ECONOMETRIC APPROACH TO DIFFERENCE EQUATIONS MODELING OF EXCHANGE RATES CHANGES

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1 Croaan Operaonal Research Revew (CRORR), Vol., ECONOMETRIC APPROACH TO DIFFERENCE EQUATIONS MODELING OF EXCHANGE RATES CHANGES Absrac Josp Arnerć Facul of Economcs Spl Mace hrvaske 3, Spl Phone: ; E-mal: Lana Kordć Facul of Economcs Spl Mace hrvaske 3, Spl Phone: ; E-mal: Tme seres models ha are commonl used n economerc modelng are auoregressve sochasc lnear models (AR) and models of movng averages (MA). Menoned models b her srucure are acuall sochasc dfference equaons. Therefore, he objecve of hs paper s o esmae dfference equaons conanng sochasc (random) componen. Esmaed models of me seres wll be used o forecas observed daa n he fuure. Namel, soluons of dfference equaons are closel relaed o condons of saonar me seres models. Based on he fac ha volal s me varng n hgh frequenc daa and ha perods of hgh volal end o cluser, he mos successful and popular models n modelng me varng volal are GARCH pe models and her varans. However, GARCH models wll no be analzed because he purpose of hs research s o predc he value of he exchange rae n he levels whn condonal mean equaon and o deermne wheher he observed varable has a sable or explosve me pah. Based on he esmaed dfference equaon wll be examned wheher Croaa s mplemenng a sable polc of exchange raes. Ke words: dfference equaons, me seres, economerc esmaon, exchange rae. INTRODUCTION Tme seres models ha are commonl used n economerc modelng are auoregresson sochasc lnear models (AR) and movng average models (MA). B her srucure, hese models are acuall sochasc dfference equaons. The objecve of evaluaon of me seres models s based on analss of dfference equaons conanng sochasc (random) componens, n order o forecas he observed phenomena n he fuure. Wheher he observed varables have a sable or an explosve me pah s closel relaed o condons of saonar me seres models. Namel, f for example AR(p) model s no saonar hen can be used n forecasng purposes. 34

2 Croaan Operaonal Research Revew (CRORR), Vol., The paper wll show ha he deermnsc par of he parcular soluons n economerc sense s he same as expecaon of sochasc processes. Whasoever, o deermne he order of dfference equaons and auoregresson models he auocorrelaon funcons and paral auocorrelaon funcons are usuall esmaed. An example wll be llusraed on he bass of he movemen of exchange raes ncludng monhl observaons. The mos basc objecve of monear polc n lberal democraces, ncludng Croaa, s prce sabl, respecvel n Croaa he sabl of exchange raes o conrol nflaonar expecaons. Based on he esmaed dfference equaon wll be examned wheher Croaa s mplemenng a sable polc of exchange raes. Evdence of mplemenaon of sable polc of exchange rae also means he successful mplemenaon of he prmar objecve of Croaan monear polc. The srucure of he paper s organzed as follows. In he second secon me seres models are descrbed as a pe of dfference equaon n her mos general form, conanng sochasc componen (random or rregular dsurbance). In he hrd secon dfference equaons and her soluons are examned wh mplemenaon on modelng of exchange rae changes n Croaa. Las secon provdes concludng remarks of emprcal fndngs.. TIME SERIES MODELS In he mos general form dfference equaons of order p can be wren as follows p x, () where componen x s he random process. In economercs x s mporan as a specal case when where s whe nose process. j x, () Lnear sochasc ARMA(p,q) model of observed me seres conans he followng analcal expresson (Enders, 4) In hs auoregresson model,.e. AR (p) j j... s a par of he model movng average model,.e. MA(q)... qq. (3) p, (4) Sochasc process s a "whe nose" process whose random varables are non-correlaed wh expecaon equal o zero, consan varances and covarance equal o zero for an lag k. 35

3 Croaan Operaonal Research Revew (CRORR), Vol., q j j. (5) jo Takng no accoun equaon (5) can be noed ha he specal case of equaon () s acuall MA( ) process. If s assumed ARMA(p,q) model n whch p = and q =, we ge. (6) I can be seen ha he equaon (6) s a dfference equaon of he frs order wh a sochasc erm. Solvng sochasc dfference equaon (6) b erave mehod (connuous subsuon for one perod n advance of sarng me ) and provdng nal value, we ge he followng soluon. (7) If he nal value s no se, hen b connuous subsuon for m perod from he momen follows m m m m. (8) If we assume ha n equaon (8), hen for a ver large shf of m perods we can oban, (9) 3 as an nfne geomerc sum... converges o, as componen m ends o zero when m (Hamlon, 994). The soluon n equaon (9) s a parcular soluon of sochasc dfference equaon of he frs order, whle he componen s deermnsc par of he parcular soluons of sochasc dfference equaons., () Specfcall, he soluon obaned n equaon (9) s no unque, because for an arbrar consan A here s a soluon So, f we ake a homogeneous par of he equaon (6) A. () Deermnsc par of parcular soluon n erms of economercs acuall corresponds o he mean value of he process (when akng mahemacal expecaon of he expresson (9)),.e.. 36

4 Croaan Operaonal Research Revew (CRORR), Vol., whch soluon s equall o, (), (3) hen can be shown ha for an consan A soluon s homogeneous soluon. 3 A, (4) Snce he A s homogeneous soluon of sochasc dfference equaon of he frs order, hen he soluon n equaon () s called a general soluon (he sum of parcular soluon and homogeneous soluon), whereb he arbrar consan A can be elmnaed b nroducng he nal condon (Chang, 994). If we nroduce he nal value of hen () mus hold for each, and also for Then he value of A equal from whch follows he soluon A. (5) A, (6). (7) 3. DIFFERENCE EQUATIONS MODELLING OF EXCHANGE RATES AND FORECASTING The mos basc objecve of monear polc n lberal democraces, ncludng Croaa, s prce sabl, respecvel n Croaa he sabl of exchange raes o conrol nflaonar expecaons. Prce sabl and growh fnancng hrough mpor foregn savngs are feaures of he ranson model of developmen. The mechansm ha allows movemen of savngs from developed counres no developng counres n he form of nvesmen s nernaonal capal mobl. As he counr mplemens reforms n he process of assocaon n EU, han apprecaon pressures, pcal for hs model, ncrease (Tca e al., 7). On apprecaon pressures, he monear polc n Croaa responded wh bung ou foregn currenc o defend exchange rae and hus smulaneousl ncreased he amoun of kuna n crculaon and currenc reserves. In he conex of he global crss, apprecaon pressures are reduced and pror of he Croaan Naonal Bank s o preven excessve exchange rae volal n order o preserve prce sabl. 3 The soluon of s acuall A because of equal A A. 37

5 Croaan Operaonal Research Revew (CRORR), Vol., Therefore, n hs paper dfference equaon wll be esmaed for descrbng he movemens of exchange raes HRK/EUR based on monhl observaons from Januar 999 o June. Fgure :. Auocorrelaon and paral auocorrelaon funcons from he sample. The order of dfference equaon s deermned accordng o he auocorrelaon and paral auocorrelaon funcons from he sample. Economerc approach n me seres modelng depends on he sgnfcance of coeffcens of auocorrelaon funcon (ACF) and paral auocorrelaon funcons (PACF). From Fgure can be shown ha ACF decreases exponenall, whle he PACF des ou afer he second me lag, so he esmaed second-order dfference equaons s evaluaed (Hamlon, 994). Therefore, usng Evews sofware he second order dfference equaon, whch corresponds o he auoregresson model AR(), was esmaed. In Table esmaed coeffcens wh usual sascal ess and dagnoscs measures are presened. Namel, second-order dfference equaon s esmaed usng mehod of leas squares. Based on esmaed resuls from Januar 999 o June, sochasc dfference equaon for he varable exchange rae can be expressed as follows (8) There are wo soluons o he homogeneous dfference equaon of he second order. Homogeneous par of equaon (8) equals (9) If possble soluons of he equaon (9) s a, hen b subsuon we ge

6 Croaan Operaonal Research Revew (CRORR), Vol., a.4543a.3468a. () Table : Esmaed coeffcens of he second-order dfference equaon wh sascal ess and dagnoscs measures Source: Croaan Naonal Bank [onlne], hp:// [Accessed3/9/.] If we dvde he equaon () b a, he problem s o fnd he values of a ha sasf a.4543a () The quadrac equaon n () s called he characersc equaon, whose soluons are characersc roos. Snce he dscrmnan d. 98 s posve han he characersc roos are wo dsnc real numbers a / ; a.84; a.443. () However, hese soluons are no unque. In fac, for an wo arbrar consans A and A he lnear combnaon of homogeneous soluons also solves second order dfference equaon.84 A. A 443. (3) Snce boh roos are less hen un n absolue value, he homogeneous soluon s convergen. From Fgure can be seen ha boh homogeneous soluons converges drecl because a, a. Therefore, he soluon s sable,.e. he sabl condons are sasfed whch means ha characerscs roos le whn he un crcle. Fgure shows he me pah of hs soluon for he case n whch he arbrar consans equal un for me uns. Terms of sabl for second order dfference equaon are: (a) he larges characersc roo mus be less han and (b) he smalles characersc roo mus be greaer han - (Enders, 4). Takng hese condons of sabl no accoun wh he assumpon of posve dscrmnan value, we ge he followng relaons. (4) 39

7 Croaan Operaonal Research Revew (CRORR), Vol., Fgure :. Convergence of he homogeneous soluons when A A. The above condons descrbe he saonar condons of second order auoregresson AR(). Ths means ha he same condons are fulflled f he roos of he characersc equaon a and a le whn he un crcle. I s eas o confrm ha he condons n (4) are sasfed for.4543 and Each sochasc dfference equaon can be wren usng he lag operaor L k. (5) Usng he lag operaor sochasc dfference equaon of second order can be wren as follows ( L L), (6) k where L k s he lag operaor polnomal of he auoregresson model. k Accordng o resuls n Table lag operaor polnomal of AR() model has he form k (.4543L.3468L ). (7) Auoregresson roos of a polnomal n he expresson (7) are L and L I s eas o confrm ha he nverse roos of auoregresson polnomal corresponds o he characersc roos of he dfference equaons n expresson () L.84; L.443. (8) Ths means ha he condons of sabl are sasfed f he roos of auoregresson polnomal le ousde he un crcle, whch s equvalen o he sae ha he roos of he characersc equaon le nsde he un crcle. Therefore, n Table nverse roos of he auoregresson polnomal n he las row (Invered AR Roos) are presened. Invered AR() roos are presened on Fgure 3. 4

8 Croaan Operaonal Research Revew (CRORR), Vol., Fgure3:. Inverse roos of AR() polnomal So, f he nverse roos of auoregresson polnomal le ousde he un crcle, hen we sa ha he saonar condon s sasfed. If he condon s me hen he saonar me seres s suable for forecasng. Specfcall, we have alread shown ha he soluon n (3) s homogeneous soluon of sochasc dfference equaon n (8), whle he expresson (9) shows how he parcular soluon can depend onl on consans and elemens of f. Therefore, parcular soluon of second order dfference equaon n whch he characersc roos are nsde he un crcle s b c c c... (9) In equaon (9) he consan erm corresponds o deermnsc par of he parcular soluon b (3) To fnd a sochasc par of he parcular soluon s necessar o check wheher a possble soluon c, (3) ndeed s a soluon (Enders, 4). Thus, afer subsuon of expresson (3) no (8) s necessar o examne under whch condons he followng equal s vald c c c c c c c c c.... (3) 3 4 Snce equal n (3) mus hold for all,,..., hen s necessar o sasf he followng relaons, c c. 4543c c. 4543c. 3468c. (33) 4

9 Croaan Operaonal Research Revew (CRORR), Vol., Coeffcens c can easl be obaned eravel, c, c. 4543, c. 8,.... In oher words, we seek he soluon of second order dfference equaon c. 4543c. 3468c wh nal condons c and c Soluon of dfference equaons s c A( 84. ) A( 4. ). (34) Inroducng nal condons n equaon (34), s eas o solve he ssem A A. (35) A. 84 A. 4 Inroducng soluons of (35) n (34) elds a soluon 9 84 c. (. ) 9. ( 4. ). (36) Thus, he sum of a lnear combnaon of homogeneous soluons and deermnsc and sochasc parcular soluons equals A. 84 A. 4 c, (37) where c s defned n equaon (36). Specfcall, n our case s neresng ha n he long erm mean value of he exchange rae equals o 7.4 HRK/EUR. I s also possble o calculae he average devaon from he expeced rae n he long run,.e. he sandard devaon of AR() model of me seres. The average devaon from he expeced value n he long run equals o.4593hrk/eur. Ths means ha n he fuure we can expec a varaon rae n he range of 6.%. The resuls confrm ha he Croaan Naonal Bank conduced a sable polc of exchange rae. Evdence of mplemenaon of sable polc of exchange rae also means he successful mplemenaon of he prmar objecve of Croaan monear polc. 4. CONCLUDING REMARKS Lnear sochasc me seres models b her srucure are acuall dfference equaons conanng random componen whch follows whe nose process. Soluons of dfference equaons are closel relaed o he saonar condons of me seres models. I can be shown ha he condons of sabl are sasfed f he roos of auoregresson polnomal le ousde he un crcle, whch s equvalen o he sae ha he roos of he characersc equaon le nsde he un crcle. To deermne wheher he observed varable has a sable or explosve me pah means ha he me seres converges or dverges n he long run. From esmaed AR() model of exchange rae changes can be seen ha boh homogeneous soluon converges drecl. We can also conclude ha deermnsc par of parcular soluon n erms of economercs corresponds o he expeced value of he process, whle he sochasc par of parcular soluon follows MA( ) process. The average devaon from he expeced value of exchange rae HRK/EUR n he range of 6.% n he long run confrms sable exchange rae polc. 4

10 Croaan Operaonal Research Revew (CRORR), Vol., REFERENCES [] Chang, A. C. (994), Osnovne meode maemačke ekonomje, reće zdanje, Mae, Zagreb. [] Enders, W. (4), Appled Economerc Tme Seres, Second Edon, Wle, New York. [3] Hamlon, J.D. (994), Tme Seres Analss, Prnceon Unvers Press, New Jerse. [4] Rodseh, A. (), Open Econom Macroeconomcs, Cambrdge Unvers Press, Cambrdge. [5] Tca, J., Smolć, Š., Bogdan, Ž. (7), "Analss of exchange rae based sablzaon effecs n conemporar economc pracce wh specal reference o Croaa" 7h Inernaonal Conference on Enerprse n Transon, Facul of Economcs, Unvers of Spl, pp