Efthymios G. Pavlidis, Ivan Paya and David A. Peel 1029The variableyt−d(though in general it could be a vector) is assumed to be stationary
and ergodic and, ford∈1, 2,...,dmaxwithdthe delay parameter, must satisfy
1 ≤d≤rforr=max(p,q). We will only consider the casep=q. We are interested
in the special case of a unit root in the linear polynomial,
∑p
j= 1 βj=1. The function
F(·)is at least fourth-order, continuously differentiable with respect toγ. There are
two common forms of the STAR model. The one we will discuss here in detail is
the exponential STAR (ESTAR) model of Granger and Teräsvirta (1993), in which
transitions between a continuum of regimes are assumed to occur smoothly.^6 The
transition functionF(·)of the ESTAR model is:
F(yt−d;γ,c)=[ 1 −exp(−γ(yt−d−c)^2 )]. (22.4)This transition function is symmetric around(yt−d−c)and admits the limits:
F(·;γ) → 1as∣
∣yt−d−c
∣
∣→+∞,F(·;γ) → 0as∣
∣yt−d−c
∣
∣→0.Parameterγcan be seen as the transition speed of the functionF(·)towards 1 (or
0) as the deviation grows larger (smaller). The variableytmoves between an AR
of the formyt=β 0 +φ 0 +
∑p
j= 1 (βj+φj)yt−j+utwhenF(·)is 1 and a unit root,
yt=β 0 +
∑p
j= 1 βjyt−j+ut, when the variable is in “equilibrium,”F(·)=0. Ifφj=−βj
∀j, the variableytwould be an AR process that moves between a white noise and
a unit root depending on the size of the deviation,
∣∣
yt−d−c∣∣.^7 Unlike in a linear
model, the speed of adjustment of these nonlinear models will depend on the size
of the deviation from PPP. They can exhibit strong persistence and near unit root
behavior.^8 Recent empirical work (e.g., Michaelet al., 1997; Tayloret al., 2001;
Payaet al., 2003; Paya and Peel, 2006a) has employed monthly real exchange rates
for the interwar and post-war float as well as a two-century span of annual rates
and showed that the ESTAR model provides a parsimonious fit to the data.
22.2.1.1 Linearity testing against STR
When testing for the existence of the nonlinear part of (22.2) an identification
problem arises. The null hypothesis of linearity corresponds toH 0 :φ′=0. Under
H 0 , the parametersγandccould take any value as they are not identified under
the null. Alternatively, if the null hypothesis wasH 0 :γ =0, then parameters
φandcwould not be identified under the null. In these cases it would not be
possible to differentiate between a linear or nonlinear process (see Davies, 1977).^9
This problem is solved by taking a Taylor series approximation ofF(·)with respect
toγevaluated atγ=0. This method was introduced by Luukkonenet al.(1988)
and adopted by Teräsvirta (1994). A third-order Taylor expansion of the logistic
function would yield:
yt=β′ ̃yt+1
4
γφ′ ̃yt(yt−d−c)+1
48
γ^3 φ′ ̃yt(yt−d−c)^3. (22.5)In the case of the exponential function a first-order Taylor approximation yields:
yt=β′ ̃yt+γφ′ ̃yt(yt−d−c)^2. (22.6)