1036 The Econometrics of Exchange Rates
estimate ofθis thenθˆ=θ(ˆˆc). Furthermore, the LS principle allows the estimation
of the, typically, unknown valuedby extending problem (22.20) to a search across
the discrete space[1,d ̄].
The hypothesisH 0 :α=bis tested as follows. Let{et}Tt= 1 be an i.i.d. sequence
ofN(0, 1)draws. Regressetonxtto obtain the residual varianceσˆT^2 and on
xt(c)to obtainσˆT^2 (c)and computeF(c)=T(σˆT^2 −ˆσT^2 (c))/σˆT^2 (c). Then compute
F=supc∈CF(c). Repeat the procedurentimes and the asymptoticp-value of the test
is given by the percentage of samples for whichFexceeds the observedFT. Hansen
(1997) also provides critical values and a method to construct asymptotically valid
confidence intervals.^32
In the last decade the basic TAR model has been extended. New unit root tests
against these TAR models have been used as tests for PPP. We detail below some
of those tests that include TAR models but differ according to the nature of the
transition variable (in levels or in differences), the symmetry or asymmetry of the
bands, the autoregressive process within each region (unit root, stationary AR),
and the number of regimes.
22.2.2.1 Unit root test versus TAR
Enders and Granger (1998) (EG) modify the Dickey–Fuller (DF) critical values of
theF-statistic in order to have the right size to test the unit root null against TAR,
or momentum-TAR (M-TAR) models. In particular, they consider theF-statistic of
the null unit root hypothesis (H 0 :ρ 1 =ρ 2 = 0 )in the following TAR model:
yt={
ρ 1 yt− 1 +ut ifyt− 1 ≤0,
ρ 2 yt− 1 +ut ifyt− 1 >0,(22.21)and also for the M-TAR model:
yt={
ρ 1 yt− 1 +ut ifyt− 1 ≤0,
ρ 2 yt− 1 +ut ifyt− 1 >0.(22.22)EG show that the power of their test improves relative to the standard ADF as the
asymmetric adjustment becomes more pronounced.^33
An alternative threshold unit root test is developed by Basci and Caner (2005)
(BC). They consider the following M-TAR model:
yt={
θ 1 ′xt− 1 +et if∣∣
yt− 1 −yt−m− 1∣∣
<λ,
θ 2 ′xt− 1 +et if∣∣
yt− 1 −yt−m− 1∣∣
≥λ,(22.23)wherext− 1 =(yt− 1 ,1,yt− 1 , ....,yt−k)fort=1, 2, ...,T.etis an i.i.d. error term,
mrepresents the delay parameter and 1≤m≤k. It is possible to rewrite the model
above as follows:
yt=θ 1 ′xt− 11 {|yt− 1 −yt−m− 1 |<λ}+θ′ 2 xt− 11 {|yt− 1 −yt−m− 1 |≥λ}+et. (22.24)