00Thaler_FM i-xxvi.qxd

The product ∆P 1 ∆P 2 can only take on two values, based upon the various
signal combinations:

(A24)

(A25)

Combining, E[∆P 1 ∆P 2 ] can be written as (1−a)X+aY, where a=p+q−
2 pq. After some calculation, the two components of this expression become:

(A26)

aY= 2 p(2q−1)(2p−1)(p−1). (A27)

Combining these two terms and a great deal of factoring produces the final
result,

(A28)

When there is no overconfidence (pC=p) this expression is zero and price
changes are uncorrelated.

A Second Noisy Public Signal

The model so far shows that overreaction can be exaggerated by a possible
rise in confidence triggered by a noisy public signal. We now add a second
noisy public signal to consider whether correction of mispricing is gradual.
Signal s 3 ′follows s 2 and can take on values Gor B. The precision of this
signal is as follows:

Pr(s 3 ′=Gθ=+1)=r=Pr(s 3 ′=Bθ=−1). (A29)

This signal does not affect confidence. If the player becomes overconfident
(and replaces p with pC) after s 2 , then the player will continue to use pCas
his measure of the precision of s 1 , regardless of whether s 3 ′confirms s 1. As
there are two possible prices after the first signal and four possible prices
after the second, there are eight possible prices after observation of the
third signal. As above, by symmetry, only half of these prices need to be cal-
culated. Using the conditional probabilities, the period three prices are:

EPP

qp p p q

pq q

C

C

[]

()( )()

()()

∆∆ 12.

22 1 1

211

= 0

−−−

−+−

>

()

()( )( )

()()

1

22 1 2 1 2

211

−=

−−−+++− −

−+−

aX

ppqpqpppqpq ppq p

pq q

CC C

C

YPP PP

p

pq

pq p

p

≡=sHsD sLsU

=−

−

+−

−−













[]==[]==

() ().

∆∆ 1212 ,,∆∆ (^1212)
21
2
21
XPP PP
p
pq
pq q
p
sHsU sLsD
C
C
≡=
=−
+−
−+−
−−






[]==[]==
()
()()
().
∆∆ 1212 ,,∆∆ (^1212)
21
1
211
21
494 DANIEL, HIRSHLEIFER, SUBRAHMANYAM