Anon

Model Estimation 283

A normal distribution has the following multivariate probability distri-
bution function:

()

()

Σ= ()()

πΣ

−−Σ−











Na, yaya− 

1

2

exp

1

2

N '

2

1

2

1

hence

loglog

loglog

LPy

TN T

t

t

T

()= ()()

=− ()− ()

=

∑

1

2

2

2

πΣ−−−()−()−

=

∑

1

2

1

1

yattya

t

T

'Σ

(13.19)

From equation (13.9), we know that ΣΨ=+BB' ; we can therefore
determine the parameters B,Ψ by maximizing the log-likelihood with
respect to these parameters.
ML is a widely used estimation method. Note, however, that the ML
method implies that one knows the form of the distribution, otherwise, one
cannot compute the likelihood.

in StrUMental VariableS

The instrumental variables (IV) estimation approach is a strategy for chang-
ing the variables of an estimation problem that cannot be solved with any
of the above methods. To understand this approach, consider that in all
regression models discussed thus far, it is imperative that regressors and
errors are uncorrelated. This condition ensures that the dependent variable
is influenced independently by the regressors and the error terms. Regres-
sors are said to be exogenous in the regression model. If this condition is not
satisfied, the OLS estimator of regression parameters is biased.
In practice, in regressions commonly used in financial modeling the con-
dition of independence of errors and regressors is often violated. This hap-
pens primarily because errors include all influences that are not explicitly
accounted for by regressors and therefore some of these influences might
still be correlated with regressors. OLS estimation is no longer suitable
because the regression problem is not correctly specified.
A possible solution to this problem is given by the IV approach, which
solves a different regression problem. Consider the usual regression equa-
tion (13.4) where we allow some regressors x to be correlated with the
errors. Suppose there is a vector of variables z of size equal to k such that