Efthymios G. Pavlidis, Ivan Paya and David A. Peel 1043Another important possibility is inference problems arising from the differ-
ent statistical properties of changes in the the spot rate and forward premium.
Baillie and Bollerslev (2000) consider a sample of monthly observations on the
DM/$ spot and one-month forward rates from January 1974 to December 1991,
realizing a total of 215 observations. They report that the monthly sample standard
deviation of percentage changes in the spot rate is 2.75 and the correspond-
ing figure for the monthly forward premium as 0.217. These figures are typical.
Changes in the spot rate usually exhibit a standard deviation at least 100 times
bigger than the forward premium. In addition, the forward premia are generally
very persistent whilst changes in the spot rate are not. In fact, Baillie and Boller-
slev (1994), Byers and Peel (1996), and Maynard and Phillips (2001) have argued
that the temporal dependencies in the forward premium can be parsimoniously
described by a fractionally integrated, or I(d), process.
Mathematically, the autoregressive fractional integrated moving average
(ARFIMA) model for a time series processytcan be written as:
φ(L)( 1 −L)d(yt−μ)=θ(L)
t, (22.46)whereφ(z)=0 andθ(z)=0 have all roots lying in the unit circle and{ (^) t}is a
martingale difference sequence. The differencing operator,( 1 −L)d, is defined as
follows:
( 1 −L)d=
∑∞
j=o
(j−d)Lj
(−d)(j+ 1 )
,
whereis the gamma or generalized factorial function. A fractionally integrated
process is one that exhibits long memory, with persistent local trends, but which
nonetheless eventually “reverts to the mean.” The degree of persistence is measured
by the real-valued parameterd, lying on the unit interval.^44 Smallwood (2005) has
an interesting discussion of the properties of the various estimators of the fractional
parameterdin the context of PPP. One important property of fractional processes
is the self-similarity property. This implies that the estimate ofdshould remain
invariant over different temporal aggregates of the process. Ohanissianet al.(2007)
have developed a statistical test based on this property. Their simulations show
that the test has good size and power properties against some alternatives such
as Markov-switching. The potential use of this test in exchange rate econometrics
seems great. We also consider that the power of this test against alternatives such
as ESTAR would be of interest.
From a purely statistical perspective the different properties of changes in the
spot rate and the forward premium suggest that the orders of integration of the
dependent and explanatory variables are not the same. One method of dealing
with the long-memory characteristics of the forward premium, as suggested by
Baillie and Bollerslev, is to regress the spot return on the fractionally differenced
forward premium (see Maynard and Phillips, 2001; Abadir and Talmain, 2006):
st+ 1 −st=λ+θ( 1 −L)d(ft−st)+ (^) t+ 1. (22.47)