00Thaler_FM i-xxvi.qxd

So the key is to calculate Et(yt+j). Define

where

where Φtis the investor’s information set at time tconsisting of the

observed earnings series (y 0 , y 1 ,..., yt), which can be summarized

as (yt, qt).

Note that

The key insight is that

qt+j=Qqt+j−^1 ,

where Qis the transpose of the transition matrix for the states

(st+j, yt+j), i.e.,

where, for example,

Pr(st+j=2,yt+j=yt
st+j− 1 =1,yt+j− 1 =yt)=λ 1 πH.

Therefore,

qQqQ

q

q

tj jt j

t

t

+ ==

−





























0

1

0

.

′=

−−− −

−− − −

−− −−

Q

() ( ) ( ) ( )

() ( ) ()( ) ( )

() ( )( ) () ( )

() ( ) ( ) ()( )

LLH H

LL H H

LLH H

12 3 4

11 11 1

21 1 1 1

31111

11 1 1

111 1

22 2 2

λπ λ π λπ λ π

λπ λπ λπ λπ

λπ λ π λ π λ π

(() 41 λπ222 2(−−−−LL) λπ ( 11 λ π)()()H H 1 λπ

Pr( )

(, , , ).

yy qq qtj+ ==+=′t t tj++tj tj+

′=

Φ 13

1010

γ

γ

qsyy

qsyy

qsyy

qsyy

tj

tj tj t t

tj

tj tj t t

tj

tj tj t t

tj

tj tj t t

1

2

3

4

1

1

2

2

+

++

+

++

+

++

+

++

===

===−

===

===−

Pr( , ),

Pr( , ),

Pr( , ),

Pr( , ),

Φ

Φ

Φ

Φ

qtj+ =′(qqqq 1234 tj tj tj tj++++,,,),

A MODEL OF INVESTOR SENTIMENT 451