Seite - 48 - in The Austrian Business Cycle in the European Context

48 This approach has the advantage that the identified common factors are invariant to their number being identified. The first common factor explains the maximum proportion of total varia- tion of the p observed time series, with all further factors showing diminishing explanatory power, till the pth common component completes the total variance of the whole data set. Forni et al. (2004) suggest an information criterion for determining the number of factors to be extracted, which concentrates on the amount of common variance that can be explained by the factors. Again, each time series is represented as the sum of one or more common factors and its residual idiosyncratic component. In the dynamic formulation, the static factors common to all series can be loaded with lead and lag structures, but all are constructed as being mutually orthogonal at all leads and lags. For the idiosyn- cratic components it can again be assumed that either they are orthogonal to all other idiosyncratic ones (strict factor model) or that some amount of correlation is allowed. Formally, this problem can be expressed72 where as z, is the N x 1 vector of observed second-order stationary time series at time t. It is represented as the sum of the vector of common components y, and the idiosyncratic l;,i. The common term y, itself is constructed by a vector of q orthogonal common components x, loaded by time-shifted weights Cq (L), where L stands for the lag operator73. 72 See Forni et al. (2000). 73 Allowing for a dynamic structure in factor models - which shows up in leading and lagging properties of the loadings - leads to a loss of information at the begin- ning and the end of the time series. As already mentioned, this endpoint problem is not a disadvantage for the present study, because no real time dating is intended.