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