start by fixing λ 1 =0.1 and λ 2 =0.3. These numbers are small to ensure that
regime switches do not occur very often and λ 2 >λ 1 to represent the in-
vestor’s belief that the world is in the Model 1 regime more often than in the
Model 2 regime.
Now that λ 1 and λ 2 have been fixed, we want to know the range of val-
ues of πLand πHfor which the conditions for underreaction and overreac-
tion both hold. Given the way the model is set up, πLand πHare restricted
to the ranges 0<πL<0.5 and 0.5<πH<1. We evaluate the conditions in
Proposition 2 for pairs of (πL, πH) where πLranges from zero to 0.5 at in-
tervals of 0.01 and πHranges from 0.5 to one, again at intervals of 0.01.
The graph at the left of figure 12.1 marks with shading all the pairs for
which the sufficient conditions hold. We see that underreaction and overre-
action hold for a wide range of values. On the other hand, it is not a trivial
result: There are many parameter values for which at least one of the two
phenomena does not hold.
The graph shows that the sufficient conditions do not hold if both πLand
πHare near the high end of their feasible ranges, or if both πLand πHare near
the low end of their ranges. The reason for this is the following. Suppose
both πLand πHare high. This means that whatever the regime, the investor
believes that shocks are relatively likely to be followed by another shock of
the same sign. The consequence of this is that overreaction certainly obtains,
although underreaction might not. Following a positive shock, the investor
on average expects another positive shock and since the true process is a ran-
dom walk, returns are negative, on average. Hence the average return follow-
ing a positive shock is lower than that following a negative shock, which is a
characterization of overreaction rather than of underreaction.
442 BARBERIS, SHLEIFER, VISHNY
πLAllowable region for πL and πHπH0 0.1 0.2 0.31.00.80.90.70.50.60.4
πLOVERREACTION onlyπH0 0.1 0.2 0.31.00.80.90.70.50.60.4
πLUNDERREACTION onlyπH0 0.1 0.2 0.31.00.80.90.70.50.60.4Figure 12.1. Shaded area in graph at left marks the [πL, πH] pairs which satisfy the
sufficient conditions for both underreaction and overreaction, when λ 1 =0.1 and
λ 2 =0.3. Graph in middle (at right) shows the [πL, πH] pairs that satisfy the condi-
tion for overreaction (underreaction) only. πL(πH) is the probability, in the mean-
reverting (trending) regime, that next period’s earnings shock will be of the same
sign as last period’s earnings shock. λ 1 and λ 2 govern the transition probabilities be-
tween regimes.