Page - 47 - in The Austrian Business Cycle in the European Context

47 A different method of estimating factor models is the principal component approach. It was originally developed in order to re- duce the variance of large data sets by singling out a common component that is capable of representing the largest part of the total variation, by constructing simple linear combinations of it. Each individual time series is represented by one or more common components and some idiosyncratic variation. In the static case, the idiosyncratic parts of all observed time series are either as- sumed to be mutually uncorrelated (this is called the "strict factor model") or they are allowed to be weakly cross-correlated and heteroskedastic ("approximate factor models")71• A proper identifi- cation of the common factors with relaxed assumptions - as it is done in the approximate factor model - requires that the number of time series considered exceeds the time dimension by far and theoretically goes to infinity. This allows for the existence of idio- syncratic common movements between business sectors, as long as they are not too dominant. In the latter case they would enter into the common component. The dynamic version of this type of factor model approach seems to be ideal for business cycle analysis. It allows the common fac- tors to move auto-regressively and the observed time series can be classified as leading, lagging or coincident according to the common component represented by them. Again, some weak cross-correlation between the idiosyncratic components is al- lowed and the factors are required to be uncorrelated among each other. Forni et al. (2000) suggested a generalised dynamic factor model where the dynamic factors are identified in the fre- quency domain. Instead of observing the cross-correlation matrix in order to identify the dynamic factors, they focus is on the spec- tral density matrix. 71 In the approximate factor model case, a weak correlation is allowed even be- tween the factors and the idiosyncratic components. See Breitung - Eickmeier (2005).