MODELLING SLOVAK UNEMPLOYMENT DATA BY A NONLINEAR LONG MEMORY MODEL

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1 Proceedigs of ALGORITMY 2005 pp MODELLING SLOVAK UNEMPLOYMENT DATA BY A NONLINEAR LONG MEMORY MODEL JOZEF KOMORNíK AND MAGDA KOMORNíKOVÁ Abstract. The mai visual feature of Slovak mothly uemploymet data (from the period Jauary 1993 August 2004) is a dramatic icrease after the electios i October 1998 which ca be expressed by a step i the determiistic level fuctio. I the residuals from this level fuctio we ca idetify a sigificat cyclical compoet ad a log memory structure (with the estimated value of the Hurst parameter close to 0.8), as well as a autoregressive short memory behavior. Key words. Time series modellig, smooth regime switchig, spectral aalysis, fractioal itegratio, ARFIMA modellig. AMS subject classificatios. 62M10, 62M15, 62M20 1. Itroductio. Aalysis of US moth uemploymet data (from the period July 1968 December 1999) based o a oliear log memory model has bee performed i [Va Dijk et al., 2000]. The cosidered oliearity (with 2 regimes) was based o the observatio that US uemploymet seems to grow faster i recessios tha it falls i expasios. The mai visual feature of Slovak mothly uemploymet data (from the period Jauary 1993 August 2004) is a dramatic icrease after the electios i October 1998 (ad followig chages i compositio of govermet coalitio parties ad i ecoomic policies, that had supported artificial employmet i uretable busiesses ad ecoomic expasio based o a dramatic icrease of foreig idebtedess). This switch of political regime correspods to a parallel switch of regime i our time series, which ca be expressed by a remarkable positive shift i its values (the average value for the period before the ed of 1998 was %, for the followig period %). Visual ivestigatio of the values of our time series aroud the ed of 1998 idicates that the icrease has bee quick but ot totally istat. Ispired by the methods of smooth trasitio regime-switchig models, we model this chage i values by a logistic class smooth step fuctio that we will call base level fuctio. I the residuals from this base level fuctio, we ca idetify a sigificat cyclical compoet ad a log memory structure (with the estimated value of the Hurst parameter close to 0.8), as well as a autoregressive short memory behavior. 2. Methods ad results of the modellig Slovak Uemploymet data. Accordig to moder priciples of time series modellig (see Frases, 1998) we adapt idividual steps of our aalysis to the mai visual properties of the studied time series of mothly values of uemploymet rates i Slovak Republic i the period Jauary 1993 August 2004 that is show below Regime switchig chage of base level. Sice the level of Slovak uemploymet have clearly rise after the electio (i ed subsequet chage of Gover- The research summarized i this paper was partly supported by the Grats GACR 402/04/1026 ad VEGA 1/0273/03. Faculty of Maagemet, Comeius Uiversity, Bratislava, Slovakia Faculty of Civil Egieerig, Slovak Uiversity of Techology, Slovakia ad UTIA AV CR Prague, Czech Republic 334

2 MODELLING SLOVAK UNEMPLOYMENT DATA 3 Fig Slovak uemploymet data from the period Jauary 1993 August 2004 met) i October 1998 (correspodig to t = ), we tried to model the cosequece of this chage by a base level fuctio (2.1) L(t) = a + bg(t, c, γ) where (2.2) G(t, c, γ) = e γ(t c) is a logistic fuctio. Applyig methods of oliear optimizatio we received the followig (least squares) parameter estimates: a = ; b = 3.738; c = 71.36; γ = The shape of the resultig base level fuctio is show at the followig Fig Fig Origial time series ad the base level fuctio. Origial data, Base level fuctio - - -

3 336 J. KOMORNÍK, M. KOMORNÍKOVÁ 2.2. Idetificatio of seasoal ad cyclical compoet. Now we proceed i aalysis of residuals of origial uemploymet data from correspodig values of the estimated base level fuctio. We retur to classical procedures of time series aalysis ad estimate the coefficiets (ad test sigificace) of the seasoal (aual) period compoet i this residuals. The cotributio of this seasoal compoet (with the amplitude close to 0.5) to the explaatio of the variability of the aalyzed time series is strogly sigificat (p < 0.01). Next we proceed by calculatig the values of periodogram for remaiig residuals (Tab. 2.1) ˆf (ω j ) = ˆγ (0) 2π N 1 + k=1 ˆγ (k) cos (k ω j) π where N is the legth of the time series (N = 140), ω j = 2π j N ad ˆγ (k) is the sample covariace fuctio for lag k. is j-th Fourier frequece Table 2.1 Periodogram for remaiig residuals after base level fuctio ad seasoal compoet N/j N/j [moths] ˆf (ωj ) [moths] ˆf (ωj ) / / / / / Applyig step-wise regressio procedure o the relatio betwee the cosidered residual time series ad periodic fuctios with the above Fourier frequecies, we coclude that the most importat periods are (approximately) 6, 4 ad 3 years. The resultig determiistic fuctio (systematic compoet) has the form ( ) ( ) X t = cos si 1 + e 2.73(t 71.36) ( ) ( ) ( ) ( ) cos 0.73 cos 0.51 cos 0.36 cos ( ) ( ) ( ) ( ) si si si si The quality of the fit of the origial time series by the combied systematic compoet (base level fuctio, seasoal ad cyclical compoets) ca be see i the Fig The graph of the autocorrelatio fuctio of residuals from the combied systematic compoet fuctio (see Fig. 2.4) idicates the existece of sigificatly ozero terms both of low ad high orders Modellig log memory features. I order to idetify ad elimiate high order (log memory) terms, we try to fid (for the cosidered time series of residuals Z t ) a represetatio of the ARFIMA form [Fouskitakis et al., 2000, Va Dijk et al., 2000]

4 MODELLING SLOVAK UNEMPLOYMENT DATA 337 Fig Origial data ad regressio fuctio. Origial data, Systematic compoet Fig Autocorrelatio fuctio of residuals (2.3) (1 B) d Z t = a t where B is the backward shift operator (i.e. BZ t = Z t 1 ), {a t } is a covariace statioary ARMA process. (2.4) The fractioal differece operator is defied by (1 B) d = k=0 = 1 db + ( ) ( B) k k d (d 1) B 2 2! d (d 1) (d 2) B 3 + 3! The series Z t is covariace statioary if d < 0.5 ad ivertible if d > 0.5. The autocorrelatio fuctio of Z t does ot declie at a expoetial rate, as is characteristic for covariace-statioary ARMA processes, but rather at a (much) slower hyperbolic rate. For 0 < d < 0.5, Z t possesses log memory i the sese that the

5 338 J. KOMORNÍK, M. KOMORNÍKOVÁ autocorrelatios r(k) are ot absolutely summable. This implies that eve though the r(k) s are idividually small for large lags k, their cumulative effect is importat. The ukow parameter d ca be estimated via so-called Hurst parameter H = d that is the slope i the regressio ( ) R log = log c + H log S for the so-called rescaled rages ( ) R S (defied below). Details of the estimatio procedures has bee take from [Mille ad Beard, 2003]. Firstly the time series must be divided ito D cotiguous sub-series of legth, where D = N, the total legth of the time series. For each of theses sub-series m, where m = 1,..., D: 1. Determie the mea E m of each sub-series. 2. Determie the stadard deviatio S m of each sub-series. 3. Normalise the data {Z i,m } by subtractig the mea from each data poit: X i,m = Z i,m E m, for i = 1,...,. 4. Usig the ormalized data create a cumulative time series by cosecutively summig the data poits: Y i,m = i X j,m for i = 1,...,. j=1 5. Usig the ew cumulative series fid the rage by subtractig the miimum value from the maximum value: R m = max{y 1,m,..., Y,m } mi{y 1,m,..., Y,m }. 6. Rescale the rage, Rm S m by dividig the rage by the stadard deviatio. 7. Calculate the mea of the rescaled rage for all sub-series of legth : ( ) R = 1 S D D m=1 R m S m. 8. The legth of must be icreased to the ect higher value where D = N ad D is a iteger value. Step 1 to 7 are the repeated util = N/2. 9. Fially, the value of H is obtaied usig a ordiary least squares regressio with log() as idepedet variable ad log ( ) R S as the depedet variable. The slope of the resultig equatio is the estimate of the Hurst expoet. The regressio is ru over values of greater tha 10, as small values of produce ustable estimates whe sample size are small. The resultig value of the estimate of Hurst parameter (as the measure of logterm persistece of shocks) i our case is H = 0.792, which is sigificatly larger tha the critical value 0.5. The estimate of the fractioal differecig parameter d = 0.292, hece the series X t is covariace statioary, ivertible ad possesses log memory. We see that the quality of fit of the time series of residuals improved cosiderably after applicatio log memory filter (see Fig. 2.5).

6 MODELLING SLOVAK UNEMPLOYMENT DATA 339 Fig Time series of residuals ad their model. Residuals, model Modellig short memory structure of residuals. As the last step of our aalyzes we applied Box Jekis methodology to the residuals of the log term filter. The best fit (with respect to AIC ad BIC criteria) has bee received for a model i the class AR(7) (the umbers i paretheses are estimates of stadard deviatios of estimates of model coefficiets): Z t = 0.92Z t Z t Z t Z t Z t Z t 7 (0.08) (0.11) (0.09) (0.09) (0.11) (0.08) The fial fit of origial data is i the Fig Fig Time series of Slovak uemploymet data ad model The diagostic check of the ultimate residuals (see Fig. 2.7 (left)) idicated o sigificat autocorrelatios (see Fig. 2.7 (right)). 3. Coclusio. We selected four steps of aalysis accordig to basic properties of the cosidered time series that have bee maifested i the graphically visible chage of the basic level after October 1998, i the periodogram of remaiig residuals, i the pox diagram ad i autocorrelatio structure of resultig residuals. The resultig model exhibits a high order of fit ad desired quality of diagostic check for residuals.

7 340 J. KOMORNÍK, M. KOMORNÍKOVÁ Fig Residuals after model (left). Autocorrelatio fuctio (right). REFERENCES [1] Artl, J., Artlová, M., Fiačí časové řady Vlastosti, metody modelováí, příklady a aplikace. GRADA Publishig, [2] Fouskitakis, G. N., Fassois, S. D., O the estimatio of log-memory time series models. Uiversity of Patras, Greece, [3] Frases, P. H., Time series models for busiess ad ecoomic forecastig. Cambridge Uiversity Press, [4] Frases, P. H., Dijk, D., No-liear time series models i empirical fiace. Cambridge Uiversity Press, [5] Grager, C. W. J., Teräsvirta, T., Modellig oliear ecoomic relatioships. Oxford Uiversity Press, [6] Mille, S., Beard, R., Estimatio of the Hurst expoet for the Burdeki River usig the Hurst-Madelbrot Rescaled Rage Statistic. Workig paper o the Uiversity of Queeslad, [7] Rose, O., Estimatio of the Hurst Parameter of Log-Rage Depedet Time Series. Report No. 137, Uiversity of Würzburg, Germay, [8] Va Dijk, D., Frases, P. H., Paap, R., A oliear log memory model for US uemploymet. Research Report EI /A, Ecoometric Istitute, Erasmus Uiversity, Rotterdam, 2000.

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