The aim of this article is to establish asymptotic distributions and consistency of subsampling for spectral density and for magnitude of coherence for non-stationary, almost periodically correlated time series. We show the asymptotic normality of the spectral density estimator and the limiting distribution of a magnitude of coherence statistic for all points from the bifrequency square. The theoretical results hold under α-mixing and moment conditions.

The main purpose of this article was to describe the asymptotic properties of subsampling procedure applied to nonstationary, periodically correlated time series. We present the conditions under which the subsampling version for the estimator of Fourier coefficient of autocovariance function is consistent. Our result provides new tools in statistical inference methods for nonstationary, periodically correlated time series. For example, it enables to construct consistent subsampling test which successfully distinguishes the period of the series. Copyright 2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd

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We propose a non-standard subsampling procedure to make formal statistical inference about the business cycle, one of the most important unobserved feature characterising uctuations of economic growth. We show that some characteristics of business cycle can be modelled in a non-parametric way by discrete spectrum of the almost periodically correlated time series. On the basis of estimated characteristics of this spectrum business cycle is extracted by ltering. As an illustration we characterise the main properties of business cycles in industrial production index for Polish economy.