THE IMPACT OF SEASONAL ADJUSTMENT ON TIME SERIES PREDICTION

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1 Review of he Air Force Acaemy No 3 (27) 2014 THE IMPACT OF SEASONAL ADJUSTMENT ON TIME SERIES PREDICTION Aela SASU Transilvania Universiy of Braşov Absrac: In his aricle we inen o comare he erformance in erm of accuracy for wo moels ARIMA(,,q)(P,D,Q) s, corresoning o wo kins of ime series: seasonally ajuse an no seasonally ajuse. The ime series use in his work are ublicly available. The measure of accuracy use is mean absolue ercenage error. Exerimenally, we show ha he reicion accuracy is beer for seasonally ajuse ime series. Key wors: forecasing, ime series, seasonally ajuse, ARIMA. 1. INTRODUCTION A ime series is a sequence of observaions y 1,, y-2, y- 1, y generae sequenially in ime. The key roeries of ime series are: he aa are no ineenenly generae, heir variance may vary in ime, hey are ofen governe by a ren an hey may have cyclic comonens. Saisical roceures ha suose ineenen an ienically isribue aa are, herefore, exclue from he analysis of ime series. Time series analysis inclues a broa secrum of exloraory an hyohesis esing mehos wih wo main goals: (a) ienifying he naure of he henomenon reresene by he sequence of observaions, an (b) forecasing, i.e. reicing fuure values of he ime series variable. Seasonal ajusmen (SA) is he rocess of esimaing an removing seasonal effecs from a ime series in orer o beer reveal cerain non-seasonal feaures. The mechanics of seasonal ajusmen involve breaking own a series ino ren-cycle, seasonal, an irregular comonens: Tren cycle: Level esimae for each monh (quarer) erive from he surrouning years of observaions. Seasonal effecs, efine as effecs ha are reasonably sable in erms of annual iming, irecion, an magniue. Possible causes inclue naural facors (weaher), aminisraive measures (saring an ening aes of he school year), an social/culural/religious raiions (fixe holiays such as Chrismas). Irregular comonens, ha is anyhing no inclue in he ren-cycle or he seasonal effecs (or in esimae raing ay or holiay effecs). Their values are unreicable as regars iming, imac, an uraion. They can arise from samling error, non-samling error, unseasonable weaher, naural isasers, srikes, ec. X-12-ARIMA is he seasonal ajusmen sofware rouce an mainaine by he Census Bureau. I is use for all official seasonal ajusmens a he U. S. Census Bureau. The original ime series is no seasonal ajuse (NSA). Mehos for seasonally ajusing ime series are also escribe in [5], [6]. 2. The ARIMA Moel Auoregressive Inegrae Moving Average (ARIMA) rocesses are a class of sochasic rocesses use in he area of ime series moeling. The alicaion of he ARIMA mehoology for he suy of ime series analysis is ue o Box an Jenkins [1]. 105

2 The Imac of Seasonal Ajusmen on Time Series Preicion Le us consier y y y, y, (shorly:, { y } ) he observaions a equally sace ime momens an le a, 1 a- 1 a, a+ 1, or { a } be a whie noise series consising of ineenen an ienically isribue ranom variables, whose isribuion is aroximaely normal 2 wih mean zero an variance σ a. Assume ha E ( y ) = µ y an we noe y - µ ~ y = y ; herefore, E( y~ ) = 0. Le us consier he general ARMA(, q) moel as in [2]: ~ y = φ ~ y 1-1 -θ a φ ~ y - - -θ a q -q + a (1) or, equivalenly φ ( B ) ~ y = θ ( B) (2) a 2 where φ( B) = 1-φ1B -φ2b - -φ B is he auoregression oeraor of orer an 2 q θ ( B) = 1+ θ1b + θ 2B + + θ q B is he moving average oeraor of orer q, wih B being he backwar shif oeraor B y = y - 1, k B y = y-k. The general moel ARIMA(,, q) is efine as in [3]: φ ( B )( 1- B) ~ y = φ( B) ~ y = θ ( B) a (3) where = 1 - B is backwar ifference, an = ( 1- B) is he backwar ifference of orer. A common assumion for many ime series echniques is ha he aa are saionary. A saionary rocess has he roery ha he mean, variance an auocorrelaion srucure o no change over ime. Saionariy can be efine in recise mahemaical erms, bu for our urose we mean a fla looking series, wihou ren, consan variance over ime, a consan auocorrelaion srucure over ime an no erioic flucuaions. If he roos of olynomial φ (B) lie ousie he uni circle, i may be shown ha an ARMA(, q) is saionary. Many ime series are no saionary. I is ofen he case ha he series of firs ifferences w = y - y- 1 = (1 - B) y is saionary. If a series { y } has o be ifference once o obain saionariy, hen he moel corresoning o he original series is calle an inegrae ARMA moel of orer, 1, q or ARIMA(, 1, q). In racice, ifferencing on he firs orer is necessary, while ifferencing on he secon orer is rarely neee. If he original series { } y is saionary, hen i is no necessary o iffereniae hese series. However, if ime series manifes a erioic flucuaion (a seasonal aern), hen he general ARIMA moel is efine following [3]: Φ ( Θ a (4) s D s B ) s y = ( B ) where s is he number of erios in a season. Le us noe he seasonal ARIMA moel ARIMA(,,q)(P,D,Q) s, where P=number of seasonal auoregressive erms, D=number of seasonal ifferences, Q=number of seasonal moving average erms. In [1], Box an Jenkins suggese ha he search for a goo moel coul be base on he following: (i) Moel ienificaion, i.e. eciing on (iniial values for) he orers ; ; q; P;D;Q, (ii) Esimaion, i.e. fiing of he arameers in he ARIMA moel, (iii) Diagnosic checking an moel criicism, (iv) Ieraion: moifying he moel (i.e. he orers ; ; q; P;D;Q) in he ligh of (iii) an reurning o (ii). The imlemenaion rovie by IBM SPSS Saisics analysis ackage version 21 was use for ARIMA. As ARIMA is elivere as an SPSS roceure a leas from version 13 of his rouc, we believe ha he curren version is error-free. The ses for roucing he mos aroriae ARIMA moel are eaile in [6]. 3. Exerimens an resuls The overall erformance of a forecasing moel is evaluae by an accuracy measure, Mean Absolue Percenage Error (MAPE) comue as: N 1 y - yˆ MAPE = 100 (7) N = 1 y where y is he esire value, ŷ is he reice value for erio, an N is he number of forecase values. MAPE is a common meric in forecasing alicaions an i measures he roorionaliy beween he forecasing error an he acual value. 106

3 Review of he Air Force Acaemy No 3 (27) 2014 The five ime series use in his secion are: US Toal New Privaely Owne Housing, US Reail Sales an Foo Services, Floria Consrucion; All Emloyees, Manufacuring Informaion Technology Inusries Toal Invenories Millions of Dollars, Manufacuring Woo Proucs Invenories o Shimens Raio, They are freely available a [8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. Each of hese ime series are given boh as seasonally ajuse (SA) an as no seasonally ajuse (NSA). For he ARIMA moel, he values for he coefficiens,, q, P, D, an Q are auomaically eermine by he IBM SPSS sofware, hrough is inernal rouines. Afer he forecas is inuce, he MAPE values are comue an he values of MAPE are groue in Table 1: Table1. The MAPE values of he consiere ime series SERIES MAPE US Toal New Privaely Owne Housing US Reail Sales an Foo Services Floria Consrucion; NSA SA CONCLUSION As shown in Table 1, all he values of MAPE for SA ime series are lower he ones for NSA. In summary, he MAPE values for he SA series are beween 39.9% an 94.3% of he corresoning NSA MAPE scores. In four ou of he five cases, he MAPE score for SA is less han 71% of he MAPE for NSA. We conclue ha seasonal ajusmen of ime series imroves he reicion erformances an i is recommene o be erforme as a rerocessing se before analyzing of ime series, esie he sulemenal comuaional resources neee. Fig. 1(a) US Toal New Privaely Owne Housing Unis Comlee; Thousans; NSA beween an [8] Manufacuring Informaion Technology Inusries Manufacuring Woo Proucs Invenories o Shimens Raio Fig. 2(a) US Reail Sales an Foo Services, Toal, NSA beween an [10] 107

4 The Imac of Seasonal Ajusmen on Time Series Preicion Fig. 3(a) Floria Consrucion; All Emloyees; Thousans; NSA beween an [12] Fig. 3(b) Floria Consrucion; All Emloyees; Thousans; SA beween an [13] Fig. 1(b) US Toal New Privaely Owne Housing Unis Comlee; Thousans; SA beween an [9] Fig. 4(a) Manufacuring Informaion Technology Inusries Toal Invenories Millions of Dollars NSA beween an [14] Fig. 2(b) US Reail Sales an Foo Services, Toal, SA beween an [11] Fig. 5(a) Manufacuring Woo Proucs Invenories o Shimens Raio NSA beween an [16] 108

5 Review of he Air Force Acaemy No 3 (27) 2014 BIBLIOGRAPHY Fig. 4(b) Manufacuring Informaion Technology Inusries Toal Invenories Millions of Dollars SA beween an [15] Fig. 5(b) Manufacuring Woo Proucs Invenories o Shimens Raio SA beween an [17] 1. Box, G.E.P., Jenkins, G.M., Time Series Analysis, Forecasing an conrol, San Francisco: Holen-Day (1976). 2. Chafiel, C., The Analysis of Time Series. An Inroucion, New York: Chaman & Hall/CRC(1995). 3. Poescu, T., Time Series Alicaion in Sysem Analysis, Technical Publishing House(2000) (in Romanian). 4. Sasu, A., An Alicaion of ARIMA Moels, Bullein of he Transilvania Universiy of Braşov 11(46) - New Series, Series B1, (2005). 5. Sasu A., Time series seasonal ajusmen meho, Bullein of he Transilvania Universiy of Brasov 14(49), Series B1. Transilvania Univ. Press-Brasov, 91-94, (2007). 6. Sasu, A., Time Series Forecasing using ARIMA Moels, 8-h Euroean Conference E-COMM-LINE, 67e, (2007). 7. Yaffee, R.,A., McGee, M., An Inroucion o Time Series Analysis an Forecasing: Wih Alicaions of SAS an SPSS, Acaemic Press(2000). 8. US Toal New Privaely Owne Housing Unis Comlee; Thousans; NSA: h:// cenc25/comua01 9. US Toal New Privaely Owne Housing Unis Comlee; Thousans; SA: h://www. economagic.com/em-cgi/aa.exe/cenc25/ comsa US Reail Sales an Foo Services, Toal, NSA: h:// 11. US Reail Sales an Foo Services, Toal, SA: h:// 12. Floria Consrucion; All Emloyees; Thousans; NSA: h://www. economagic.com/em-cgi/aa.exe/blssm/ SMU Floria Consrucion; All Emloyees; Thousans; SA: h://www. economagic.com/em-cgi/aa.exe/blssm/ SMS

6 The Imac of Seasonal Ajusmen on Time Series Preicion 14. Manufacuring Informaion Technology Inusries Toal Invenories Millions of Dollars NSA: h:// aa.exe/cenm3/uiii 15. Manufacuring Informaion Technology Inusries Toal Invenories Millions of Dollars SA: h:// exe/cenm3/aiii 16. Manufacuring Woo Proucs Invenories o Shimens Raio NSA: h:// cenm3/u21sis 17. Manufacuring Woo Proucs Invenories o Shimens Raio SA: h:// cenm3/a21sis 110