Efthymios G. Pavlidis, Ivan Paya and David A. Peel 104922.4.2 Smooth transition autoregressive target zone
Lundbergh and Teräsvirta (2006) propose a flexible parametric target zone model
that nests the model of Krugman but also allows estimation of an implicit tar-
get zone if it exists. A feature of their model is that it allows joint modeling
of both the conditional mean and the conditional variance. Their model builds
on an earlier contribution of Bekaert and Gray (1998). They call their model the
Smooth Transition Autoregressive Target Zone (STARTZ) model. The STARTZ model
is a parameterization of the first and second moments ofst, the deviation of the
exchange rate from the central parity. The STARTZ model is given by the following
equation:
st=λt+ (^) t, (22.60)
with (^) t=
√
ztht, where{zt}∼i.i.d.(0, 1)andhtis the conditional variance of (^) t.
The conditional meanλtis defined by:
λt=φ′xt+(μsl−φ′xt)Gl(st− 1 ;γa,θa,μsl)+(μsu−φ′xt)Gu(st− 1 ;γa,θa,μsu),
wherext=(1,st− 1 ,...,st−n)′andφ=(φ 0 ,φ1,...,φn)′is the corresponding parameter
vector. The two transition functionsGlandGuhave the form:
Gl(st− 1 ;γ,θ,c)=
[
1 +exp
(
−γ(c−st− 1 )
)]−θ
,γ>0,θ>0,
Gu(st− 1 ;γ,θ,c)=
[
1 +exp
(
−γ(st− 1 −c)
)]−θ
,γ>0,θ>0,
wherest− 1 is the transition variable, andγ,candθare the slope, location and
asymmetry parameters, respectively. The lower and upper bounds of the zone are
defined byslandsu, so thatc=μslandc=μsuare location parameters.^55
The target zone literature implies that the conditional variance of the exchange
rate must be very small near the edges of credible, implicit or explicit, bands.
Lundbergh and Teräsvirta (2006) parameterize this feature by assuming that the
conditional variance has a parametric specification similar to the conditional mean
and given by:
ht=η′wt+(δ−η′wt)Gl(st− 1 ;γb,θb,μsl)+(δ−η′wt)Gu(st− 1 ;γb,θb,μsu),
whereη=(α 0 ,α 1 ,...,αq,β 1 ,...,βp)′,wt=(1,ε^2 t− 1 ,...,ε^2 t−q,ht− 1 ,...,ht−p)′.Itis
assumed thatδ>0 which, together with the restrictionsα 0 >0,αi≥0,i=1,...,q,
βj≥0,j=1,...,p, is sufficient to ensure the conditional variance is positive.
Because (^) t=st−λt, such thatφis assumed not to depend onη, the conditional
variance is a nonlinear function of the elements ofwt.^56
Lundbergh and Teräsvirta obtain their parameter estimates by maximizing the
log-likelihood under the assumption that{zt}is a sequence of independent stan-
dard normal errors. Under this assumption the (quasi) log-likelihood function
equals:
lt=const−0.5 lnht−0.5
(^2) t
ht
.