Anon

388 The Basics of financial economeTrics

Given a square n × n matrix A, the matrix dgA is the diagonal matrix
extracted from A. The diagonal matrix dgA is a matrix whose elements are
all zero except the elements on the diagonal which coincide with those of
the matrix A:

=

⋅⋅⋅

⋅⋅⋅

⋅⋅⋅⋅

⋅⋅ ⋅⋅

⋅⋅ ⋅⋅

⋅⋅⋅





      





      

⇒=

⋅⋅⋅

⋅⋅⋅

⋅⋅⋅⋅

⋅⋅ ⋅⋅

⋅⋅ ⋅⋅

⋅⋅⋅





      





      

aa a

aa a

aa a

a

a

a

AAdg

00

00

00

n

n

nn nn nn

11 12 1

21 22 2

12

11

22

The trace of a square matrix A is the sum of its diagonal elements:

=∑

=

trA aii

i

n

1
A square matrix is called symmetric if the elements above the diago-
nal are equal to the corresponding elements below the diagonal: aij = aji. A
matrix is said to be a skew-symmetric if the diagonal elements are zero and
the elements above the diagonal are the opposite of the corresponding ele-
ments below the diagonal: aij = –aji.
The most commonly used symmetric matrix in financial econometrics
and econometrics is the covariance matrix, also referred to as the variance-
covariance matrix. For example, suppose that there are N risky assets and
that the variance of the excess return for each risky asset and the covari-
ances between each pair of risky assets are estimated. As the number of
risky assets is N, there are N^2 elements, consisting of N variances (along the
diagonal) and N^2 − N covariances. Symmetry restrictions reduce the number
of independent elements. In fact the covariance between risky asset i and
risky asset j will be equal to the covariance between risky asset j and risky
asset i. Hence, the variance-covariance matrix is a symmetric matrix.

Determinants

Consider a square, n × n, matrix A. The determinant of A, denoted A, is
defined as follows:

A=−()()

=

∑^1 ∏

1

1

tj j

ij

i

n

,...,n a