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Lecture 13 – Threshold Autoregressions: I
References –
 Enders, Applied Economic Time Series, Chapter
7
 B. Hansen, Inference in TAR models,
Studies in Nonlinear Dynamics and
Econometrics, 1997.
 B. Hansen, Testing for Linearity, Journal of
Economics Surveys, 1999.
 Threshold Autoregressions
(A class of nonlinear time series models)
Example –
yt = β0(1) + β1(1)yt-1 + εt if yt-1 < γ
= β0(2) + β1(2)yt-1 + εt if yt-1 > γ
or, equivalently,
yt = β0,1+β0,211t + β1yt-1 + β21tyt-1 + εt
where
 β1(1) and β1(2) are less than one in a.v.
 εt is a white noise process
 γεR
 1t = 0 if yt-1 < γ ; 1t = 1 if yt-1 > γ
 β0,1 = β0(1), β0,2=β0(2)-β0(1), β1 = β1(1),
β2 = β1(2) – β1(1)
 [or,
1t = 1 if yt-1 < γ ; 1t = 0 if yt-1 > γ
β0,1 = β0(2), β0,2 = β0(1)-β0(2)
β1 = β1(2), β2 = β1(1) – β1(2)]
γ is called the threshold parameter.
Note that var(εt) can also be allowed to be
regime-dependent, although we will ignore that
aspect of TAR models.
Applications?
 yt = real GDP growth rate; real GDP
dynamics may be different when recent
growth rates have been low vs. when
they have been large.There is a literature
that argues that economic expansions
are smoother and last longer than
economic contractions. This kind of
asymmetry can be captured through a
TAR representation of real GDP growth
rates. (Potter, Journal of Applied
Econometrics, 1995)
 yt = inflation rate; Fed policy and,
therefore, inflation rate dynamics may
be different when recent inflation rates
have been low vs. when they have been
large.
The example above is a special case of a SETAR
model:Self-Exciting Threshold Autoregression”
“Self-Exciting”? The threshold variable is a
lagged value of y itself.
An alternative to the “self-exciting” TAR?
yt = β0,1 + β0,21t + β1yt-1 + β21tyt-1 + εt
1t = 0 if xt-s < γ
= 1 if xt-s > γ
for some time series xt and nonnegative
integer, s. (E.g., inflation dynamics depend on
whether current or past unemployment is/was
low or high.)
The example is also a “two-regime” SETAR,
since the value of the autoregressive
parameters depend on whether at time t the
system is in regime 1 (yt-1 < γ) or regime 2
(yt-1 > γ). More generally, we can imagine an
r-regime SETAR such that
yt = β0(1) + β1(1)yt-1 + εt if yt-1 < γ1
= β0(2) + β1(2)yt-1 + εt if γ1 < yt-1 < γ2
…
= β0® + β1(r)yt-1 + εt if γr-1 < yt-1 < γr
where -∞ < γ1 < … < γr = ∞
In practice, two-regime and three-regime
threshold models are the most commonly
encountered SETAR models. (Why three
regimes?)
Note: linear autoregressions are one-regime
SETAR models.
The SETAR model can also be generalized to
the p-th order autoregressive case –
yt = β0(1) + β1(1)yt-1+…+ βp(1) + εt if yt-d < γ1
= β0(2) + β1(2)yt-1 +…+ βp(2) +εt if γ1<yt-d < γ2
…
= β0(r) + β1(r)yt-1 +…+ βp(r) + εt if γr-1<yt-d < γr
where -∞ < γ1 < … < γr and d ε{1,…,p}. d is
called the “delay parameter”. (There is no
compelling theoretical reason to constrain d to
be less than or equal to p, but in every
application I have seen or worked on, that
constraint is imposed.)
So, the form of the SETAR is determined by
three parameters –
1. the lag length, p
2. the number of regimes, r
3. the delay parameter d
So, we sometimes write that a process has a
SETAR(p,r,d) form. [Our initial example –
SETAR(1,2,1).]