00Thaler_FM i-xxvi.qxd

In the case of the parameters used for the table in section 4.2.,

q

• 
  

  =0.28 and q–=0.95.
  There is no loss of generality in restricting the support of qtto the
  interval [q

  
  

   ,q–]. Certainly, an investor can have prior beliefs that lie

  
  

  outside this interval, but with probability one, qtwill eventually be-
  long to this interval, and will then stay within the interval forever.
  We are now ready to begin the main argument of the proof. Un-
  derreaction means that the expected return following a positive
  shock should exceed the expected return following a negative shock.
  In other words,

  
  

Et(Pt+ 1 −Pt
yt=+y)−Et(Pt+ 1 −Pt
yt=−y)>0.

Overreaction means that the expected return following a series of

positive shocks is smaller than the expected return following a series

of negative shocks. In other words, there exists some number J≥1,

such that for all j≥J,

Et(Pt+ 1 −Pt
yt=yt− 1 =...=yt−j=y)

−Et(Pt+ 1 −Pt
yt=yt− 1 =...=yt−j=−y)<0.

Proposition 2 provides sufficient conditions on p 1 and p 2 so that

these two inequalities hold. A useful function for the purposes of

our analysis is

f(q)=Et(Pt+ 1 −Pt
yt=+y, qt=q)−Et(Pt+ 1 −Pt
yt=−y, qt=q).

The function f(q) is the difference between the expected return

following a positive shock and that following a negative shock,

where we also condition on qtequaling a specific value q. It is sim-

ple enough to write down an explicit expression for this function.

Since

we find

E( , ) ()

()() ()

()(()())

ttPPy yqqtt t

y

yp q

y

yp pq yp q yp q

ypq p yp q q

+ −=+== +













+−− − − +













=−+ +

12

12 2 2

21

1

2 2

1

2

1

2

22

δ

δ

∆

∆∆

∆∆

PP

y

yyppqypqq

yypqq

tt

t

tt tttt

tttt

+

+

++

++

−= + − − − −

−− −

1

1

112 21

121

δ

()( )()

()(),

454 BARBERIS, SHLEIFER, VISHNY