Fortified financial forecasting models: non-linear searching approaches

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1 0 Inernaional Conference on Economic and inance Reearch IPEDR vol.4 (0 (0 IACSIT Pre, Singapore orified financial forecaing model: non-linear earching approache Mohammad R. Hamidizadeh, Ph.D. Profeor, Mg. and Accouning aculy, Shahid Behehi Univeriy, USB Eveen, Tehran, Iran Mohammad E. adaeinejad, Ph.D. Aociae Profeor, Mg. and Accouning aculy, Shahid Behehi Univeriy, SBU Eveen, Tehran, Iran bu.ac.ir Abrac- When reearcher are going o foreca daa wih variaion involving a ime erie daa, hey believe he be forecaing model i he one which realiically conider he underlying caual facor in a iuaional relaionhip and herefore ha he be rack record in generaing daa for forecaing. aϊve forecaing model may be adjued for variaion in relaed daa. When a ime erie procee a grea deal of randomne, moohing echnique may improve he accuracy of he financial foreca. Bu neiher aϊve model nor moohing echnique are capable o idenify he major fuure change in he direcion of a iuaional daa erie. Hereby, onlinear echnique, like direc and equenial earch approache, overcome hoe horcoming can be ued. Thi paper i o manipulae how o prepare and inegrae heuriic non-linear earching mehod o erve calculaing adjued facor o produce he be foreca daa. Keyword; aϊve forecaing model; Smoohing echnique; ibonacci and Golden ecion earch; Line earch by curve fi. I. I TRODUCTIO The aricle aim i o conruc, give a new mehod and o preen a proper analyi of he facor which will effec he forecaing equaion. The new calculaing proce i baed on nonlinear inelligen mehod which earching o find a new poin in ime erie procee for reidual minimizaion. We will ry o preen how o forify forecaing financial model wih nonlinear echnique characeriic baed on line earch mehod for economic, finance and buine forecaing echnique. ou may udy many inereing nonlinear echnique on he one hand on Luenberger (989 and Bazaraa (993, bu on he oher hand, wih analyi of forecaing echnique ha i preened by McGuigan (999, Gourieroux and Jaiak(00, hi idea go o preen ielf o improve ignificanly i predicive power by applying nonlinear mehod advanage of generaing aionary poin in differen way o advocae he adjumen facor of he forecaing echnique o exen ha minimize difference e = ( or ( - - []. A in he univariae cae, he pure vecor auoregreive proce of order [VAR (] provide a imple framework for exploring he muliplier effec and forecaing. Moreover, hi mulivariae auoregreive proce can accommodae quie complex dynamic of individual componen erie. In empirical reearch, he VAR ( model ofen provide a aifacory fi o mulivariae reurn erie. Therefore, he proce (y ; <Z i inegraed, denoed I(, if and only if i aifie he recurive equaion y =y - +e ; where (e i a weak whie noie [3]. Thi paper rie o minimize e [, 9]. II. IITIAL PREREQUISITE Reearcher ofen deal wih procee ha vary a ime pae. Time erie are analyzed o beer underand, decribe, conrol, and predic he underlying proce. The analyi uually involve a udy of he componen of he ime erie (TS uch a ecular rend, cyclical variaion, eaonal effec, and ochaic variaion- he paricular componen of inere endency o vary from one problem o anoher [3]. The foreca i merely a predicion concerning he fuure and i required in virually all area of he operaion []. Obviouly, good forecaing i eenial o reduce he uncerainy of he TS in which mo deciion are made. The level of ophiicaion required in forecaing echnique varie direcly wih ignificance of he problem being examined [, 8]. The forecaing echnique ued in any paricular iuaion depend on wo major facor: aggregaion facor uch a inernaional / regional economic; naional economic; Indury economic; individual firm economic; he inernal of he firm. Diagnoic facor are co and poenial gain; he complexiy of he relaionhip; ime period; accuracy required; lead ime [4]. III. ORECASTIG TECHIQUES I. aϊve forecaing model are baed on TS obervaion of he value of he variable being foreca. Le ŷ denoe he foreca daa of he variable of inere, y denoe an acual oberved daa of he variable and he ubcrip idenify he ime period. The imple model ae ha he foreca value of he variable for he nex period will be he ame a he value of ha variable for he preen period [7]. If a recognizable paern exi, i may be incorporaed by adjuing relaed equaion. Two poible cae are a linear rend and nonlinear rend. A linear rend follow a conan rae of growh paern [3]. Since he relaed equaion i a nonlinear relaionhip, he parameer canno be eimaed direcly wih he lea-quare mehod. When eaonal variaion are inroduced ino a naive forecaing model, i may be poible o improve ignificanly i hor-run predicive power. The be known of hee i he raio-o-rend mehod. Trend projecion equaion are mo ueful for inermediae and long-erm forecaing [, 9]. 65

2 Anoher model ha i ofen very effecive in generaing foreca when here i a ignifican eaonal componen i he exponenially weighed moving-average (EWMA forecaing model. A each poin in ime, he EWMA model eimae a moohed average from pa daa, an average from pa daa, an average rend gain, and he eaonal facor. Thee hree componen are hen combined o compue a foreca [3]. II. Smoohing echnique are higher form of naϊve forecaing model which aume ha an underlying paern can be found in he TS of a variable ha i being foreca. I i aumed ha hee hiorical obervaion repreen no only he underlying paern bu alo random variaion. By aking ome form of an average of po obervaion, moohing echnique aemp o eliminae he diribuion ariing from random variaion in he erie and o hoe he foreca on a moohed average of everal pa obervaion [8, 9]. A. Moving average. If a daa erie poee a large random facor, in an effor o eliminae he effec o hi randomne, a erie of recen obervaion can be average o arrive a a foreca. Thi i he moving average mehod. A number of oberved value are choen, heir average i compued, and hi average erve a a foreca for he nex period.. Simple MA may be defined a: Ŷ+ = = + [ ] = + Δ = (9 = + + Where i Δ [7].. Weighed moving average may be viewed a a cenered MA, where he obervaion in a equence don receive equal weigh. Ŷ + =[Σw y ], Where i he revere of he um of he weigh, and i he number of obervaion in a moving average. The greaer he number of obervaion ued in moving average, he greaer he moohing effec becaue each new obervaion receive le weigh (/ a increae. Hence, generally, he greaer he randomne in he daa erie and he lower he change in he underlying paern, he more preferable i i o ue a relaively large number of pa obervaion in developing he foreca. The choice of an appropriae MA period, ha i, he choice of, hould be baed on a comparion of he reul of he model in forecaing pa obervaion. One crieria ha i ofen ued for uch comparaive purpoe i he minimizaion of he average foreca error, or roo mean quare error. RMSE = m [ Σd ] = m = ( y y Where m i he number of ime period [8]. B. ir-order exponenial moohing. To foreca a aionary ime erie wih he exponenial moohing model, we fir mooh he ime erie wih a MA imilar o hoe decribed in order o iolae he yemaic or mooh componen of he erie. We hen projec hi mooh componen ino he fuure. The MA employed by he exponenial moohing model for aionary TS i a pecial ype of weighed MA. The mooh componen of a aionary ime erie may be conidered a a ucceion of eimae of he underlying mean level of he aionary proce. Updae he moohed eimae: = y + ( = + ( = + ( = + d + ( 3 + ( k =, k =,,3,... Where i he moohed eimae for he curren period, i he moohed eimae for he proceeding period -, y i he obervaion for he curren period, i he moohing conan, and Ŷ +k he foreca for k period ahead. Thee model weigh he mo recen obervaion by 0<<, and he pa forecaing by (-. A large indicae ha a heavy weigh i being placed on he mo recen obervaion [8]. C. Double exponenial moohing Model. Daa ha are colleced over ime frequenly exhibi a linear rend when a linear rend i no apparen in he daa, ingle exponenial moohing i well uied a a forecaing echnique. How if a rend i preen in he daa, DES i more appropriae for obaining foreca han i ingle ES. The DES i cloely relaed o he rend analyi echnique of imple linear regreion, in which he foreca for +, given TS up o and including ime. In DES, he coefficien a and b are conidered o be funcion of ime and are updaed a each ucceive obervaion become available. Thee updaed coefficien are given by he following expreion: ŷ + =a +b (+ b =[(-](S -S ( ( a =S - -b S S =y +(-S - ( =S +(- S S ( Where i a moohing conan choen o minimize he um of he quared foreca error, S i he ingle moohed aiic compued recurively, S ( i he double moohed aiic alo compued recurively, and he iniial value S 0 and S ( 0 are obained uing he relaed equaion a follow: S 0 = 0 -[(-/]b 0 S ( 0 = 0 -[(-/]b 0 The quaniie 0 and b 0 are iniial value of he regreion coefficien. If daa are available, ay, o ime,

3 and b 0 are imply regreion coefficien compued uing he model = 0 +b 0. If no daa are available, value for 0 and b 0 are aigned ubjecively, or if hi i no poible S 0 and S ( 0 are boh aigned he iniial value of he erie, 0 [8,3]. D. ES for TS wih rend ecular. Sep. Updae he moohed eimae: = + ( ( + + d Sep.Updae he rend eimae: d = b ( + ( b d,0<b< Sep3. oreca for period +k: + k = + kd Where i he rend eimae for he curren period, d - i he rend eimae for he preceding period -, b i he rend adjumen conan, and i he foreca k + k period ahead for fuure period + k []. E. ES for TS wih rend and eaonal componen. Sep. Updae he moohed eimae: = ( + ( ( + d, p 0<a< Sep. Updae he rend eimae: d = b( + ( b d Sep 3. Updae he moohed index eimae: S = ( + ( S p Sep 4. oreca for period +k: k = ( + kd S +, Where i he eaonal index eimae for curren period, S -p i he eaonal index eimae for period -p, one eaonal cycle earlier, and i he eaonal index adjumen conan, 0<< [8, 9]. IV. OLIEAR TECHIQUES I i ime o urn o a decripion of he nonlinear echnique ued for he value of adjued facor baed on ieraively olving unconrained variaion minimizaion. Thee echnique are, of coure, imporan for pracical projecion ince hey ofen offer he imple, mo direc alernaive for obaining oluion; bu perhap heir greae imporance i ha hey eablih cerain reference plaeau wih repec o difficuly of implemenaion and peed of convergence. There i a fundamenal underlying rucure for almo all he decen algorihm we dicu []. One ar a an iniial obervaion deermine, according o a fixed rule, a direcion of movemen; and hen move in ha direcion o a (relaive minimum of he TS error on ha line. A he new prediced value a new direcion i deermined and he proce i repeaed. Once he elecion i made, all algorihm call for movemen o he minimum foreca oberve on he correponding componen [5]. The proce of deermining he minimum on a given variaion i called variaion earch [9]. or general nonlinear funcion ha can no be minimized analyically, hi proce acually i accomplihed by earching, in an inelligen manner, along he variaion for he minimum foreca value error [0]. Thee line earch echnique form he backbone of nonlinear programming algorihm, ince higher dimenional problem are ulimaely olved by execuing a equence of ucceive line earche [8]. I. IBOACCI AD GOLDE SECTIO SEARCH A. ibonacci earch. The mehod deermine he minimum value of a ime erie over a cloed inerval [C,C ]. In projecion, a TS may in fac be defined over a broader domain, bu for hi mehod a fixed inerval of earch mu be pecified [6]. To develop an appropriae earch raegy, ha i, a raegy for elecing obervaion baed on he previouly obained value, we poe he following proce: ind how o ucceively elec obervaion o ha, wih explici knowledge of TS, we can deermine he malle poible region of uncerainy in which he minimum reidual mu lie. Thu, afer value are known a obervaion y, y y n wih C < < < n- < n C The region of uncerainly i he inerval [ k-, k+ ] where k (=d k i he minimum error among he, and we define 0 =C, n+ =C for coniency. The minimum of d k mu lie ome where in hi inerval. The derivaion of he opimal raegy for ucceively elecing obervaion o obain he malle region of uncerainy i fairly raigh forward bu omewha ediou. Le =C -C, The iniial widh of uncerainly k = widh of uncerainly afer k meauremen. Then, if a oal of obervaion are o be made, we have ( W n k + k = = n,where he ineger k are member of he ibonacci equence generaed by he recurrence relaion: = - + -, 0 = +, The reuling equence i,,,3,5,8,3, The procedure for reducing he widh of uncerainy o i hi; The fir wo obervaion are made ymmerically a a diance of ( = W from he end of he iniial inerval; according o which of hee i a leer value, an uncerainy inerval of widh = (W i deermined, he hird obervaion i placed ymmerically in hi new inerval of uncerainy wih repec o he meauremen already in he inerval. The reul of hi ep give an inerval of uncerainy 3 =( - / =w. In general, each ucceive obervaion i placed in he curren inerval of uncerainy ymmerically wih he poin already exiing in ha inerval [6]. 67

4 B. Search by Golden ecion. If he number of allowed obervaion in a ibonacci earch i made o approach infiniy, we obain he golden ecion mehod. I can be argued, baed he opimal properly of he finie ibonacci mehod, ha he correponding iniial verion yield a equence of inerval of TS whoe widh end o zero faer han ha which would be obained by oher mehod. The oluion o he ibonacci equaion = = f +b f i o finding, where f and f are roo of he characeriic equaion f =f+. Explicily f.68 and f Therefore,.68 i known a he golden ecion raio. or large he fir erm on he righ ide of f equaion dominae he econd and hence lim = 0.68 I follow from (6 ha he inerval of TS a any obervaion in he TS ha widh =.( f + And from hi i follow ha = = f Therefore, we conclude ha, wih repec o he widh of he TS inerval, he earch by golden ecion cover linearly [4, 6]. II. LIE SEARCH B CURVE ITTIG. A. ewon mehod. Suppoe ha he TS error of a ingle variable i o minimized, and uppoe ha a a value y k where an obervaion i made i i poible o evaluae he foreca, fir derivaive ( and econd derivaive ( of he prediced value on uni of ime : ŷ, ŷ, ŷ. Hereby, i i poible o conruc a quadraic ime erie q which a y agree wih TS up o econd derivaive (, ha i q(y= Ŷ + Ŷ (y 0 -y + Ŷ ( ow, we can calculae an eimae +k of minimum reidual of TS by finding he reidual where he derivaive of q vanihe. Thu, eing 0= Ŷ + Ŷ ( + - Or Ŷ + Ŷ. =0 We hen find foreca for period +k: Ŷ +k = +, k=,, where = / Ŷ. B. Mehod of fale poiion. ewon mehod for reidual minimizaion i baed on fiing a quadraic on he bai of informaion a a ingle obervaion; by uing more poin, le informaion i required a each of hem.. Thu, uing Ŷ, Ŷ, Ŷ - i i poible o fi he quadraic q(= Ŷ + Ŷ ( 0 - Δ + Δ ( 0 +. which ha he ame correponding value. An eimae +k can hen be deermined by finding he obervaion where he derivaive of q vanihe; Thu, we find foreca for period +k: Ŷ +k = -. Ŷ, Where =[ - - ]/( Ŷ - - Ŷ. Thu, uing Ŷ, Ŷ, Ŷ - i i poible o fi he quadraic which ha he ame correponding value. An eimae +k can hen be deermined by finding he obervaion where he derivaive of q vanihe; Thu, we find foreca for period +k. Comparing hi equaion wih ewon mehod, we ee again ha he value Ŷ doe no ener; hence, our fi could have been paed hrough eiher Ŷ or Ŷ -. Alo he model can be regarded a an approximaion o ewon model where he econd derivaive i replaced by he difference of wo fir derivaive[]. C. Cubic i. Given he obervaion, - and wih correponding foreca and derivaive value Ŷ -, Ŷ -, Ŷ, Ŷ, I i poible o fi a cubic model o he obervaion having correponding value. The nex oberve + can hen be deermined a he relaive minimum poin of hi cubic. Thi lead o find foreca for period +k; Ŷ +k = -( - - = -. wih =[ Ŷ +u -u ]/[ Ŷ - Ŷ - +u ] Where u = Ŷ - + Ŷ -3([Ŷ - Ŷ ]/( - - u =[u - Ŷ. - Ŷ ] / I can be hown ha he order of convergence of he cubic fi mehod i. Thu, alhough he mehod i exac for cubic model indicaing ha i order migh be hree; i order i acually only wo[4, 5]. D. Quadraic i. Thi mehod i ofen mo ueful in line earching by fiing a quadraic hrough hree given acual and foreca obervaion. Thi ha he advanage of no requiring fir and econd derivaive informaion. If we have poin,, 3 and correponding foreca value Ŷ, Ŷ, Ŷ 3. We conruc he quadraic paing hrough hee poin: q ( y = 3 i i = Where =[ j±i ( 0 - j ]/[ j±i ( i - j ] and deermine a new obervaion 4 a he poin where he derivaive of q vanihe. Thu b =. + b + b a3 + a 3 + a3 bij i j Where a ij = i - j, = [6]. III. COCLUSIO Thi aricle could make connecion beween wo iolaion area: forecaing echnique and nonlinear programming echnique. The aim of hi ineracion wa o 68

5 be ued nonlinear opimizaion mehod for finding forecaing value of inere variable baed on line earching mehod. The iniial difference of forecaing mehod i mainly on adjuing he daa mechanim for opimal predicion of variable; hereby, he aricle fir analyzed differen kind of forecaing echnique epecially on adjuing variaion componen power, hen i followed ome baic decen mehod o minimize he ime erie reidual in relaion o acual value of variable. Becaue of making aionary ime erie advanage of hee mehod, i ha ried o how how o inegrae hi characeriic for generaing forified forecaing daa. REERECES [] J. Sco, Armrong, Reearch of forecaing: a quarer cenury review, Inerface, January-ebruary 986, Pp [] M. S., Bazaraa, H.D. Sherali and C.M. Shey, onlinear programming: heory and algorihm,..: John Wiley & Son, 993. [3] C. Gourieroux, & J. Jaiak, inancial economeric, ew Jerey: Princeon Univeriy Pre, 00. [4] M.R. Hamidizadeh, onlinear programming, Tehran: SAMT Publihing Co., 00. [5] J., Kowalik, and M.R. Oborne, Mehod for unconrained opimizaion problem,..: Elevier, 968. [6] D. G. Luenberger, Linear and nonlinear programming, Maachue: Addion-Weley Publihing Co., Reading,989. [7] J.R. McGuigan, & R.C. Moyer, Managerial economic, S. Paul: We Publihing Co., 989. [8] J.R. McGuigan, R.C. Moyer, and.h. Harri, Managerial economic: applicaion, raegy, and acic, Cincinai: Souh-Weern College Publihing, 999. [9] J. eer, W. Waerman & G.A. Whimore, Applied aiic, Boon: Allyn and Bacon, Inc., 988. [0] J.. Traub, Ieraive mehod for he oluion of equaion,.j.: Prenice-Hall Englewood Cliff, 964. [] D. j. Wilde, and C.S. Beighler, oundaion of opimizaion,.j.: Prenice-Hall Englewood Cliff, 967. [] W.I. Zangwill, onlinear programming via penaly funcion, Managemen Science vol. 3, no 5, 967, pp [3] V. Zarnowiz, Recen work on buine cycle in hiorical perpecive, Journal of Economic Lieraure, June 985, pp

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