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.