Lecture 15 – Threshold Autoregressions: III
 Inference on the threshold parameters, γ
 Testing for TAR effects and # of regimes
As noted in the previous lecture, standard
inference procedures regarding the intercept and
slope parameters in the TAR model are
asymptotically valid, so long as they are
estimated using a consistent estimator of the
threshold parameter vector γ. [Estimate γ as
above; for that γ, estimate β by OLS; apply OLS
inference methods to construct CIs for and test
hypotheses concerning the components of β.]
Suppose that we believe that yt follows a
SETAR(p,2,d) process, where p and d can be
known or unknown. Following the procedures
discussed earlier, we estimate β, γ (and, if they
are unknown, p and d).
 How do we test H0: γ = γ0? (e.g., γ0 = 0)
 How do we construct interval estimates for
γ?
 How do we test the null of a one-regime
TAR (i.e., a linear AR) against the
alternative of a two-regime TAR?
 How do we test the null of a two-regime
TAR against the alternative of a threeregime TAR?
Testing H0: γ = γ0
Hansen (1997) showed that under the null
hypothesis, the sequence of LR-like
statistics
LRT ( 0 )  T (
SSRT ( 0 )  SSR(ˆ )
)
SSR(ˆ )
converges in
distribution to a r.v., ξ. The distribution of ξ
can be derived by simulation methods
(HOW?). Simulated percentiles of that
distribution are provided in Hansen’s paper
(Table 1, row 2).
So, for example, consider testing H0: γ = 0
for the two-regime TAR model of real GDP
growth rates. First, we estimate the model
treating γ as unknown to get USSR. Second,
we estimate the model treating γ as known
and equal to 0 to get RSSR. Third, compute
LR and compare its value to the percentiles
in Hansen’s Table 1. E.g., the 95th percentile
of the distribution of ξ is 7.35. So reject H0:
γ=0 at the 5-percent level if LR > 7.35.
Constructing CIs for γ
Asymptotically valid confidence intervals
for γ can be constructed by “inverting the
LR statistic.”
That is, to construct, e.g., a 95-percent CI
for γ, find the set
Г0.95 = {γ: LRT(γ) < ξ0.95}
= {γ: LRT(γ) < 7.35}
where ξ0.95 means the 95-th percentile of the
distribution of ξ.
1. Go through the set of candidate γ’s:
y[τT], y[τT]+1,…,y[(1-τ)T]. For each of these
candidate γ’s compute LR(γ) to
determine if that γ is or is not in Г0.95
2. If, say, y(j) ε Г0.95, then place γε[y(j),y(j+1))
in Г0.95.
One unappealing aspect of this construction is
that the resulting CIs can be the union of a
collection of disconnect intervals. E.g.,
Г0.95 = [0.20,0.60] U [0.80, 0.90]
Hansen suggests that define use the more
conservative procedure of using the
“convexified” interval
0c.95  [ˆ1 , ˆ2 ]
where
ˆ1  min  0.95
and
ˆ2  max  0.95
So, if Г0.95 = [0.20,0.60] U [0.80, 0.90], then
0c.95  [0.20,0.90] .
(Why is this called a “conservative procedure?)
Notes
1. In a paper that Walt Ender, Pierre Siklos, and
I are revising for publication in Studies in
Nonlinear Dynamics and Econometrics we use
Monte Carlo methods to evaluate the finite
sample coverage properties of several methods
of constructing CIs for the threshold parameter.
- Use normal approximation
{(γ-hat + 2*S.E.) for 95% CI}
- Use Hansen’s approach
- Use bootstrap methods (t and percentile
methods)
Our conclusion – none of these procedures have
good coverage properties in samples of size 100.
As part of our revision, we are considering
sample size 250.
2. The asymptotic distribution of β-hat is the
same (and normal) whether γ is known or
replaced by a consistent estimator. Hansen
points out (and my work with Walt and Pierre
confirms) that when γ is unknown, this approach
does not tend to work well in finite samples
(even samples as large as 500-1000!). In
particular, the CIs generated using the normal
approximation tend to be much too small (too
“liberal”). E.g., actual coverage probabilities for
nominal 95% intervals are often on the order of
50%-80%!
Hansen suggests an approach that takes the
union of CIs for β’s using a set of γ’s around γhat rather than simply the CI implied by γ = γhat. This provides more conservative intervals,
but the choice of an appropriate set of γ’s that
does not lead to overly conservative intervals is
problematic.
The message – Much work remains to be done
on developing reasonably reliable inference
methods for the parameters in TAR models.
Testing H0: r = 1 (vs. HA: r =2)
The idea: Fit the model with r = 1 and r = 2 and
see how much the SSR increases under the
restriction that r = 1, i.e., construct the test
statistic
F12 = T*(SSR1-SSR2)/SSR2
If γ is known, F12 is asymptotically χ2(k), where
k is the number of parameters that are allowed to
vary across regimes (usually, p or p+1).
The problem:
When γ is unknown, the nuisance parameter, γ,
is not identified under the null. (Where have we
seen this problem before?) So, standard
distribution theory will not apply to this (or
other natural) statistic(s). [Applying the χ2(k)
distribution in this case will lead to rejecting the
null too often. I.e., the usual chi-square test’s
actual size will be much larger than its nominal
size.]
Hansen’s solution (1996, Econometrica; see
also Hansen 1999, Journal of Economic Surveys
for a simpler exposition):
Hansen showed that under the null F12 does
converge in distribution to a random
variable, say G, under appropriate side
condition on {yt}. However, the distribution
G depends on the moments of the data, so
that asymptotic critical values and p-values
would have to be tabulated for each and
every application. A “bootstrap-like”
algorithm to compute these values is
provided in Hansen (1996, 1997).
An alternative, that requires about the same
amount of effort and may work better is to
simply use a bootstrap procedure.
“Although there is no Monte Carlo study
comparing the bootstrap and asymptotic
approximations in the context of testing for
SETAR models, Diebold and Chen (1996)
present an analogous study of the Andrews
structural change test in AR(1) model. They
find that the bootstrap yields and excellent
approximation, a certain improvement over
the asymptotic distribution.”
 How would the bootstrap distribution of F12
be constructed in this setting?
 Testing H0: r = 2 (vs. HA: r = 3) is a similar
problem. The natural test statistic is F23,
F23 = T*(SSR2-SSR3)/SSR3
The asymptotic distribution of F23 is
dependent on the data in a complicated way.
Bootstrapping F23’s may be the better
approach.
 Both interval estimation and hypothesis
testing in the TAR framework remain open
and active areas of research in econometrics.