News Contents: Factor Regression Model

2011-05-26

Factor Regression Model

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.

The 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.

Reference

Retrieved from : http://en.wikipedia.org/wiki/Factor_regression_model