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.
