Efthymios G. Pavlidis, Ivan Paya and David A. Peel 1033The unrestricted ESTAR model considered here has the following form:yt∗=∑p
j= 1φjyt∗−j+(p
∑
j= 1φj∗y∗t−j)[
1 −exp(
−γ(
yt∗−d) 2 )]
+ut, (22.15)wherey∗tdenotes de-meaned, detrended or in-deviation data.^18 In the estimation
of the nonlinear model,γis estimated by scaling it by the variance of the transition
variable. This scaling is suggested for two reasons. One is to avoid problems in the
convergence of the algorithm. Second, it makes it easier to compare speeds of
adjustment across different studies (see Teräsvirta, 1994).^19
Since thet-ratio of the estimated coefficientγin (22.15) does not provide a
valid significance test in the usual way, its critical values must be obtained by
simulation.^20 The estimation technique can be nonlinear least squares (NLS) or
maximum likelihood. Under the assumption thatutis normally distributed, NLS is
equivalent to maximum likelihood, otherwise, NLS estimates can be interpreted as
quasi-maximum likelihood estimates. Wooldridge (1994) and Pötscher and Prucha
(1997) discuss regularity conditions that allow consistent and asymptotically
normal estimators.
The adequacy of the estimated STR model can be evaluated employing the LM-
type diagnostic tests for the hypothesis of no error autocorrelation, (the customary
portmanteau test has an unknown asymptotic null distribution), nonlinearity and
parameter constancy of Eitrheim and Teräsvirta (1996). The last two tests address
important issues of misspecification due to neglected nonlinearity and possible
parameter instability.
The nonlinear models reported in empirical work have been estimated on data
sampled at different levels of aggregation, namely monthly, quarterly and annual
(see, e.g., Michaelet al., 1997; Tayloret al., 2001; Taylor and Kilian, 2003; Paya
et al., 2003).^21 As noted by Taylor (2001), if the true DGP is nonlinear, the tempo-
rally aggregated data could exhibit misleading properties regarding the adjustment
speeds if a linear model is estimated. Paya and Peel (2006c) complement this work
by examining the effects of different levels of temporal aggregation of an ESTAR
DGP on aggregate estimates of ESTAR models.^22 They show that ESTAR type non-
linearities are usually preserved under the temporal aggregation schemes examined.
However, the dynamic structure of the best fitting models changes and tends to
take the form researchers have found to fit well on actual data of the same fre-
quency. Furthermore, comparison of the measured speed of response to shocks
with models estimated on the temporally aggregated data and the true DGP shows
that the measured speed of adjustment declines the more aggregated the data.
22.2.1.3 Time-varying equilibrium real exchange rate
A variety of theoretical models, such as those of Balassa (1964), Samuelson (1964),
Lucas (1982), and Backus and Smith (1993), imply a non-constant equilibrium in
the real exchange rate and estimates, including proxies for the equilibrium determi-
nants, appear significant (see, e.g., Lothian and Taylor, 2000; Hegwood and Papell,
2002). Paya and Peel (2004, 2006a), employing various proxies, show the estimated