Factor regression model

Within statistical factor analysis, the factor regression model, or hybrid factor model, is a special multivariate model with the following form:
:$\mathbf_n= \mathbf\mathbf_n+ \mathbf\mathbf_n +\mathbf+\mathbf_n$
where,

:$\mathbf_n$ is the $n$-th $G \times 1$ (known) observation.

:$\mathbf_n$ is the $n$-th sample $L_x$ (unknown) hidden factors.

:$\mathbf$ is the (unknown) loading matrix of the hidden factors.

:$\mathbf_n$ is the $n$-th sample $L_z$ (known) design factors.

:$\mathbf$ is the (unknown) regression coefficients of the design factors.

:$\mathbf$ is a vector of (unknown) constant term or intercept.

:$\mathbf_n$ 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 ($\mathbf_n= \mathbf\mathbf_n+ \mathbf+\mathbf_n$) and regression model ($\mathbf_n= \mathbf\mathbf_n +\mathbf+\mathbf_n$).

Alternatively, the model can be viewed as a special kind of factor model, the hybrid factor model
:$\begin & \mathbf_n = \mathbf\mathbf_n+ \mathbf\mathbf_n +\mathbf+\mathbf_n \\ = & \begin \mathbf & \mathbf \end \begin \mathbf_n \\ \mathbf_n\end +\mathbf+\mathbf_n \\ = & \mathbf\mathbf_n +\mathbf+\mathbf_n \end$
where, $\mathbf=\begin \mathbf & \mathbf \end$ is the loading matrix of the hybrid factor model and $\mathbf_n=\begin \mathbf_n \\ \mathbf_n\end$ are the factors, including the known factors and unknown factors.

Software

Open source software to perform factor regression is available

References

{{Reflist

Factor analysis
Latent variable models
Regression models