The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 323 Wednesday, February 4, 2004 12:59 PM

Financial Econometrics: Model Selection, Estimation, and Testing 323

Suppose that one wants to estimate a regressive model Y = aX + b +

ε from a sample of n pairs (yi, xi). The linear regressive model is charac-

terized by the two parameters a and b, which can be estimated with the

Ordinary Least Square (OLS) method. The OLS computes the straight

line that minimizes the sum of the squares of the distances of the sam-

ples from that straight line.

In a probabilistic setting, the estimates aˆ , bˆ of the two parameters a

and b depend on the sample. They obey a distribution that depends on

the distribution of the errors ε. It can be demonstrated that, if the errors

are normally distributed IID sequences than the OLS estimators aˆ , bˆ are

unbiased ML estimators. They are therefore efficient estimators. If the

errors are IID variables with finite variance but are not normally distrib-

uted, then the OLS estimators aˆ , bˆ of the two parameters a and b are

unbiased estimators but not necessarily ML estimators.

The OLS estimation procedure is very general. It can be demon-

strated that any linear unconstrained autoregressive model with normal

innovations can be estimated with OLS estimators and that the ensuing

estimators are unbiased ML estimators and thus efficient estimators.

One can also estimate directly the moments of a distribution. In par-

ticular, in a multivariate environment we have to estimate the variance-

covariance matrix Ω. It can be demonstrated that the variance-covari-

ance matrix can be estimated through empirical variances and covari-

ances. Consider two random variables X,Y.

The empirical covariance between the two variables is defined as

follows:

n

σˆ XY

1

, = ---∑(Xi – X)(Yi – Y)

ni = 1

where the empirical means of the variables are:

n

1

X = ---∑ Xi

ni = 1

n

1

Y = ---∑ Yi

ni = 1

The correlation coefficient is the covariance normalized with the

product of the respective empirical standard deviations: