Exact Limit of the Expected Periodogram in the Unit-Root case

We derive the limit of the expected periodogram in the unit-root case under general conditions. This function is seen to be independent of time, thus sharing a fundamental property with the stationary case equivalent. We discuss the consequences of this result to the frequency domain interpretation of filtered integrated time series.

Exact Limit of the Expected Periodogram in the Unit-Root Case

We derive the limit of the expected periodogram in the unit-root case under general conditions. This function is seen to be time-independent, thus sharing a fundamental property with the stationary case equivalent. We discuss the consequences of this result to the frequency domain interpretation of filtered integrated time series.Periodogram, Unit root

A Multivariate Band-Pass Filter

We develop a multivariate filter which is an optimal (in the mean squared error sense) approximation to the ideal filter that isolates a specified range of fluctuations in a time series, e.g., business cycle fluctuations in macroeconomic time series. This requires knowledge of the true second-order moments of the data. Otherwise these can be estimated and we show empirically that the method still leads to relevant improvements of the extracted signal, especially in the endpoints of the sample. Our filter is an extension of the univariate filter developed by Christiano and Fitzgerald (2003). Specifically, we allow an arbitrary number of covariates to be employed in the estimation of the signal. We illustrate the application of the filter by constructing a business cycle indicator for the U.S. economy. The filter can additionally be used in any similar signal extraction problem demanding accurate real-time estimates.

A Multivariate Band-Pass Filter

We develop a multivariate filter which is an optimal (in the mean squared error sense) approximation to the ideal filter that isolates a specified range of fluctuations in a time series, e.g., business cycle fluctuations in macroeconomic time series. This requires knowledge of the true second-order moments of the data. Otherwise these can be estimated and we show empirically that the method still leads to relevant improvements of the extracted signal, especially in the endpoints of the sample. Our filter is an extension of the univariate filter developed by Christiano and Fitzgerald (2003). Specifically, we allow an arbitrary number of covariates to be employed in the estimation of the signal. We illustrate the application of the filter by constructing a business cycle indicator for the U.S. economy. The filter can additionally be used in any similar signal extraction problem demanding accurate real-time estimates.

Business Cycles: Cyclical Comovement Within the European Union in the Period 1960-1999. A Frequency Domain Approach

This paper provides a descriptive analysis of the business cycles of the European Union countries and of the two main industrialised countries outside the Union, the United States and Japan. We use the spectral analysis to identify three main features of the business cycles: ��1- The duration of the business cycle. ��2- The degree of correlation, in the frequency domain, of the business cycles. ��3- The identification of leading and lagging countries with respect to the business cycles of a reference series. We conclude that the United States, Italy and Greece have the shortest cycles, with an average duration around eight years. Japan, Spain and Austria have the longest cycles, lasting more than ten years. All the other countries lie in between with an average duration ranging from eight to nine years. By comparing the business cycles of the various countries with the Euro Area business cycle we conclude that Sweden, Finland, Great Britain and the United States lead the Euro Area by more than one year. The Netherlands, Italy, Japan and Spain are also leading countries but with a lead of no more than one year. There is evidence of counter-cyclical behaviour for Denmark in a sub-period of the sample and no reliable conclusions can be stated for Greece and Ireland. The remaining countries exhibit a high degree of correlation with the Euro Area business cycles and with a lag of no more than three-quarters, with the exception of Austria.

Tracking Growth and the Business Cycle: a Stochastic Common Cycle Model for the Euro Area

This paper proposes a new model-based method to obtain a coincident indicator for the business cycle. A dynamic factor model with trend components and a common cycle component is considered which can be estimated using standard maximum likelihood methods. The multivariate unobserved components model includes a stationary higher order cycle. Also higher order trends can be part of the analysis. These generalisations lead to a business cycle that is similar to a band-pass one. Furthermore, cycle shifts for individual time series are incorporated within the model and estimated simultaneously with the remaining parameters. This feature permits the use of leading, coincident and lagging variables to obtain the business cycle coincident indicator without prior analysis of their lead-lag relationship. Besides the business cycle indicator, the model-based approach also allows to get a growth rate indicator. In the empirical analysis for the Euro area, both indicators are obtained based on nine key economic time series including gross domestic product, industrial production, unemployment, confidence indicators and interest rate spread. This analysis contrasts sharply with earlier multivariate approaches. In particular, our more parsimonious approach leads to a growth rate indicator for the Euro area that is similar to the one of EuroCOIN. The latter is based on a more involved approach by any standard and uses hundreds of time series from individual countries belonging to the Euro area.

Finite Sample Performance of Frequency and Time Domain Tests for Seasonal Fractional Integration

Testing the order of integration of economic and financial time series has become a conventional procedure prior to any modelling exercise. In this paper, we investigate and compare the finite sample properties of the frequency domain tests proposed by Robinson (1994) and the time domain procedure proposed by Hassler, Rodrigues and Rubia (2008) when applied to seasonal data.

Interpretation of the Effects of Filtering Integrated Time Series

We resort to a rigorous definition of spectrum of an integrated time series in order to characterise the implications of applying linear filters to such series. We conclude that in the presence of integrated series the transfer function of the filters has exactly the same interpretation as in the covariance stationary case, contrary to what many authors suggest. This disagreement leads to different conclusions regarding the link of the original fluctuations with the transformed fluctuations in the time series data, embodied in various unjustified criticisms to the application of detrending filters. Despite this, and given the frequency domain characteristics of filtered macroeconomic integrated series, we acknowledge that the choice of a particular detrending filter is far from being a neutral task