The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 197 Wednesday, February 4, 2004 3:00 PM

Concepts of Probability 197

THE REGRESSION FUNCTION

Given a probability space (Ω,ℑ,P), consider a set of p +1 random variables.

Let’s suppose that the random vector {XZ 1 ... Zp} ≡{XZ}, Z = {Z 1 ... Zp}

has the joint multivariate probability density function:

fxz( 1 ...zp)= fx( ,z), z = {z 1 ...zp}

Let’s consider the conditional density

fx( z 1 , ...,zp) = fx( ,z)

and the marginal density of Z,

∞

fz()z = ∫ fx( ,z)dx

• ∞

Recall from an earlier section that the joint multivariate density f(x,z)

factorizes as

fx( ,z) = fx( z)fz ()z

Let’s consider now the conditional expectation of the variable Xgiven Z

= z = {z 1 ... zp}:

∞

g()z = EX[ Z = z] = ∫ vf v( z)dv

• ∞

The function g, that is, the function which gives the conditional expec-

tation of Xgiven the variables Z, is called the regression function. Oth-

erwise stated, the regression function is a real function of real variables

which is the locus of the expectation of the random variable Xgiven

that the variables Z assume the values z.

Linear Regression

In general, the regression function depends on the joint distribution of

[XZ 1 ... Zp]. In financial econometrics it is important to determine

what joint distributions produce a linear regression function. It can be