Factor regression model

Within statistical factor analysis, the factor regression model,^[1] or hybrid factor model,^[2] is a special multivariate model with the following form:

${\displaystyle \mathbf {y} _{n}=\mathbf {A} \mathbf {x} _{n}+\mathbf {B} \mathbf {z} _{n}+\mathbf {c} +\mathbf {e} _{n}}$

where,

${\displaystyle \mathbf {y} _{n}}$ is the ${\displaystyle n}$-th ${\displaystyle G\times 1}$ (known) observation.

${\displaystyle \mathbf {x} _{n}}$ is the ${\displaystyle n}$-th sample ${\displaystyle L_{x}}$ (unknown) hidden factors.

${\displaystyle \mathbf {A} }$ is the (unknown) loading matrix of the hidden factors.

${\displaystyle \mathbf {z} _{n}}$ is the ${\displaystyle n}$-th sample ${\displaystyle L_{z}}$ (known) design factors.

${\displaystyle \mathbf {B} }$ is the (unknown) regression coefficients of the design factors.

${\displaystyle \mathbf {c} }$ is a vector of (unknown) constant term or intercept.

${\displaystyle \mathbf {e} _{n}}$ is a vector of (unknown) errors, often white Gaussian noise.

The factor regression model can be viewed as a combination of factor analysis model (${\displaystyle \mathbf {y} _{n}=\mathbf {A} \mathbf {x} _{n}+\mathbf {c} +\mathbf {e} _{n}}$) and regression model (${\displaystyle \mathbf {y} _{n}=\mathbf {B} \mathbf {z} _{n}+\mathbf {c} +\mathbf {e} _{n}}$).

Alternatively, the model can be viewed as a special kind of factor model, the hybrid factor model ^[2]

${\displaystyle {\begin{aligned}&\mathbf {y} _{n}=\mathbf {A} \mathbf {x} _{n}+\mathbf {B} \mathbf {z} _{n}+\mathbf {c} +\mathbf {e} _{n}\\=&{\begin{bmatrix}\mathbf {A} &\mathbf {B} \end{bmatrix}}{\begin{bmatrix}\mathbf {x} _{n}\\\mathbf {z} _{n}\end{bmatrix}}+\mathbf {c} +\mathbf {e} _{n}\\=&\mathbf {D} \mathbf {f} _{n}+\mathbf {c} +\mathbf {e} _{n}\end{aligned}}}$

where, ${\displaystyle \mathbf {D} ={\begin{bmatrix}\mathbf {A} &\mathbf {B} \end{bmatrix}}}$ is the loading matrix of the hybrid factor model and ${\displaystyle \mathbf {f} _{n}={\begin{bmatrix}\mathbf {x} _{n}\\\mathbf {z} _{n}\end{bmatrix}}}$ are the factors, including the known factors and unknown factors.

Open source software to perform factor regression is available.