# ⓘ Factor regression model. The factor regression model, or hybrid factor model, is a special multivariate model with the following form: y n = A x n + B z n + c + ..

## ⓘ Factor regression model

The factor regression model, or hybrid factor model, is a special multivariate model with the following form:

y n = A x n + B z n + c + e n {\displaystyle \mathbf {y} _{n}=\mathbf {A} \mathbf {x} _{n}+\mathbf {B} \mathbf {z} _{n}+\mathbf {c} +\mathbf {e} _{n}}

where,

y n {\displaystyle \mathbf {y} _{n}} is the n {\displaystyle n} -th G × 1 {\displaystyle G\times 1} known observation. x n {\displaystyle \mathbf {x} _{n}} is the n {\displaystyle n} -th sample L x {\displaystyle L_{x}} unknown hidden factors. A {\displaystyle \mathbf {A} } is the unknown loading matrix of the hidden factors. z n {\displaystyle \mathbf {z} _{n}} is the n {\displaystyle n} -th sample L z {\displaystyle L_{z}} known design factors. B {\displaystyle \mathbf {B} } is the unknown regression coefficients of the design factors. c {\displaystyle \mathbf {c} } is a vector of unknown constant term or intercept. e n {\displaystyle \mathbf {e} _{n}} is a vector of unknown errors, often white Gaussian noise.

## 1. Relationship between factor regression model, factor model and regression model

The factor regression model can be viewed as a combination of factor analysis model y n = A x n + c + e n {\displaystyle \mathbf {y} _{n}=\mathbf {A} \mathbf {x} _{n}+\mathbf {c} +\mathbf {e} _{n}} and regression model y n = B z n + c + e n {\displaystyle \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

y n = A x n + B z n + c + e n = {\displaystyle \mathbf {f} _{n}={\begin{bmatrix}\mathbf {x} _{n}\\\mathbf {z} _{n}\end{bmatrix}}} are the factors, including the known factors and unknown factors.

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