here are and λ 1 =0.1<0.3=λ 2. Note again that the earn-
ings stream is generated using the true process for earnings, a random walk.
In periods 0–4, positive shocks to earnings alternate with negative shocks.
Since Model 1 stipulates that earnings shocks are likely to be reversed in the
following period, we observe an increase in qt, the probability that Model 1
is generating the earnings shock at time t, rising to a high of 0.94 in period 4.
From periods 10 to 14, we observe five successive positive shocks; since this
is behavior typical of that specified by Model 2, qtfalls through period 14 to
a low of 0.36. One feature that is evident in the above example is that qtrises
if the earnings shock in period thas the opposite sign from that in period t− 1
and falls if the shock in period thas the same sign as that in period t−1.
5.Model Solution and Empirical Implications
5.1. Basic ResultsWe now analyze the implications of our model for prices. Since our model
has a representative agent, the price of the security is simply the value of the
security as perceived by the investor. In other words
Note that the expectations in this expression are the expectations of the in-
vestor who does not realize that the true process for earnings is a random
walk. Indeed, if the investor did realize this, the series above would be sim-
ple enough to evaluate since under a random walk, Et(Nt+j)=Nt, and price
equals Nt/δ. In our model, price deviates from this correct value because the
investor does not use the random walk model to forecast earnings, but rather
PENN
tt
= tt
++
++⋅⋅⋅
++ 12
1 δ () 1 δ^2.ππLH=<=^1334 ,A MODEL OF INVESTOR SENTIMENT 439Table 12.1tyt qt tyt qt
0 y 0.50
1 −y 0.80 11 y 0.74
2 y 0.90 12 y 0.56
3 −y 0.93 13 y 0.44
4 y 0.94 14 y 0.36
5 y 0.74 15 −y 0.74
6 −y 0.89 16 y 0.89
7 −y 0.69 17 y 0.69
8 y 0.87 18 −y 0.87
9 −y 0.92 19 y 0.92
10 y 0.94 20 y 0.72