suppose that investors view the growth rate of dividends as a parameter
that is not only unknown but also changing over time. The optimal esti-
mate of the parameter closely resembles a distributed lag on past one-
period dividend growth rates, with declining weights. If dividends rise
steadily over several periods, the investor’s estimate of the current dividend
growth rate also rises, leading him to forecast higher dividends in the future
as well. Analogously, in our model, a series of positive shocks to earnings
leads the investor to raise the probability that earnings changes are cur-
rently being generated by the trending regime 2, leading him to make more
bullish predictions for future earnings.
4.2. A Formal ModelWe now present a mathematical model of the investor behavior described
above, and in section 5, we check that the intuition can be formalized. Sup-
pose that earnings at time tare Nt=Nt− 1 +yt, where ytis the shock to
earnings at time t, which can take one of two values,+yor −y. Assume that
all earnings are paid out as dividends. The investor believes that the value
of ytis determined by one of two models, Model 1 or Model 2, depending
on the “state” or “regime” of the economy. Models 1 and 2 have the same
structure: they are both Markov processes, in the sense that the value taken
by ytdepends only on the value taken by yt− 1. The essential difference be-
tween the two processes lies in the transition probabilities. To be precise,
the transition matrices for the two models are:
Model 1 yt+ 1 =yyt+ 1 =−y Model 2 yt+ 1 =yyt+ 1 =−y
yt=y πL 1 −πL yt=y πH 1 −πH
yt=−y 1 −πL πL yt=−y 1 −πH πH
The key is that πLis small and πHis large. We shall think of πLas falling
between zero and 0.5, with πHfalling between 0.5 and one. In other words,
under Model 1 a positive shock is likely to be reversed; under Model 2, a
positive shock is more likely to be followed by another positive shock.
The investor is convinced that he knows the parameters πLand πH; he is
also sure that he is right about the underlying process controlling the switch-
ing from one regime to another, or equivalently from Models 1 to 2. It, too,
is Markov, so that the state of the world today depends only on the state of
the world in the previous period. The transition matrix is
st+ 1 = 1 st+1= 2
st= 11 −λ 1 λ 1
st= 2 λ 2 1 −λ 2
The state of the world at time tis written st. If st=1, we are in the first
regime and the earnings shock in period t, yt, is generated by Model 1;
A MODEL OF INVESTOR SENTIMENT 437