Anon

238 The Basics of financial economeTrics

Explicit form. We can write a factor model explicitly as a set of N

equations:

yabf bf

yabf

ttqqtt

it iit

11 11 11 1

11

=+ ++ +

=+ ++







ε

bbf iNtT

yabf

iqqt it

Nt NN

+= =

=+

ε 11

1

, ...,, , ...,



11 tN++ bfqqtN+ε t

(12.2)

The ai are the constant terms, the coefficients bjt are called factor load-

ings, the fjt are the hidden factors, and the εit are the error terms or the

residuals.

Vector form. We can write a factor model more compactly in vector

form as follows:

yatt=+Bf +=εt,,tT...,1 (12.3)

where yytt=[] 1 yNt ' is the N × 1 vector of observed variables at time t

aa=[] 1 aN' is the N × 1 vector of constant terms

fftt= 1 fqt' is the q × 1 vector of factors at time t

=

























B

bb

bb

q

NNq

11 1

1







is the N × q matrix of factor loadings

and εεtt=[] 1  'εNt is the N × 1 vector of residuals

Matrix form. Equations (12.2) and (12.3) represent a factor model in

terms of variables. However, we can also represent a factor model in

terms of realizations of sample data in the following matrix form which

is analogous to the matrix form of regression, equation (12.2):

Y = FC + E (12.4)