00Thaler_FM i-xxvi.qxd

more confident in his private signal. If the new public signal disconfirms his
private signal, the investor revises the estimated precision downwards, but
not by as much. Thus, the specific updating rule that we implement is:

(18)

where σΦ,tis the standard deviation of Φat time t.We impose the restriction
that The ratio is an index of the investor’s attribu-
tion bias.^18

B. 2 the equilibrium

Since the investor is risk-neutral and the risk-free rate is zero, at each point
in time the stock price is the expectation of its terminal value:

(19)

Define vθ=1/σ^2 θ, and vη=1/σ^2 η. The price of the security at time tis
given by:

(20)

Recall that the precision of Φtis (t−1)vη.

B. 3 simulation results and empirical implications

For the simulation we use the parameters σθ^2 =σε^2 =1,
and ση^2 =7.5. We also make the investor’s initial estimate of his precision
equal to the true precision of his private signal. We perform this simulation
50,000 times, each time redrawing the value θ, the private signal s 1 =θ+ε,
and the public information set φt, for t=2,...T.
It is useful to first illustrate the dynamic price path implied by the model
for specific realizations of s 1 and θ. Figure 13.2 shows the average price path
following a private signal of s=1 when θ=0, so that the informed investors’
signal is unduly favorable. The price initially jumps from 0 up to 0.5, a ra-
tional assessment. On average, the price continues moving up, reaching a
maximum of 0.7366 in period 16. The average price then declines, and even-
tually asymptotes to zero. Thus, there is an initial overreaction phase in

kk== 075 ., 01 .,

̃ [ ̃ ,]() ,.

,

PE s

tv vs

tC t vvv

tCt

Ct

==

−+

++

θ

η

θη

 1

(^11)
Φ
Φ
PE stC==[θφ ̃ 12 ,,..., ]φt CE s[θ ̃ 1 ,].Φt
kk>>0. ()/() 11 +−kk
if
()()and 2
then ( ) otherwise
() ,
,
,,
,,
sign s sign s
vkv
vkv
ttt tt
Ct Ct
Ct Ct
11 1 11
1
1
1
1
−= − −
=+
=−






−−−
−
−
ΦΦΦφσ< Φ
482 DANIEL, HIRSHLEIFER, SUBRAHMANYAM
(^18) Several alternative ad hocupdating rules consistent with this intuition all lead to roughly
equivalent results. For tractability, we assume that the investor forms beliefs as if, at each
point in time, he knows his exact signal precision. Rationally he should allow for the fact that
vC,tis an estimate. We expect that the essential results are not sensitive to this simplification.