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

$\begin{array}{rl}& {\mathbf{y}}_n=\mathbf{A}{\mathbf{x}}_n+\mathbf{B}{\mathbf{z}}_n+\mathbf{c}+{\mathbf{e}}_n\\ =& \left[\begin{array}{cc}\mathbf{A}& \mathbf{B}\end{array}\right]\left[\begin{array}{c}{\mathbf{x}}_n\\ {\mathbf{z}}_n\end{array}\right]+\mathbf{c}+{\mathbf{e}}_n\\ =& \mathbf{D}{\mathbf{f}}_n+\mathbf{c}+{\mathbf{e}}_n\end{array}$

where, $\mathbf{D}=\left[\begin{array}{cc}\mathbf{A}& \mathbf{B}\end{array}\right]$ is the loading matrix of the hybrid factor model and ${\mathbf{f}}_n=\left[\begin{array}{c}{\mathbf{x}}_n\\ {\mathbf{z}}_n\end{array}\right]$ are the factors, including the known factors and unknown factors.

Software

Open source software to perform factor regression is available.