DEM_2015_27to48 © 2015 Nicolaus Copernicus University Press. All rights reserved. http://www.dem.umk.pl/dem D Y N A M I C E C O N O M E T R I C M O D E L S DOI: http://dx.doi.org/10.12775/DEM.2015.002 Vol. 15 (2015) 27−47 Submitted October 25, 2015 ISSN (online) 2450-7067 Accepted December 15, 2015 ISSN (print) 1234-3862 Łukasz Lenart* Discrete Spectral Analysis. The Case of Industrial Production in Selected European Countries∗∗ A b s t r a c t. The aim of this paper is to show the usefulness the discrete spectral analysis in identification cyclical fluctuations. The subsampling procedure was applied to construct the asymptotically consistent test for Fourier coefficient and frequency significance. The case of monthly production in industry in European countries (thirty countries) was considered. Using proposed approach the frequencies concerning business fluctuations, seasonal fluctuations and trading-day effects fluctuations were recognized in considered data sets. The comparison with existing procedures was shown. K e y w o r d s: discrete spectral analysis, almost periodic function, frequency identification, graphical test. J E L Classification: C14, C46, E32. Introduction The main part of monthly or quarterly macroeconomic time series con- cerning industry, trade, service, national accounts, prices, etc. exhibit both: seasonal fluctuations and business cycle fluctuations. The problem of analys- ing these cyclical fluctuations is widely considered in the literature using different statistical tools. One popular nonparametric approach is based on a spectral analysis. * Correspondence to Łukasz Lenart, Cracow University of Economics, Department of Mathematics, e-mail: lenartl@uek.krakow.pl ∗∗ This research was supported by Research Grant DEC-2013/09/B/HS4/01945 from the National Science Centre Łukasz Lenart DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 28 The spectral analysis of macroeconomic time series is considered mainly in continuous counterpart (see Ftiti, 2010; Metz, 2009; Orlov, 2006; Orlov, 2009; Pakko, 2004; McAdam and Mestre, 2008; Uebele and Ritschl, 2009). Under stationarity assumption the continuous function called spectral den- sity function is defined. Based on spectral density function (and the defini- tion of harmonizable time series) the popular spectral characteristics are defined: modulus of coherency function, dynamic correlation, dynamic cor- relation on a frequency band, cohesion, cohesion within the frequency band, phase shift (see Priestley, 1981; Hamilton, 1994; Croux, 2001 for more de- tails). These measures are broadly used in analysing the business cycle fluc- tuations. They are estimated under fundamental assumption that time series is zero mean. But this assumption is not supported by any formal statistical test in most empirical macroeconomic real data analysis. In this paper the more general assumption is formulated concerning non- trivial mean function. This general assumption was considered in Lenart, 2011; Lenart and Pipień, 2013a; Lenart and Pipień, 2013b (in univariate case) and recently in Lenart and Pipień, 2015 (in multivariate case) with application to macroeconomic time series. In this paper we show that the parameters of the discrete spectrum can be identified not only with the sea- sonal and business cycle fluctuations but additionally with the trading-day effect. In the section 2, based on Lenart and Pipień, 2013a and Lenart and Pipień, 2015 the model was formulated and illustrative example was pre- sented. In the next part the empirical analysis was presented. The production in industry – monthly data (mining and quarrying; manufacturing; electric- ity, gas, steam and air conditioning supply) from Feb. 2000 to Dec. 2014 was considered. In the first subsection the graphical methods to iden- tify/recognize the frequency (concerning to business fluctuations, seasonal fluctuations and trading-day effect fluctuations) was presented. Such graphi- cal methods to identify ’periodic phenomena’ in the autocovariance function in class of Almost Periodically Correlated time series were presented in Hurd and Gerr, 1991 and recently in Lenart, 2011. Finally in the second subsection formal statistical test for frequency significance was applied to data sets. 1. Model specification Let tY be macroeconomic time series (index, gross data) with possible: seasonal fluctuations with period T , business fluctuations and trading-day Discrete Spectral Analysis. The Case of Industrial Production… DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 29 fluctuations. Let us denote the natural logarithm: )(ln= ~ tt YY . Based on Le- nart and Pipień, 2013a we assume that ),(),(=) ~ ( ttfYE t µβ + (1) where )(tµ is almost periodic function (ap in short) of the form tiemt ψ ψ ψµ )(=)( ∑ Ψ∈ , where .)( 1 lim=)( 1= ti n tn et n m ψµψ − →∞ ∑ We assume that the set of frequencies 0}=|)(:|)[0,2{= /∈Ψ ψπψ m is finite and unknown. This set Ψ can be decompose in natural way via: 321= Ψ∪Ψ∪ΨΨ , where 1Ψ corresponds to business fluctuations, 1}0,1,=,/{22 −⊂Ψ TkTkπ corresponds to seasonal fluctuations and 3Ψ is a set of remaining frequencies (corresponding to interaction between sea- sonal and business fluctuations and frequencies corresponding to trading-day effects). Equivalently, the model (1) can be written via: ),()(),(=) ~ ( 21 tttfYE t µµβ ++ (2) where )(1 tµ is a periodic function with period T which represents the sea- sonal fluctuations and tiemt ψ ψ ψµ )(=)( 2\ 2 ∑ ΨΨ∈ . The function )(1 tµ can be equivalently represented by the vector (sequence of seasonal values) ' 121 ][= −Tµµµ Kµ , and )(= 121 −+++− TT µµµµ K . The sequence 1 ~~ = −− ttt YYX represents the monthly log growth rate. If ),( βtf is a polynomial of order one then ),(~)(~=)( 21 ttXE t µµ + (3) where 1)()(=)(~ 111 −− ttt µµµ is periodic function that corresponds to sea- sonal pattern and tiemttt ψ ψ ψµµµ )(~=1)()(=)(~ 2\ 222 ∑ ΨΨ∈ −− . The sequence 1 ~~ =' −− ttt YYX represents the annual log growth rate. If ),( βtf is a polyno- mial of order one then ),('~=)'( 2 tXE t µ (3) Łukasz Lenart DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 30 where tiemttt ψ ψ ψµµµ )('~=12)()(=)('~ 2\ 222 ∑ ΨΨ∈ −− . In future work we as- sume that the autocovariance function of the time series }:{ Z∈tX t (or }:'{ Z∈tX t ) is a periodic function with the same period T . This is a natural generalization of the assumption concerning second order stationarity. This assumption follows from the natural hypothesis that for monthly data at some months the variability can be higher than at another month. With no loss of generality we assume that exists natural m such that msn = . Then the time series }:{ Z∈tX t can be represented as a second order stationary T -valued time series with almost periodic mean function. More precisely the time series Ttsststt XXXY ][= 1)(21)(1 K−+−+ is T values second order station- ary time series with almost periodic mean function. This mean function can be decomposed (in natural way) to two main parts: periodic function (that corresponds to seasonal frequencies 2Ψ ) and almost periodic function (that corresponds to frequencies 31 Ψ∪Ψ ) The natural estimator of kµ (k=1,2,...,T–1) based on sample },,,{ 21 nXXX K where mTn = has the following form . 1 =ˆ 1)( 1= , Tjk m j nk X m −+∑µ (4) The estimator '1,2,1, ]ˆˆˆ[=ˆ nTnnn −µµµ Kµ is asymptotically normally distrib- uted with known variance covariance matrix. Theorem 2.1 Assume that there exist constants 0>δ , ∞∆ < and ∞