 0.0561 6.8185 0.0000 1.3631 0.1714 6.1711 0.0000 1.2095 0.2767 0.5886 0.9177 0.9657 0.5203
Jordan 4.1323 0.0000 1.0124 0.4695 1.3588 0.1852 2.5959 0.0031 1.8016 0.0462 2.6355 0.0030 1.7245 0.0617 1.4924 0.1293
Lebanon 3.0217 0.0006 2.7228 0.0018 1.5550 0.1004 2.7360 0.0019 2.3258 0.0080 2.0722 0.0193 1.3408 0.2007 1.0384 0.4458
Morocco 2.2538 0.0063 1.6871 0.0551 0.7847 0.7350 1.1058 0.3678 0.9977 0.4831 1.0118 0.4605 1.7016 0.0555 0.5928 0.9144
Saudi Arabia 1.5242 0.0978 5.0275 0.0000 1.7105 0.0512 3.4924 0.0001 1.9334 0.0232 2.1622 0.0101 1.6452 0.0677 1.7608 0.0456
Sudan 5.9825 0.0000 4.9738 0.0000 4.7007 0.0000 3.5242 0.0002 2.4682 0.0054 1.2362 0.2707 0.9342 0.5590 1.9439 0.0328
Syria 18.1856 0.0000 7.7545 0.0000 6.5657 0.0000 9.6093 0.0000 5.6520 0.0000 4.2087 0.0000 0.3157 0.9969 0.4271 0.9804
Tunisia 0.7835 0.7369 5.7108 0.0000 0.5504 0.9421 1.2031 0.2803 1.9349 0.0231 0.5498 0.9416 1.0104 0.4695 0.4875 0.9691
Turkey 4.1825 0.0000 3.3173 0.0001 2.0208 0.0161 1.6934 0.0551 2.8153 0.0008 1.0061 0.4739 1.6807 0.0598 1.8474 0.0335
a
Notes: The F statistic tests the null hypothesis of linearity against the alternative of ESTR-type nonlinearity and is constructed as described in the text
1
assuming a lag length of four; d denotes the delay parameter. Marginal significance levels are calculated using the appropriate F-distribution and reported in parentheses, up to the fourth decimal point.
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. Sarno Economics Letters 66 2000 127 –136
Then, applying the standard information criteria due to Akaike and Schwartz to a linear ECM for Ds 1 p 2 p , the lag length n was set equal to 4 for each country in order to execute the linearity
t
tests F . Panel A of Table 1 reports p-values of the test statistics F for d [ h1, 2, . . . , 8j. Linearity is
1 1
rejected most strongly and with very low p-values when d 5 1 for 8 out of 11 cases and when d 5 2 for the three remaining cases, suggesting a rather fast response to shocks for all of these real exchange
rates. Next, I estimated an ESTR of the form 1 by nonlinear least squares, and executed tests of the
restrictions described in Section 2 using likelihood ratio LR tests in order to choose between the ESTR 1 and its nested univariate alternative considered in this letter, the ESTAR model 2. The LR
tests, whose p-values are reported in Table 2, are very interesting in that for eight out of 11 countries the restrictions are rejected at conventional nominal levels of significance, hence implying that, for
those countries, an ESTAR model for q is a misspecified model. For Iran, Lebanon and Turkey,
t
however, the restrictions are not rejected, hence suggesting adequateness of an ESTAR model. Assuming n 54 and with d set according to the linearity tests results, I then estimated smooth
transition models of the type implied by the restrictions tests results. In each case I followed the ¨
recommendation of Granger and Terasvirta 1993 of standardizing the transition parameter by dividing
u by the sample variance of the dependent variable and using a starting value of u 5 1 for the estimation algorithm. Parsimonious models were obtained for each of the real exchange rates after
imposing the restrictions r 5 0, r 5 2 1 for ESTR models and r9 5 0, r9 5 2 1 for ESTAR models,
in addition to various exclusion restrictions see the LR tests in Table 3, and applying the conventional general to specific procedure. These restrictions imply an equilibrium value in the
neighborhood of which q is a unit root process, with the adjustment toward equilibrium becoming
t
faster with the absolute size of the deviation from equilibrium. While space considerations preclude me from reporting each of the estimated equations, the equations reported in Table 3, namely the
estimated ESTR for Egypt and the estimated ESTAR model for Turkey may be viewed as reasonably representative for the two nonlinear models considered here.
The resulting models display very high coefficients of determination and insignificant diagnostics. The ratio of the residual variance of the estimated nonlinear model to the residual variance of the best
fitting alternative linear model V suggests that each estimated nonlinear model often leads to a very
Table 2
a
Restrictions tests results Bahrain
Egypt Iran
Jordan Lebanon
Morocco Saudi Arabia
Sudan Syria
Tunisia Turkey
d 2
1 1
1 1
1 2
1 1
2 1
p-value 0.0000
0.0001 0.0751
0.0024 0.1757
0.0271 0.0000
0.0043 0.0012
0.0450 0.0869
a
Notes: d is the delay parameter used in estimation of the unrestricted ESTR 1 for each country; p-value refers to the marginal significance level from executing likelihood ratio tests of the restrictions described in Section 2, where the null
hypothesis tested is that the restrictions hold – implying that the ESTR 1 becomes the ESTAR model 2 – against the alternative hypothesis that the restrictions do not hold – implying that an ESTAR model 2 for the relevant country’s real
2
exchange rate would be a misspecified model. The likelihood ratio tests are distributed as central x
under the null hypothesis, with degrees of freedom equal to the number of restrictions.
L . Sarno Economics Letters 66 2000 127 –136
133 Table 3
a
Estimated smooth transition models Estimated ESTR for Egypt
ˆ ˆ
ˆ ˆ
ˆ ˆ
Dq 5 a 2 a Dp 1 a Ds 2 a Dp
2 a Dp
t 2
t 31
t 23 41
t 21 42
t 22 2
ˆ ˆ
ˆ ˆ
ˆ 1 2q
2 a Dp 1 a Ds
2 a Dp
2 a Dp
[1 2 exp h 2 uq
j]
t 21 2
t 32
t 22 41
t 21 42
t 22 t 21
ˆ ˆ
ˆ where a 5 0.473 11.996, a 5 1.799 4.139, a
5 0.509 2.927,
2 31
ˆ ˆ
ˆ ˆ
a 5 3.908 3.934, a
5 6.340 6.518, a 5 4.877 3.346, a 5 0.487 2.440,
41 42
2 32
ˆ ˆ
ˆ ˆ
a 5 4.076 2.488, a 5 7.383 4.609, u 5 0.917 14.116, u 5 7.732
41 42
u 2
R 5 0.915 ET 1 5 0.448 ET 2 5 0.376 ARCH12 5 0.721 LR 5 0.846 V 5 0.536 Estimated ESTAR for Turkey
2
ˆ ˆ
ˆ ˆ
ˆ ˆ
ˆ Dq 5 b 1 b Dq
1 b Dq 1 b Dq
1 2q 1
b Dq [1 2 exp
h 2 uq j]
t 1
t 21 2
t 22 3
t 23 t 21
2 t 22
t 21
ˆ ˆ
ˆ where b 5 0.873 7.956, b 5 2.090 4.312, b 5 0.457 2.246,
1 2
ˆ ˆ
ˆ ˆ
b 5 0.658 2.724, b 5 2 1.824 23.158, u 5 0.197 5.401, u 5 8.189
3 2
u 2
R 5 0.954 ET 1 5 0.577 ET 2 5 0.472 ARCH12 5 0.830 LR 5 0.463 V 5 0.739
a
Notes: Figures in parentheses next to coefficient estimates are t-ratios; the restrictions r 5 0, r 5 2 1 and for the ESTR,
and r9 5 0, r9 5 2 1 for the ESTAR model were imposed in estimation. u is the unstandardized speed of adjustment
u 2
2 2
2 2
coefficient u, R is the coefficient of determination, V 5 s s
where s
and s
denote the residual variance from the
N L
N L
estimated smooth transition model and the residual variance from the estimated best fitting alternative linear model. ET 1 and ET 2 are test statistics for no residual serial correlation and for no remaining nonlinearity respectively, constructed as
¨ suggested by Eitrheim and Terasvirta 1996; ARCH12 is a test statistic for autoregressive conditional heteroscedasticity in
2
the residuals up to order 12, distributed as x 12. LR is a likelihood ratio test statistic for the restrictions implicit in the
2
estimated equation against the corresponding unrestricted nonlinear model with a lag length of four, and is distributed as x
with degrees of freedom equal to the number of restrictions. For ET 1, ET 2, ARCH12 and LR statistics, only the p-values are reported.
significant reduction – e.g. 47 for Egypt – of the residual variance relative to the best alternative linear model.
Removing the standardization on u yields unstandardized values of the speed of adjustment, say u ,
u
between 1.381 Lebanon and 8.189 Turkey, suggesting rather fast – albeit varying across countries
6
– speed of adjustment. This is made very clear by Fig. 1, which displays the plots of the estimated transition functions against q
2 c and q
2 c for ESTR and ESTAR models respectively for
t 2d 1E
t 2d 2E
each of the real exchange rates examined and shows that the limiting case of F ? 5 1 is attained for a number of countries. Also, the observed strong nonlinearity in real exchange rates and their local
instability is consistent with and presumably largely explains the difficulty of rejecting the hypothesis
6
Note that the t-ratio for u should be interpreted with caution since, under the null hypothesis that u 5 0, the real exchange
rate follows a unit root process. Hence, the presence of a unit root under the null hypothesis complicates the testing procedure analogously to the way in which the distribution of a Dickey–Fuller statistic cannot be assumed to be Student’s t.
In the present case, however, given that the deviation from the Student’s t is generally found to be relatively moderate, the size of the t-ratios implied by the estimates in Table 3 may be regarded sufficiently large to make us confident of their
statistical significance. Also, the statistical significance of
u is, in a sense, not questionable once one rejects the linearity ¨
hypothesis against the alternative of an ESTAR see Terasvirta, 1994, 1998.
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. Sarno Economics Letters 66 2000 127 –136
Fig. 1.
Estimated transition
functions.
L . Sarno Economics Letters 66 2000 127 –136
135
of unit root behavior encountered by Bahmani-Oskooee 1998 using linear nonstationarity tests for
7
the same group of countries.
4. Conclusion
