386 The Basics of financial economeTrics
Here are two examples. Suppose that we let wn be a risky asset’s weight
in a portfolio. Assume that there are N risky assets. Then the following vec-
tor, w, is a row vector that represents a portfolio’s holdings of the N risky
assets:
w = [w 1 w 2... wN]As a second example of a row vector, suppose that we let rn be the
excess return for a risky asset. (The excess return is the difference between
the return on a risky asset and the risk-free rate.) Then the following row
vector is the excess return vector:
r = [r 1 r 2... rN]If the vector components are arranged in a column, then the vector is
called a column vector.
For example, we know that a portfolio’s excess return will be affected
by what can be different characteristics or attributes that affect all asset
prices. A few examples would be the price-earnings ratio, market capitaliza-
tion, and industry. Let us denote for a particular attribute a column vector,
a, that shows the exposure of each risky asset to that attribute, denoted ai:
=
a
aaaN1
2
Matrices
An n × m matrix is a bi-dimensional ordered array of n × m numbers. Matri-
ces are usually indicated with boldface uppercase letters. Thus, the generic
matrix A is an n × m array of the form
=
⋅⋅
⋅⋅⋅⋅⋅
⋅⋅
⋅⋅⋅⋅⋅
⋅⋅
aaaaaaaaaA
jmiijimnnjnm1,11,1,,1 ,,,1 ,,