Factor regression model

Within statistical factor analysis, 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 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 ($$ \mathbf{y}_n= \mathbf{A}\mathbf{x}_n+  \mathbf{c}+\mathbf{e}_n $$) 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

\begin{align} & \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{align} $$ where, $$ \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.

Software
Open source software to perform factor regression is available.