Page - 45 - in The Austrian Business Cycle in the European Context

45 ing, in monetary policy analysis and for business cycle analysis69• Their big advantage is that they try to extract common dynamics from a large number of different time series without running out of degrees of freedom. Even if the number of time series observations is lower than the number of variables, this does not constitute a problem. By filtering out common dynamics, the time series are cleaned for idiosyncratic shocks and cycles at the same time. Fur- thermore, this can be done without making prior assumptions about a special structure of the common or idiosyncratic dynam- ics. However, the estimation and identification of the number of factors or components is not straightforward and may lead to less than robust results in some cases. In its simplest form, several time series are to be represented by one common factor and an idiosyncratic part for each of the se- ries. Here p represents the number of time series, each ranging over the time domain 1, ... , T. The common factor F and an idio- syncratic part u;, represent each of the p time series z;, which can be formalised as (15) z;,, =I; F, +u;,, i = l, ... ,p I; is the so-called loading of the factor showing the contribution of the factor to the common driving force F, represented in time se- ries i. For identification the factor is supposed to be uncorrelated with the idiosyncratic parts u;,. and the u;.,-, are also uncorrelated among each other. This single common factor approach can eas- ily be extended to the identification of more than one common factor7o_ 69 Breitung - Eickmeier (2005) give a rather complete overview of applications of factor models in recent times. 70 An example for this method is Pena -Box ( 1987).