00Thaler_FM i-xxvi.qxd

From Eq. (A.9), it follows that the roots xof

1 −φx+φxk+^1 = 0

must lie outside the unit circle (e.g.,
x
>1). It follows that

 1 −φx
=
φ

x
k+^1. (A.10)

Hence, as kincreases, it follows that
φ
decreases for equation (A.10) to
hold. The stated result follows for arbitrary j. QED
We use this result to characterize a number of properties of a conjectured
covariance stationary equilibrium.

Proof of Lemma 1.We show that φ>0 in a covariance stationary

equilibrium by contradiction. Suppose it is not, so that φ≤0. It is

easy to verify from equation (A.4) and from Lemma A.I that

αk≥ 0 ∀k →α 2 +α 3 +...+αj+ 1 > 0

implying that φ>0, leading to a contradiction. QED

C. Existence and Numerical Computation

An equilibrium φsatisfying the convariance stationary condition in Lemma
A.1 does not exist for arbitrary parameter values. It is easy to verify however
that a covariance stationary equilibrium does exist for sufficiently small γ.

Lemma A.2.For γsufficiently small, there exists a covariance station-

ary equilibrium.

Proof.It is easy to show that for γsufficiently small, we can apply

Brouwer’s fixed point theorem. QED

In general, the equilibrium needs to be solved numerically. For the case of
j=1, we can always verify the resulting φleads to covariance stationarity.
For arbitrary j, we only have a necessary condition although the calcula-
tions for the autocovariances would likely explode for a φthat does not
lead to a covariance stationary process. So, we always begin our calcula-
tions for j=1 and γsmall and use the resulting solutions to bootstrap our
way to other regions in the parameter space. The solutions are gotten easily.
When we move outside the covariance stationary region of the parameter
space, autocovariances take on nonsensible values such as negative values
for the unconditional variance or autocovariances that do not satisfy the
standard property that

αk
<
α 0 
, k> 0

534 HONG AND STEIN