A generalization of the model discussed above is the factor analysis model. It is written, in a somewhat different notation, as
Уп = Л£ n + Є n,
where £n is a vector of (common) factors, yn is a vector of M indicators of these factors, Л is an M x k matrix of factor loadings, and єn is a vector of M errors. It is assumed that E(£n) = 0, E(£n£’n) – Ф, Е(єn) = 0, ¥ – Е(єпєn) is diagonal, and
Е(£пє n) = 0. Under this model, the covariance matrix of the observations is
2 – E(ynyn) = ЛФЛ’ + ¥.
The diagonality of ¥ implies that any correlation that may exist between different elements of y is solely due to the common factors £.
The unrestricted model is not identified, but frequently, identifying restrictions on the parameters are available from substantive theory. This is the case when an economic theory forms the base of the model and for every concept in that theory (e. g. productivity of a worker) several well-chosen variables are used that should reflect the concept in question as well as possible. In that case, the loadings of the indicators with respect to the factors (concepts) that they are supposed to reflect are free parameters, whereas the other loadings are fixed to zero. For a given set of restrictions (i. e. a given model), the program IDFAC of Bekker et al. (1994) can be used to check whether the model is identified.