The factor regression model, or hybrid factor model, is a special multivariate model with the following form.
where,
is the -th (known) observation.
is the -th sample (unknown) hidden factors.
is the (unknown) loading matrix of the hidden factors.
is the -th sample (known) design factors.
is the (unknown) regression coefficients of the design factors.
is a vector of (unknown) constant term or intercept.
is a vector of (unknown) errors, often 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

Factor analysis

Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved, uncorrelated variables called factors. In other words, it is possible, for example, that variations in three or four observed variables...

 model () and regression model ().

Alternatively, the model can be viewed as a special kind of factor model, the hybrid factor model
where, is the loading matrix of the hybrid factor model and are the factors, including the known factors and unknown factors.