DYNAMIC ECONOMETRIC MODELS Vol. 8 Ncolaus Coperncus Unversy Toruń 2008 Marola Płaowsa Ncolaus Coperncus Unversy n Toruń The Economerc Models Sasfyng he Congruence Posulae an Overvew. Non-saonary he Key Problem of Dynamc Modelng The non-saonary of economc processes should be reaed as he ey problem of dynamc modelng. Three man approaches o non-saonary may be dsngushed: () classcal decomposon, (2) negraed processes approach, (3) new decomposon of economc processes as an agreemen of rend saonary versus dfference saonary debae. () Tll he early sevenes of he 20 h cenury he classcal decomposon of economc processes was domnaed,.e. Y = P + S + C + η, where P a deermnsc rend componen, S a deermnsc seasonal componen, C deermnsc busness cycle flucuaons, η a sochasc componen whch s assumed o be saonary. Ths decomposon assumes ha an economc process s non-saonary n mean. In oher words, flucuaons of deermnsc me funcon are saonary wha mached he prevalng vew amongs economss ha flucuaons (ncludng busness cycle) around deermnsc funcon of me varable are ransory. To elmnae non-saonary n mean he funcon of me no he model for levels was nroduced or was subraced from a gven process. Only rarely he ransformaon of process o s dfferences or growh raes was used. As a consequence he prevalng sraegy n dynamc modelng was always ae levels (or devaons from deermnsc rend), f economc processes were non-saonary. (2) Snce hree semnal wors,.e. he publcaon by Box and Jenns (976) whch brough he ARIMA modelng no general use, he paper by Granger and Newbold (974) focusng on he danger of spurous regresson for non-saonary processes and manly paper by Nelson and Plosser (982) referrng o dsngushng he ype of non-saonary for macroeconomc processes, Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House
54 Marola Płaowsa he alernave decomposon of non-saonary processes was preferred,.e. Y = μ + γ + ξ + η, where μ sochasc rend componen, γ sochasc seasonal componen, ξ saonary sochasc componen. Ths decomposon assumes ha an economc process s non-saonary n varance, whch means ha random shocs have a permanen effec on he sysem,.e. here s no endency for fuure values of a process o rever o a rend lne (a fall of process oday brngs abou ha forecass wll fall n he ndefne fuure). The non-saonary n varance was elmnaed by calculang he frs dfferences (or n general dfferences of d-order). As a resul he prevalng sraegy n dynamc modellng was always ae dfferences f economc processes were non-saonary. Snce ha me he non-saonary of economc processes sared o be auomacally referred only o non-saonary n varance 2. (3) In eghes and frs half of nnees of he 20h cenury he lenghy and vas debae on he rend saonary versus dfference saonary oo place. A he early sage of ha debae he hypohess of dfference saonary ouperformed hose of rend saonary. In favour of he former hypohess spoe he more serous effecs of underdfferencng han effecs of overdfferencng on esmaon and sascal nference and low power of un roo ess whch oo ofen dd no rejec he hypohess of dfference saonary. A he laer sage of ha debae some balance beween rend saonary and dfference saonary hypohess was obaned wha resuled from he accepance of dfferen properes of rend saonary processes and dfference saonary processes (n erms of mean, varance, auocorrelaon funcon, mean reverng propery and perssence) and general agreemen ha economc processes can be non-saonary as well n mean as n varance. Therefore snce second half of nnees of he 20h cenury afer he debae on he rend saonary versus dfference saonary (none of approaches ganed an advanage) a new decomposon sared o be domnang,.e. Y = P + S + C + μ + γ + ξ + η.ths decomposon ncludes he wde class of non-saonary processes (as well n mean, e.g. producon, ncome, sale, as n varance, e.g. exchange raes, soc ndexes). From ha debae resuled some suggesons for economerc modelng. The dsncon of rend saonary or dfference saonary processes s mporan especally wh regard o economc forecasng because each ype of nonsaonary can assume que dfferen dynamcs and hence dfferen forecass. The un roo ess can be useful as a dagnosc ool n model specfcaon for forecasng purposes when he pon s no o fnd he rue model bu o fnd Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House Ths was suggesed by he resuls of un roo ess whch have a low power and herefore preferred he null of non-saonary n varance. I should be clearly emphaszed ha dfference beween sraeges always ae levels and always dfference does no only refer o he way of removal of non-saonary bu s much deeper and refers o he economc nerpreaon (Płaowsa, 2003). 2 I s apparen especally n Polsh exboos n economercs.
The Economerc Models Sasfyng he Congruence Posulae an Overvew 55 he model whch gves more accurae forecass. And fnally he man purpose n economerc modelng s no he choce beween he sraegy: always ae levels (TS) or always dfference (DS), bu raher o buld he dynamc model whch has he requred properes. The laer s obaned when he model sasfes he congruence posulae. The purpose of he paper s o overvew he dfferen approaches o dynamc economerc modelng sasfyng he congruence posulae whch was nroduced by Granger (98). 2. The Congruence Posulae n Dfferen Approaches o Dynamc Modelng The dea of congruence was nroduced by Granger (98) and was frsly oulned n he conex of frequency doman. In he me frequency (n a lnear model) hs posulae says ha f dependen varable Y has some domnan feaures (srong auocorrelaon, seasonaly, rend n mean or n varance) hen he explanaory varables X have o conan smlar feaures o explan Y and o sasfy a condon for he model o be sasfacory 3. The model sasfyng hs condon s called balanced (Granger, 992). In he case of unbalanced model he domnan feaures of dependen process no explaned by domnan feaures of explanaory processes wll have o appear n he resdual, whch wll hen have undesrable feaures for esmaon and nference. In dfferen approaches o dynamc modelng he congruence posulae s realzed n dfferen way and wha s worh nocng he reference o hs posulae s no always explcly. To approaches realzng he congruence posulae belong: () he concep of congruen modelng accordng o Zelńs (984), (2) general o specfc modelng accordng o Hendry (2000), (3) conegraon and error correcon model (Engle, Granger, 987), (4) he VAR modelng (Sms, 980). The Concep of Congruen Modelng Accordng o Zelńs The dea of congruence oulned by Granger was he sarng pon o develop he concep of dynamc congruen model n me doman by Zelńs (984). A model s congruen accordng o Zelńs f he harmonc srucure of dependen process Y s he same as he harmonc srucure of explanaory processes X (=,2,,) and he resdual process. The model for whe noses of gven processes Y and X,.e. ε = = ρ ε + ε, () Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House y x 3 Such defned congruence posulae can be exended no condonal varances (Fszeder, 2006).
56 Marola Płaowsa s congruen because he harmonc srucure of ε y and harmonc srucure of ε x are dencal. Hence he resdual process has whe nose properes. The model () s a sarng pon o buld he congruen model for observed processes Y and X. Frs sep o do s o specfy he domnan feaures (nernal srucure) of gven processes. I s realzed by buldng me seres models: rend/seasonaly models, auoregressve models Y = P + S + η, B( u) η = ε, (2) X y y y y y = P + S + η, A ( ) η = ε, =, 2, K,, (3) x x x u x x where P y, P x denoe polynomal funcons of me varable, S y, S x seasonal componens (consan or changng flucuaon amplude), η y, η x saonary auoregressve processes for Y and X respecvely, B(u), A (u) auoregressve bacshf operaors wh all roos lyng ousde he un crcle, ε y, ε x whe noses of respecve processes. In he nex sep, subsung for ε y and ε x n () from models (2) and (3) he sarng specfcaon 4 of congruen model s obaned: B( u) Y ρ = A ( u) X = + P + S + ε, where A ( u) = A ( u), P = B( u) P A ( u) P, S or equvalenly Y q y s= qx = y = Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House x = B( u) S y = (4) A ( u) S β sy s + α, s X, s + P + S + ε. (5) = s= 0 I should be noced ha he resdual process n he model (5) s he same as n model (3). Ths means he congruence condon of domnan feaures (nernal srucures) of boh sde of equaon s sasfed. To chec he congruence he msspecfcaon ess for whe nose errors, condonally homoscedasc errors, normally dsrbued errors, uncondonally homoscedasc errors, consan parameers are conduced for sarng and fnal model. The specfcaon of model (5) ncludes four componens: lagged dependen process, curren and lagged explanaory processes, rend/seasonaly componen P + S and resdual process, x 4 The esmaed nal specfcaon of model (5) has n general excessve nsgnfcan varables whch are elmnaed by selecon mehods and a he very end he fnal congruen model, reduced o sgnfcan varables, s obaned. The concep of congruen modellng can be appled o he case of non-saonary n mean and n varance.
The Economerc Models Sasfyng he Congruence Posulae an Overvew 57 each havng a dfferen meanng. Lagged dependen processes should be nerpreed as subsue elemens whch appear n model f: (a) mporan explanaory varables are omed, (b) he dependence of Y on X for dfferen frequency componens s no he same,.e. regresson parameer of Y on X for low frequences componens s dfferen han hose for hgh frequences componens. Curren and lagged explanaory processes are reaed as economc facors (wh causal nerpreaon), bu some of hem play he role of subsue facors for omed varables wha leads o he model beng balanced. The ncluson of rend/seasonaly componen o model means ha from each process he nonsaonary n mean was elmnaed and herefore parameers β s and α s refer o he dependence on saonary level. General o Specfc Modelng Accordng o Hendry In general o specfc modelng an emprcal model s congruen f parsmonously encompasses he local DGP (.e. he generang process n he space of he varables under analyss) and acheves he pre-assgned crera: whe nose errors, condonally homoscedasc errors, normally dsrbued errors, uncondonally homoscedasc errors, consan parameers (see Mzon, 995; Bonemps and Mzon, 200). The emprcal analyss commences from he formulaon of general unresrced model GUM, (Hendry, 2000), ang smlar form as n he concep of congruen model accordng o Zelńs (model (5)), whch afer esng for msspecfcaons and f none s apparen, s smplfed o parsmonous congruen model, each smplfcaon sep beng checed by dagnosc esng. The dfference n formulaon of nal model specfcaon n general o specfc modelng and he concep of dynamc congruen model (accordng o Zelńs) consss n dfferen approach o he recognon of nernal srucure of gven processes. In general o specfc modelng usually he lnear rend s nroduced o elmnae a poenal non-saonary n mean, and he lags are se as a maxmal lag lengh accordng o avalable evdence (number of observaons and varables), he same for all varables n queson, o manan he congruence. Whle n he dynamc congruen model (accordng o Zelńs) he nal model specfcaon s esablshed hrough he recognon of domnan feaures (nernal srucure) separaely for gven processes. As a resul he lag lengh for dfferen processes (dependen and explanaory) s no he same, and also he degree of polynomal rend may be hgher han one. The nex dfference n boh approaches refers o he selecon rules wh regard o nsgnfcan varables. In he concep of dynamc congruen model (accordng o Zelńs) he erave selecon mehod (a poseror) s preferred, whle n general o specfc modelng he auomac selecon procedure, PcGes (Hendry, Krolzg, 999). Ths procedure consss n elmnang nsgnfcan varables by selecon ess, boh n blocs and ndvdually. Moreover Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House
58 Marola Płaowsa many reducon pahs are searched o preven he algorhm from becomng suc n a sequence ha nadverenly elmnaes a varable ha maers, and hereby reans oher varables as proxes. If several models sasfyng he congruence posulae are seleced, encompassng ess and model selecon crera resolve he choce. Conegraon Error Correcon Model The conegraon dea formulaed by Engle, Granger (987) assumes ha a combnaon of processes nonsaonary n varance, each negraed of order one, I() and rend n mean s no apparen, s saonary,.e. negraed of zero order 5, I(0). Ths means ha exss -dmensonal vecor θ such ha a lnear combnaon Z = Y θ X, where X -dmensonal vecor of explanaory processes, s saonary. Conegrang vecor θ elmnaes a sochasc rend (non-saonary n varance) and a he same me s a vecor measurng relaonshp beween Y and X on saonary level. The relaonshp for conegraed processes can be expressed n he form of error correcon model (Engle, Granger, 987): ΔY = p q j j j j = j= = j= 0 α ΔY + β ΔX, + δ ( Y θ X, ) + η, (6) where EC = Y = θ X, s saonary error correcon erm represenng he devaon of Y from he long-run equlbrum 6. The error correcon coeffcen δ measures he speed of convergence o equlbrum. Parameers θ are long-run coeffcens for he response of Y o a un change n X. The remanng coeffcens α j and β j relae o he shor-run dynamcs of he model s convergence o equlbrum. In fac model (6) sasfes he congruence posulae, however s realzed n dfferen way han n prevous approaches. The error correcon model s bul for frs dfferences,.e. for processes ransformed by he dfference fler whch elmnaes wde band of low frequences referrng o long-run componens and as a resul elmnaes non-saonary n varance, bu also n mean. Therefore coeffcens β j are he measure of relaonshp for saonary dfferences of processes. Ino model (6) he lagged dfferences of dependen varable and explanaory varables and also error correcon erm EC - are necessary o be n- 5 The frs noe concernng he conegraon appeared earler n Granger (98), n whch he dea of congruence posulae was oulned. 6 I should be remembered ha despe he saonary of lnear combnaon of nonsaonary processes sll exss he danger of spurous regresson effecs because of he omsson of auoregressve srucure of Y and X n conegrang relaon. See resuls of smulaon wh regard o spurous regresson for ndependen and dependen auoregressve processes, e.g. Granger, Hung, Jeon (998); Płaowsa (2003). Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House
The Economerc Models Sasfyng he Congruence Posulae an Overvew 59 cluded, oherwse he auocorrelaon of resduals wll appear as a resul of applyng he dfference fler. Hence he error correcon erm no only measures he speed of convergence o equlbrum bu also enables o manan he congruence of model. The VAR Modelng The vecor auoregregresson model VAR(p) has he form: Y = AY + A2Y 2 + K + AY p + ε, (7) where Y s an K x vecor of jonly deermned varables, he A are fxed (K x K) coeffcen marxes of coeffcens, ε s K-dmensonal whe nose, ha s, E(ε ) = 0, E( ε ε ) = Σ, for all, E( ε ε s ) = 0 for s. The covarance marx Σ s assumed o be nonsngular f no oherwse saed. The model (7) can be exended by he deermnsc erm, A 0 D (nercep, deermnsc rend, seasonal dummes), where D denoes a vecor of deermnsc varables, A 0 a coeffcen marx. In such a way he non-saonary n mean can be aen no accoun. In he case of non-saonary n varance he model (7) s wren for dfferences of varables n neres. Some explanaory varables can also be added no model (7). In he case of conegraon he VAR model s a sarng pon o buld vecor error correcon model, VECM. The VAR model wh deermnsc erm realzes he congruence posulae by amng a such a model specfcaon ha he resdual process has requred properes. The number of lags n he VAR model resuls from he seng of maxmal lag lengh accordng o avalable evdence (number of observaons and varables) whch furher s reduced by he means of approprae ess. Whle n he mulvarae congruen model (accordng o Zelńs) he number of lags s separaely esablshed by he deecon of auoregresson order for all varables n neres. Moreover he mulvarae congruen model enables o conemporaneous relaonshps among varables, whle he VAR does no gve such possbly. 3. Fnal Remars The congruence posulae, amed a buldng he model whch aes no accoun he domnan feaures of endogenous and explanaory processes, s an dea whch lns all presened conceps of economerc modelng. The aracveness of hs dea consss n benefs resulng for he esmaon and sascal nference (e.g. avodng he danger of spurous regresson) and for forecasng because he models sasfyng he congruence posulae gve n general beer forecass. However should be emphaszed ha he models wll dffer wh regard o economc nerpreaon and forecasng behavour, especally models for levels and models for dfferences. Hence he dea of congruence does no solve he dchoomy beween model selecon and forecasng sll remans. Therefore Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House
60 Marola Płaowsa s sensble o accep he coexsence of models wh dfferen specfcaon raher han search for he only one rue model. References Bonemps, C., Mzon, G. E. (200), Congruence and Encompassng, n: Sgum, B. (red.), Sudes n Economc Mehodology, Cambrdge, Mass., MIT Press. Engle, R. F., Granger, C. W. J. (987), Co-negraon and Error Correcon Represenaon: Esmaon and Tesng, Economerca, 55, 25 276. Fszeder, P. (2006), Consequences of Congruence for GARCH Modellng, n: Zelńs, Z. (ed.), Dynamc Economerc Models, Wydawncwo UMK, Toruń, 43 49. Ganger, C. W. J. (98), Some Properes of Tme Seres Daa and her Use n Economerc Model Specfcaon, Journal of Economercs, 6, 2 30. Granger, C. W. J. (992), Where Are he Conroverses n Economerc Mehodology?, w: Granger, C. W. J. (red.), Modellng Economc Seres, Clarendonpress, Oxford. Granger, C. W. J., Hyung, N., Jeon, Y. (992), Spurous Regressons wh Saonary Seres, Dscusson Paper 98-25, Unversy of Calforna, San Dego. Granger, C. W. J., Newbold, P. (974), Spurous Regresson n Economercs, Journal of Economercs, 2, 20. Hendry, D. F. (2000), Economercs: Alchemy or Scence?, Oxford Unversy Press, Oxford. Hendry, D. F., Krolzg, H.-M. (999), Improvng on Daa Mnng Reconsdered by K. D. Hoover and S.J. Peres, Economercs Journal, vol. 2. Mzon, G. E. (995), Progressve Modelng of Macroeconomc Tme seres: he LSE Mehodology, n: Hoover, K. D. (red.), Macroeconomcs: Developmens, Tensons and Prospecs, Dordrech: Kluver Acadmc Press. Nelson, C. R., Plosser, C. I. (982), Trends and Random Wals n Macroeconomc Tme Seres: Some Evdence and Implcaons, Journal of Moneary Economcs, 0, 39 62. Płaowsa, M. (2003), Modelowane nesacjonarnych procesów eonomcznych. Sudum meodologczne (Modelng of Non-saonary Economc Processes. Mehodologcal Sudy), Wydawncwo UMK, Toruń. Sms, C. A. (980), Macroeconomcs and Realy, Economerca, 48, 48. Zelńs, Z. (984), Zmenność w czase sruuralnych paramerów modelu eonomerycznego (Tme Varably of Srucural Parameers n Economerc Model), Przegląd Saysyczny (Sascal Survey), z. /2, 35 48. Zelńs, Z. (2002), Analza wybranych oncepcj modelowana dynamcznego w eonomer (Analyss of Dfferen Conceps of Dynamc Modelng), n: Zelńs, Z., Analza eonomcznych procesów sochasycznych. Psma wybrane (Analyss of Economc Sochasc Processes), Wydawncwo UMK, Toruń. Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House