242 The Basics of financial economeTrics
between the monthly returns series for the 500 stocks and some macroeco-
nomic variables such as the change in gross domestic product, the rate of
inflation, and the yield spread between high-grade and low-grade corporate
bonds. This is a multiple regression.
Properties of Factor Models
Let’s now discuss some properties of factor models. Consider a factor model
as defined in equation (12.3) with the assumptions given by equation (12.7).
We will adopt the following notation:
- Ω denotes the covariance matrix of factors, Ω=cov'()fEtt= ()fft.
- Σ=Ey()()tt−ay()−a' denotes the covariance matrix of the observed
variables. - Ψ=E()εεtt' denotes the covariance matrix of the residuals.
The three covariance matrices of factors, observed variables, and residu-
als are supposed to be constant matrices, independent of t. From the defini-
tion of covariance we can write:
Σ=Ey()()tt−ay()−aE''= ()()()BfBftt+ Ef()tt'2 ε++E()εεtt'Because we assume that factors and residuals are uncorrelated (i.e.,
Ef()tt'ε = 0 ), we can write:
(^) ΣΩ=+BB' Ψ (12.8)
Equation (12.8) is the fundamental relationship that links the covari-
ance matrix of the observed variables to the covariance matrices of factors
and residuals. It is sometimes referred to as the fundamental theorem of
factor models.
From equation (12.3) we can see that a factor model is not identified.
This means that observed variables do not univocally determine factors and
model parameters. To see this, consider any nonsingular q × q matrix T.
From matrix algebra we know that a nonsingular matrix T is a matrix that
admits an inverse matrix T–1 such that T–1T = I. Given that multiplication
by the identity matrix leaves any matrix unchanged and given that matrix
multiplication is associative, we can write
yatt=+Bf+=εεttaB++If tt=+aBTT−−^11 fa+=εt +()BT (()Tftt+ε
Hence, given any set of factors ft, if we multiply them by any nonsingu-
lar q × q matrix T, we obtain a new set of factors gt = Tft. If we multiply the