数理解析研究所講究録 1215 巻 2001 年 14-29

14

On Ergodicity of Some TAR(2) Processes*

Naoto Kunitomo
Faculty of Economics, University of Tokyo
December 2000

Summary

We give aset of sufficient conditions for the geometrical ergodicity and the non-explosivenessof the solutions in the second-0rder threshold autoregressive (TAR) processes. We also
discuss some conditions for the geometrical ergodicity of the second-0rder simultaneous
switching autoregressive (SSAR) processes. Unlike the linear autoregressive processes andthe first rder TAR processes, the ergodic regions and non-explosive regions become quite
complicated even in some special TAR processes.

1. Introduction
In the statistical time series analysis several nonlinear time series models have been
proposed in the past decade. In particular, considerable attention has been paid to theclass of threshold autoregressive (TAR) processes, which was systematically investigatedby Tong (1990) and some applications have been reported. The statistical properties of thefirst order TAR processes have been investigated first by Petruccelli and Woolford (1984),
and later by Chen and Tsay (1991) in more details. Unlike the linear autoregressivemodels, however, the statistical properties of the second order TAR processes have notbeen fully investigated mainly due to some technical problems involved. It seems that thenecessary and sufficient conditions for the ergodicity have been known only for the firstorder TAR processes. The main purpose of my work is to investigate the basic propertiesof the second-0rder TAR processes and the second-0rder SSAR (simultaneous switching
autoregressive) processes.
Let
(1.1)

$\{y_{t}\}$be asequence of scalar time series satisfying$a_{1}y_{\mathrm{t}-1}+a_{2}y_{t-2}+\sigma_{1}v_{t}$if$y_{t-d}\geq 0$$b_{1}y_{t-1}+\mathrm{b}y_{t-2}+\sigma_{2}v_{t}$if$y_{t-d}<0$$y_{t}=\{$’My report was based on arevised version of Discussion Paper CIRJE-F-55, Faculty of Economics,
University of Tokyo. Ithank Dr. Seisho Sato of the Institute of Statistical Mathematics for the help of
preparing some figures based on the simulations reported in my work

15
where is apositive (finite) integer parameter, and $\alpha.,b.\cdot,\sigma.\cdot(>0)(i=1,2)$ are unknown parameters, andare asequence of independently and identicaly distributed(i.i.d.) random variables having an absolutely continuous density $f(v)$ with respect to
the Lebesgue measure and $E[v_{t}]=0$ . We assume that $f(v)$ is continuous and everywhere
positive inthe second order threshold autoregressive model given by (1.1) will bedenoted as TAR(2:d) and the integer-valued parameter is cffid the delayed parameter.
(We also use the notation as TAR(2) for the second order TAR processes. )Petrucelli
and Woolford (1984) have considered the first-0rder TAR process when $a_{2}=4$ $=0$ and$d=1$ , which is denoted as TAR(I:I). They have shown that the necessary and sufficientconditions for the geometrical ergodicity are given by$d$$\{v_{t}\}$$\mathrm{R}$

$d$

,

(1.2)

,

$a_{1}<1$ $b_{1}<1$ $a_{1}b_{1}<1$.

Chen and Tsay (1991) have extended their results to the TAR(l:d) processes whentakes an arbitrary positive integer. The conditions they have obtained for the geometricalergodicity include (1.2) as aspecial case when $d=1$ .

$d$On the other hand, Kunitomo and Sato $(1996, 2000)$ , and Sato and Kunitomo (1996)
have proposed the class of simultaneous switching autoregressive (SSAR) processes, which
can be regarded as anatural extension of the TAR processes for some econometric applications. The second-0rder SSAR model can be written as(1.3)$a_{1}y_{t-1}+a_{2}y_{t-2}+\sigma_{1}v_{t}$if$b_{1}y_{t-1}+hVt-2$

if$y_{t}\geq y_{t-1}$$y_{t}=\{$$+\sigma_{2}\mathrm{v}_{t}$$ye$$<yt-i$where $a:$ , $b\dot{.},\sigma.\cdot(>0)(i=1,2)$ are unknown par ameters, andare asequence of independently and identically distributed (i.i.d.) random variables having an absolutely
continuous density $f(v)$ with respect to the Lebesgue measure and $E[v_{t}]=0$ . We also
sume that $f(v)$ is continuous and everywhere positive in R. The discrete time series model
given by (1.3) will be denoted by SSAR(2). By imposing the restrictions on parameters
given by$\{v_{t}\}$$\mathrm{a}*$(1.4)$\frac{1-a_{1}}{\sigma_{1}}=\frac{1-b_{1}}{\sigma_{2}}=r_{1}$,$\frac{a_{2}}{\sigma_{1}}=\frac{b_{2}}{\sigma_{2}}=r_{2}$,this time series model can be written as(1.5)$a_{1}y_{t-1}+\mathrm{Q}\mathrm{a}\mathrm{V}\mathrm{t}-2+\sigma_{1}v_{t}$if$v_{t}\cdot\geq r_{1}y_{t-1}-r_{2}y_{t-2}$$b_{1}y_{t-1}+\mathrm{h}\mathrm{V}\mathrm{t}-2+\sigma_{2}v_{\ell}$if$v_{t}<r_{1}y_{t-1}-r_{2}y_{t-2}$$y_{t}=\{$

where $r_{i}(i=1,2)$ are unknown parameters. We should note that the class of simultaneous
switching autoregressive (SSAR) process is different from the TAR models given by (1.1).
The two phases at are determined by both the past time series and the present innovation
at , we do not have any delayed parameter in the SSAR processes. Then the SSAR(2)process has the Markovian representation with the state vector
by using
the relation
$t$

$t$

$y_{\acute{t}}=(y_{t}, y_{t-1})$(1.6)$y_{t}=y_{t-1}+[\sigma_{1}I(v_{t}\geq r_{1}y_{t-1}-r_{2}y_{t-2})$$+\sigma_{2}I(v_{t}<r_{1}y_{t-1}-r_{2}y_{t-2})][v_{t}-(r_{1}y_{t-1}-\mathrm{r}2\mathrm{y}\mathrm{t}-2)$,

16, then this SSAR process becomesis the indicator function. When
where
the standard $AR(2)$ model by a -parametrization. Kunitomo and Sato (1996) haveshown that even the first order SSAR process (denotd as SSAR(I)) When $a_{2}=b_{2}=0$
gives us some explanation and description on an important aspect of the asymmetricalmovement of time series in two different (up ward and down-ward) phases. The ergodicity
condition for the SSAR(I) process is the same as (1.2) with the coherency conditions
implied by (1.4).
$\sigma_{1}=\sigma_{2}=\sigma$$I(\cdot)$$\mathrm{r}\triangleright$

2. Some Preliminaries
The first important property of any statistical time series model is whether it is ergodic
or not. For the Markovian time series models with continuous state spaces and discrete
time intervals, the geometrical ergodicity and the related concepts have been developedin the nonlnear time series analysis. For the sake of completeness, we mention to its
definition and the drift criterion. For the more precise definitions of related concepts
including irreducibility, aperiodicity, and ergodicity of the Markov chains with the generalstate spaces, see Nummelin (1984) or the Appendix of Tong (1990).
Markovian process
be anDefinition 1(Geometrical Ergodicity) :Let.
with the state space ofif there exists aprobability measure on $(R^{m},B(R^{m}))$ ,ergodicis(i)anon-negative constant $\rho<1$ and -integrable non-negative measurable function
such that,(2.1)
vector, and $P(x, \cdot)$ isdenotes the total variation norm, $x=(x:)$ is anwhere
the transition probability.
, $A\in B(R^{m})$ with $\psi(A)>0(\psi(\cdot)$ is a -finite
ucible if for any(ii)ismeasure), and$m\mathrm{x}1$$\{\mathrm{y}_{\mathrm{t}}\}$$R^{m}$$\{\mathrm{y}_{\mathrm{t}}\}$$\pi$$\mathrm{g}\infty \mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{y}$$h(\cdot)$$\pi$$||P^{n}(x, \cdot)-\pi(\cdot)||_{\tau}\leq\rho^{\mathfrak{n}}h(x)$$m\mathrm{x}1$$||\cdot||_{\tau}$$\{\mathrm{y}_{\mathrm{t}}\}$$x$ $\in R^{m}$$\psi- \mathrm{i}\mathrm{r}\mathrm{r}^{\wedge}\mathrm{d}$(2.2)$\sigma$$. \sum_{*=1}^{\infty}P^{n}(x,A)>0$.

For the geometrical ergodicity of the Markov Chains with the continuous general(1990) has given adrift criterion, which will be useful for our purpose
states,and thus reported as Lemma 1. It is an extension of the well- nown drift criterion on theMarkov chain with the general states due to Tweedie (1975).
$\mathrm{T}\mathrm{j}\phi \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{m}$is an aperiodic -irreducible Markov:Assume that
Lemma 1and is a non-negative continuous (measurable) funcChain with the state space ofis geometrically ergodic if there exist a positive integer , a srrunll set $C$
tion. Then
, $M<+\infty$ , and $r>1$ such that$\psi(C)>0$ , positive constants
satisfying (2.2)$(T\ovalbox{\tt\small REJECT}.

$\{y_{t}\}$s\theta\iota eim (\mathit{1}\mathit{9}\mathit{9}\mathit{0}))$$R^{m}$$\psi$$g$$h$$\{y_{t}\}$$\epsilon>0$$uri\theta\iota$(2.3)$rE[g(y_{t+h})|y_{t}=y]\leq g(y)-\epsilon$

for any positive constant(2.4)$\epsilon$

if $y\in C$ ,and
$E[g(y_{t+h})|y_{t}=y]\leq M$

where C’ is the complement set of $C$ .

if $y\in C$ ,’

17
We note that when the density function of disturbances $f(v)$ is continuous and everywhere positive in we can take acompact set as the small set in Lemma 1. See Theorem
A.1.7 of Tong (1990) on the related problems.
Now we introduce another concept on the stability of the solution, which is slightly
different ffom the geometrical ergodicity. Because its conditions are slightly weaker thanthose in Lemma 1and they are necessary for the geometrical egodicity, we use the ter. We shall use this concept in this paper due tominology of Near Geometricalthe technical reason indicated in later sections. However the detail of its mathematicalproperties has not been fully explored and is still under investigation.
. Then we partition the state
Let $Q$ be the state space of the stochastic process$Q^{:}(i=1,\cdots, k)$ such that $Q=$
space $Q$ into afinite number of disjoint subspaces$(\psi(Q^{:})>0, Q^{:}\cap Q^{j}=\phi ;i\neq j)$ md for any$\mathrm{R}$$E\eta d\dot{\iota}city$$\{y_{t}\}$$t$$\bigcup_{i=1}^{k}Q^{:}$

(2.5)

where

$1=. \cdot\sum_{=1}^{k}I(y_{t}\in Q^{:})$$I(\cdot)$,is the indicator function and k is apositive integer.

We say that the solution:
Definition 2 Near Geometricalis near geometrically ergodic if (i) for any $i,i’(\acute{\iota},\iota=1, \cdots, k)$ and (sufficiently large)
positive integers $h_{j}(j=j(i,i’)\geq 1)$ , there exist asequence of positive constants and$r_{j}(r_{j}>1)$ such that$\{y_{t}\}$$($$\mathrm{E}\mathrm{r}\mathrm{g}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{y})_{/}$$\epsilon_{\mathrm{j}}$(2.6)$r_{j}E[g(y_{t+h_{\mathrm{j}}})I(y_{t+h_{\mathrm{j}}}\in Q^{i’})|y_{t}=y]\leq g(y)-\epsilon_{\mathrm{j}}$if$y\in Q^{:}\cap C^{\mathrm{c}}$,where $C$ is asmall set satisfying $\psi(C)>0$ in (2.2),and (ii) there exists apositive constant $M$ such that
$E[g(y_{t+h_{\mathrm{j}}})I(y_{t+h_{\mathrm{j}}}\in Q^{\cdot}.’)|y_{t}=y]\leq Mif$ $y\in Q^{:}\cap C$(2.7)

for any , $i^{l}(i,i’=1, \cdots, k)$ and positive integers $h_{j}(hj\geq 1)$ .
$i$

We notice that there is adecomposition(2.8)$I(y_{t+h} \in Q^{i’})I(y_{\ell}\in Q^{:})=\sum I(y_{t+h}k\in Q^{i^{l}})I(y_{t+h_{1}}\in Q^{l})I(y_{t}\in Q^{:})$$l=1$, . Then it is apparent that (2.6) is reducd to (2.3)
for any $h>h_{1}\geq 1$ and $i,i’=1,2$ ,can be deleted in (2.6) with the common $r=r_{j}$ and $h=h_{\mathrm{j}}(j\geq 1)$ with
if the termafinite $j’s$ . Hence the near geometrical ergodicity we introduce in this section coincides
with the geometrical ergodicity in the nonlinear time series analysis if we can take theand the common constant $r=r_{j}(j\geq 1)$ . It is acommon positive integer
kind of the stability property or non-explosiveness of the sample paths of the imderlying
stochastic process, whose behavior is near to the geometrical ergodicity.
$\cdots$$k$$I(\cdot)$$h=h_{\mathrm{j}}$3. Ergodic conditions for TAR(2) in the leading caseIn this section we consider the conditions for the geometrical ergodicity of the TAR(2)processes in aspecial case. For this purpose we shall utilize the Markovian representatio$\mathrm{n}$18’be a2 x1 vector of the time series generatedof the TAR(2) processes. Let
by. The second order TAR process given by (1.1) can be represented by$1^{\mathrm{F}_{*}\ovalbox{\tt\small REJECT}}(\ovalbox{\tt\small REJECT}_{\mathrm{t}_{\mathrm{t}}}y_{t-1})$$\{\mathrm{y}_{\mathrm{t}}\}$(3.1)$Ay_{t-1}+D\sigma_{1}v_{\mathrm{t}}$if$e_{k}’y_{t-1}\geq 0$$By_{t-1}+D\sigma_{2}v_{t}$if$d_{k}y_{t-1}<0$$y_{t}=\{$

where $e_{1}=(1,0)’,e_{2}=(0,1)’$ (forwith k $=1,$ 2), and D $=(1,0)’$ are 2 x1 constantvectors, and A,B are 2 x2 coefficient matrices given by$e_{k}$

A(3.2)$=(\begin{array}{ll}a_{1} a_{2}1 0\end{array})$, B$=(\begin{array}{ll}b_{1} \mathrm{h}\mathrm{l} 0\end{array})$In the rest of this section we shall consider the leading case for the TAR(2) processes
when $4=0$ . This is simply because we can obtain general characterizations on thegeometrical ergodic regions in the leading case and the necessary and sufficient conditions
for the geometrical ergodicity can be mostly obtainable. we have to stress that even inthis leading case our conditions sometimes become quite complicated and non-standard
in comparison with the results known for the TAR(I) processes. In order to obtain theconditions for the geometrical ergodicity and state our results, we partition the parameter
space ofinto four different regions given by$C_{1}=\{a_{1}\geq 0,02\geq 0\}$ , $C_{2}=\{a_{1}<0,a_{2}\geq 0\}$ , $C_{3}=\{a_{1}<0, a_{2}<0\}$ , and$C_{4}=\{a_{1}\geq 0,a<0\}$ , respectively.
Because we set $4=0$ , we need to consider two cases whenand $b_{1}>0$ . Then
it is intuitively obvious that in the latter case we have to restrict the conditions for thegeometrical ergodicity when $0<b_{1}<1$ in order to avoid the possible explosion of thesample paths of the solutions. Because we have the density function $f(v)$ over $R$, thiscan be proven in the rigorous way.
$(a_{1},a_{2})$$b_{1}\leq 0$

3.1 TAR(2:1) Processes
First, we consider the TAR(2:1) process when $b_{1}>0$ and $4=0$ . This is the simplest
case in the TAR(2) process in terms of our conditions on the coefficients. Although thecharacteristic roots of $B$ are $b_{1}(0<b_{1}<1)$ and 0, we have some complications due tothe behaviors of two characteristic roots of $A$ . Let(3.3)$D(A)=a_{1}^{2}+4a_{2}$,which is the discriminant of the characteristic equation for the first phase in (3.1) givenby$g_{A}(\lambda)=\lambda^{2}-a_{1}\lambda-a_{2}=0$(3.4).
Then we can present the necessary and sufficient conditions for the geometrical ergodicity
in the present case. Some of proofs and derivations of the propositions in this section are
given in Section 5. We consider aset of conditions :$C_{1}$

Condition I:: $a_{1}+a_{2}<1,0<b_{1}<1$ ,19$C_{2}$$C_{3}$$C_{4}$$0<4$ $<1$ ,$0<4$ $<1$ ,[either$a_{1}+a_{\mathit{2}}<1$(0$\ovalbox{\tt\small REJECT}$$a_{1}<2,D(\mathrm{A})\ovalbox{\tt\small REJECT}$0) or $D(\mathrm{A})<0$ ], $0<6_{1}<1$ .

Proposition 1:For the TAR(2:1) process with $b_{1}>0$ and $4=0$ , the necessary andare given by Condition $I$ .
sufficient conditions for the geometrical ergodicity of$\{y_{t}\}$Second, we consider the TAR(2:1) process whenand $4=0$ . The conditions inthis case become far more complicated than in the first case. This is mainly because thestochastic process
can be ergodic when $b_{1}<0$ and its absolute value is greater than
1. In order to deal with some complications involved, we define $\rho(A^{k}B)$ be the non-zerocharacteristic root of the matrix $A^{k}B$ for any positive integer . In the present case$b_{1}\leq 0$$\{y_{t}\}$$k$$\rho(A^{k}B)=e_{1}^{l}A^{k}b$,where $b’=(b_{1},1)$ .
Then we give aset of conditions for the geometrical ergodicity and the near geometricalergodicity of the sample paths of the solutions. The concept of the near geometricalergodicity of the solutions has been given in Definition 2in Section 2. We note that it is
not needed for the linear time series processes and TAR(I) processes. We consider asetof conditions :Condition $II$ :: $a_{1}+a_{2}<1$ ,: $\rho(AB)<1$ ,: $\rho(A^{\mathrm{j}}B)<1(j=1,2)$ ,: either $[a_{1}+a_{2}<1(0\leq a_{1}<2,D(A)\geq 0)]$ or [$j(j\geq 1)$ such that $0\leq\rho(A^{j}B)<1]$ .
$C_{1}$$C_{2}$$C_{3}$$C_{4}$$\mathrm{D}(\mathrm{A})<0$

and there exists some

Proposition 2:For the TAR(2:1) process withand $b_{2}=0$ , (i) the necessary andare given by Condition $II$
sufficient conditions for the geometrical ergodicity of$with$, , and , and (ii) the sufficient conditions for the near geometri$cal$ ergodicity
are given by Condition $II$ with .
of the solution$b_{1}\leq 0$$\{y_{t}\}$$C_{1}$$C_{2}$$C_{3}$$\{y_{t}\}$$C_{4}$As an illustration, we present two figures of the ergodic regions and non- xplosive
regions for the TAR(2:1) processes, which are based on the simulations of the stochastic
processes. Contrary to the ergodic regions for the linear
models which havebeen known in the statistical time series analysis (see Brockwell and Davis (1991) for
instance), they are often unbounded as we see in these figures. Some of the conditions
above can be written more explicitly by using$\{y_{t}\}$$\mathrm{A}\mathrm{R}(2)$$\rho(AB)=a_{1}b_{1}+a_{2}$, $\rho(A^{2}B)=a_{1}(a_{1}b_{1}+a_{2})+a_{2}b_{1}$ .

The most complicated region in the TAR(2:1) process isin Condition II and wehave found some strange shapes as the non-explosive regions for the sample paths ofthe solutions depending upon the parameter values of Aand $B$ . Although we canno$C_{4}$

20
give the complete characterizations of the ergodic regions on the present case, we may
conjecture that the conditions are necessary and sufficient for the geometrical ergodicity
in all cases. As an immediate corolary of the above two propositions, we can obtain the
result originaly derived by Petruccelli and Woolford (1984) for the TAR(I:I) process.
Corollary 1 : For the TAR(l:l) mdel, the necessary and sufficient conditions for theare given bygeometrical ergodicity of$\{y_{t}\}$(3.5)$a_{1}<1$, h $<1$ , $a_{1}b_{1}<1$ .

3.2 TAR(2:2) Processes
and $4=0$ . Contrary to the
Next, we consider the TAR(2:2) process whenTAR(2:1) process, this is simpler than the case when $4>0$ in the TAR(2:2) process. In
the present case we can give the necessary and sufficient conditions :$b_{1}\leq 0$$C_{1}$$C_{2}$$C_{3}$$C_{4}$

Condition $III$ :: $a_{1}+a_{2}<1$ , $\rho(AB^{2})<1$ ,: $\rho(AB)<1$ ,, $\rho(AB^{2})<1$ ,:$[a_{1}+a_{2}<1(0\leq
a_{1}<2,D(A)\geq 0)]$ or $[D(A)<0,\rho(AB^{2})<1]$ .
: either$\mathrm{p}(\mathrm{A}\mathrm{B})<1$

Proposition 3:For the TAR(2: 2) process with $4\leq 0$ and $4=0$ , the necessary andare given by Condition $III$ .
for the geometrical ergodicity ofsufficient$\omega n\ \cdot t\cdot.ons$$\{y_{t}\}$Second, we consider the TAR(2:2) process when $b_{1}>0$ and $4=0$ . It is far more
complicated than in the previous case as for the TAR(2:1) process when $b_{1}\leq 0$ and$4=0$ . In this case we give aset of conditions for the geometrical ergodicity and theof the solution.
$\mathrm{n}\mathrm{o}\mathrm{n}\ll \mathrm{x}\mathrm{p}\mathrm{l}\mathrm{o}\mathrm{e}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}\infty$$C_{1}$$C_{2}$$C_{3}$$C_{4}$

Condition $IV$ :: $a_{1}+a_{2}<1,0<b_{1}<1$ ,: $a_{1}+a_{2}<1$ and there exists some $j(j\geq 2)$ such that $0\leq\rho(A^{\mathrm{j}}B)<1$ ,: $\rho(A^{2}B)<1,0<b_{1}<1$ ,: either $[a_{1}+a_{2}<1(0\leq a_{1}<2, D(A)\geq 0)]$ or $[D(A)<0]$ , $0<b_{1}<1$ .

$b_{1}>0$ and $4=0$ ,the necessary andProposition 4:For the TAR(2:2) process
are given by Condition $IV$
sufficient conditions for the geometrical ergodicity of, and , and (ii) the sufficient conditions for the near geometrical ergodicity
are given by Condition $IV$ with .
of the solution$uri\theta\iota$$(\dot{l})$$\{y_{t}\}$$uri\theta\iota$$C_{1},C_{3}$$C_{4}$$\{y_{t}\}$$C_{2}$As an illustration, we present two figures of the ergodic and the non-explosive regions
for the TAR(2:2) processes, which are based on simulations. Some of the conditions above
can be written more explicitly by usin$\rho(AB^{2})=b_{1}(a_{1}b_{1}+a_{2})$.

21
We note that some of the conditions for the TAR(2:2) process in the leading case when$D(A)<0$ have been partially discussed by Tong (1990) without the disturbance terms.
From Page 70 of Tong (1990) we set $b_{1}=-0.9,a_{1}=1.8,a_{2}=-0.9$ as Example 1and$b_{1}=-1.1,a_{1}=0.6,a_{2}=-0.1$ as Example 2. By using Proposition 3, we can conclude
that Example 1is not geometricaly ergodic while Example 2is geometrically ergodic.
processes, some of them are unContrary to the ergodic regions for the linear
bounded as we see in these figures. The most complicated region in the TAR(2:2) process
in Condition IV although it may be difficult to judge this finding directly from theisdefigure. We have found some strange shapes as the noDhexplosiveness regions in$B$$A$involved
pending upon the parameter values of and . It seems that the complications
for the TAR(2:1)
in this case is different from the corresponding one in the region. However, we have not succeeded in the complete characterization ofprocess when
this situation except the present conditions we have obtained.
As an immediate corollary of the above two propositions, we have the result obtainedby Chen and Tsay (1991) for the firat-0rder threshold model TAR(l:d) when $d=2$ .
$\mathrm{A}\mathrm{R}(2)$$C_{2}$$C_{2}$$C_{4}$$b_{1}\leq 0$

Corollary 2:For the TAR(1:2) process, the necessary anti sufftcient conditions for theare given bygeometrical ergodidty of$\{y_{t}\}$(3.6)$a_{1}<1$, $b_{1}<1$ , $a_{1}h<1$ ,$a_{1}^{2}b_{1}<1$,$a_{1}b_{1}^{2}<1$.

We should mention that Chen and Tsay (1991) have given the necessary and sufficientconditions for the TAR(l:d) processes with any positive integer-valued parameter d.

4. Discussions
4.1 SSAR(2) Processes
By using (1.3)-(1.5), the SSAR(2) process have the Markovian representation, which
be a 2is simlar to the one given in (3.1) for the TAR(2) processes. Let
vector of time series. Then the SSAR(2) process in (1.3)-(1.5) can be represented by$y_{t}=(\mathrm{y}\mathrm{t},y_{t-1})’$(4.1)$Ay_{t-1}+D\sigma_{1}v_{t}$if$e’y_{t}\geq 0$$By_{t-1}+D\sigma_{2}v_{t}$if$e’y_{t}<0$$\mathrm{x}1$$y_{t}=\{$

where $e=(1, -1)’$ , $D=(1,0)’$ , and the coefficient matrices Aand $B$ are given by(3.2) with the coherency conditions (1.4). We first note that due to the structure of the
SSAR processes, it is not possible to investigate the leading case as in Section 3. The
determination of phases at time depends on afinite number of past time series and thepresent innovation as given by (1.6), the stability of the paths of the solution become
andquite complex and it depends on the complicated products of matrices$0\leqk,l$in general. As the sufficient conditions, we consider the restrictions :for$t$$A^{k}B^{l}$

Condition V :(4.2)$a_{1}+|a_{2}|<1$, $b_{1}+|b_{2}|<1$ , $(a_{1}-|a_{2}|)(b_{1}-|b_{2}|)<1$ , $\min\{|a_{2}|, |b_{2}|\}<1$ .

$B^{k}A^{l}$

22
By using the coherency conditions (1.4) on 04, (i 1,2) for the SSAR(2) process, theabove conditions can be rewritten in the parameter space of (i 1,2) and (i 1,2)$\ovalbox{\tt\small REJECT}$$b_{\ovalbox{\tt\small REJECT}}$$o_{i}$$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$7_{\mathrm{j}}^{\ovalbox{\tt\small REJECT}}$as

Condition(4.3)$V^{0}$:$r_{1}>|r_{2}|$,$r_{1}+|r_{2}|< \frac{1}{\sigma_{1}}+\frac{1}{\sigma_{2}}$,$|r_{2}|< \dot{\mathrm{m}}\mathrm{n}\{\frac{1}{\sigma_{1}}, \frac{1}{\sigma_{2}}\}$.

Presently it is only possible to show that the first three conditions are necessary for thegeometrical ergodicity by investigating possible many cases which would be occurred andwe omit their lengthy proofs. The last condition in (4.3) is quite strong one, which is notnecessary, but we expect that Condition are sufficient for the geometrical ergodicityofin the SSAR(2) process. They are slightly weaker than the sufficient conditions.

$\{y_{t}\}$(4.4)$\rho=\mathrm{m}\mathrm{i}$$\mathrm{V}$$\{|a_{1}|+|a_{2}|, |b_{1}|+|b_{2}|\}<1$.

This type of conditions for the TAR(p:l) processes [has been obtained by Chanand Tong (1985). However, they are not necessary even for the TAR(2:1) processes as wehave shown in Section 3. As an illustration, we present one figure of the non-explosive
regions for the SSAR(2) process, which is based on the simulations and drawn in thephase. The geometrically ergodic regions in this case seem to be different fromthe corresponding ones in the TAR(2) processes and the ergodic region is bounded inthe-space due to the coherency conditions given by (1.4). The conditions for thegeometricaly ergodicity of the SSAR(I) process is considerably simpler than the SSAR(2)process, which can be summarized as the next proposition.
$p$$\geq 1)$$(r_{1},r_{2})$$(r_{1},r_{2})$Proposition 5:For the SSAR(l) process, the necessary and sufficient conditions forthe geometrical ergodicity ofare given by$\{y_{t}\}$(4.3)$a_{1}<1$, $b_{1}<1$ , $a_{1}b_{1}<1$ .

By using the coherency conditions given by (1.4) for the SSAR(2) process, these conditions can be furtheras$\mathrm{r}\triangleright \mathrm{r}\mathrm{e}\mathrm{w}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{n}$(4.6)$0<r_{1}< \frac{1}{\sigma_{1}}+\frac{1}{\sigma_{2}}$.

Rom this representation it is clear that (4.2) and (4.3) are natural generalizations of (4.5)
and (4.6) in sense. We should note that the ergodic region in terms of thespace
is bounded for the SSAR(I) model.
$(r_{1},r_{2})$4.2 TAR(2) Processes
In principle it would be possible to develop the conditions for the geometrical ergodicity
and the near geometrical ergodicity for the general case of the TAR(2) processes. However,they become substantially more complicated than in the leading case (i.e. $b_{2}=0$ ) wehave discussed in Section 3. In particular, it is quite tedious to write down the explici$\mathrm{t}$23sufficient
expressions in terms of the coefficient matrices Aand $B$ as the necessary
conditions of the geometrical ergodicity and the near geometrical ergodicity. Thus for theillustrative purpose we have done some simulations. Among many simulations we only
present several figures having complex shapes in some regions. However, they have somesimilar aspects essentially to those in the leading cases when the TAR(2:1) process with$b_{1}\leq 0$ and the TAR(2:2) process with $b_{1}>0$ as we have discussed in Section 3.
For
illustrations we also give some figures suggesting the mathematical complexities involvedin our situations. Also we have confirmed that the analytical characterizations of those
figures are not easy tasks.
$\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$4.3 Concluding Remarks

Since the conditions for the geometrical ergodicity are the basic property of the MarkOvian stochastic processes, they have some implications for statistical inferences and themodelling procedure of the TAR(2) processes. In general, from our findings in this paper
we expect that the higher order TAR(p) processes with an arbitrary delayed parameter
have quite complicated ergodic conditions. Although it is possible to use the least squares
estimation method 1 for the consistent estimation of the unknown parameters in the TAR
processes, we need amore careful investigation on the properties of the estimation resultsin empirical studies. Because of our results reported in the previous sections, however, it
is far beyond the scope of this paper to have acomplete characterization of the stochastic
processes of the TAR(p) and the SSAR(p) processes for practical usages and there still
remains many statistical problems to be solved.
$d$5. The Method of Proofs
In this section, we discuss the our method of derivations and proofs of our results in
Section 3. Due to the lack of space, the interested readers should ask the author to send
the revised version of the full paper presented. The method of proofs is basically similar tothe one developed by Chen and Tsay (1991) for the TAR(I) processes. For the theoreticalresults on Markov chain with the general state space, see Nummelin (1984), or
be the -field(1990). Here we prepare some notations used in this section. Let$\{y_{s},
s\leq
t\}$
and we use the notation forgenerated by asequence of random variables. Also define asequence of time dependent
the conditional expectation,,asphases for the stochastic process$Q_{t}^{3}=\{y_{t}\leq 0,y_{t-1}\leq 0\}$ , and.
,decomposewecan
1into the indicator functions
theindicator
function
By usingwith four different phases as $1=I(Q_{t}^{1})+I(Q_{t}^{2})+I(Q_{\ell}^{3})+I(Q_{t}^{4})$ . Then we can further
as $I(Q_{t}^{1})=I(Q_{t}^{1}Q_{t-1}^{1})+I(Q_{t}^{1}Q_{t-1}^{4})$ , for instance. The most importantdecomposetechnical finding in our derivations and proofs of our resutls lies in the fact that we canignore many terms when we evaluate the growth condition (2.3) as long as for the leadingcases $(i.e. 4 =0)$ .
$\dot{\mathrm{u}}\mathrm{n}$$\mathrm{T}\mathrm{j}\phi \mathrm{t}\mathrm{h}\mathrm{e}$$\mathcal{F}_{t}$$\sigma$$E_{t}[\cdot]=E[\cdot|\mathcal{F}_{t}]$$\{y_{t}\}$$Q_{t}^{1}=\{y_{t}>0,\mathrm{y}\mathrm{t}-\mathrm{i}>0\}$$Q_{t}^{2}=\{y_{t}\leq 0,\mathrm{y}\mathrm{t}-\mathrm{i}>0\}$$Q_{t}^{4}=\{y_{t}>0,\mathrm{y}\mathrm{t}-\mathrm{i}\leq 0\}$$I(\cdot)$$I(Q_{t}^{1})$1 For theSSAR processes, however, Sato and Kunitomo (1996) have shown that the standard leastsquares method does not give us any reliable estimation results

24
We illustrate the details of our method by using the case of TAR(2:1) when $b_{1}>0$ .
[1] In this stochastic process we notice that for $Q_{t+h}^{4}(h\geq 1)$ we have $y_{t+h-1}\leq 0$ and$y_{t+h}=b_{1}y_{t+h-1}+\tau k+h>0$ . It implies that $fl_{+h-1}[|y_{t+h}|I(Q_{t+h}^{4})]$ and $|y_{t+h-1}|$ are boundedbecause we have $b_{1}>0$ , $v_{t+h}>-b_{1}y_{t+h-1}\geq 0$ , and $2|v_{t+h}|>|y_{t+h}|$ . Then we have used
the relation$\frac{1}{b_{1}}ffl_{+h-1}[|v_{t+h}|]>|y_{t+h-1}|\geq 0$ .
(5.1)
imWe notice that the boundedness of the conditional expectations of $y_{t+h-1}$ andshowplies the boundedness of the conditional expectation of $y_{t+h+1}$ . Sequentially we can. Hence we can find aposithat fl $[|yt+h+\iota|I(Q_{t+h}^{4})]$ are bounded for any integer
such that ffl $[||y_{t+h+k}||I(Q_{t+h}^{4})]\leq c_{1.1}$ for any (positive) integers $h\geq 1$
tive constant. By using this boundedness relation, we have several consequences. For inandstance, since we can decompose $I(Q_{t+h}^{1})=I(Q_{t+h}^{1}Q_{t+h-1}^{1})+I(Q_{t+h}^{1}Q_{t+h-1}^{4})$ and $I(Q_{t+h}^{2})=$$I(Q_{t+h}^{2}Q_{t+h-1}^{1})+I(Q_{t+h}^{2}Q_{t+h-1}^{4})$ , the conditional expectations ofand fl $[||y_{t+h}||I(Q_{t+h}^{2}Q_{t+h-1}^{4})]$ are bounded for any integer $h\geq 1$ .
By this consideration on the present case, we only need to evaluate the conditional expectation terms associated with the four phases on the process :$I(Q!+hQ_{t+h-1}^{1})$ ,
.
, and
,
Then we shall consider the ergodic conditions for four regions of the parameter values, and , separately.
$y_{\mathrm{t}+h}$$h$$\geq 1$$c_{1.1}$$k$$\geq 1$

$E_{t}[||y_{t+h}||I(Q_{t+h}^{1}Q_{t+h-1}^{4})]$$I(Q_{t+h}^{2}Q_{t+h-1}^{1})$$C_{1},C_{2},C_{3}$$I(Q_{t+h}^{3}Q_{t+h-1}^{3})$$I(Q_{t+h}^{3}Q_{t+h-1}^{2}Q_{t+h-2}^{1})$$C_{4}$: In this case we first notice that $0\leq e_{1}’Al$ $<1$ implies $0\leq e_{1}’A^{2}l<1$ for
vector $l=(1,1)’$ . Define the indicator functions by $I_{t+2}^{(1)}=I(y_{t+2}\geq y_{t+1})$ anda2$I_{t+2}^{(2)}=I(y_{t+2}<\Re_{+1})$ . Then there exists apositive
such that[2]$C_{1}$$\mathrm{x}1$$c_{1S}$$\leq fl[(e_{1}’ A^{2}lI_{t+2}^{(1}’+e_{1}’ AlI_{t+2}^{(2)})||y_{t}||(I(Q_{t+2}^{1}Q_{t+1}^{1})+I(Q_{t+2}^{2}Q_{t+1}^{1}))]+c_{1.2}fl[||y_{t+2}||(I(Q_{t+2}^{1}Q_{t+1}^{1})+I(Q_{t+2}^{2}Q_{t+1}^{1}))]$

Hence we can find positive constants$c_{13}$and$\delta_{1.1}(0\leq\delta_{1.1}<1)$.

such that$E_{t}[||y_{t+2}||(I(Q_{t+2}^{1}Q_{t+1}^{1})+I(Q_{t+2}^{2}Q_{t+1}^{1}))]\leq\delta_{1.1}||y_{1}||ffl[I(Q_{t+2}^{1}Q_{t+1}^{1})+I(Q_{t+2}^{2}Q_{\ell+1}^{1})]+c_{1.3}$.

, we substitute (3.1) for $||y_{t+2}||=-y_{t+2}I_{t+2}^{(1)}-y_{t+1}I_{t+2}^{(2)}$ and we
For the phasesuch thatandtake positive constants$Q_{t+2}^{3}Q_{t+1}^{2}$$c_{1.4}$$=$$\leq$

ffl
ffl$\delta_{12}(0\leq\delta_{1.2}<1)$$[||y_{t+2}||I(Q_{t+2}^{3}Q_{t+1}^{2})]$$[(b_{1}|e_{1}’Ay_{t}|I_{t+2}^{(1)}+|e_{1}’Ay_{t}|I_{t+2}^{(2)}+|v_{t+2}+b_{1}v_{\ell+1}|I_{t+2}^{(1)}+|v_{t+1}|I_{t+1}^{(2)})I(Q_{t+2}^{3}Q_{t+1}^{2})]$$\delta_{1.2}||y_{t}||fl[I(Q_{t+2}^{3}Q_{t+1}^{2})]+c_{1.4}$,

where we have taken apositive constant $612= \max\{b_{1}(a_{1}+02),a_{1}+a_{2}\}$ . Similarly, for, we can find apositive constant Cl.l such thatthe phase$Q_{t+2}^{3}Q_{t+1}^{3}$

ffl$[||y_{t+2}||I(Q_{t+2}^{3}Q_{t+1}^{3})] \leq\max\{b_{1}^{2}, b_{1}\}(-y_{t})$ffl$[I(Q_{\ell+2}^{3}Q_{t+1}^{3})]+c_{1.5}$.

By summarizing the above inequalities on four phases, we can find positive constants, which leads to (2.3) inand $\delta_{1.3}(0\leq\delta_{1.3}<1)$ such that$M$
and acompact set $C(M)$Lemma 1. This is because we can find asufficiently large$c_{1.6}$$E_{t}[||y_{t+2}||]\leq\delta_{1.3}||y_{t}||+\mathrm{C}1.1$

25depending on $M$ such that $C(M)=\{||y||\leq M\}$ and the growth condition (2.3) in Lemma
1can be satisfied.
In the following derivations for other cases we need to use the above type of arguments
repeatedly in each case. Because the arguments are quite similar and tedious, however,
we shaU try to discuss the essential differences and not to repeat the same arguments.
: By repeating the procedure we have used in [2] and taking the conditional[3], we can find apositive integer $h(\geq 2)$ and apositive constant
expectations givensuch that$C_{2}$$c_{1.7}$$F_{t}$

fl(5.2)$[||y_{t+h}||]$$\leq E_{t}[||y_{t+h}||(I(Q_{t+h}^{1}Q_{t+h-1}^{1}Q_{t+h-2}^{1})+I(Q_{t+h}^{2}Q_{t+h-1}^{1}Q_{t+h-2}^{1})$.

$+I(Q_{t+h}^{3}Q_{t+h-1}^{2}Q_{t+h-2}^{1})+I(Q_{t+h}^{3}Q_{t+h-1}^{3}Q_{t+h-2}^{3}))]+c_{1.7}$

is bounded, we need to evaluate its first four terms. Let $\lambda_{:}(i=1,2)$ be thein , we have thecharacteristic roots of $g_{A}(\lambda)=0$ in (3.4). Because $a_{1}<0$ and. Then there exists apositive integer such
relation that
and
become eventualy
that $e_{1}’A^{h_{1}}y_{t}<0$ given
because each component of. If we write
negative for asufficiently largeSince$c_{1.7}$$a_{2}\geq 0$$\lambda_{1}>0>\lambda_{2}$$C_{2}$$h_{1}$$|\lambda_{2}|>|\lambda_{1}|$$e_{1}’A^{h_{1}}$$y_{t}\in Q_{t}^{1}$$h_{1}$.

(5.3)$y_{t+h_{1}}=e_{1}’A^{h_{1}}y_{t}+ \sigma_{1}(.\cdot\sum_{=1}^{h_{1}}e_{1}A^{:-1}e_{1}v_{t+}.)’$, then we have the condition $w_{t+h_{1}}>$and we denote the second term of (5.4) as$-e_{1}A^{h_{1}}y_{t}’\geq 0$such thatwhen. Hence there exists apositive constant.
For the last term of (5.3), we use the relation that $y_{t+h_{2}}=b_{1}y_{t+h_{2}-1}+v_{t+h_{2}}$ whensuch thatfor any positive integer
and we can take apositive$w_{t+h_{1}}$$y_{t+h_{1}}\in Q_{t+h_{1}}^{1}$$c_{1.8}$$E_{t}[||y_{t+h_{1}}||I(\mathrm{I}\mathrm{L}_{=0}^{h_{1}}.Q_{t+}^{1}.\cdot)]\leq c_{1.8}$$y_{t+h_{2}}\in Q_{t+h_{2}}^{3}$$h_{2}$$c_{1.9}$$E_{t}[||y_{t+h_{2}}||I(\Pi_{arrow-0}^{h}.Q_{t+h_{2}-:}^{3})]\leq fl[(b_{1}^{h_{2}}I_{t}^{(1)}+b_{1}^{h_{2}-1}I_{t}^{(2)})(-y_{t})I(\mathrm{I}\mathrm{L}_{=0}^{h_{2}}.Q_{t+h_{2}-:}^{3})]+c_{1.9}$.

By repeating the substitution of each phase on the right hand side of (5.4), we take asufficiently large $h_{3}(\geq h_{1}+h_{2}+2)$ and we can reduce the second and the third terms of(5.4) to their first and last terms. By using the condition $0\leq b_{1}<1$ we can find positive
constants.
andsuch that$c_{1.10}$$\delta_{1.4}(0\leq\delta_{1.4}<1)$$E_{t}[||y_{t+h_{3}}||]\leq\delta_{1.4}||y_{t}||+c_{1.10}$: In this case we notice that when $y_{t+3}\in Q_{t+3}^{1}$ and $y_{t+2}\in Q_{t+2}^{1}$ , the equation$y_{t+3}=a_{1}y_{t+2}+a_{2}y_{t+1}+v_{t+3}$ implies that $v_{\iota+\mathrm{s}}>(-a_{1})y_{t+2}+(-a_{2})y_{t+1}\geq 0$ . Then by applying the same argument as [2] and [3] to, we can show thatis bounded. By using the facts that we can decopmpose[4]$C_{3}$$E_{t}[||y_{t+3}||I(Q_{t+3}^{1}Q_{t+2}^{1}Q_{t+1}^{1})]$$y_{t+1}$$I(Q_{t+3}^{2}Q_{\ell+2}^{1}Q_{t+1}^{1})=I(Q_{t+3}^{2}Q_{t+2}^{1}Q_{t+1}^{1}Q_{t}^{1})+I(Q_{t+3}^{2}Q_{t+2}^{1}Q_{t+1}^{1}Q_{t}^{4})$,and the conditional expectations, weandare bounded for the phasefind thatis bounded and henceis alsobounded. Also by using that $0<y_{t+1}=a_{1}y_{t}+a_{2}y_{t-1}+v_{t+1}<v_{t+1}$ for the phase, we have thatis bounded. Then ffl
and, we canare bounded. For the term fortake apositive constantsuch that$y_{t+2}$$y_{t+1}$$E_{t}[|y_{t+2}|I(Q_{t+2}^{1}Q_{t+1}^{1})]$$Q_{t+3}^{2}Q_{t+2}^{1}Q_{t+1}^{1}Q_{t}^{1}$$Q_{t+1}^{1}Q_{t}^{1}$$E_{t}[||y_{t+3}||I(Q_{t+3}^{2}Q_{t+2}^{1}Q_{t+1}^{1})]$$E_{t}[y_{t+1}I(Q_{t+1}^{1}Q_{t}^{1})]$$E_{t}[||y_{t+3}||I(Q_{t+3}^{3}Q_{t+2}^{2}Q_{t+1}^{1}Q_{t}^{1})]$$[||y_{t+2}||I(Q_{t+2}^{1}Q_{t+1}^{1})]$$I(Q_{t+3}^{3}Q_{t+2}^{3})$$c_{1.11}$$E_{t}[||y_{t+3}||I(Q_{t+3}^{3}Q_{t+2}^{3})]\leq E$ $[(b_{1}^{2}I_{t}^{(1)}+b_{1}I_{t}^{(2)})(-y_{t})I(Q_{\ell+3}^{3}Q_{t+2}^{3})]+c_{1.11}$.

26
Because $0<b_{1}<1$ , we can find positive constants$E_{t}[||y_{t+3}||]\leq\delta_{1B}||y_{t}||+c_{1.12}$$c_{1.12}$.

and$\delta_{1.5}(0\leq\delta_{1.5}<1)$such thatand: We need to consider the terms involving the phases$(h\geq 2)$ in particular. Since $a_{2}<0\leq a_{1}$ in, there are two different cases depending on$g_{A}(\lambda)=0$
in (3.4) are real or complex, separately.
whether the characteristic roots of$0\leqa_{1}+a_{2}<1$$D(A)\geq0$, the characteristic roots are real and theirandWhenabsolute values are less than one. In this case we immediately confirm the conditions
that $0\leq e_{1}’Al$ $<1$ and $e_{1}’A^{2}l<1$ . Then we have the same inequality as in [2]. When
$D(A)<0$ , on the other hand, there exists apositive integer such that (5.6) holds. Then we can reduce the problem
because the angle of two roots is in $(0, \pi)$ orin [3]. Hence we find positive constants
into the one within [3].
andandsuch that the last inequalty holds instead of,
we can use the same argument
For the remaining term involving the phasebecause of the condition $0<b_{1}<1$ .
as [3] for[5]$Q_{t+h}^{1}Q_{t+h-1}^{1}$$C_{4}$$Q_{\ell+h}^{2}Q_{t+h-1}^{1}$$C_{4}$$h$$(\pi, 2\pi)$$\theta$$\delta_{1.6}(0\leq\delta_{1.6}<1)$$Q_{t+h}^{3}Q_{t+h-1}^{3}$$\delta_{1.4}$$c_{1.13}$$c_{1.10}$$Q_{t+h}^{3}Q_{t+h-1}^{3}$$C_{2}$[6] Necessity :For proving the necessity of our conditions, we modify the similararguments used by Petruccell and Woolford (1984). Because they become quite lengthy
and tedious, we consider the case when . $>0(i=1,2)$ and $a_{1}+a_{2}>1$ as an illustration.
In this case we have two real characteristic roots $\lambda_{:}(i=1,2)$ , which satisfy the condition$\lambda_{1}>1>0>\lambda_{2}$ . Then we have two cases depending on the relative magnitudes of two
matrix. By defining anonsingulax 2and (ii)
roots, that is, (i)$\alpha$$|\lambda_{1}|>|\lambda_{2}|$$\mathrm{x}2$$|\lambda_{1}|<|\lambda_{2}|$$\mathrm{A}=(\begin{array}{ll}\lambda_{1} \lambda_{2}1 \mathrm{l}\end{array})$. If we write
vectorand tranforming the original system as 2
$U_{t}=(u_{1t},u_{21})’=\mathrm{A}^{-1}(\sigma(t)v_{t},0)’$, we have awith.
. For the case of (i), we
new representation of the process
considerand it is not difficult to show that its solution explodes with apositive
probability by the same method used of Petruccelli and Woodford (1984). For the case of(ii), we consider
and it is also not difficult to show that its solution explodes withapositive probabilty.
$\mathrm{x}1$$x_{t}=(x_{1t},x_{2\mathrm{C}})’=\mathrm{A}^{-1}y_{t}$$\mathrm{a}(\mathrm{t})=\sigma_{1}I(y_{t-1}>0)+\sigma_{2}I(y_{t-1}\leq 0)$$x_{u}$$=\mathrm{A}.x_{\mathrm{u}-1}.+u_{\mathrm{u}}(i=1,2)$$\{x_{u}\}$$\{x_{\mathfrak{U}}\}$REFERENCES
Brockwell, P. and R.Davis (1991), Time Series :Theory and Methods, 2nd Edition,

Springer.
Chan, K.S. and H. Tong (1985), “On the Use of the Deterministic Lyapunov Rinction for
the Ergodicity of Stochastic Difference Equation,” Adv. Appl. Prob., 17, 666-678.
Chen, R. and R S.Tsay (1991), “On the Ergodicity of TAR(I) Processes,” Annals of Applied Pmbabd.ity, 1, 613634.

Kunitomo, N. and S. Sato (1996), “Asymmetry in Economic Time Series and Simultaneous Switching Autoregressive Model,” Structural Change and Economic Dynamics, 7,27
Kunitomo, N. and S. Sato (2000), “On Simultaneous Switching Autoregressive Models,”in Nonlinear Statistical Inference, edited by C.Hsiao, Cambridge University Press, forthcoming.
Nummelin, E. (1984), “General Irreducible Markov Chains and Non-negative Operators,”
Cambridge University Press.
$\mathrm{P}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{c}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{i},\mathrm{J}$. and S.W.Woolford (1984), “A Threshold$\mathrm{A}\mathrm{R}(1)$Probability, 17, 66&678.

Model,” $Jou$ rnal of AppliedSato, S. and N. Kunitomo (1996), “Some Properties of the Maximum Likelihood EsSeries
timator in the Simultaneous Switching Autoregressive Model,” Journal ofAndysis, 17, 287-307.
$T\dot{\iota\pi}oe$$\mathrm{T}\mathrm{j}\phi \mathrm{s}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{m}$, D. (1990), “Nonlinear Time Series and Markov Chains,” Adv. Appl. Prob.,

22, 587-611.

Tweedie, R.L. (1975), “Sufficient Conditions for Ergodicity and Recurrence of Markov
Chains on aGeneral State Space,” Stock. Proc. Appl. Prob., 3, 385-403.

Tong, H. (1990), Non-linear Time Series, Oxford University Press, Oxford.
Appendix :Some Figures

In this appendix, we give several figures. All figures reported here are the results of thebased on the TAR(2) processes andsimulations on the sets of 10,000 realizations of
the SSAR(2) processes without any disturbance terms. We also have checked the nonexplosiveness of the sample paths of the solutions and basically confirmed the adequacy ofthe same regions by the corresponding simulations for the TAR(2) and SSAR(2) processes
with disturbances. It was all we could do because the criteria of convergence in simulations
are more difficult and subtle when there are noise terms.
All figures for the TAR(2:d) processes are denoted by TAR(2) with the delayed paspace while the figure for the SSAR(2) process are
rameter and drawn in thespace. The shaded areas in figures are the non-explosive regions indrawn in the
our simulations$\{y_{t}\}$$d$$(a_{1}, a_{2})$$(r_{1},r_{2})$28
Appendix : Some Figures

TAR (2) :$\mathrm{b}’\overline{-}0.6$.

$\mathrm{b}2=$$0$.

$\mathrm{d}=$’TAR (2).

$\mathrm{b}’--\triangleleft$$l$.

$\mathrm{b}2=$$0$.

Fi ure 2

Figure 1

TAR(2)$\mathrm{F}\mathrm{l}\mathrm{e}\mathrm{u}\cdot*3$::$\mathrm{b}\prime_{-}^{-}0.2$.

Figure 4$\mathrm{b}2=$$0$.

$\mathrm{d}=$$2$$\mathrm{d}--$’29
Appendix :Some Figures (Continued)

SSAR (2) :$\mathrm{s}\mathrm{i}.’--$

’

.

.

.

$j.2–0$ ’Fi ure 5$\mathrm{Y}\mathrm{A}\mathrm{R}(2)$:$\mathrm{b}’---0.2$

Figure 7

TAR (2) :$\mathrm{b}\mathrm{t}--0.2$.

Figure 6.

$\mathrm{b}\mathrm{z}_{-}^{-}\mathrm{o}.$

’

. d–,

bh

. 5.

$\triangleleft$$\#$’