The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 198 Wednesday, February 4, 2004 3:00 PM

198 The Mathematics of Financial Modeling and Investment Management

demonstrated that joint normal distributions produce a linear regression

function. Consider the joint normal distribution

-^1 ---
1
f()v = [( 2 π)
n
ΣΣΣΣ]

2

exp – ---(v – μμμμ)

T

ΣΣΣΣ

• 1
  (v – μμμμ)
  2

where parameters are those defined in an earlier section in this chapter.

Let’s partition the parameters as follows:

 x  μx σxx, σz,x

v= , μ= , Σ=  

 z  μz  σx,z Σz 

where μx, μz are respectively a scalar and a p-vector of expected values,

σx,x, σx,z, σz,x, and Σz are respectively a scalar, p-vectors and a p×p

2

matrix of variances and covariances and σxx= σ

2

, x , σzi,zi = σzi. It can

be demonstrated that the variable (X|Z = z) is normally distributed with

the following parameters:

(XZ = z) ∼N[μ –^1

x – (Σ

– (^1) σ
z z,x)' (μz – z), σxx, – σx,zΣz σz,x +]
From the above expression we can conclude that the conditional
expectation is linear in the conditioning variables. Let’s call
α= μx – (Σ– z^1 σz,x)' μz and β = Σ–z^1 σz,x
We can therefore write
g()z = EX[ Z = z]= α β′+ z
If the matrix Σis diagonal, the random variables (X,Z 1 ,...,Zp) are
independent, such that σz,x= 0 and β= Σz

• 1
  σz,x = 0 and therefore the
  regression function is a constant that does not depend on the condition-
  ing variables. If the matrix Σz is diagonal but σx,z, σz,xdo not vanish,
  then the linear regression takes the following form

p σ p σ

xz, i xz, i

g()z = EX[ Z = z]= μx – ∑-----------μ + ∑-----------zi

i= 1 σ^2

zi

zi i=^1 σ^2 zi