Advances in Risk Management

260 MONITORING COVARIANCES OF ASSET RETURNS

Proof of Theorem 2

In this proof we make use of the same notation as in proving Theorem 1. Here the tilde
means that the corresponding quantities are calculated for the process with a covariance
matrix# ̃instead of#. Furthermore, from Proposition 1 of Bodnar (2004) it follows that

bˆ ̃−^1 ∼W−^1
q (n−p+q, ̃b−^1 ).
(a) Hence it follows that

E(vˆ ̃)=E(

√

n−p

√

Hˆ ̃(−)

22

bˆ ̃−^12 (wˆ ̃

M;q−wM;q))

From the independency ofHˆ ̃

(−)

22 and

bˆ ̃(Proposition 1 of Bodnar (2004)) and independency

ofHˆ ̃

(−)

22 andwˆ ̃M;q(see the proof of Theorem 1) the last equation transforms to:

E(vˆ ̃)=E







√n−p

√

Hˆ ̃(−)

22

√

n− 1

√

H ̃( 22 −)





E(

√

n− 1

√

H ̃( 22 −)b ̃ˆ−^12 (w ̃ˆM;q−wM;q)) (13.19)

Note thatH ̃( 22 −)/Hˆ ̃

(−)

22 ∼χ^2 n−p, and it holds that:

E







√n−p

√

Hˆ ̃(−)

22

√

n− 1

√

H ̃( 22 −)





=

√n−p

√

2



(n−p− 1

2

)



(n−p

2

)

Let consider the second term in the product (13.19). From the proof of Theorem 1 it follows
that

w ̃ˆM;q|(n−1)−^1 ˆ ̃b∼N



w ̃M;q,(n−1)

− (^1) bˆ ̃
H ̃( 22 −)


Thus
(w ̃ˆM;q−wM;q)|(n−1)−^1 bˆ ̃∼N

w ̃M;q−wM;q,(n−1)
− (^1) bˆ ̃
H ̃( 22 −)


Hence
√
H ̃( 22 −)√n− 1 ˆ ̃b−^12 (wˆ ̃M;q−wM;q)|(n−1)−^1 bˆ ̃ (13.20)
∼N(
√
n− 1
√
H ̃( 22 −)b ̃ˆ−^12 (w ̃M;q−wM;q),I)
As a result
E(
√
n− 1
√
H ̃( 22 −)b ̃ˆ−^12 (w ̃ˆM;q−wM;q))=
√
n− 1
√
H ̃( 22 −)b ̃ˆ−^12 (w ̃M;q−wM;q)
wherew ̃M;q=# ̃−^11 / 1 ′# ̃−^11. To calculate the unconditional density we make use of
Theorem 3.2.14 of Muirhead (1982), for example the fact thatˆ ̃b−
(^12)
= ̃b−
(^12)
T, where
T=(tij)i=1,...,q,j=1,...,iis aq×qlower triangular matrix witht^2 ii∼χ^2 n−p+q−i+ 1 i=1,...,q,