Efthymios G. Pavlidis, Ivan Paya and David A. Peel 1037The expression above is estimated using OLS for eachλ∈%and the OLS estimate^34
ofσ^2 is then̂σ^2 (λ)=T−^1
∑T
t= 1 ̂et(λ)(^2). The estimated threshold parameterλis the
one that minimizes the error variance:̂λ=arg minλ∈%̂σ^2 (λ). Testing linearity
reduces toH 0 :θ 1 =θ 2. BC propose the following test:
sup
λ∈%
WT(λ)=sup
λ∈%
T
(
̂σ 02
̂σ^2 (̂λ)
− 1
)
, (22.25)
wherêσ 02 is the residual variance from simple OLS estimation of the null linear
model. Following Caner and Hansen (2001), BC resort to a bootstrap approxi-
mation of the distribution ofWTto obtainp-values. They tested the nonlinear
behavior of 14 OECD real exchange rates and found that 11 of them display a unit
root inside the band and mean reversion outside the band.^35
This section has examined recent developments in linearity testing and the
autoregressive modeling of real exchange rates. The overall conclusion is that non-
linear models provide considerably greater support for the PPP hypothesis and that
the PPP puzzle is largely resolved by them.
22.3 International parity conditions
22.3.1 Covered interest parity (CIP)
In the absence of frictions such as transactions costs or limits to arbitrage
funds, riskless arbitrage should ensure that the covered interest differential on
assets of identical characteristics should be equal to zero. Employing the usual
approximations, we have that:
it−i∗t=ft−st, (22.27)
whereit,i∗tare the interest rates on the domestic and foreign assets concerned,ftis
the logarithm of the forward exchange rate (the rate at which the future exchange
of currencies is agreed at timet) of the same term to maturity as the assets, andst
is the spot exchange rate (domestic price of foreign currency).
Whether CIP holds is of interest for at least three reasons. First, absence of CIP,
given its riskless nature in principle, wouldceteris paribusimply that the efficient
markets assumption approach to modeling exchange rates (or asset prices in gen-
eral) had very serious limitations. Second, CIP forms a basis with uncovered interest
arbitrage (see below) to determining the properties of the forward rate as a predictor
of future movements in the spot rate. Uncovered interest arbitrage could hardly be
expected to hold in the absence of CIP. Finally, it is assumed in numerous models
that covered and uncovered parity hold. Many empirical tests of the covered inter-
est rate condition have been undertaken.^36 Taylor (1987, 1989), unlike in most
previous studies, employs high frequency contemporaneously sampled data for
spot and forward dollar–sterling and dollar–Mark exchange rates and correspond-
ing euro-deposit interest rates for a number of maturities and makes allowance for
bid-ask spreads and brokerage costs in his calculations for the 1980s and selected