Factor regression model

The factor regression model or hybrid factor model is a special multivariate model with the following form.

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

where,

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

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

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

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

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

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

$\mathbf {e} _{n}$ is (unknown) error or white Gaussian noise.

Factor Regression Model, Factor Analysis Model and Regression Model

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

Alternatively, the model can be viewed as a special kind of factor model, the hybrid factor model, where, part of the factor matrix are already known.

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

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