The factor regression model, or hybrid factor model, is a special multivariate model with the following form.
y n = A x n + B z n + c + e n where,
y n is the
n -th
G × 1 (known)
observation.
x n is the
n -th sample
L x (unknown) hidden factors.
A is the (unknown) loading matrix of the hidden factors.
z n is the
n -th sample
L z (known) design factors.
B is the (unknown) regression coefficients of the design factors.
c is a
vector of (unknown)
constant term or intercept.
e n is a vector of (unknown) errors, often white Gaussian noise.
Relationship between factor regression model, factor model and regression model
The factor regression model can be viewed as a combination of factor analysis model ( y n = A x n + c + e n ) and regression model ( y n = B z n + c + e n ).
Alternatively, the model can be viewed as a special kind of factor model, the hybrid factor model
y n = A x n + B z n + c + e n = [ A B ] [ x n z n ] + c + e n = D f n + c + e n where, D = [ A B ] is the loading matrix of the hybrid factor model and f n = [ x n z n ] are the factors, including the known factors and unknown factors.
Factor regression software is available from here.
