Appendix: Proofs

A. ARMA Representation of the Return Process

Let us begin by recalling Eq. (5) from the text (suppressing constants):

(A.1)

It follows that

(A.2)

Assuming that φsatisfies proper conditions to be specified, ∆Ptis a cov-
anance stationary process. Let

αk≡E[∆Pt∆Pt−k]

(i.e., the unconditional autocovariance lagged kperiods). When k=0, we
have the unconditional variance. The autocovariances of this process sat-
isfy the following Yule-Walker equations.

(A.3)

And for k>0, we have

(A.4)

It is not hard to verify that for k>z−1,

(A.5)

And for k≤z−1, we have

(A.6)

E
z

P

zk

z

E

z

PE

z

P

ti

i

z

tk

ti

i

z

tk

ti

i

z

tkj

ε σ

φ

ε

φ

+ ε

=

−

−

+

=

−

−+

+

=

−

∑∑∑−++

















=

−

+

















−















0 

(^12)
2
0
1
1
0
1
∆∆∆ 1
()
() ( )
E
z
P
ti
i
z
tk
ε+

−

∑ −

















=

1

1

∆ 0.

α

ε

k φα φα

ti

i

z

=E z Ptk k k j

















+−

+

=

−

∑ −−−+

0

1

∆ 11 ().

α

ε

0 φα φα

0

1

= 11

















+−

+

=

−

E∑ z P +

ti

i

z

∆tj.

∆∆∆P

z

t PP

i ti

z

=+−ttj

= +

−

−−+

∑ ε

φφ

0

1

11 ().

PD

z

zz

tt t tz Pti

i

j

=+

−

++− −+⋅⋅⋅+ +

=

∑

() 11

11

1

εεφ∆

532 HONG AND STEIN