STATIONARY AND NON-STATIONARY TIME SERIES

Transcription

1 The Aals of The "Şefa cel Mare" Uvers of Suceava. Fasccle of The Facul of Ecoomcs ad Publc Admsrao Vol., No. (, STATIONARY AND NON-STATIONARY TIME SERIES Assocae Professor Ph.D. Elsabea R. ROŞCA Uvers Şefa cel Mare of Suceava, Romaa Absrac The paper Saoar ad No-saoar Tme Seres preses a heorecal approach, he cocep of me seres, s characerscs whch are: varabl, homogee, perodc ad erdepedece of me seres erms, from whch resul he mehods of esmao ad aalss of me seres, specfc for each compoe (red, perodcal compoe-cclcal or seasoal, radom compoe. As me seres erms have or hav a evoluo me, s saoar ad o-saoar. I he case of saoar me seres he paper are defed he whe-ose processes ad he auoregressve processes. Sce mos processes are o-saoar b rasformaos we oba saoar me seres ad he paper are preseed he resrcos for whch a auoregresve process of movg average ARMA ( p, q s saoar. I he paper are approached also o-saoar processes b deermsc pe ad b sochasc pe as he wa o acheve saoar whch mea he red esmao ad s elmao from he al me seres. Kewords: saoar me seres, o-saoar me seres, whe-ose process, auoregressve movg average process, auocorrelao fuco. JEL Classfcao: C INTRODUCTION The me seres s a se of umercal observaos regsered dffere me momes or ervals such as: aual profs, quarerl produco, weel sales, dal exchage rae a.s.o. The aalss of me seres meas frs he aalss of he aure of varao hese seres. The resuled formao offer a mage cocerg wh varables behavour me ad serve o mag forecasg of he erms values of me seres. The aalss of me seres begs wh her graphc represeao whch shows a grea vare of movemes, some of hem wh ssemac aure such as produco growh, ohers wh a more or less ssemac varao coeco wh a ear perods ad ohers wh a radom occurrece. I he aalss of me seres s ecessar o measure he ssemac movemes of daa, o def a cera edec her evoluo, a cera perodc whch occurs o log erm-cclc or wh a ear-seasoal, a cera rregular whch mea o aalss he four compoes of seres whch are he red ad cclcal, seasoal ad radom varaos. Amog he mehods used for he aalss of he compoes of me seres, he mehod based o he movg averages has a specal mporace boh for sascal ad ecoomercal approach. I hs paper we wa o def he characerscs of me seres, he alread meoed compoes ad, depedg o he exsece or o-exsece of he red, he characerscs of saoar ad o-saoar me seres. CHARACTERISTICS OF TIME SERIES The me seres cosss from wo parallel rows of umbers whch he frs row shows he varao of me characersc ad he secod he chage of he aalsed varable from a me u o aoher. So we defe he me seres as a successo of real values gve b he relao:,,..., T or ( Y ( =,..., T where represes he level of Y varable a a gve mome or for a me perod oed wh ad T s he umber of erms from s bacgroud. Each value of seres s he acheveme of a radom varable Y. All he radom varables ( Y defed a radom process. A me seres s dscree f =,..., T Z or s couous f R. 77

2 The Aals of The "Şefa cel Mare" Uvers of Suceava. Fasccle of The Facul of Ecoomcs ad Publc Admsrao Vol., No. (, I he esmao ad aalss of me seres are mpora her characerscs whch are: varabl, homogee, perodc ad erdepedece of me seres erms. I he aalss of socal ad ecoomcal me seres we mus assure he followg codos: he homogee of me seres, he erdepedece wh he seres, he sascal comparabl of seres erms. The me seres erms varabl resuls from ha each erm s obaed b he ceralzao of some dvdual dffere daa as a developme level because he aco of some esseal causes ad also because of a grea eough umber of o-esseal causes whose assocao usuall chage from oe perod o aoher. So, he esmao ad aalss of me seres eed he measureme of he degree ad he form of esseal facors fluece ad he degree of devao from he geeral edec deermed b he o-esseal, radom facors fluece. The me seres erms homogee shows ha such d of seres cosss from he same pe of daa whch are he resul of he same laws aco. The aalss of me seres eeds for each seres o verf s erms homogee whch mea all he me seres erms o be expressed he same u of measure ad for he value dcaors o ae o accou also b he prces evoluo. I he case of some log ecoomcal ad socal me seres s possble o appear breas of slope or level as resul of some esseal modfcaos of he aalsed pheomeo he cosdered me erval. I such cases, he average esmao he all daa seres s releva because he average s a sgfca value ol f he seres erms are qualavel smlar. Whe he prelmar aalss here s more qualavel dffere subperods s recommeded o dvde he seres subseres, o deerme frs averages o subperods ad subsequel a average for he whole seres. We cosder a seres wh erms, whch s smbolzed b ( E, ad whch preses breas of level. So we have he seres: ( E, p =. We defe he followg relaos: E = U E = p ( The aalss of me seres whch preses breas of level eeds he followg seps: he defcao of subseres ad he esmao of her shec dcaors ad f s possble, he defcao of he perodc of appearace he breas of level wh he seres. So, he me seres wll be suded cocerg wh her sabl comparg wh a cera rajecor (as a rule, shows eres he aalss of seres sabl comparg wh a sragh le parallel wh he me axs or wh he seres red. A grea value of he varao coeffce ca resul from he exsece of breas of slope ad/or level wh he seres ad hs case s recommeded o dvde he seres subperods or ca shows ha he daa seres dd preses a regular red s evoluo. I case whch from he graphc represeao of he daa seres resuls breas of slope or level, he queso s he aalss of he degree of smlar amog subperods. I hs purpose for each subperods ( E, p, = are esmaed: mea, varace, sadard devao ad, f s ecessar, varao coeffce. For he whole seres ( E, s calculaed, sarg from he seres erms or from he subperods dcaors, he approprae averages. I he am o esablsh f amog subperods here are subsaal dffereces ca be used more mehods such as: - decomposo of oal varace of he seres paral varaces (ad he esmao of her average ad he varace amog subperods (due o brea facors ad he esablshg of he wegh of varace due o brea facors (as much as hs s bgger he dffereces amog subperods are bgger - usg he aalss of varace b esmao he varao deermed b he brea facors, varao wh he perod, oal varao ad he erpreao of he value of sascs F. The me seres erms perodc resuls from he choce of me u a whch he daa are regsered, depedg o he coe ad he measureme possbles of each dcaor. So, some varables are recommeded o be measured small us of me (for example, exchage rae s dal regsered, flao rae, emplome mohl a.s.o. ad ohers grea us of me (for example, urover, compa's e come, macroecoomc dcaors of resuls such as: gross 78

3 The Aals of The "Şefa cel Mare" Uvers of Suceava. Fasccle of The Facul of Ecoomcs ad Publc Admsrao Vol., No. (, domesc produc, e domesc produc, e expors a.s.o. are earl regsered. Depedg o he aure of fluece facors ha deerme he appearace of oscllaos he evoluo of he me seres erms ad he perods dmeso a whch s mafesed repeaabl are ow cclcal ad seasoal varaos. So, some me seres hghlgh suaos whe macroecoomc varables prese oscllaos aroud he red, usuall susodal pe, caused b eral facors of ecoom, b he errelaoshps amog he dffere sdes of hs, repeaed a uequal me ervals ad bgger ha a ear, called cclcal oscllaos (. The me seres whch hghlghs he suaos whe some macroecoomc varables are flueced b he chage of seasos are esmaed ad aalsed as me seres wh seasoal (for example, oursm demad, level of harves a agrculural crops, cosruco acv a.s.o. are recorded mohl or quaerl. Frederc C. Mlls has preseed hs wor Sascal Mehods wo mehods of deermao he cclcal oscllaos whch are: resdual mehod ad Mehod of he Naoal Bureau of Ecoomc Research (USA. The applcao of he resdual mehod he aalss of cclcal oscllaos volves he followg seps: - correc chose of log erm red for he aalsed seres - elmao he seasoal - deermao he red - deermao of he devao from he red of acual daa, beg elmaed he seasoal varao - deermg he "ccles" or perceages of devaos from he red of he acual daa. Mehod eeds he deermao of wo dcaors whch are: - he perceage of red ( T %, obaed b he reporg he acual values of he seres ( Y o he adjused values ( Y ˆ =,..., T, usg he formula: =,..., T Y T % = (3 Y ˆ - resdue cclcall relave expresso ( RC %, whch expresses he perceage of devao from red of each values of seres, compued wh he formula: Y Yˆ RC % = (4 Y ˆ Graphcal represeao of perceage of red gves a suggesve mage of cclcal compoe devao from red le. Resdual mehod do allows a predco of cclcal varao, bu ol he aalss of evoluo of real seres. Whe mohl or quarerl daa are avalable ad graphc represeao of he seres suggess he presece of seasoal pheomeo, he applcao of resdual mehod requres he elmao advace from he acual daa of seasoal varaos usg oe of he ow mehods for deermao of mohl (quarerl seasoal dces ad he fluecg he emprcal values wh hese dces. I subseque calculaos s used he deseasoalzed me seres. Mehod of he Naoal Bureau of Ecoomc Research (USA s a more complex mehod havg he am o aswer a wo quesos whch are: - f he cocree me seres exss a form of varao ha repeas (wh smaller or larger devaos he cosecuve ecoomcal ccles of he geeral ecoomc acv (commercal or dusral ad whch are he characerscs of hs varao. Ths queso cocers he behavor of varous me seres he cosecuve phases of he elveg ad depresso of ecoomc suao ae as a whole. - f he cocree me seres exss a ow udulao moveme ad whch are s characerscs. Ths queso refers o he deermao of he cclcal compoe from he me seres ad ca be solved usg he resdual mehod. Esseall, he applcao of hs mehod supposes he acheveme of a referece model b s usg o be esablshed, her hsorcal evoluo, he lower ad upper pos of he dusral ad commercal acv. Subsequel, he specfc ccles overlap he referece ccles, 79

4 The Aals of The "Şefa cel Mare" Uvers of Suceava. Fasccle of The Facul of Ecoomcs ad Publc Admsrao Vol., No. (, wh he am o esablsh he complace or lac of complace amog he phases of geeral ecoomc ccle ad specfc courses. There are hree ma reasos o sud he seasoal whch are:. deermao he seasoal allow o esablsh he prevous developme model of he pheomeo ad b hs we oba valuable formao coecg wh he predced evoluo of a pheomeo or process, o a prese perod.. deermao he seasoal s useful for he acheveme of he forecasg calculao of he seasoal varaos, esseal for he shor-erm decsos. 3. oce esablshed he exsece of he seasoal compoe we ca elmae s effecs from me seres, operao called deseasoalsao. The deermao of he seasoal effecs allows herefore or he elmao of hese flueces ad he sud of he remaed varaos or he sud separae of he seasoal varaos. The basc mehod of aalss he seasoal s ha afer we have esablshed ha he seasoal facors are mafesed addve or mulplcave, he values of compoes are esmaed for each seaso of he aalsed perod, b deermao of he seasoal devaos or he seasoal dces. I he case of o-saoar me seres, such as here are he mos ecoomcal ad socal me seres, he esmae of seasoal devaos eeds prevousl red deermao wh a aalcal model or wh mehod of leas squares ad defcao subseque defcao of seasoal oscllaos. The me seres erms erdepedece resuls from he fac ha he are successve values of he same varables recorded a cera ervals or momes of me, whch mae ha he value of each erm depeds b he level from he pas. Such a depedece ca be surprsed b he fucoal relao: = f (,...,, (5 where represe he umber of erms from pas whch have a sgfca fluece o he value. I he aalss of a me seres we mus mae he dsco bewee: - emporal coeco, whch supposes he orderg of a varable values b her emporal successo - causal coeco, whch supposes ha f a eve A occurred afer he eve B, he he elmao of he eve B wll deerme ecessarl he elmao of he eve A. Aalss of me seres s based o he smulaeous exsece of he emporal ad causal coeco. Are defed wo eves whch are: A - value of varable Y o be a he mome B - value of varable Y o be a he mome. We cosder P ( A ad P ( B he probables of acheveme he eves A ad B ad P ( A / B codoal probabl, so ha A o be preceded me b B. We descrbe he probablsc causal coeco b he followg formula: P ( Y = Y = f P( Y = (6 Ierdepedece amog he me seres erms s show b he auocorrelao of me seres erms whch explas he objecve edec of developme he pheomea. Auocorrelao amog me seres erms creaes dffcules esmag parameers of he adjusme equao, herefore s exsece should be defed from he graphc represeao of he me seres ad he esablshg of s level b esmag he auocorrelao coeffce. The calculus relao s: 8

5 The Aals of The "Şefa cel Mare" Uvers of Suceava. Fasccle of The Facul of Ecoomcs ad Publc Admsrao Vol., No. (, ρ = ( ( + = cov = σ (, + ( σ ( + ( ( + = = where represes he umber of perods whch are separaed he pars of values ad ad are calculaed wh he relaos: = x = ad = + = (8 I he case of saoar me seres or for large eough, case whch =, he auocorrelao coeffce s gve b he relao: = = ( ( ( = + ρ (9 If he values ρ are graphc represeed depedg o, he graphc so obaed s amed correlogram ad llusraed he prcpal characerscs of he auocorrelao pheomeo of he me seres erms. STATIONARY TIME SERIES Cosderg he gve values of me seres ( wh =, ha:. hs seres defed a saoar process a resrced sese, f E( = cos a for a ad he correlao marx of seres + τ, + τ,..., + τ do deped o τ. hs seres s amed whe-ose f s a saoar process wh ull average ad he, f = auocorrelao coeffces sasf he proper ρ =., f Cosderg he me seres gve b he relao ( we roduce he followg defos: - a process { Y } Z s srcl saoar f for a (,..., T, wh p p... p R defg he me axs ad a h T so ha h T have he +, he processes { Y } + h ad { Y } same dsrbuo - a process { Y } Z s a saoar process b m order f for a (,..., T wh p p... p ad a h T so ha h T verf equal averages: m m ( ( ( ( m m Y,... Y = E Y + h,..., Y h E + +, for he followg wo processes, { Y } = = (7 ad { Y = h } + ( * where he parameers m,..., m N verf equal m m m. I he case of saoar of frs order, whch m = ad m = obas equal averages: E ( Y = E( Y +h = µ. I a saoar process, s characerscs (average, varace a.s.o. are cosa me. Y, where T = R, N or Z have he followg properes: A saoar process oed { } T. ( Y m E =, T. var( Y σ =, T 3. ( Y, Y = E( Y Y m = γ ( s cov, (, s T s s. 8

6 The Aals of The "Şefa cel Mare" Uvers of Suceava. Fasccle of The Facul of Ecoomcs ad Publc Admsrao Vol., No. (, The hrd proper of he saoar process leads a a ew defo of he seasoal of a sochasc process whch s: a process { Y } Z s saoar of he secod order or s slgh saoar f hs sasfes he followg hree properes:. E( Y = m, T = σ, T. var( Y 3. cov ( Y, Y = γ ( h, (, h T + h. I he case of a saoar process of secod order he achevemes of covarace fuco are deermed b he dfferece me ad o b he acual values of me seres. Saoar of secod order s called also saoar covarace (. I he case of o-saoar seres b daa rasformaos we ca oba a saoar me seres. If he seres s o-saoar ad o-homogeeous boh he mea ad varace are vared over me. I such a d of seres frs recourse o he rasformao of he seres b logarhm, whch s a specal case of Box-Cox rasformao. I he case of seres gve b he relao ( Y hs s: = T Y λ, λ ( (, T Y = ( λ λ log Y, λ = Choosg he opmal value of he parameer rasformao eeds he usg of more values of hs ad choose he oe ha esures he lower value of he sum of squares resdues or ca use he maxmum verosml mehod. Trasformg daa hrough logarhm appl f he varao coeffces of he orgal seres are cosa me, so he average ad he sadard devao have proporoal szes o me. Afer logarhm, for he seres so oba appl he proceedg of saoalzg appled o a homogeeous o-saoar seres. For example, he me seres { Y, R}, defed b he relao Y = b + a + ε, where ε s a whe-ose s saoar because he average of process defed b EY = b + a s depede b. To oba a saoar seres appl he frs order dffereal operaor, resulg he seres: V = Y Y = a + ε ε ( I he case of rasformed seres we have he followg characerscs:. average of rasformed seres s me depede, because EV = a. varace s depede of he me varable because var( V = var( a + ε ε = σ ε 3. seres covarace s me depede, because: cov( Y, Y + h = E( ε ε ( ε + h ε + h = E( ε ε + h E( ε ε + h σ ε, h {, } = σ ε, h = (3, h Z {,, } Seres obaed from hs rasformao has o he characerscs of whe-ose. σ s he varace of hs process. So le s cosder ha ( ε s a whe-ose process ad ( Whe we refer a wo or more whe-ose process we use smbols such as ( wh hese oaos we ca buld a model of me seres wh he form: x = q ε ad ( ε β ε (4 For each perod, x cosruc from he values q. So, ε, ε,..., ε mulpled b he value assocaed β. A sequece formed hs wa s amed movg average of q order ad s oed MA ( q. B combg a movg average process wh a lear dffereal equao we oba a auoregresve movg average model. Cosderg he dffereal equao of p order wh he form: 8

7 The Aals of The "Şefa cel Mare" Uvers of Suceava. Fasccle of The Facul of Ecoomcs ad Publc Admsrao Vol., No. (, ad { } p + a x (5 = a + x a process of movg average of q order, ha s MA ( q = + p + q a a ε, we ca wre: β (6 We follow he coveo of ormalzg he u so ha β s alwas equal wh he u. If he gve characersc roos of he relao (6 are all sde a u crcle ha { } s called auoregresve movg average model ( ARMA for. The auoregresve par of he model s he dffereal equao, gve b homogeeous poro (5 ad he par of movg average s he sequece { x }. If he homogeeous dffereal equao coas p lags ad he model for x q lags, he model s amed ARMA ( p, q. If q =, he process s called pure auoregresve, oed AR ( p ad f p =, he process s a pure movg average process, oed MA ( q. Le s r o def he ecessar ad suffce codos for a auoregresve process of order AR ( o be oe saoar process. We cosder he process: = a + a + ε (7 Also we cosder ha he process sared he perod zero, so ha s he al deermsc al codo. The soluo of such equaos s: = a a + a + a ε (8 If we ca use he erm lm value (8, ha he sequece { } wll be saoar. For a gve ad a p, resuls ha mus be eough. Ideed, f a surve s geeraed b a process whch has recel begu, he recorded values ca o be saoar ad from hs reaso researchers based o he assumpo ha he daa geerao process oo place a fel log me. Whou he orgal value, he sum of homogeeous ad parcular soluos for s: ( a + a A( a a / + ε (9 = = where A s a arbrar cosa. Sabl codos are he followg:. homogeeous soluo mus be zero. The sequece mus beg a f he pas or he process mus alwas be balaced (so ha arbrar cosa o be zero. characersc roo a mus be less ha u, absolue value. p q ARMA,, resuls ha a ecessar codo ha hese processes o be saoar s he homogeeous soluo o be zero. Also, s demosraed ha for a movg average process MA, gve b he relao (6, Geeralzg hese ow codos for all he processes ( ecessar ad suffce codos for o be saoar are ha: ( ( ( ( β β β + β β... o be zero ( s s+ s+ β ad Cosderg a auoregressve movg average process, gve b he relao (5, wh x gve b he relao (4, f roos of homogeeous form (5 are sde of a u crcle ad he sequece { x } s saoar, ha he process { } s saoar, hs beg he codo of saoar of he auoregresve coeffces. NON-STATIONARY AND NON-HOMOGENEOUS TIME SERIES Mos of ecoomc daa seres are o saoar, he hav cosa average ad dspla phases of relave sabl followed b perods of hgh varabl. Ma of prese ecoomercal 83

8 The Aals of The "Şefa cel Mare" Uvers of Suceava. Fasccle of The Facul of Ecoomcs ad Publc Admsrao Vol., No. (, research are based o he exeded Box-Jes mehodolog he behavor aalss of hese pe seres. Usuall, he usg of auoregressve mehods eeds hese seres saoalzao. Also he mos macroecoomc varables are o-saoar. The sample averages are o cosa ad/or here s o srog edec of heerosedasc. We ca def same echaracerscs of he grea vare of me seres from hs doma whch are:. ma me seres of macroecoomc varables coa a clear red (for example, gross domesc produc ad s compoes, prof raes, flao raes a.s.o. have posve reds whch ma be followed b perods of decle ad herefore s dffcul for such varables have a cosa mea, so he are o saoar. some seres seem o oscllae (for example, exchage raes do show a clear edec of crease or decrease: some perods currec s assessg, ohres are deprecag ad hs radom moo s specfed o-saoar seres 3. a mpac regsered b a seres has a hghl degree of perssece 4. ma me seres varabl s cosa me. Such d of seres are codoall heerosedasc f he ucodoal varace (or o log-erm s cosa, bu here are perods whch varace s relavel hgh. 5. some seres are smlar developme wh ohers. Such smulaeous movemes should o surprse bu we mus udersad ha he forces whch fluece he aoal ecoom have a smlar fluece o he eraoal ecoom. Aalss of a me seres defes wo pes of o-saoar seres whch are:. processes of deermsc o-saoar pe - TS (Tred Saoar. processes of o-saoar sochasc pe - DS (Dfferece Saoar. Each of hese wo processes have cera characerscs ad red removal ad coverso o saoar processes requre dffere echques. A o-saoar process b deermsc pe s reprezeed b he relao: = + ε ( P where P s a polomal of a cera degree relao o varable me ad ε s a saoar process. If he polomal s frs-degree, case whch P = α + β ad ε s a whe-ose, he he process s characerzed b he relao: = α + β + ε ( Such a process has he followg characerscs:. he process average s depedeed b he me, depedec characerzed b he relao: E( = α + β. he process varace s cosa me, hs beg equal wh he varace of wheose whch s: var( var( α + β + ε = = σ ε 3. covarace of hs process s zero: cov (, ' =. I he case of such a process he edec has a deermsc aure ad he varace s cosa over me. Passg hs deermsc o-saoar process pe b a saoar process s doe b esmag he red ad elmao from he orgal seres. Therefore, he saoalzao s acheved wo phases amel:. are esmaed polomal parameers P b usg he mehod of leas squares. If he red has obaed a lear model: ˆ ˆ ˆ = α + β.. saoar seres s obaed b removg he esmaed red he frs sage, amel: ˆ ε = ˆ ( ˆ ˆ = α β. Such a saoar seres ca be represeed b a auoregressve process of a parcular pe. A o-saoar sochasc process has he represeao: = ρ + β + ε, where ε s a saoar process ad ρ, β are real cosa. 84

9 The Aals of The "Şefa cel Mare" Uvers of Suceava. Fasccle of The Facul of Ecoomcs ad Publc Admsrao Vol., No. (, Whe ρ = o-saoar sochasc process allows he represeao: = β + ε. As we show mos ecoomc seres are o saoar, havg he followg characerscs of o-saoar: he seres has a average whch s o cosa over me, hs followg usuall a lear pah wh posve or egave slope (seres s o-saoar homogeeous pe, such seres characerzg b cosa chages from oe perod o aoher here are seres whch have varao o-cosa from oe perod o aoher, case whch he seres s o-saoar b o-homogeeous pe, boh varace ad average beg varables over me. For esablshg wheher he me seres s saoar are appled dffere sascal ess bewee he mehods mos commol used beg he aalss of auocorrelao fuco (4. If he seres respecs oe of he followg crera, s o-saoar ad mus be saoalzed b dffereao:. auocorrelao fuco values are he vc of oe or are egave. a umber of successve values of auocorrelao fuco are relavel equal. Whe oe of he wo cases s doe, ha he seres { } s dffereaed ad we oba = T =, =,..., T. For he ew seres s resumed he aalss of saoal o he base of esmao he auocorrelao fuco. Amog he crera of choosg he mos adecquae auocorelao model we clude: classcal crera, b whch s chose ha model whch has he hghes value for R adjused or he lowes varace or dsperso resdues dcaors based o formao heor, provdg he auoregressve model seleco for whch he crero Aae or Schwarz has he lowes value. The formal defcao of he red presece me seres s realzed b sascal asg, as we saw or b graphc represeaos. CONCLUSIONS We see from hs bref overvew of he characerscs of me seres he mporace of defg hem, aalcal or graphcal, her compoes ad he he presece or o he red. Varabl me seres resuls from he fac ha o he aalsed pheomeo ac esseal causes whch deermes he geeral edec of he me seres erms ad o-esseal radom causes, whch deermes he degree of devao from red. Varabl has deermed he applcao of some ess for defcao he pe seres (saoar or o-saoar, frequel used o deerme he saoar beg he aalss of auocorrelao fuco. If he seres s osaoar, he saoalzao s he frs sep selecg he mos approprae model for he red. For saoar or saoarzed me seres are defed he parameers of auoregressve models, s chose he mos approprae model ha ca be used laer aalss ad forecasg. I hs paper we red o esablsh codos for saoar of a me seres a auoregressve movg average process. ENDNOTES ( Cclcal oscllaos are deermed b he perodc of successo of dffere ecoomc processes such as he perodc reewal of he produco u of soce, he formao ad he cclcal reproduco of prmar resources (fxed capal, power resources ad raw maerals, labor, he roduco he exploao of some ew resources, he developme leaps of scece ad echologcal ovao, he perodc resrucurg of aoal ecoom, he socal revoluos, wars a.s.o. (source: Bj, E. M., Llea, E., Roşca, R. E., Vău, M., Sască aplcaă î ecoome, Eded b Uversal Dals, Buchares,, p., 36. ( A example of saoar seres s he whe-ose ε, for whch are vald he hree properes: E ε, T. ( =. var( ε = σ T ε, 3. cov (, = ε, h ε h Bourboas, R., Ecoomere, Eded b Edura Ecoomca, Buchares, 8, p. 34. (source: Adre, T., 85

10 The Aals of The "Şefa cel Mare" Uvers of Suceava. Fasccle of The Facul of Ecoomcs ad Publc Admsrao Vol., No. (, (3 If he codo ( s vald for all values of s ad β, ha he codo ( = s reduda ad a movg average process of fe order MA wll be alwas saoar. For a process of fe order, he codo ( s rue for all values s, where s s he umber of perods. (source: Eders, W., Appled ecoomerc me seres, s edo, Eded b Joh Wle & Sos, Ic., USA, 995, p (4 Saoalzao of me seres s he frs sage usg he Box-Jes procedeure for selecg he mos approprae model of red. Brefl, he seps of hs procedure are: calculae dcaors for saoalzed aalss (whe he me seres s o-saoar wll be saoalzed b proper rasformaos, for example, logarhm he seres values or appl a Box-Cox rasformao, ha dffereae he daa seres hus obaed. If s a saoar seres s defg he pe of model, he parameers of model are esmaed usg a auoregressve ow mehod such as: he mehod of leas squares, he maxmum lelhood mehod, he mehod of momes a.s.o., are esed he characerscs of esmaed auoregresve models, are choosed he bes model, are acheved varous aalss ad forecass. (source: Adre, T., Bourboas, R., Ecoomere, Eded b Edura Ecoomcă, Buchares, 8, p BIBLIOGRAPHY. Adre, T., Bourboas, R., Ecoomere, Eded b Edura Ecoomca, Buchares, 8.. Adre, T., Sacu, S., e al., Sască. Teore ş aplcań, Eded b Edura Ecoomca, Buchares,. 3. Baro, T., Bj, E. M., Tövss, e al., Sască eorecă ş ecoomcă, Eded b Edura Ddacca s Pedagogca R. A. Buchares, Bj, E. M., Llea, E., Roşca, R. E., Vău, M., Sască aplcaă î ecoome, Eded b Edura Uversal Dals Buchares,. 5. Bj, M., Bj, E. M., Sască eorecă, Eded b Edura Ddacca s Pedagogca, Buchares, Eders, W., Appled ecoomerc me seres, s Edo, Eded b Joh Wle & Sos, Ic., USA, Flemg, C. M., Nells, G. J., The Essece of Sascs for Busess, s Edo, Eded b Prece Hall Ieraoal (UK, Isac-Mau, Al., MruŃ, C., Voeagu, V., Sasca peru maagemeul afacerlor, Eded b Edura Ecoomca, Buchares, Jaba, E. (coord., Bala, C., e al., Ecoomere aplcaă, Eded b Edura Uversa Al. I. Cuza Ias, 8.. Jaba, E., Sască, Eded b Edura Ecoomca, Buchares, Keller, G., Warrac, B., Barel, H., Sascs for Maageme ad Ecoomcs. A Ssemac Approach, Wadsworh Publshg Compa, Belmo, Calfora, Kora, M., Begu, L. S., Tuşa, E., Maole, C., Bazele sasc peru ecoomş, Eded b Edura Trbua Ecoomca, Buchares, 5 3. Lecallo, J., Labrousse, C., Sasque descrpve, Édos Cujas, Pars, Lev, I. R., Sasc for Mageme, 4 h Edo, Eded b Prece-Hall, Eglewood Clffs, New Jerse, Mlls, C., F., Meode sasce (vol. I ş II, Dreca Cerala de Sasca, Buchares, Nculescu-Aro, I. G., Mazurecu-Marescu, M., Meode ecoomerce peru afacer, Eded b Edura ASE Buchares, Roşca, R. E., Modelarea serlor croologce, he volum Sasca î cercearea ecoomco-socală, Eded b Edura Jumea Ias,. 8. Roşca, R. E., Modele ecoomerce î aalza compoeelor serlor de mp, he volum of Scefc Sesso wh Ieraoal Parcpao, vol. III, Eded b Edura Uversa Suceava, łarcă, M., Traa de sască aplcaă, Eded b Edura Ddacca s Pedagogca R. A. Buchares,