Palgrave Handbook of Econometrics: Applied Econometrics

1028 The Econometrics of Exchange Rates

The power of both ADF and PP tests was around 45% for a sample size of 100. Caner

and Kilian (2001) did a similar experiment but with the null of stationarity. They

employ the Kwiatkowskiet al.(1992) (KPSS) and Leybourne and McCabe (1994)

tests. They found that, if real exchange rates were a linear autoregressive process

with half-life of about three to five years (β=0.5(^1 /^60 )=0.9885 for monthly data,

andβ=0.5(^1 /^20 )=0.9659 for quarterly), they would reject the null of stationarity

with about 60% probability for quarterly data and 40% for monthly data. Together,

these results pointed out a serious problem with unit root and stationarity tests due

to their lack of power and size, respectively.^4

A number of authors have employed a multivariate cointegration methodology

to test for a long-run relationship between exchange rates and foreign and domestic

price levels in the recent floating exchange rate period (MacDonald, 1993; Baum

et al., 2001). The cointegrating relationship is usually specified as:

st=α+β 1 p∗t+β 2 pt+ut. (22.1)

The standard empirical findings employing these methods are that cointegration

cannot be rejected but the assumption of proportionality and symmetry between

the nominal exchange rate and domestic and foreign prices (β 1 =−β 2 =1) is not

supported by the data.^5

Theoretical analysis of purchasing power parity deviations demonstrate how

transactions costs or the sunk costs of international arbitrage induce nonlinear but

mean reverting adjustment of the real exchange rate (see, e.g., Dumas, 1992; Sercu

et al., 1995; O’Connell and Wei, 1997). Whilst globally mean reverting, these non-

linear processes have the property of exhibiting near unit root behavior for small

deviations from PPP, since small deviations are left uncorrected if they are not large

enough to cover transactions costs or the sunk costs of international arbitrage. The

nonlinearity postulated can be captured by a set of parametric nonlinear auto-

regressive models. We can classify this set of nonlinear models according to the

way the real exchange rate switches between regimes: first, with smooth transition

models; second, with threshold autoregressive models.

22.2.1 Smooth transition (STR) models

Consider the following STR model:

yt=β′ ̃yt+φ′ ̃ytF(zt−d;γ,c)+ut, (22.2)

where ̃yt=(1,yt− 1 ,...,yt−p,wt− 1 ,...,wt−q), withwta vector of exogenous vari-
ables, andut is a white-noise sequence with mean zero and varianceσu^2 .If
̃yt= (1,yt− 1 ,...,yt−p), the model is called a smooth transition autoregressive
(STAR). We will concentrate on this case hereafter as results can easily be gener-
alized to the STR. The variablezt−dis the transition variable, the one that drives
the dependent variable to move between regimes. We will consider the case that
zt−d=yt−d. The STAR model can then be written as:

yt=β 0 +

∑p

j= 1

βjyt−j+

(

φ 0 +

∑q

j= 1

φjyt−j

)

F(yt−d;γ,c)+ut. (22.3)