Efthymios G. Pavlidis, Ivan Paya and David A. Peel 1067
poor estimates, on average, of the cointegrating vector, with a range of values that include
those reported in the literature.
- Conventional maximum likelihood theory is therefore not applicable. If one were to use
a maximum likelihood estimator, the testing procedure would be analogous to the one
described below but using the partial derivatives of the likelihood function evaluated
under the null (see Granger and Teräsvirta, 1993, Ch. 6).
- That is,yt=β′ ̃yt+γφ′ ̃yt(yt−d−c)^2 +γφ′ ̃yt(yt−d−c)^4. Note that even powers of the
Taylor approximation of the logistic function are all zero while odd powers of the Taylor
approximation of the exponential are all zero. The logistic function has one inflection
point while the exponential possesses two, which is the point of using a second-order
Taylor expansion.
- In particular, they propose to use the White heteroskedasticity consistent covariance
matrix in the LM test. Their analysis is based on MacKinnon and White (1985). See also
Wooldridge (1990).
- See Lundbergh and Teräsvirta (1998) for the specification, estimation and evaluation of
models with nonlinear behavior in the mean (STAR) and in the conditional variance
(STGARCH), the STAR-STGARCH model.
- See below for a description and references of this methodology.
- However, its distribution is the same when the transition variabley∗t− 1 is substituted by
yt∗−dor moving averages ofyt∗− 1 (see Venetiset al., 2005). What changes is the form of the
auxiliary regressions, which generalize to:
y∗t=δy∗t− 1 y∗t−^2 d+error,
y∗t=
∑p
j= 1
ajy∗t−j+δy∗t− 1 y∗t−^2 d+error.
The corollary results in Venetiset al.are particularly important since it is possible to
generalize KSS tests against wider alternatives that assume longer adjustment periods.
- The linear unit root hypothesis against an ESTAR has also been tested using a different
methodology to that of a Taylor approximation. Kiliç (2003) overcomes the identification
problems forγandcin his test by using a grid search over the space of values for the
parametersγandcto obtain the largest possiblet-value forφin the following regression:
yt∗=φy∗t−^31
[
1 −exp(−γ(zt−c)^2 )
]
+error,
whereztis the transition variable, which in our framework would bey∗t− 1. He tests the
null hypothesis of H 0 :φ=0 (unit root case) against H 1 :φ<0. Kiliç (2003) claims that
the advantages of his procedure over KSS is twofold. First, it computes the test statistic even
when the threshold parameter needs to be estimated in addition to the transition param-
eter. Second, it claims to have higher power. Using quarterly data for 17 real exchange
rates of developed countries against the dollar for the floating period, Kiliç finds strong
evidence of nonlinear ESTAR behavior.
- This approach is suggested by Vogelsang (1998).
- PPP has also been tested using nonlinear cointegration. Kapetanioset al.(2003b) (KSSb)
propose a testing procedure to detect the presence of a cointegrating relationship that
follows a STAR process. Venetiset al.(2005) and Paya and Peel (2006a) find evidence of
nonlinear cointegration between the real exchange rate and productivity proxies. Further
support for nonlinear cointegration has been found with a cointegration method that
uses a transformation of the variables. Breitung (2001) suggests a rank test procedure
based on the difference between the sequences of ranks of the variables involved in the
cointegrating relationship. Haug and Basher (2005) also apply the Breitung test for the
dollar and DM based real exchange rates of the G10 countries and only found evidence
of nonlinear long-run PPP in two of them, the pound/dollar and Belgian franc/DM.
