1046 The Econometrics of Exchange Rates
whereG(·)is the transition function of an LSTAR or ESTAR model andkis the delay
parameter. Under the null hypothesis,γ=0, the time series process is distributed
as a long-memory ARFIMA(p,d,0). We outline Smallwood’s test for joint fractional
and ESTAR nonlinearity, which seems more appropriate in the context of exchange
rate econometrics where issues of transactions bands and limits to arbitrage suggest
ESTAR (or threshold) as the parametric form of nonlinearity.^50 The Smallwood test
procedure is similar to that outlined in section 22.2.1 and consists of a first-order
Taylor series expansion of model (22.53):
( 1 −L)dyt=⎛
⎝α1,0+
∑pj= 1α1,j( 1 −L)dyt−j⎞
⎠+⎛
⎝
∑pj= 1α2,j( 1 −L)dyt−jyt−k⎞
⎠+⎛
⎝∑pj= 1α3,j( 1 −L)dyt−jy^2 t−k⎞
⎠+et. (22.54)Assuming the error term is Gaussian, the null hypothesis of a linear fractional
process is given byH 0 :α2,j=α3,j=0,j=1,...,p. Smallwood illustrates that
the existence of the fractional differencing parameter complicates the construc-
tion of the LM-type test statistic based on (22.54). However, he shows that aχ^2
andFversion can be calculated as follows. One first estimates an ARFIMA(p,d,0)
model and obtains the estimate ofd(̂d), and the set of residualŝεt. The sum
of squared errors, denotedSSRR, is then constructed from the residualŝεt. Sec-
ond, a regression of̂εtis run on
∑t− 1
j= 1 ̂εt−j/j,1,(^1 −L)̂d
yt− 1 ,...,( 1 −L)̂d
yt−p,( 1 −L)
̂d
yt− 1 yt−k,...,( 1 −L)̂d
yt−pyt−k, and( 1 −L)̂d
yt− 1 yt^2 −k,...,( 1 −L)̂d
yt−py^2 t−k.
The unrestricted sum of squared residuals,SSRUR, is formed from this regression.
Theχ^2 version of the LM test statistic is calculated as LMχ 2 =T(SSRR−SSRUR)/SSRR,
and is distributed as aχ^2 ( 2 p). TheFversion of the LM test statistic is calculated as
LMF=
[
(SSRR−SSRUR)( 2 p)−^1][
SSRUR(T− 3 p− 1 )−^1]− 1
, and is distributed as an
F( 2 p,T− 3 p− 1 ).
In practice, of course, the long-memory parameter is generally unknown. Differ-
ent methods have been employed to obtain a consistent estimate in the first step
of the test. Smallwood prefers the estimator of Beran (1995) based on the condi-
tional likelihood function of the time series process. Baillie and Kapetanios (2006)
employ the local Whittle semiparametric estimator.^51 We would also suggest that
the test of Ohanissianet al.(2007) mentioned above would be worth exploring in
this context. Its size properties given data generated from a FI-STAR model would be
of interest and, similarly, its power properties for data generated from an ESTAR.^52
22.4 Target zone models
There have been a large number of papers that have examined the behavior of
exchange rates in target zones. The basic theory is due to Flood and Garber (1983),
with the particular application to target zones by Krugman (1991). The model of