Further, it is easily checked that
Et(Pt+ 1 −Ptyt=+y, qt=q)=−Et(Pt+ 1 −Ptyt=−y, qt=q)
and hence that
f(q)= 2 y(p 2 q−p 1 )+yp 2 (∆- (q)+∆–(q)).
First, we show that a sufficient condition for overreaction is f(q- )<0.
If this condition holds, it implies
Et(Pt+ 1 −Ptyt=+y, qt=q
)<Et(Pt+1−Ptyt=−y, qt=q
).
Now as j→∞,
Et(Pt+1−Ptyt=yt− 1 =...=yt−j=y) →Et(Pt+ 1 −Ptyt=+y, qt=q
- )
and
Et(Pt+ 1 −Ptyt=yt− 1 =...=yt−j=−y)
→Et(Pt+ 1 −Ptyt=−y, qt=q- ).
Therefore, for ∀j≥Jsufficiently large, it must be true that
Et(Pt+ 1 −Ptyt=yt− 1 =...=yt−j=y)
<Et(Pt+ 1 −Ptyt=yt− 1 =...=yt−j=−y),
which is nothing other than our original definition of overreaction.
Rewriting the condition f(q
)<0 as
2 y(p 2 q
−p 1 )+yp 2 (∆
(q
)+∆–(q
))<0,
we obtain(A.1)which is one of the sufficient conditions given in the proposition.
We now turn to a sufficient condition for underreaction. The defini-
tion of underreaction can also be succinctly stated in terms of f(q)as
Eq(f(q))>0,
where Eqdenotes an expectation taken over the unconditional distri-
bution of q. Rewriting this, we obtain:
2 yp 2 E(q)− 2 yp 1 +yp 2 Eq(∆- (q)+∆–(q))>0,
and hence,pp q (A.2)
q qq
12 <+ 2 +
E( ) E( () ())
.∆∆ppqq
12 >+ 2
∆()A MODEL OF INVESTOR SENTIMENT 455