Seite - 49 - in The Austrian Business Cycle in the European Context

49 Assuming orthogonality between the common and the idiosyn- cratic parts allows - in the frequency domain - representing the total spectral density matrix ~(ro) as the sum of the one responsible for the common component and the one of the idiosyncratic part where ro is a certain frequency within the interval [-n and +n]. This enables the common part y, to be identified by analysing the ei- genvalues of the matrix ~(ro). According to the method proposed by Forni et al. (2000), the eigen value solutions of the covariance matrix are ranked by their magnitude, whereby the largest corre- sponds to the first principal component explaining the largest part of full data variation, the second principal component explains the second largest part of total variation and so on. As the com- mon factor loadings appear with some lead or lag structure, the practical approach for solving this requires a fixing of the number of frequencies at which the spectral density matrix is to be evalu- ated. Forni et al. (2000) recommended the number of frequencies to be observed within the interval [0,2n] as the rounded square root of the time series length divided by four. This should reasona- bly settle the trade-off between the length of the time series and the quality of the identification output. The identification of each of the common components is sup- ported by demanding that their variance at several leads and lags inherent in each of the time series is maximised if it is summed up over all time series II (21) max ~:>ar(rj,,) 1(1...m) j= I with n being the number of observed time series and q is again one specific common component. The essential process of identi- fying one or more common components consists of looking at the eigen values and eigen vectors of the spectral density matrix of