Anon

42 The Basics of financial economeTrics

The Multiple Linear Regression Model

The multiple linear regression model for the population is of the form

y = β 0 + β 1 x 1 + β 2 x 2 +... + βkxk + ε (3.1)

where we have β 0 =constant intercept
β 1 ,... , βk=regression coefficients of k explanatory or
independent variables
β=model error

In vector notation, given samples of dependent and explanatory or inde-
pendent variables, we can represent equation (3.1) as

y = Xβ + ε (3.2)

where y is an n × 1 column vector consisting of the n observations of the
dependent variable, that is,

= 





















y

y

yn

1

(3.3)

where X is a n × (k + 1) matrix consisting of n observations of each of the
k independent variables and a column of ones to account for the vertical
intercepts β 0 such that

...

 

...

=





























X

xx

xx

xx

1

1

1

k

k

n nk

(^111)
21 2
1

(3.4)

The (k + 1) regression coefficients including intercept are given by the k + 1
column vector:

β= 

β

β





















k

0

(3.5)

Each observation’s residual is represented in the column vector ε:

ε= 

ε

ε





















n

1

(3.6)