APPLICATION OF ADAPTIVE NEURO-FUZZY INFERENCE SYSTEM IN INTEREST RATES EFFECTS ON STOCK RETURNS

Transcription

1 Elefheros Govas / Ida Joural of Compuer Scece ad Egeerg (IJCSE) APPLICATION OF ADAPTIVE NEURO-FUZZY INFERENCE SYSTEM IN INTEREST RATES EFFECTS ON STOCK RETURNS Absrac ELEFTHERIOS GIOVANIS * Deparme of Ecoomcs, Royal Holloay Uversy of Lodo Egham, Surrey, TW0 0EX, Ued Kgdom * Elefheros.Govas.00@lve.rhul.ac.uk hp://.rhul.ac.uk I he curre sudy e exame he effecs of eres rae chages o commo sock reurs of Greek bakg secor. We exame he Geeralzed Auoregressve Heeroskedascy (GARCH) process ad a Adapve Neuro-Fuzzy Iferece Sysem (ANFIS). The coclusos of our fdgs are ha he chages of eres raes, based o GARCH model, are sgfca o commo sock reurs durg he perod e exame. O he oher had, h ANFIS e ca ge he rules ad each case e ca have posve or egave effecs depedg o he codos ad he frg rules of pus, hch formao s o possble o be rereved h he radoal ecoomerc modellg. Furhermore e exame he forecasg performace of boh models ad e coclude ha ANFIS ouperforms GARCH model boh -sample ad ou-of-sample perods. Keyords: ANFIS, Ieres raes, Sock reurs. Iroduco The ssue of eres rae sesvy remas emprcally uresolved. Mos of he sudes use a varey of shorerm ad log-erm bod reurs as he eres rae facor hou provdg ay raoale for her use. Ye, here s o cosesus o he choce of he eres rae facor ha should be used esg he o-facor model. There s a broad cosesus amog he pracoers ad academcs ha eres raes have a sgfca effec o share prces, bu also here s a lle agreeme as o heher or o eres raes affec he sock reurs. The problem h ecoomercs s ha are based o probables ad sascs ad o o possbles ad membershp. To be specfc mpossble o fd sgfca esmaed coeffces over a specfc perod e exame ad herefore e coclude ha he pheomeo e exame s reeced, our cases he effecs of eres raes chages o sock reurs. Ths s o absolue logcal ad correc because radoal ecoomercs are o able o capure mprecso ad o leares. Wh fuzzy logc e ca rereve he rules he a specfc codo s fred, so here ll be alays posve ad egave effecs, our case, based o specfc rules ad behavour of pus. For example h coveoal ecoomerc modellg e ca fd ha here are o sgfca effecs a specfc perod, bu here are a sub-perod. Ad he queso s ho ca e fd hs perod? Eve f e apply rollg regressos hs s o very helpful for facal praccal purposes. Addoally, coveoal ecoomerc modellg s based o sascal properes, here a log sample s eeded. Also, msspecfcao errors, heeroskedascy, ARCH effecs ad auocorrelao resduals are some problems of ecoomerc esmaos. Wh fuzzy logc ad eural eorks e ca ake all he pus ad exame her mporace egh he deermao of he oupu h shor or log sample as log as fuzzy rules have bee defed. Furhermore, euro-fuzzy modellg, because s o based o sascal ad ecoomerc properes, auocorrelao ad heeroskedascy resduals, amog oher problems, are o examed as he dsurbace erm s o cluded fuzzy regressos ad euro-fuzzy sysem herefore hese problems have o meag. A proposal for furher research sudy ad applcaos s o roduce he dsurbace erm fuzzy ad euro-fuzzy modellg. Addoally, eural eorks have bee crczed ha are black boxes bu are able o descrbe very ell he oleares. O he oher had fuzzy logc s o alays able o descrbe oleares appropraely, bu s he mos effce mehod o approach mprecso, ad especally face, because s deermed by huma behavour ad hs s exacly he rue mprecso. More specfcally, ecoomerc mehodology reas huma behavor as a compuer based o bary logc h oly o possble values, rue or false, yes or o, expasve or recessve. To be correc he real values ha a huma expresses are maybe rue, maybe false, or rue f ad false f. For hs reaso e use Neuro-Fuzzy sysem ad e beleve ha s he fuure ecoomcs ad ecoomercs, as arfcal ellgece procedures are already used face. ISSN : Vol. No. 4

2 Elefheros Govas / Ida Joural of Compuer Scece ad Egeerg (IJCSE) The srucure of he paper has as follos: I seco e prese a shor leraure reve, seco 3 e prese he mehodology for he GARCH process ad he o-dex facor model, as also he mehodology of ANFIS s descrbed. I seco 4 e prese he perod examed ad e descrbe he daa frequecy, hle seco 5 e prese he emprcal resuls ad e dscuss abou hem. Fally, he fal seco e repor our cocludg remarks of our fdgs.. Leraure Reve There s a grea umber of research sudes examg he effecs of eres rae chages o sock prces ad o reurs. Fama [98] documes a srog posve correlao beee commo sock reurs ad real ecoomc varables lke capal expedures, dusral produco, real GNP, moey supply, lagged flao ad eres raes. Hardouvels [987] pos ou ha a verse relaoshp beee sock prces ad chages of eres rae exss ad hs ca be raoalzed erms of moey supply surprses. Che e al. [999] exame he effec of dscou rae chages o he volaly of sock prces ad o radg volume ad hey foud ha uexpeced dscou rae chages corbued o hgher, hough shor-lved, volaly ad radg volume. Sock reurs sesvy o eres raes as heorecally advocaed by Mero [973], Log [974] ad Soe [974]. Esseally, rsk averse vesors demad hgher compesao for exposure o facors, oher ha he marke porfolo, ha are correlaed h eremporal chages he vesme opporuy se. Soe [974] has offered aoher meas of expadg he marke model. He has proposed a o-dex model cossg of he radoal equy marke dex ad a deb marke dex ad he usfed he model by argug ha dvdual equy secures exhb varyg degrees of sesvy o eres raes ad ha he opporuy o ves rsky deb secures may represe a aracve alerave o rskless asses ad rsky equy secures. Booh ad Offcer [985] ad Bae [990] es he effec of curre ad uacpaed chages eres rae. Fraser e al., [00[ exame he effec of uacpaed rae chages. All hese sudes, as also oher research emprcal evdeces [Fama ad Scher (977); Chrse, (98)], foud srog suppor for a egave effec of boh curre ad uacpaed eres chages o bak sock reurs. Whle some sudes have foud he eres rae facor o be a mpora deerma of commo sock reurs of baks, o he coras Chace ad Lae [980] have foud he reurs o be sesve or oher supporg ha sock reurs oly margally explaed by he eres rae facor, so hese sudes fd o cremeal explaaory poer for eres rae chages [Lloyd ad Shck, 977)]. Research sudes employg fuzzy logc, ANFIS ad geerally arfcal ellgece procedures have o ye bee made. 3. Mehodology 3. To facor model The proposed geeralzed formulao of he o-facor model [Soe, (974)] s as follos: R p R ΔI 0 m (), here 0 s he cosa R p deoes he eekly reurs of a equally eghed porfolo of sock eek, R m s he eekly reur o he marke porfolo eek, ΔI s a defaul free deb dex as proxy of eres raes perod, s a saoary y sochasc process h zero mea for each porfolo, ISSN : Vol. No. 5

3 Elefheros Govas / Ida Joural of Compuer Scece ad Egeerg (IJCSE) The oe moh, hree, sx ad elve mohs Treasury bll raes have bee esed as he eres rae varable equao (), bu e prese radomly oly he resuls for hree mohs Treasury bll raes because he coclusos are he same all cases ad here s o dfferece usg shor-erm or loger-erm eres raes. Because h ordary leas squares e foud auocorrelao ad ARCH effecs e esmae h symmerc GARCH (p,q) process, hch s maly used facal ecoomerc leraure. GARCH model as proposed by Bollerslev [986]. The mea equao remas he same as equao () bu GARCH (,) process s:, here ~ (0, ) () 0 (3) u Addoally e have esed asymmerc GARCH models [Nelso, (99); Glose e al., (993)] bu he resuls are o preseed, as are exacly he same h hose of symmerc GARCH process. 3. Adapve Neuro-Fuzzy Iferece Sysem (ANFIS) We follo a smple ANFIS sysem order o mprove s forecasg performace ad o make much more useful. We corporae o lgusc erms {posve, egave}. More lgusc erms ca be roduced, as very posve ad very egave, bu he forecasg performace s almos he same, dcag ha e ca smplfy he procedure by akg less lgusc erms ad less rules. O he oher had more lgusc erms mgh be more useful, bu he case e exame facal professoals ad raders are eresg maly o posve ad egave reurs. The rules are 4 because e have o pus h o lgusc erms ad s *=4. These rules are: IF RET s egave AND IR s egave THEN f =p x + q x + r IF RET s egave AND IR s posve THEN f =p x + q x + r IF RET s posve AND IR s egave THEN f 3 =p 3 x + q 3 x + r 3 IF RET s posve AND IR s posve THEN f 4 =p 4 x + q 4 x + r 4, here RET deoes he Geeral sock dex reurs ad IR deoes he eres rae chages. We choose he AND operaor so e ll ake he produc sead o m operaor o avod moooc resuls. Also each rule has parameers plus he cosa hece here ll be 3*4= parameers. Jag [993] ad Jag ad Su [995] roduced he adapve eork-based fuzzy ferece sysem (ANFIS). Ths sysem makes use of a hybrd learg rule o opmze he fuzzy sysem parameers of a frs order Sugeo sysem. A example of a o pu h o rules frs order Sugeo sysem ca be graphcally represeed by Fg.. Fg. Example of ANFIS archecure for a o-pu, o-rule frs-order Sugeo model ISSN : Vol. No. 6

4 Elefheros Govas / Ida Joural of Compuer Scece ad Egeerg (IJCSE), here he cosequece parameers p, q, ad r of he h rule corbue hrough a frs order polyomal of he form: f p x q x r (4) The ANFIS archecure s cossed of o raable parameer ses, he aecede membershp fuco parameers ad he polyomal coseque parameers p,q,r. The ANFIS rag paradgm uses a grade desce algorhm o opmze he aecede parameers ad a leas squares algorhm o solve for he coseque parameers. Because uses o very dffere algorhms o reduce he error, he rag rule s called a hybrd. The coseque parameers are updaed frs usg a leas squares algorhm ad he aecede parameers are he updaed by backpropagag he errors ha sll exs. The ANFIS archecure cosss of fve layers h he oupu of he odes each respecve layer represeed by O l, here s he h ode of layer l. Because e have four rules ad o pus he case e exame he seps for ANFIS sysem compuao are: I he frs layer e geerae he membershp grades O x ), ( x ) (5) A ( B, here x ad x are he pus. I layer e geerae he frg sreghs or eghs O ( ( x ), ( )) ( ( ), ( )) x admehod x x A B A B (6) produc ( ( x ) ( x )) A m B I layer e use he AND relao, as as meoed prevously, so e ake he produc operaor. I layer 3 e ormalze he frg sreghs ad ll be: 3 O (7) 3 4, here s for =,,3,4. I layer 4 e calculae rule oupus based o he coseque parameers. I layer 5 e ake O 4 y f p x q x r ) (8) ( f O 5 f (9) I he las layer he coseque parameers ca be solved for usg a leas square algorhm as: Y X (0), here X s he marx X [ x x... 9 x 9 ] () ad θ s a vecor of uko parameers as: p q r p q r p q r T,,,,,,,..., 9, 9, 9 (), here T dcaes he raspose. Because he ormal leas square mehod leads o sgular vered marx e use he Sgular Value Decomposo (SVD) h Moore-Perose pseudoverse of marx [Moore, (90); Perose, (955); Perou ad Bosdoga, (000)]. Oher mehods ha ca be used o elmae he parcular problem s Loer Tragular, Upper Tragular ad Orhogoal decomposo (QR) amog ohers. More specfcally he Sgular Value Decomposo (SVD) procedure s: T X USV (3) ISSN : Vol. No. 7

5 Elefheros Govas / Ida Joural of Compuer Scece ad Egeerg (IJCSE) The sgular values S are posve ad arraged decreasg order. Ther magude s relaed o he formao coe of he colums of U -prcple compoes- ha spa X. Therefore, o remove he ose effecs o he soluo of he egh marx, e smply remove he colums of U ha correspod o small dagoal values S. The egh marx s he solved for usg he follog: T VS U Y (4) For he frs layer ad relao (5) e use he Tragular, Gaussa ad sgmodal shape membershp fucos. We have descrbed he compuao procedure for he coseque parameers by usg leas squares algorhm h Moore-Perose pseudoverse marx. The ex sep s o descrbe he rag procedure for he aecede parameers h he error backpropagao algorhm ad he smple seepes desce mehod [Tsoukalas ad Uhrg, (996); Dea e al., (004); Kha e al., (00)]. The ragular fuco s defed as: x a b, f x a ( x ; a, b ) b / (5) 0, oherse, here α s he peak or ceer parameer ad b s he spread or suppor parameer. The symmercal Gaussa membershp fuco s defed as: ( x c ) ( x ; c, ) exp (6), here c s he ceer parameer ad σ s he spread parameer. The las membershp fuco e exame s he Sgmod shape fuco such as: ( x ; a, c ) (7) exp( a ( x c ), here c locaes he ceer of he curve ad a s he spread parameer. The aecede parameer c updae s: c E c ( ) c ( ) (8) p c,here η c s he learg rae for he parameer c, p s he umber of paers ad E s he error fuco hch s: E ( y y ) (9), here y s he arge-acual ad y s ANFIS oupu varable. The cha rule order o calculae he dervaves used o updae he membershp fuco parameers are: E c E y y y y c (0) The paral dervaves for o pus are derved belo: E y For he oupu s: y, hece ll be y y y y y e () () (3) ISSN : Vol. No. 8

6 Elefheros Govas / Ida Joural of Compuer Scece ad Egeerg (IJCSE), hece ll be : y ( p x q x r ) (4), he ll be y ( p x q x r ) m y (5) (6) The las paral dervave, Eq. 7 depeds o he membershp fuco e exame. The updae equaos for aecede c, ad σ parameers of Gaussa fuco are: ( p ) x - c x q x r y c ( ) c( ) c e σ ( p x ) ( ) e q ) x -c x r y σ ( σ 3 The updae equaos for a are, b are respecvely ( x ( x) ) (7) (8) (9) ( p xr) y x a ( ) ( ) e (30) α ( x) b ( px r ) y ( x) b( ) b( ) b e ( x) b (3) I a smlar fasho he updae equaos for sgmodal shape fuzzy membershp fuco ca be derved. The ex sep s o defe he al values for aecede parameers. I all cases e ge as al values for eer ad bases parameers he mea ad sadard devao. To be specfc e ge oe sample here he reurs o asses are egave ad oe sample here he reurs are posve. The same procedure s folloed for cash flo. So for ceer parameers a, c ad c of ragle, Gaussa ad sgmod respecvely e ake he average for egave ad posve samples. O he oher had for he base parameers, b, σ ad a e ake he sadard devaos of he respecve samples. For he ceer ad coseque RHS parameers e se up he value 0.05 as he learg rae ad for base parameers e se up he learg raes a Daa The sample perod e exame he curre sudy s We exame 5 Greek baks ad he daa are o eekly frequecy. Addoally, he perod s used as he -sample or rag daa perod ad he remag perod 009 s used as he ou-of-sample or esg daa perod. The oo of porfolo heory ad ISSN : Vol. No. 9

7 Elefheros Govas / Ida Joural of Compuer Scece ad Egeerg (IJCSE) sysemac rsk as o developed a ha me, ad as ul laer he Soe [97] exeded he marke model by corporag he effecs of deb dces. To assess he effec of he marke yeld so e have cosruced equally eghed sock porfolos for he follog secors he Greek The Geeral dex of Ahes sock marke s used as proxy for he Greek baks, hle he Lbor of oe ad hree mohs s used as he eres varable equao (). Addoally, e exame f he equally eghed porfolos reurs, he Geeral dex of Ahes sock exchage marke reurs ad eres rae chages are saoary. To be specfc e cofrm hs assumpo by applyg Augmeed Dckey-Fuller-ADF [Dckey ad Fuller, (979)] u roo es ad KPSS saoary es [Kakosk e al., (99)]. The ADF es s defed from he follog relao: y γy y... p y p (3), here y s he varable e exame each me. I he rgh had of regresso (3) he lags of he depede varable are added h order of lags equal h p. Addoally, regresso (3) cludes he cosa or drf μ ad red parameer β. The dsurbace erm s defed as ε. I he ex sep e es he hypoheses: H 0 : φ=, β=0 agas he alerave hypohess H : φ < We accep ha a varable s saoary f e reec he ull hypohess of u roo es. O he oher had KPSS es seres s assumed o be saoary uder he ull hypohess. The seres s dereded by regressg y o a radom alk process x.e., x = x - + u ad a deermsc erm β y x (33) KPSS sasc s based o he resduals for he ordary leas squares regresso (33). Le he paral sum seres of ε be s. I s: The KPSS sasc s he defed as: KPSS s e T T s ^ (34) / ( p) (35) ^ (, here T s he umber of sample ad p) s he log-ru varace of ε ad ca be cosruced from he resduals ε as: ^ T p T ( p) ( p) (36) T T, here p s he rucao lag, ( p) s a opoal eghg fuco ha correspods o he choce of a specal do. Uder he ull hypohess of level saoary, KPSS V ( r dx (37), here V (x) s a sadard Broa brdge: V (r) = B(r) rb() ad B(r) s a Broa moo (Weer process) o r [0, ]. Because relao (37) s refereed esg oly o he ercep ad o he red ad as e are esg h boh ercep ad red e have he secod-level Broa brdge V (x) ad s: 0 ) KPSS V ( r dx (38) 0 ) ISSN : Vol. No. 30

8 Elefheros Govas / Ida Joural of Compuer Scece ad Egeerg (IJCSE), here V (x) s gve by: 0 V ( r) W ( r) (r 3r ) W ( 6r 6r ) Ws ( s) ds (39) I able he resuls of u roo ad saoary ess are repored. We coclude ha all he varables are saoary, I(0), based o ADF es, as e reec he ull hypohess all levels of sascal sgfcace. O he oher had e observe ha he equally eghed sock porfolo reurs ad he Geeral dex reurs are saoary all levels of sascal sgfcace, bu he eres rae chages are saoary α=0.05 ad α=0.0. Table. ADF u roo ad KPSS saoary ess 4. Emprcal Resuls Varables ADF -sa. KPSS LM-sa. 3 moh eres rae chages Geeral dex reurs Equally eghed porfolo reurs of Greek Baks. MacKo, (996),. Kakosk e al., (99) Crcal values for ADF es α= 0.0, α= 0.05, -3.4 α= 0.0, Crcal values for KPSS es α= 0.0, 0.6 α= 0.05, 0.46 α= 0.0, 0.9 I able e prese he correlao coeffces ad her assocaed -sascs beee eres rae chages ad Geeral Idex reurs of Ahes sock marke. We observe ha he correlao coeffces are lo excep from some sub-perods, as 005 ad 008, ad sascally sgfca, bu sll o eve close o or -. Also he correlao he overall perod e exame s very close o zero ad he case of moh eres rae chages he correlao coeffce s sascally sgfca. Mulcolleary ca sll be a problem eve he par-se correlaos are small. Aoher ay o deec mulcolleary such suaos s o calculae he varace flaoary facors (VIF s). There s a dffere VIF for each depede varable. Each depede varable s VIF measures ho much he varace of s coeffce esmae has bee flaed by mulcolleary. The deal VIF for a varable s, ad values hgher ha 0, or 4-5 proposed by oher researchers, dcae he exsece of mulcolleary. Aoher measure s he olerace hch s defed as / VIF, so he closer s he olerace o zero he greaer he degree of colleary of ha varable h he oher regressors ad he closer olerace s o, he greaer he evdece ha he varable s o collear h he oher regressors. The VIF s smply compued by fdg he verse of he correlao marx ad akg he dagoal elemes. I becomes clear ha from he resuls of ables ad 3 he VIF values are almos or very close o, dcag he reeco of mulcolleary. Furhermore, olerace s very close o so here s evdece ha he varables are o correlaed. So here s o reaso o ake ay addoal procedure o solve for mulcolleary. I able 3 he esmao resuls h GARCH (,) ad hree moh eres raes are repored. We do o prese he resuls h oher eres raes as oe, sx or elve mohs as he resuls are smlar ad he coclusos are exacly he same. From he resuls of able 3 e observe ha oly coeffce β s posve ad sascally sgfca. So f he Geeral markes reurs are creased he he commo bak sock reurs are creased oo. Based o he dagosc ess e reec he auocorrelao ad ARCH effecs. Our resuls are cosse h hose of Chace ad Lae [980] ho foud ha feer ha per ce of he baks exhbed sgfca eres rae sesvy o a shor-, medum- or log-erm reasury dex ad h hose of Lloyd ad Shck [977] ho suppor ha sock reurs oly margally explaed by he eres rae facor ad hey fd o cremeal explaaory poer for eres rae chages. Addoally our fdgs cofrm he argumes of oher auhors [Cho ad Elyasa, (996); Alle ad Jaga, (997); Bek ad Wolff, (000)], ho coclude ha eres rae sesvy has decreased he lae 980's ad early 990's due o he avalably of eres rae dervaves coracs ha ca be used for hedgg purposes. Also our emprcal fdgs are cosse h he resuls of Bere e al. [009] ho exame 3 Europea coures, cludg Greece ad Germay, as also exame USA ad Japa ad fd ha eres rae chages have sgfca effecs o sock reurs, hle fd sgfca effecs, oly Sede h boh shor-erm ad log-erm eres ISSN : Vol. No. 3

9 Elefheros Govas / Ida Joural of Compuer Scece ad Egeerg (IJCSE) raes, Irelad ad Neherlads, oly h log-erm eres raes ad fally Ialy oly h shor-erm eres raes. I able 4 he esmaed fuzzy aecede ad coseque parameers for ANFIS h ragle ad Gaussa membershp fucos are repored, hle he esmaed coeffces for ANFIS h sgmodal shaped fuco are provded able 5. Table. Correlao ad VIF beee eres raes chages ad Geeral dex Perod Correlao coeffces 3-moh eres rae chage VIF 3-moh eres rae chage (-3.584) (-4.77) (-.364) * -sascs pareheses Table 3. GARCH modelg for he o-facor dex model ad hree mohs eres raes Mea equao Varace equao Dagosc ess β 0 β β [-0.784] [8.943]* [.94]* γ 0 γ γ e-05 [3.799]* [3.76]* [3.89]* Log-Lkelhood LBQ (8) ARCH-LM (5) (0.999) 0.5 (0.9875) * deoes sascal sgfca α=0.0, -sascs brackes, p-values pareheses,. ARCH-LM s for Auoregressve codoal heeroskedascy es of resduals h 5 lags,. LBQ s he Lug-Box es o squared sadardzed resduals h 8 lags. Table 4. Fal fuzzy aecede ad coseque parameers afer he rag process for ANFIS h ragle ad Gaussa membershp fucos ANFIS h ragle membershp fuco Aecede parameers for Geeral dex reurs Aecede parameers for eres rae chages ANFIS h Gaussa membershp fuco Aecede parameers for Geeral dex reurs Aecede parameers for eres rae chages α α α α α α α α b b b b b b b b Coseque parameers Coseque parameers p p p 3 p 4 p p p 3 p q q q 3 q 4 q q q 3 q r r r 3 r 4 r r r 3 r Table 5. Fal fuzzy aecede ad coseque parameers afer he rag process for ANFIS h sgmodally shaped membershp fuco ISSN : Vol. No. 3

10 Elefheros Govas / Ida Joural of Compuer Scece ad Egeerg (IJCSE) ANFIS h sgmodally shaped membershp fuco Aecede parameers for Geeral dex reurs Aecede parameers for eres rae chages α α α α b b b b Coseque parameers p p p 3 p q q q 3 q r r r 3 r From ables 4 ad 5 e ca derve varous rules, hch ca help us o dra he effecs of eres rae chages ad Geeral Idex reurs o Greek bak sock reurs ad herefore o vesgae he behavor of sock reurs. For example h ANFIS ad ragle fuco e have he follog: If he Geeral dex reurs are egave ad eres rae chages are egave he e have f = 0.59x x If he Geeral dex reurs are egave ad eres rae chages are posve he f = -0.06x x If he Geeral dex reurs are posve ad eres rae chages are egave he f 3 = x.355x If he Geeral dex reurs are posve ad eres rae chages are posve he f 4 = -0.06x x Le suppose for example ha boh Geeral dex reurs ad eres rae chages are posve h values ad respecvely, ad he e ll have: f 4 = -0.06x x = =0.055 So, he boh pus are posve ad h he specfc values e ca have posve sock reurs. Because here are dffere values he he posve reurs ca be made egave. For example e cosder Geeral dex reurs ad eres rae chages are posve h values ad respecvely he ll be: f 4 = -0.06x x = = So based o he e values he posve eres rae chages ad sock dex reurs have a egave mpac o commo bak sock reurs. Le us ake aoher example, here e cosder posve Geeral dex reurs egave eres rae chages, h values ad respecvely. f 3 = x.355x = =0.053 No le s ake e values here Geeral dex reurs are posve, bu eres rae chages are egave, h values 0.04 ad f 3 = x.355x = =-0.0 ISSN : Vol. No. 33

11 Elefheros Govas / Ida Joural of Compuer Scece ad Egeerg (IJCSE) Hece, hs case e ge egave sock reurs sead of posve e foud prevously. The ANFIS echology s much more useful for he pracoers because hey are able o deerme more effcely her porfolo as also hey ca be more flexble. For example based o he regresso e esmaed e fd ha here are o sgfca effecs. Furhermore, f for example e had foud sgfca ad egave effecs, hs s o useful for he facal maagers ad raders, because he sg s chaged each perod, a leas o hgh frequecy, as daly or eekly, based o he behavor ad rules e have defed for he pus. Addoally, f he sg remas cosa for he log erm as yearly, hs s o aga useful or pracoers ad raders facal dusry, because hey ork ad rade daly frequecy ad shor erm perods. So hs case he coveoal ecoomerc modelg s o very helpful for praccg purposes. I ables 6 ad 7 he Roo Meas Squared Error (RMSE), Mea Absolue Error (MAE) ad he correc perceage sg of sock reurs are repored. We observe ha ANFIS RMSE ad MAE are loer o he respecve values of GARCH process boh -sample ad ou-of-sample perods. Furhermore, ables 6 ad 7 e prese he perceage of he correc sg of sock reurs. The las oe s preferred because researchers have argued ha he sole use of mea squared error ad mea absolue error may be approprae for facal daa ad parcularly for marke pracoers ad facal raders ho may be more eres he sg of forecass raher ha he magude of forecas errors, as he former provdes formao h respec o buy ad sell sgals. Besdes he fac ha f s o mpossble, s very dffcul o predc he acual value, pracce he ages ad raders facal markes are eresg more he ably of models o gve he correc sgalg. More specfcally, based o he predco of he correc sg o sock reurs ANFIS ouperforms sgfca GARCH modelg, as for sample perod e predc correcly he sg a 93.33, ad per ce h sgmodal shape, Gaussa ad ragle membershp fucos respecvely, hle he respecve perceage s h GARCH process. Addoally, he ou-of-sample perod, hch s of greaes eres ad mporace for he facal rsk maagers ad raders, e predc he bes ad ors cases 8.69 ad per ce respecvely h ANFIS, hle h GARCH model e predc he correc sock reurs sg a 7.5 per ce. Table 6. Forecasg performace for GARCH ad ANFIS h ragle fuco GARCH ANFIS h ragle fuco I-sample Ou-of-sample I-sample Ou-of-sample RMSE MAE Correc Perceage Sg Table 7. Forecasg performace for ANFIS h Gaussa ad Sgmodal fuco ANFIS h Gaussa fuco ANFIS h Sgmodal fuco I-sample Ou-of-sample I-sample Ou-of-sample RMSE MAE Correc Perceage Sg Coclusos We examed he curre effecs of eres rae chages sock reurs of he bakg secor Ahes sock exchage marke. We cocluded ha e reec he effecs of eres rae chages based o GARCH model as e foud hem o be sascally sgfca. Addoally, e proposed ANFIS echology h hree membershp fucos, he ragle, he Gaussa ad he sgmodal shaped fuco. We propose ANFIS order o overcome he problems ad he resrcos of ecoomerc approaches. Ths s because here s o alays a uque sg or effecs of depede varables o he depede because here s o a sragh forard ay o defy he effecs of pus o oupus. To be specfc s o ecessary ha here ll be cosa sgs or effecs durg a specfc perod e exame bu hese are chaged each observao ad each perod. So he case e exame, here e have eekly daa, each eek, dffere effecs ll be repored based o he values ad he rules of he sysem ISSN : Vol. No. 34

12 Elefheros Govas / Ida Joural of Compuer Scece ad Egeerg (IJCSE) Refereces [] Alle, L., Jaga, J. (997): Rsk ad Marke Segmeao Facal Iermedares Reurs. Joural of Facal Servces Research,, pp [] Bae, S.C. (990): Ieres Rae Chages ad Commo Sock Reurs of Facal Isuos: Revsed. Joural of Facal Research, 3, 7-79 [3] Bere, J., Caporale, M., Spagolo, N. (009): Marke, Ieres Rae ad Exchage Rae Rsk Effecs o Facal Sock Reurs: A GARCH-M Approach. Quaave ad Qualave Aalyss Socal Sceces, 3(), pp [4] Bek, H., Wolff, C. (000): Survey Daa ad he Ieres Rae Sesvy of US Bak Sock Reurs. Ecoomc Noes, 9(), pp. 0-3 [5] Bollerslev, T. (986): Geeralzed Auoregressve Codoal Heeroskedascy. Joural of Ecoomercs. 3, pp [6] Booh, J.R., Offcer, D. T. (985): Expecaos, Ieres Raes, ad Commercal Bak Socks. Joural of Facal Research, 8, pp [7] Chace, D.M., Lae, W. R. (980): A Re-examao of Ieres Rae Sesvy he Commo Sock of Facal Isuos. Joural of Facal Research, 3, pp [8] Che, C.R., Moha, N. J., Seer, T.L. (999). Dscou rae chages, sock marke reurs, volaly, ad radg volume: Evdece from raday daa ad mplcaos for marke effcecy. Joural of Bakg ad Face, 3(6), [9] Cho, J., Elyasa, E. (996): Dervave Exposure ad he Ieres Rae ad Exchage Rae Rsks of US Baks. The Wharo School, Uversy of Pesylvaa, orkg paper, 53 [0] Chrse, A.A. (98): The Sochasc Behavor of Commo Socks Varaces: Value, Leverage ad Ieres Rae Effecs. Joural of Facal Ecoomcs, 0, pp [] Dea, M.A., Pals, F., Zeghbb, A. (004). ANFIS Based Modellg ad Corol of No-Lear sysems: A Tuoral. IEEE Coferece o Sysems, Ma, ad Cyberecs, 4, pp [] Dckey, D. A., Fuller, W. A. (979): Dsrbuo of he Esmaors for Auoregressve Tme Seres h a U Roo. Joural of he Amerca Sascal Assocao, 74, pp [3] Fama, E.F., Scher, W. G. (977): Asse reurs ad Iflao. Joural of Facal Ecoomcs, 4, pp [4] Fama, E. (98). Sock Reurs, Real Acvy, Iflao ad Moey. Amerca Ecoomc Reve,7, pp [5] Fraser, D.R., Madura, J., Wegad, R. A. (00): Sources of Bak Ieres Rae Rsk. The Facal Reve, 37, [6] Glose, L. R., Jagaaha, R., Rukle, D. E. (993): O he Relaoshp Beee he Expeced Value ad he Volaly of he Nomal Excess Reurs o Socks. Joural of Face, 48, pp [7] Hardouvels, G.A (987): Macroecoomc Iformao ad Sock Prces. Joural of Ecoomcs ad Busess, 39, 3-40 [8] Jag, J.-S.R. (993): ANFIS: Adapve-Neork-based Fuzzy Iferece Sysems. IEEE Trasacos o Sysems, Ma, ad Cyberecs, 3(3), pp [9] Jag, J.-S. R., C.-T. Su, (995): Neuro-fuzzy Modelg ad Corol, Proceedgs of he IEEE, 83(3), , March [0] Kha, L., Aum, S., Bada, R. (00): Sadard Fuzzy Model Idefcao usg Grade Mehods, World Appled Sceces Joural, 8(), pp [] Kakosk, D., Phllps, P. C. B., Schmd, P., Sh, Y. (99): Tesg he Null Hypohess of Saoary agas he Alerave of a U Roo. Joural of Ecoomercs, 54, pp [] Log, J. (974): Sock Prces, Iflao ad he Term Srucure of Ieres Raes. Joural of Facal Ecoomcs,, pp [3] Llyod, W.P., Shck, R. (977): A Tes of Soe's To Idex Models of Reurs. Joural of Facal ad Quaave Aalyss,, pp [4] MacKo, J. G. (996): Numercal Dsrbuo Fucos for U Roo ad Coegrao Tess. Joural of Appled Ecoomercs,, pp [5] Mero, R. (973): A Ieremporal Capal Asse Prcg Model. Ecoomerca, 4, pp [6] Moore, E. H. (90): O he recprocal of he geeral algebrac marx, Bulle of he Amerca Mahemacal Socey, 6, pp [7] Nelso, D. B. (99). Codoal heeroskedascy asse reurs: A e approach. Ecoomerca, 59, pp [8] Perose, R. (955). A geeralzed verse for marces, Proceedgs of he Cambrdge. Phlosophcal Socey, 5, pp [9] Perou, M., Bosdoga, P. (000): Image Processg: The Fudameals, Joh Wley [30] Soe, B. (974). Sysemac Ieres Rae Rsk a To-Idex Model of Reurs. Joural of Facal ad Quaave Aalyss, 9, pp [3] Tsoukalas, L.H., Uhrg, R. E. (997): Fuzzy ad Neural Approaches Egeerg. s ed., Joh Wley & Sos ISSN : Vol. No. 35