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\begin{document}
\baselineskip=24pt

\title{\ttiL REPARAMETERIZATION OF THE BOUNDARY OF THE ADMISSIBLE REGION FOR ARMA MODELS}
\author{{\tti A.I. McLeod and Y. Zhang} \\
{\ttiSL The University of Western Ontario } }
\date{October 23, 2003}
%\thispagestyle{empty}
\maketitle
\newpage
\strut
\hbox{Mailing Address for Corresponding Author: \hfill }

{\baselineskip 12pt

\medskip
\noindent\mbox{\tti Dr. A.I. McLeod \hfill }
\linebreak
\noindent\mbox{\tti Department of Statistical and Actuarial Sciences \hfill }
\linebreak
\noindent\mbox{\tti The University of Western Ontario  \hfill }
\linebreak
\noindent\mbox{\tti London, Ontario N6A 5B7 \hfill }
\linebreak
\noindent\mbox{\tti Canada \hfill }


\vskip 1in
\hbox{A.I. McLeod \hfil}

\hbox{Email \& Phone:  aimcleod@uwo.ca \quad 519 661-3611\hfil}

\medskip
\hbox{Y. Zhang \hfil}

\hbox{Email \& Phone:  yzhangli@stats.uwo.ca \quad 519 661-3821}

\vskip 1.5in
\strut
{\hbox {\it Running Head:\/} {\tti Reparameterization of the Boundary}\hfil}
}
\newpage


The theorem of Barndorff-Neilsen and Schou (1973, Theorem 2)
stating that the admissible region of autoregressive process may
be obtained as the image of the admissible region defined by the
partial autocorrelations under the Durbin-Levinson recursion is
derived from first principles in the case of the boundary.  
Barndorff-Neilsen and Schou (1973, Theorem 2) is also extended
to the case of the subset autoregressive model. 
The boundary transformation is useful for obtaining maximum likelihood
estimates on the moving-average boundary as well as for constructing
diagrams of the admissible region.
The subset result suggests
a new approach to subset autoregression which has been extensively
developed in another article.

\medskip
{\it Keywords and phrases:}
Admissible region for the autoregressive and moving-average time series;
Maximum likelihood estimation on the moving-average boundary;
Subset autoregression model.

\newpage
\hbox{\tti 1. INTRODUCTION\hfill}
\bigskip
The $\AR(p)$ model with mean zero may be written,
$\phi(B)z_t = a_t$, where $\phi(B)=1-\phi_1 B-\ldots \phi_p B^p$
and $a_t$ is Gaussian white noise with variance $\sigma_a^2$.
The admissible region for stationary-causal processes is defined
by the region in $\Re^p$ for which all roots of $\phi(B)=0$ lie outside the
unit circle (Brockwell and Davis, 1991).

Let $\phi_{j,k}$ denote the coefficient of $z_{t-j}$ in the minimum-mean-square-error
predictor of $z_t$ given $z_{t-1},\ldots,z_{t-k}$.
Then the partial autocorrelations, denoted by $\zeta_i$ for $i=1,\ldots,p$
are given by $\zeta_i=\phi_{i,i}$, where the $\phi_{i,i}$ are
determined by the Durbin-Levinson recursion,
\begin{equation}
\phi_{j,k+1}=\phi_{j,k}-\phi_{k+1,k+1} \phi_{k+1-j,k},\quad j=1,\ldots,k
\newcounter{DLRecursion}
\setcounter{DLRecursion}{\value{equation}}
\end{equation}
where $k=1,\ldots,p$. Barndorff-Neilsen and Schou (1973) showed
that eq. ({\theDLRecursion}) can be used to define a
transformation, ${\cal B}: (\zeta_1,\ldots,\zeta_p)\Rightarrow
(\phi_1,\ldots,\phi_p),$ and that this transformation is
continuously differentiable and has a continuously differentiable
inverse inside the admissible region. 
It will be convenient in
\S 3, to use the notation $\phi_i={\cal B}_i(\zeta)$ to refer to
each parameter $i=1,\ldots,p$, where
$\zeta=(\zeta_1,\ldots,\zeta_p)$. It should be noted that since
the determinant Jacobian of the transformation (Barndorff-Neilsen
and Schou, 1973, p.414) is zero on the boundary, the
transformation is not 1:1 there. For example, in the $\AR(2)$ case
${\cal B}(\zeta_1, \zeta_2)= (\zeta_1(1-\zeta_2),\zeta_2)$ so all
the points on the boundary of the form $(\zeta_1, 1)$ are mapped
into the point $(0,1)$.

\newpage
\hbox{ 2. {\smcaps Boundary of\/} $\AR(p)$ {\smcaps admissible region\/}\hfill}
\bigskip

Let ${\cal D}_\phi$ denote the admissible region for an
$\AR(p)$ in the parameter space $(\phi_1,\ldots,\phi_p)$. The
admissible region for the transformed parameters
$(\zeta_1,\ldots,\zeta_p)$ is the interior of the unit cube,
\begin{equation}
{\cal D}_\zeta=\{(\zeta_1, \ldots ,\zeta_p) \in \Re^p : |\zeta_i|<1
, i=1,\ldots,p \}.
\newcounter{UnitCube}
\setcounter{UnitCube}{\value{equation}}
\end{equation}
Denote the boundary sets of ${\cal D}_\phi$ and ${\cal D}_\zeta$
by $\partial_\phi$ and $\partial_\zeta$ respectively.
Barndorff-Nielsen and Schou (1973, Theorem 2) showed that
${\cal B}$ maps ${\cal D}_\zeta$ onto ${\cal D}_\phi$ and
that this mapping is one-to-one.
But ${\cal B}$ is no longer one-to-one on the boundary $\partial_\zeta$.
It is non-trivial to show that 
${\cal B}$ maps $\partial_\zeta$ onto $\partial_\phi$.
For completeness we have included an elementary proof using real
analysis of this result.

{\smcaps Theorem 1.\/} $\partial_\phi={\cal B}(\partial_\zeta)$

{\smcaps Proof.\/} First we show that ${\partial_\phi} \subset
{\cal B} ({\partial_\zeta})$.  Let $\bar {\cal D}_\phi$ and $\bar
{\cal D}_\zeta$ denote the closures of ${\cal D}_\phi$ and ${\cal
D}_\zeta$ respectively.  Since $\cal B$ is a polynomial and hence
continuous on $\bar {\cal D}_\zeta$, ${\cal B}( \bar{\cal
D}_\zeta) \subset \overline{{\cal B}({\cal D}_\zeta)} ={\bar {\cal
D}_\phi}$.  Meanwhile, ${\cal B}(\bar {\cal D}_\zeta)={\cal
D}_\phi \cup {\cal B}(\partial_\zeta)$ is closed since $\bar {\cal
D}_\zeta$ is compact and therefore ${\cal B}(\bar {\cal
D}_\zeta)\supset \bar {\cal D}_\phi$.  Hence ${\cal B}(\bar {\cal
D}_\zeta)=\bar {\cal D}_\phi$, i.e., ${\cal D}_\phi \cup {\cal
B}(\partial_\zeta)={\cal D}_\phi \cup \partial_\phi$. Since ${\cal
B}^{-1}$ is continuous, it follows that ${\cal D}_\phi$ is open
since ${\cal D}_\zeta$ is open.  ${\cal D}_\phi \cap\partial_\phi
= \emptyset$.  Hence ${\partial_\phi} \subset {\cal
B}({\partial_\zeta})$.

Next, we show $\partial_\phi \supset {\cal B}(\partial_\zeta)$.
Let $\vartheta \in \partial_\zeta$. There exists a sequence
$\{\vartheta_n \}\in {\cal D}_\zeta$ such that $\vartheta_n
\rightarrow \vartheta$.  Hence ${\cal B}(\vartheta_n)=\varphi_n
\in {\cal D}_\phi \rightarrow {\cal B}(\vartheta)=\varphi$ by the
continuity of ${\cal B}$ on $\bar {\cal D}_\zeta$. If $\varphi \in
\cal D_\phi$, ${\cal B}^{-1} ({\varphi_n})=\vartheta_n \rightarrow
{\cal B}^{-1} (\varphi)=\vartheta \in {\cal D}_\zeta $ by the
continuity of ${\cal B}^{-1}$ on ${\cal D}_\phi$, which causes a
contradiction with $\vartheta \in
\partial_\zeta$.  Therefore, ${\cal B}(\vartheta)\notin \cal D_\phi$.
It follows that ${\cal B}(\vartheta) \in
\partial_\phi$ since ${\cal D}_\phi \cup {\cal
B}(\partial_\zeta)={\cal D}_\phi \cup
\partial_\phi$.  Hence $\partial_\phi \supset {\cal B}(\partial_\zeta)$.
$\Box$

For the $\AR(4)$ model, $z_t = \phi_1 z_{t-1} + \phi_2 z_{t-2}+
\phi_3 z_{t-3}+ \phi_4 z_{t-4}$. In this case the admissible
region is 4-dimensional and we have,
\begin{eqnarray}
\phi_1 &=& {{\zeta }_1} - {{\zeta }_1}\,{{\zeta }_2} - {{\zeta }_2}\,{{\zeta }_3} -
  {{\zeta }_3}\,{{\zeta }_4},\nonumber\\
\phi_2 &=& {{\zeta }_2} - {{\zeta }_1}\,{{\zeta }_3} +
  {{\zeta }_1}\,{{\zeta }_2}\,{{\zeta }_3} - {{\zeta }_2}\,{{\zeta }_4} +
  {{\zeta }_1}\,{{\zeta }_3}\,{{\zeta }_4} -
  {{\zeta }_1}\,{{\zeta }_2}\,{{\zeta }_3}\,{{\zeta }_4},\nonumber\\
\phi_3 &=& {{\zeta }_3} - {{\zeta }_1}\,{{\zeta }_4} +
  {{\zeta }_1}\,{{\zeta }_2}\,{{\zeta }_4} +
  {{\zeta }_2}\,{{\zeta }_3}\,{{\zeta }_4}\nonumber\\
\phi_4 &=& \zeta_4.
\newcounter{ARFourBNS}
\setcounter{ARFourBNS}{\value{equation}}
\end{eqnarray}

Three dimensional projections of this admissible region may be visualized using
standard graphics software such as MatLab or {\it Mathematica\/} (Zhang, 2002).

Another application of Theorem 1 is to maximizing the likelihood function
for moving-average models over the region formed by the admissible region
and its boundary.
As noted by Monahan (1984) the eq. ({\theDLRecursion}) may be used to
reparameterize ARMA models as well by applying the transformation
separately to the autoregressive and moving-average components.
Similarly Theorem 1 shows that we may re-parameterize for the parameters
on moving-average boundary of the invertible region.
It is well-known that there is a positive probability that the moving-average
parameter may lie on the non-invertible boundary (Cryer and Ledholter, 1981)
in the case of the MA$(1)$ model.
Moreover, Davis and Dunsmuir (1996) showed that the asymptotic distribution for maximum
likelihood estimate in the closed region which included the boundary differed
from that of the estimate restricted to be inside the admissible region.

We conducted a simulation experiment to see how frequently unit roots occur
in the MA$(2)$ case with exact maximum likelihood estimator defined over
the closed admissible region obtained by taking the closure of the
invertible region.
Reparameterizing the MA$(2)$ model,
$z_t = a_t - \theta_1 a_{t-1} -\theta_2 a_{t-2}$
using
$(\theta_1, \theta_2) \longleftrightarrow (\zeta_1, \zeta_2)$
and with series of lengths $n=15,30,50,100,200$ and
with parameters $(\zeta_1,\zeta_2)$ chosen at random and uniformly distributed
inside the unit square,
$10^3$ simulations of each series were generated and fit by an exact maximum likelihood
algorithm modified for the closed admissible region,
${\bar{\cal D}}_{\zeta} = \{ |\zeta_1|\le 1$ and $|\zeta_2| \le 1$ \}.

For computational purposes the maximum in each of the nine subregions is determined:
${\cal D}_{\zeta} = \{ |\zeta_1|<1$ and $|\zeta_2|<1$ \},
${\partial}_{\zeta,1} = \{ \zeta_1 =1$ and $|\zeta_2|<1$ \},
${\partial}_{\zeta,2} = \{ \zeta_1 =-1$ and $|\zeta_2|<1$ \},
${\partial}_{\zeta,3} = \{ |\zeta_1|<1$ and $\zeta_2 =1$ \},
${\partial}_{\zeta,4} = \{ |\zeta_1|<1$ and $\zeta_2 =-1$  \},
${\partial}_{\zeta,5} = \{ \zeta_1 =1$ and $\zeta_2 =1$ \},
${\partial}_{\zeta,6} = \{ \zeta_1 =1$ and $\zeta_2 =-1$ \},
${\partial}_{\zeta,7} = \{ \zeta_1 =-1$ and $\zeta_2 =1$  \},
and
${\partial}_{\zeta,8} = \{ \zeta_1 =-1$ and $\zeta_2 =-1$ \}.
Then the overall maximum is selected to obtain the global maximum likelihood estimate.
As shown in Table 1, the probability of a unit root is quite high when $n$ is small but decreases as $n$ increases.

\bigskip
\centerline{ [ Table 1 here ] }
\bigskip

\hbox{ 2. {\tti EXTENSION TO SUBSET MODELS\/}\hfill}
\bigskip
Subset autoregressive provide a parsimonious alternative to the full
$\AR(p)$ model which is especially useful
in modelling seasonal or periodic time series.
Consider the subset $\AR(4)$ model, $z_t = \phi_1 z_{t-1} + \phi_4 z_{t-4}$.
Then the transformation,
\begin{equation}
{\cal B}^{-1}(\phi_1,0,0,\phi_4)=(\zeta_1,\zeta_2,\zeta_3,\zeta_4)
\newcounter{InverseBNS}
\setcounter{InverseBNS}{\value{equation}}
\end{equation}
is one-to-one, continuous and differentiable in the admissible region.
However the admissible region in the four-dimensional space ${\cal D}_\zeta$ is
no longer simply the interior unit 4D-cube.  Instead it is a complicated two-dimensional
subspace of the 4D-cube.
Setting the $\phi_2=0$ and $\phi_3=0$ in the second and third equations in (\theARFourBNS)
these equations may be solved for $\zeta_2$ and $\zeta_3$ to determine ${\cal B}$.
Since $\zeta_1 \ne 0$, $\zeta_4 \ne 0$, $| \zeta_1 | < 1$ and $\zeta_4 > -1$,
\begin{eqnarray}
\zeta_2 &=& \frac{1 + 2\,{{{\zeta }_1}}^2\,{{\zeta }_4} -
    {\sqrt{1 + 4\,{{{\zeta }_1}}^2\,{{\zeta }_4}\,
         \left( 1 + {{\zeta }_4} \right) }}}{2\,
    \left( -1 + {{{\zeta }_1}}^2 \right) \,{{\zeta }_4}}\nonumber\\
\zeta_3 &=&
\frac{-1 + {\sqrt{1 + 4\,{{{\zeta }_1}}^2\,{{\zeta }_4}\,
         \left( 1 + {{\zeta }_4} \right) }}}{2\,{{\zeta }_1}\,
    \left( 1 + {{\zeta }_4} \right) }.
\newcounter{zetaTwoThree}
\setcounter{zetaTwoThree}{\value{equation}}
\end{eqnarray}
Note that not all solutions of (\thezetaTwoThree) are admissible.
For example taking $\zeta_1=0.9$ and $\zeta_4=-0.9$, (\thezetaTwoThree) produces,
$\zeta_2 \doteq -0.88$ and $\zeta_3 \doteq -3.8$.
Since (\theInverseBNS) is also a one-to-one, continuous and differentiable transformation,
these equations establish that the transformation $(\phi_1,\phi_4) \rightarrow  (\zeta_1,\zeta_4)$
is one-to-one, continuous and differentiable with the appropriate admissible regions.
The admissible region for $\zeta_1$ and $\zeta_4$ is given by,
\begin{equation}
{\cal D}_\zeta=\{(\zeta_1,\zeta_4)\in (-1,1)\times(-1,1) : |\zeta_2|<1 \wedge |\zeta_3|<1 \}
\end{equation}
The admissible region for $\zeta_1$  and $\zeta_4$ is shown in Figure 1 below.
Applying the transformation
${\cal B}:(\zeta_1,\zeta_2,\zeta_3,\zeta_4) \rightarrow (\phi_1,0,0,\phi_4)$,
the admissible region, Figure 2, for $\phi_1$ and $\phi_4$ is obtained.

\medskip
\begin{center}
[ Figures 1 and 2 ]
\end{center}
\medskip
\strut
In the general case of a subset $\AR(p)$ model, let
\begin{equation}
z_t = a_t + \phi_{i_1} z_{t-i_1} + \ldots + \phi_{i_m} z_{t-i_m}
\newcounter{SubsetAR}
\setcounter{SubsetAR}{\value{equation}}
\end{equation}
where $i_1<\ldots<i_m$.
Let $\dot{\cal B}_{i_1,\ldots,i_m}^{-1}$ denote the transformation,
\begin{equation}
\dot{\cal B}_{i_1,\ldots,i_m}^{-1}:(\phi_{i_1},\ldots,\phi_{i_m})\rightarrow(\zeta_{i_1},\ldots,\zeta_{i_m})
\newcounter{InducedTransformation}
\setcounter{InducedTransformation}{\value{equation}}
\end{equation}
obtained by selecting the ${i_1,\ldots,i_m}$ elements from,
\begin{equation}
{\cal B}^{-1}:(\phi_{1},\ldots,\phi_{p})\rightarrow(\zeta_{1},\ldots,\zeta_{p}).
\newcounter{InverseTransformation}
\setcounter{InverseTransformation}{\value{equation}}
\end{equation}

{\smcaps Theorem 2.\/}
The transformation and its inverse defined in eqn. (\theInducedTransformation) is
one-to-one, continuous and onto in the admissible region.

{\smcaps Proof.\/}
The transformation given in eqn. (\theDLRecursion) may be written,
\begin{equation}
\phi_i = \zeta_i + b_i(\zeta_1,\ldots,\zeta_p),
\newcounter{Multilinear}
\setcounter{Multilinear}{\value{equation}}
\end{equation}
where $b_i(\zeta_1,\ldots,\zeta_p)$ is a multilinear function of the variables
$\zeta_1,\ldots,\zeta_p$ with the coefficient
of each term being $1$ or $-1$.  This holds for $p=1$.
Suppose it is true also for $p=k$.
Then eqn. (\theDLRecursion) ensures that the new values of $\phi_1,\ldots,\phi_{k+1}$ are
suitable multilinear functions of the variables $\zeta_1,\ldots,\zeta_{k+1}$.

Consider a subset $\AR(p)$ with only one parameter $\phi_h=0$ constrained, where $h$ is
a fixed value $0<h\le p$.
Using (\theMultilinear), an explicit solution of the equation $\phi_h=0$ may be found,
\begin{equation}
\zeta_h = g_h(\zeta^{(-h)}),
\newcounter{MultilinearSolution}
\setcounter{MultilinearSolution}{\value{equation}}
\end{equation}
where $\zeta^{(-h)}$ denotes the vector $(\zeta_1,\ldots,\zeta_p)$ with the $h$-th element removed.
We can subsitute for $\zeta_h$ in each of the $p-1$ equations
$\phi_i={\cal B}_i(\zeta_1,\ldots,\zeta_p)$ to obtain an explicit solution
$\phi_i={\cal B}_{i,h}(\zeta^{(-h)})$ which defines the required transformation.

In the more general case, there are $p-m$ constraints.
Denote the indices of those parameters $\phi_1,\dots\phi_p$ which are constrained
to zero by $j_1,\ldots,j_{p-m}$.
The $p-m$ constraint equations in $p-m$ unknowns may be written
$0=\zeta_{j_k}+b_{j_k}(\zeta_1,\ldots,\zeta_p), k=1,\ldots,(p-m)$.
Starting with the equation corresponding to $\phi_{j_1}=0$, the solution
for $\zeta_{j_1}$
can be found and substituted in the remaining $p-1$ equations.
The next equation, $\phi_{j_2}=0$, is by the multilinear property,
a quadratic with two solutions for $\zeta_{j_2}$.
Continuing this process, there are $(p-m)!$ solutions to the
constraint equations.
The theorem of Barndorff-Neilsen and Schou (1973) guarantees that at most only
one of these solutions is admissible.
$\Box$

Denote the admissible boundaries for the parameters $(\phi_{i,1},\ldots,\phi_{i,m})$ and
$(\zeta_{i,1},\ldots,\zeta_{i,m})$ by
$\partial_{\phi_{i,1},\ldots,\phi_{i,m}}$ and
$\partial_{\zeta_{i,1},\ldots,\zeta_{i,m}}$ respectively.

{\smcaps Theorem 3.\/}
$\partial_{\phi_{i,1},\ldots,\phi_{i,m}}=\dot{\cal B}_{i_1,\ldots,i_m}(\partial_{\zeta_{i,1},\ldots,\zeta_{i,m}})$

{\smcaps Proof.\/}
Since $\dot{\cal B}_{i_1,\ldots,i_m}$ is a one-to-one, continuous and differentiable transformation,
Theorem 3 follows making the necessary changes to the argument given by Theorem 1.
$\Box$

Theorem 2 indicates that the parameter space of the subset autoregression
is spanned by the corresponding partial autocorrelations.
This suggests an alternative parameterization for the subset autoregessive
model in which some partial autocorrelations are set to zero and
the others are estimated.
This new approach to subset autoregression offers many advantages
and is developed in McLeod and Zhang (2003, in press).

\newpage
%\bigskip
\hbox{\ttiSL ACKNOWLEDGEMENTS \/\hfill}
\medskip
%Drs. Dan Christensen and Andr\'e Boivin 
The authors would like to thank Dr. Andr\'e Boivin for 
his suggestions and comments on Theorem 1.

%\newpage
\bigskip
\hbox{\ttiSL REFERENCES \/\hfill}
\medskip

\hind
{\smcaps Barndorff-Nielsen, O. and Schou G.\/} (1973).
On the Parametrization of Autoregressive Models by Partial Autocorrelations
{\it Journal of Multivariate Analysis\/}  3, 408-419.

\hind
{\smcaps Brockwell, P.J. and Davis, R.A.\/} (1991).
{\it Time Series: Theory and Methods\/}.
(2nd edn.). New York: Springer-Verlag.

\hind
{\smcaps Cryer, J. D. and Ledolter, J.\/} (1981).
Small-sample properties of the maximum likelihood estimator in the first-order moving average model. {\it Biometrika\/} 68, 691-694.

\hind
{\smcaps Davis, R.A. and Dunsmuir, W.T.M.\/} (1996).
Maximum likelihood estimation for MA(1) processes with a root on or near the
unit circle.
{\it Econometric Theory\/} 12, 1--29.

\hind
{\smcaps McLeod, A.I. and Zhang, Ying\/} (2003).
Subset Autoregression: A New Approach.
Submitted for publication.

\hind {\smcaps Monahan, J.F.\/} (1984). A note on enforcing
stationarity in autoregressive-moving average models, {\it
Biometrika\/} 71, 403-404.

%\hind {\smcaps Vick, J.W.\/} (1994).
%{\it Homology Theory: An Introduction to Algebraic Toplogy\/}.
%(2nd edn.). New York: Springer-Verlag.

\hind {\smcaps Zhang, Y.\/} (2002).
{\it Topics in Autoregression\/}.
Ph.D. Thesis,  The University of Western Ontario.

\newpage
\strut

Table 1. Estimated probability, $\hat \pi$, of the maximum likelihood estimator
of the parameters in a series of length $n$ being on the boundary for the MA$(2)$ model
with true parameters randomly chosen within the invertible region.

$$\vbox{\halign{\strut
 \hfill#\quad    %push to right
&\hfill#\quad
&\hfill#\quad
&\hfill#\quad
&\hfill#\quad
&\hfill#\quad    %push to right
\cr             %END OF TEMPLATE
\noalign{\hrule}
\noalign{\smallskip}
\noalign{\hrule}
\noalign{\smallskip}
 &$n=15$&$n=30$&$n=50$&$n=100$&$n=200$
\cr
\noalign{\smallskip}
\noalign{\hrule}
\noalign{\smallskip}$\pi$&$0.592$&$0.425$&$0.319$&$0.207$&$0.139$
\cr
$Est.Sd(\pi)$&$0.016$&$0.016$&$0.015$&$0.013$&$0.011$
\cr
\noalign{\smallskip}
\noalign{\hrule}
}}$$

\newpage
\strut
%Figure 1
\begin{figure}
\centerline{\epsfig{figure=ar14zeta.eps,width=3in,height=3in}}
\caption{Admissible region, $z_t = \phi_1 z_{t-1} + \phi_4 z_{t-4}$
for reparameterized model with $\zeta_1$ and $\zeta_4$.
}
\end{figure}

\newpage
\strut
%Figure 32
\begin{figure}
\centerline{\epsfig{figure=SubsetAR14PhiSpace.eps,width=3in,height=3in}}
\caption{Admissible region, $z_t = \phi_1 z_{t-1} + \phi_4 z_{t-4}$. }
\end{figure}




\end{document}
