Efthymios G. Pavlidis, Ivan Paya and David A. Peel 1069
exercise he shows that standard information criteria (AIC, SC, HQ) have an important
role to play in model selection for nonlinear threshold models. Others (GIC, ICOMP) are
less reliable.
- This particular type of TAR model is called a self-exciting threshold autoregressive (SETAR)
model.
- Peel and Taylor (2002) adopted a similar strategy in a threshold model under the null
of RW. See Duarteet al.(2005, Appendix B) for the case where the errors display
autocorrelation.
- Estimate the model using the actual data for a set of values ofcin the rangeCand in
each case calculate the likelihood ratio (LR) statisticLR(c)for that value ofcagainst the
value of the likelihood obtained by unrestricted LS, i.e.,LR(c)=T(σˆT^2 (c)−ˆσT^2 (cˆ))/σˆT^2 (cˆ).
Notice that forc=cˆwe getLR(c)=0. Then plotLR(c)againstcand draw a flat line that
corresponds to theβ-level critical valuec∗(β)given in Hansen (1997, table 1, p. 5). For
β=5%,c∗(β)=7.35. The confidence intervalLRcis given byLRc={c:LR(c)≤c∗(β)}.
- This is generalized for the case where the attractor might be different than 0, saya 0 , and
also for the case of a linear trend,a 0 +a 1 t.
- The threshold valueλshould be between the 15th (λ 1 )and 85th percentile (λ 2 )ofyt.
λ∈%=[λ 1 ,λ 2 ].
- Further support for the TAR behavior of real exchange rates is found in De Jonget al.
(2007). They consider an extension of the simple TAR model above: an asymmetric TAR
model where the adjustment in the upper part of the “action band” might differ from the
one in the lower part of the “action band.” The asymmetric TAR can be written as:
yt=
⎧
⎪⎨
⎪⎩
ut ifc 1 ≤yt− 1 ≤c 2 ,
ρ 1 (yt− 1 −c 1 )+ut ifyt− 1 <c 1 ,
ρ 2 (yt− 1 −c 2 )+ut ifyt− 1 >c 2 ,
(22.26)
whereρ 1 ,ρ 2 ∈(−2, 0). De Jonget al.(2007) propose a three-regime threshold unit root
(TUR) test that is robust to errors that are not i.i.d. They propose an asymptotically pivotal
test statistic that optimizes over the parameter that is unidentified under the null and
allows for weakly dependent errors. De Jonget al.(2007) apply their statistic to monthly
real exchange rates of six developed countries against the dollar. While ADF and PP could
not reject the null of a unit root, their test statistic found evidence against unit root real
exchange rates.
- See Taylor (1995) and Sarno and Taylor (2002) for numerous references and further
discussion.
- Canjelset al.(2004) employ threshold autoregression to analyze the efficiency of inter-
national arbitrage under the gold standard from 1879 to 1913. A theoretical model is
developed and estimated employing continuous daily data.
- For example, Monoyios and Sarno (2002) employ an ESTAR model to describe the mis-
pricing between spot and futures. Employing daily data, they report a lag structure of five
days. The impulse response functions reported by them imply an arbitrage process taking
weeks for small shocks and days for large shocks. However, other researchers using minute
by minute data for the same market, and ESTAR or TAR adjustment mechanisms, report
adjustment mechanisms taking several minutes (Dwyeret al., 1996; Tayloret al., 2000).
- This estimator retains the properties of the FM-OLS estimator in that it can deal with
non-stationary data and serial dependence in the equation errors. However, the FM-LAD
estimator is based on a “fully modified” extension of the LAD regression estimator, which
is a robust procedure well known in the treatment of non-normality of error terms when
the regressors are stationary (Bassett and Koenker, 1978; Phillips, 1991). The FM-LAD
estimator developed by Phillips has all the features of the LAD estimator but is applicable
in models where the regressors are non-stationary, there is endogeneity in the regressors
and serial dependence in the errors. In addition, the FM-LAD estimator is valid even when
the data do not have finite variances.
