The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 321 Wednesday, February 4, 2004 12:59 PM

Financial Econometrics: Model Selection, Estimation, and Testing 321

likelihood of the sample under the model. This idea is highly intuitive: If

one throws a coin and obtains 75 heads out of 100 trials, one believes

that the probabilities of head and tail are ³
₄ and ¹⁄₄ respectively and not

that one is experiencing a very unlikely run of heads.

Suppose that an n-sample x = (x 1 ,...,xn) with a joint density function

f(x/θ) is given. Suppose also that the density depends on a set of parame-

ters θ. The likelihood function is any function L(θ) proportional to f(x/θ):

L ()θ ∝f(x θ)

computed on the given sample. The MLE prescribes to choose those

parameters θthat maximize the likelihood. If the sample is formed by

independent draws from a density, then the likelihood is the product of

individual likelihoods:

n

f(x ⁄ θ)= ∏fx( iθ)

i = 1

n

L ()θ ∝∏fx( iθ)

i = 1

In this case, in order to simplify calculations, one normally com-

putes the log-likelihood defined as the logarithm of the likelihood, so

that the product is transformed into a sum. As the logarithm is an

increasing function, maximizing the likelihood or the log likelihood

gives the same results.

The MLE is an estimation method which conforms to general scientific

principles. From a statistical point of view, it has interesting properties. In

fact, it can be demonstrated that a ML estimator is an efficient estimator

(that is, an estimator which attains the minimum possible variance).

In the case of independent samples, the classical theory of ML esti-

mators can be resumed as follows. Let Yi, i = 1,2,...,n be n independent

variables with probability density functions fi(yi|θθθθ), where θθθθis a k-vector

of parameters to be estimated. Let the joint density of n independent

observations y = (yi) of the variables Yi be

n

f(y θ)= ∏fi(yi θ) = L(y θ)

i = 1

The log-likelihood function of the sample is