00Thaler_FM i-xxvi.qxd

model.) The figure shows two possible date 1 prices, and the paths for ex-
pected price conditional on the date 1 move. It can be seen that with
outcome-dependent confidence, there are smooth overreaction and correc-
tion phases. Pairs of returns drawn from these phases will be positively cor-
related, whereas the pair which straddles the extremum will be negatively
correlated. The overall autocorrelation involving contiguous price changes
will be positive if the extremum-straddling negative correlation is suffi-
ciently small. However, price changes that are separated by long lags are
likely to straddle the extremum of the impulse-response function, and will
therefore exhibit negative autocorrelations. Thus, the pattern of momen-
tum at short lags and reversal at long lags arises naturally from the model.
We present two models with dynamic confidence that capture this intu-
ition. The model presented in subsection A is tractable but highly stylized.
The model presented in subsection B allows us to develop more complex
implications, but can only be solved by simulation.

A. The Simple Model with Outcome Dependent Confidence

We modify the basic model of section 2 as follows. We still allow for, but
no longer require, initial overconfidence, so σC^2 ≤σ
^2. For tractability, the
public signal is now discrete, with s 2 =1 or −1 released at date 2. We as-
sume that the precision assessed by the investors at date 2 about their ear-
lier private signal depends on the realization of the public signal in the fol-
lowing way. If

sign(θ+
)=sign(s 2 ), (9)

confidence increases, so investors’ assessment of noise variance decreases to
σC^2 −k, 0<k<σC^2. If

sign(θ+
)≠sign(s 2 ), (10)

confidence remains constant, so noise variance is still believed to be σC^2.
The probability of receiving a public signal +1 is denoted by p. For a high
value to be a favorable indicator of value, pmust tend to increase with θ.
However, allowing pto vary with θcreates intractable non-normalities. We
therefore examine the limiting case where the signal is virtually pure noise,
so that pis a constant. (Appendix C of Daniel, Hirshleifer, and Subrah-
manyam [1998] provides a discrete model which derives similar results
using an informative public signal.)
Given normality of all random variables, the date 1 price is

(11)

The date 0 price P 0 =0, the prior mean. If sign (θ+
)≠sign(s 2 ), then confi-
dence is constant. Since the public signal is virtually uninformative, the

]().=

+

+

σ

σσ

θ θ

θ

2

22

C

PE^1 =+C[θθ

478 DANIEL, HIRSHLEIFER, SUBRAHMANYAM