Palgrave Handbook of Econometrics: Applied Econometrics

Efthymios G. Pavlidis, Ivan Paya and David A. Peel 1035

22.2.2 Threshold autoregressive (TAR) models

If the transition between regimes is assumed abrupt rather than smooth the dynam-
ics of PPP adjustment can be captured by the TAR model of Tong (1983). Empirical
studies that use TAR models for deviations from PPP include Obstfeld and Taylor
(1997) and Sarnoet al.(2004a). One of the advantages of this methodology is the
direct estimation of the transaction cost band or threshold band. For illustrative
purposes we start by describing the estimation of a simple symmetric thresh-
old TAR model of order one employed in some of the empirical work previously
mentioned:^29

yt=

{

a 0 +a 1 yt− 1 +ut ifzt−d≤c,

b 0 +b 1 yt− 1 +ut ifzt−d>c,

(22.18)

wherezt−dis the transition variable, in our casezt−d=yt−d.^30 The integerdis
called the delay lag and is typically unknown, so it must be estimated. As we will
shortly explain, the least squares principle allowsdto be estimated along with the
other parameters. Parametercis the “threshold” that distinguishes two regimes:
(i) transition variablezt−dis belowc(lower regime); (ii) transition variablezt−d

is abovec(upper regime). Then, parameter vectorsα=(a 0 ,a 1 )

′

andb=(b 0 ,b 1 )

′

determine the response of the real exchange rate to changes in its last period’s
value.
If the threshold value,c, was known, then to test for threshold behavior all one
needs is to test the hypothesisH 0 : α=b. Unfortunately, the threshold value is
typically unknown and, under the null hypothesis, parametercis not identified.
The second difficult statistical issue associated with TAR models is the sampling dis-
tribution of the threshold estimate. Hansen (1997) provides a bootstrap procedure
to testH 0 , develops an approximation to the sampling distribution of the thresh-
old estimator free of nuisance parameters, and also develops a statistical technique
that allows confidence interval construction forc.^31 In particular, we can write the
TAR model (22.18) compactly as:

yt=xt(c)

′

θ+ut, (22.19)

wherext(c)=(x

′

t^1 {zt−d≤c},x

′

t^1 {zt−d>c})

′

withxt=(1,yt− 1 )

′

, 1 {·}the indicator

function andθ=(α

′

,b

′

)

′

. For a given value ofcthe least squares (LS) estimate
ofθis:

θ(ˆc)=

(∑

xt(c)xt(c)

′)− (^1) (∑
xt(c)yt
)
,
with LS residualŝu(c)tand LS residual varianceσT^2 (c)=( 1 /T)
∑T
t= 1
uˆ^2 (c)t. Then the
LS estimate ofcis the value:
cˆ=arg min
c∈C
σT^2 (c), (22.20)
whereCis an interval (usually trimmed) that covers the sample range of the tran-
sition variable. Problem (22.20) can be solved by a direct search overC. The LS