Anon

166 The Basics of financial economeTrics

We can conclude that all regression slope estimates are highly significant; the
intercept estimates are insignificant in all cases. There is a considerable differ-
ence between the robust (0.40) and the nonrobust (0.45) regression coefficient.

Robust Estimation of Covariance and Correlation Matrices

Variance-covariance matrices are central to financial modeling. In fact, the
estimation of the variance-covariance matrices is critical for portfolio man-
agement and asset allocation. Suppose the logarithm of returns is a multi-
variate random vector written as

rt = μ + εt

The random disturbances εt is characterized by a covariance matrix Ω.
The correlation coefficient between two variables X and Y is defined as:

ρ

σ

σσ

XY

XY

X

XY

XY

XY

,

,

(,)

(,)

() ()

=

==

corr

cov

varvar YY

The correlation coefficient fully represents the dependence structure of
multivariate normal distribution. More in general, the correlation coeffi-
cient is a valid measure of dependence for elliptical distributions (i.e., distri-
butions that are constants on ellipsoids). In other cases, different measures
of dependence are needed (e.g., copula functions).^3
The empirical covariance between two variables, X and Y, is defined as

σˆXY, ()i ()

i

N

=N− XX−−YYi

=

∑

1

(^11)
where Xi and Yi are N samples of the variables X and Y and:

X

N

XY

N

iiY

i

N

i

N

==

= =

∑∑

11

1 1

are the empirical means of the variables.

(^3) Paul Embrechts, Filip Lindskog, and Alexander McNeil, “Modelling Dependence
with Copulas and Applications to Risk Management,” in Handbook of Heavy
Tailed Distributions in Finance, ed. S. T. Rachev (Amsterdam: Elsevier/North-
Holland, 2003).