00Thaler_FM i-xxvi.qxd

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Proof of Proposition 2:Before we enter the main argument of the
proof, we present a short discussion of the behavior of qt, the prob-
ability assigned by the investor at time tto being in regime 1. Sup-
pose that the earnings shock at time t+1 is of the opposite sign to
the shock in period t. Let the function ∆–(qt) denote the increase in
the probability assigned to being in regime 1, that is,


Similarly, the function∆–(q) measures the size of the fall in qtif the
period t+1 earnings shock should be the same sign as that in period
t, as follows:

By checking the sign of the second derivative, it is easy to see that
both ∆


  • (q) and ∆–(q) are concave. More important, though, is the sign
    of these functions over the interval [0, 1]. Under the conditions
    πL<πHand λ 1 +λ 2 <1, it is not hard to show that ∆


  • (q) ≥0 over an
    interval [0, q–], and that ∆–(q)≥0 over [q




  • , 1], where q




  • and q–satisfy
    0 <q




  • <q–<1.
    The implication of this is that over the range [q




  • , q–], the following
    is true: if the time tearnings shock has the same sign as the time
    t+1 earnings shock, then qt+ 1 <qt, or the probability assigned to
    regime 2 rises. If the shocks are of different signs, however, then
    qt+ 1 >qt, and regime 1 will be seen as more likely.
    Note that if qt∈ [q




  • ,q–], then qτ∈[q




  • ,q–] for ∀τ> t. In other
    words, the investor’s belief will always remain within this interval. If
    the investor sees a very long series of earnings shocks, all of which
    have the same sign, qtwill fall every period, tending towards a limit
    of q




  • . From the updating formulas, this means that q




  • satisfies




Similarly, suppose that positive shocks alternate with negative
ones for a long period of time. In this situation, qtwill rise every pe-
riod, tending to the upper limit q–, which satisfies

q

qq
qq q q

=

−+− −
−+− −++− − −

(( ) ( ))( )
(( ) ( ))( ) ( ( )( ))( )

L.
LH

111
111 111

12
12 1 2

λλ π
λλ πλ λ π

q

qq

qq q q

=

−+−

−+− ++− −

(( ) ( ))

(( ) ( )) ( ( )( ))

.

L
LH

11

11 11

12
12 1 2

λλ π

λλ πλ λ π

∆()
(( ) ( ))
(( ) ( )) (( ( )( ))

.

,
L
LH

qqq

q
qq
qq q q

=−ttyyqqttt

=−
−+−
−+− + +− −

+== (^1) +
12
12 1 2
1
11
11 11

λλ π
λλ πλ λ π
∆()
(( ) ( ))( )
(( ) ( ))( ) (( ( )( ))( )
.
,
L
LH
qq q
qq
qq q q
q
=−ttyyqqttt


−+− −
−+− −+ +− − −

+=−= (^1) +
12
12 1 2
1
111
111 111

λλ π
λλ π λ λ π
A MODEL OF INVESTOR SENTIMENT 453

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