1032 The Econometrics of Exchange Rates
against nonlinearity. Rejection of the null is therefore indicative of nonlinearity
and not that the DGP follows a different linear process.
The HL test consists of two steps. First is the test of linearity. Second, the order of
integration of the linear or nonlinear process is determined. Consider the linearity
test (22.7) described above for the case of the null of linearI(0). In the case of
anI(1) variable, the Taylor expansion would become (considering the transition
variable isyt− 1 ):
yt=φ 0 yt− 1 +φ 1 (yt− 1 )^2 +φ 1 (yt− 1 )^3 +εt.In order to combine both possibilities,I(0) andI(1), HL propose the following
regression model:
yt=α 0 +α 1 yt− 1 +α 2 y^2 t− 1 +α 3 y^3 t− 1 +α 4 yt− 1 +α 5 (yt− 1 )^2 +α 6 (yt− 1 )^3 +εt. (22.13)The null hypothesis of linearity would beH 0 L:α 2 =α 3 =α 5 =α 6 =0. The alter-
native hypothesis (nonlinearity) would be that at least one of thoseαsis different
from zero. This test is carried out using the Wald statistic:
WT=
RSS 1 −RSS 0
RSS 0 /T,where the restricted residual sum of squares (RSS 1 )comes from an ordinary least
squares (OLS) regression ofyton a constant,yt− 1 , andyt− 1. As HL point out,
the distribution ofWTunder the null differs depending on whether the process
followed byytisI(0) orI(1). In order to make the limiting distribution ofWThomo-
geneous under the null they multiply it with a correction that is the exponential
of a weighted inverse of the absolute value of the ADF statistic:^16
WT∗=exp(
−b∣
∣DFT
∣
∣−^1)
WT, (22.14)where an expression for the value ofbis provided such that, for a given significance
level, the critical value ofWT∗coincides with that from aχ^2 ( 4 ). They also prove
that, underH 1 ,WT∗is consistent at the rateOp(T).
The second step is to test whether the series is anI(0) or anI(1) process. HL use
the Harriset al.(2003) statistic to distinguish betweenI(0) orI(1) processes regard-
less of whether nonlinearity is present or not. HL apply their testing methodology
to the monthly real exchange rates of 15 European countries against the dollar and
find evidence of PPP (linearI(0) or nonlinearI(0)) in only two of them, Finland and
the Netherlands. Notwithstanding, from a theoretical point of view, evidence that
the real exchange rate follows a nonlinearI(1) process would not be considered
evidence in support of PPP.
22.2.1.2 Nonlinear STR estimation
Once the functional form of the real exchange rate has been determined using
linearity testing, the next step is to review the nonlinear estimation procedure.^17