00Thaler_FM i-xxvi.qxd

Appendix A: Covariance and Variance Calculations

for the Basic Model

Covariances and Variances of Section 2.B,

and Proof of Proposition 3

The calculations of the covariances and variances presented in Section 2.B,
and the proof of Proposition 3 follow by routine application of the proper-
ties of multivariate normal variables. For details, see Daniel, Hirshleifer,
and Subrahmanyam (1998).

Proofs of Some Claims in Section B.3

Part 1 of Proposition 4:Denote the date 2 mispricing as M 2 .Suppress-

ing arguments on PR 2 (s 2 ) and PC 2 (s 2 ), we have that M 2 =PR 2 −PC 2 =

−E[θ−PC 2 (s 2 )s 1 , s 2 ]. By the properties of normal random variables,

this implies that the variable x=θ−PC 2 +M 2 , which is the residual

from the regression of θ−PC 2 on s 1 and s 2 , is orthogonal to s 1 and

s 2. Suppose we pick a variable y=f(s 1 , s 2 ) which is orthogonal to

M 2. Such a variable will be orthogonal to x, so that we have cov(θ−

PC 2 +M 2 , y)=0. Since cov(M 2 , y)=0 by construction, it follows from

the linearity of the covariance operator that cov(θ−PC 2 , y)=0. A con-

verse argument shows that if we pick a variable y′=g(s 1 , s 2 ) which is

orthogonal to the post event return θ−PC 2 then cov(M 2 , y′)=0.

Thus, all functions of s 1 and s 2 are orthogonal to M 2 if and only if

they are orthogonal to the post event return θ−PC 2.

For the specific case when the event depends linearly on s 2 , by (6),

(A1)

Since s 2 ≡θ+η, from the above expression, it immediately follows

that cov(P 3 −P 2 , s 2 )=0, thus showing that events that depend only

on s 2 are non-selective.

Part 2 of Proposition 4:By standard results for calculating conditional

variances of normal variables (Anderson (1984)),

(A2)

which is positive under overconfidence (σC^2 >σ^2 
).

cov( , ) cov( , )

()

[( ) ]( )

,

PPss PPsPP

pC

Cp p

3221 32210

24 2 2

22 2 22 2 2

−=−−

=

−

++ +



σσ σ σ

σσ σ σσ σ σ

θ

θθ θ

PP Cp p C

Cp Cp

32

22 22 22

−= 22 2 22

−−

++

σσθ σσ σση

σσ σ σσ

θθ

θ

()

.

490 DANIEL, HIRSHLEIFER, SUBRAHMANYAM