00Thaler_FM i-xxvi.qxd

where σis the standard deviation of the ε’s. Solving the Yule-Walker equa-
tions reduces to solving a system of j+2 linear equations. Next, the opti-
mal strategies of the momentum traders are given by

(A.7)

where

Pt+j−Pt=∆Pt+j+...+∆Pt+ 1.

In equilibrium,

ζMt =φ∆Pt− 1. (A.8)

Finally, it follows that

Cov(∆Pt− 1 , Pt+j−Pt)=αj+ 1 +...+α 2 ,

and

Var(Pt+j−Pt)=jα 0 +2(j−1)α 1 +...+2(j−(j−1)αj− 1.

Using these formulas, the problem is reduced to finding a fixed point in φ
that satisfies the equilibrium condition (A.8). Given the equilibrium φ, we
then need to verify that the resulting equilibrium ARMA process is in fact
covariance stationary (since all of our formulas depend crucially on this as-
sumption).

B. Stationarity

We next provide a characterization for the covariance stationarity of a con-
jectured return process. This condition is just that the roots of

1 −φx+φxj+^1 = 0 (A.9)

lie outside the unit circle (see, e.g., Hamilton 1994).

Lemma A.1.The return process specified in equation (A.2) is a covari-

ance stationary process only if 
φ
<1.

Proof.Proof is by induction on j. For j=1, the return process follows

an ARMA(2, z). So, the conditions for covariance stationarity are:

− 2 φ<1; and − 1 <φ<1, (see, e.g., Hamilton 1994). The stated re-

sult follows for j=1. Apply the inductive hypothesis and assume the

result holds for j=k.

ζ

γ

t

M tj t t

tj t t

EP P P

Var P P P

=

−

−

+−

+−

[]

[]

,

∆

∆

1

1

A UNIFIED THEORY OF UNDERREACTION 533