DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus University Toruń Mariola Piłatowska Nicolaus Copernicus University in Toruń

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1 DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus Universiy Toruń 26. Inroducion Mariola Piłaowska Nicolaus Copernicus Universiy in Toruń The Effecs of he Incorrec Idenificaion of Non-saionariy of Economic Processes for Predicion Mean Square Error Analysing he relaionships among non-saionary economic processes he idenificaion of he wo ype of non-saionariy i.e. in mean and in variance is of grea imporance considering he negaive effecs for esimaion and saisical inference being appeared in he case of incorrec idenificaion of nonsaionariy. These negaive effecs can be avoided if he informaion abou he inernal srucure of economic processes (rend periodiciy auoregression) will be used when building he economeric model. Such approach ensures ha he residual process will have he whie noise properies which are he mos desired and he congruence posulae (in Granger sense (98) or in Zieliński sense (984)) will be saisfied. If he congruence posulae is realized he coexisence of models wih differen specificaion as well for levels as for differences is possible. Hence he wo compeiive model specificaion i.e. for levels (sraegy always ake levels ) and for differences (sraegy always difference ) when economic processes are non-saionary can be reconciled. However i should be remembered ha he esimaes of parameers in models for levels and models for differences are in general differen. These models will be saisically accepable bu will differ wih regard o he specificaion economic inerpreaion of parameers and behavior in forecasing. The purpose of he paper is o evaluae he effecs of incorrec idenificaion of processes non-saionary in mean (saionary flucuaions around deerminis- Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House There are he effecs of overdifferencing and underdifferencing. See: Piłaowska (23a) (23b) (24).

2 94 Mariola Piłaowska ic rend rend saionary (TS)) and processes non-saionary in variance (inegraed processes difference saionary (DS)) for he behavior in forecasing of economeric models specified for levels (he TS model sraegy always ake levels ) and for differences (he DS and EC (error correcion) model sraegy always difference ). The evaluaion of hese effecs will be curried ou wih he use of he Mone Carlo experimens for processes generaed wih a given srucure (under differen assumpions) 2. The Descripion of Simulaion Experimen The scenario of experimens assuming he relaionship beween processes Y and which are non-saionary in mean or in variance is depiced he Table. The experimens A and 2A are aimed a evaluaing he effecs of incorrec idenificaion of processes non-saionary in mean and in variance respecively considered he same relaionship in he whole frequency band. This means ha he parameers measuring relaionships among differen frequency componens of Y and (for example low and high frequency componens) are he same. Then he parameers in models for differences measuring he relaionship in high frequency band (because he difference filer eliminaes he low frequency band) will be he same as he parameers in models for levels measuring he relaionship in he whole frequency band. Daa generaing model in experimen A assumes ha he linear combinaion of wo processes Y and which are non-saionary in mean (hey are rend saionary) i.e. Y α = μ + ε is saionary (does no have deerminisic rend). This means ha he vecor [ α] eliminaes he non-saionariy in mean and a he same ime reflecs he relaionship beween Y and on a saionary level (because he relaionship in he whole frequency band is he same). Then i is said ha Y and are co-rended. While in experimen A he co-rending effec is observed in experimen B he daa generaing model has he deerminisic rend componen. This means ha he co-rending does no occur i.e. he parameers measuring relaionship beween Y and on a saionary level can no be used in eliminaing he deerminisic rend from Y and. The lack of co-rending may be inerpreed in erms of omied imporan variable 2. Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House 2 The resuls of simulaion experimens in Kufel (22) pp indicae ha in he congruen model reduced o significan variables he deerminisic rend plays he role of balancing he srucure of model when he imporan variable including in he daa generaing model of Y was omied and he case of non-saionariy in mean was occurred. See also Kufel Piłaowska Zieliński (996).

3 The Effecs of Incorrec Idenificaion of Non-saionariy of Economic Processes 95 Table. Scenario of experimens Type of non-saionariy non-saionariy in mean non-saionariy in variance Relaionship of low and high frequency componens of generaed processes Y and he same for low and high differen for low and high frequencies frequencies Experimen A Experimen B Y = 3 + μ + ε low high Y = 3η + 2η + γ y + γ y + ε = γ x + γ x + η = γ x + γ x + η η x = β xη + ε η β η + ε = x Parameer values aken in experimens: β x = ( ) ε ~ N( σ ) σ = 3 n = significance level in he selecion mehod:..5. Experimen 2A Experimen 2B Y = 3 + ε low high Y = ε = AR s AR = β x AR + ε AR 3 = AR s AR = β x AR + ε AR Parameer values aken in experimens: β x = ( ) ε ~ N ( σ ) σ = 3 n = significance level in he selecion mehod:..5. low low Low frequency componens i.e. η in he B experimen and in he 2B experimen are obained hrough he filraion of process η and process respecively by he means of he Spencer moving average 3 and high frequency componens will be high low high low calculaed as: η = η η =. The parameer values γ γ y of deerminisic rend are generaed from he symmeric disribuion wih parameers (.5;.2) (.75;.25) respecively and γ x = γ y are equal. The purpose of experimens B and 2B is o evaluae he effecs of incorrec idenificaion of processes non-saionary in mean and in variance considered he differen relaionship for differen frequency componens. Then he parameers in models for differences are no he same as he parameers in models for levels which as before measure he relaionship in he whole frequency band bu his ime hey are averaged wih weighs equal o he proporion of variance of differen frequency componens in he oal variance of. In experimen 2A he daa generaing model assumes ha he processes Y and are coinegraed i.e. he combinaion Y α = ε is saionary alhough processes Y and are firs order inegraed processes (have a rend in vari- Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House 3 The Spencer moving average in he form: /35[5] 2 [7][ 2...] eliminaes high frequency componens (see: Yule Kendall (966)).

4 96 Mariola Piłaowska ance). This means ha he vecor [ α] eliminaes he non-saionariy in variance and a he same ime measures he relaionship beween processes Y and on a saionary level. In experimen 2B he daa generaing model also assumes he exisence of coinegraion bu addiionally he differen relaionship for low and high frequency componens of processes Y and (α α 2 ). As a resul he parameers measuring he relaionship beween processes Y and are averaged wih appropriae weighs. In all experimens A B 2A 2B he OLS mehod was used o esimae he following models: he TS model wihin he sraegy always ake levels he DS and EC models wihin he sraegy always difference. The models are of he form: sraegy always ake levels : qy q x he TS model: Y = β Y + β + δ + δ + ε () ys s sraegy always difference : Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House xs q y q x he DS model: ΔY = β ΔY + β Δ + μ + η (2) qy ys s he EC model: ΔY = β ΔY + β Δ + θec + μ + η. (3) ys s qx When he case of he relaionship among processes non-saionary in mean occurring (experimen A B) model () is reaed as a rue model and he models for differences (models (2) and (3)) are reaed as alernaive ones wih s xs xs s s EC regard o model () for levels. In model (3) denoes he error correcion which in experimen A assuming he same relaionship in he whole frequency band is equal EC = Y ˆ α and in experimen B assuming differen * * relaionships for differen frequency componens EC = Y ˆ α where * * Y and sand for processes afer eliminaion of linear rend. When he relaionship among inegraed processes (non-saionary in variance) is considered (comp. experimen 2A 2B) models (2) and (3) are reaed as rue models and model () for levels is reaed as an alernaive one. The resuls of he Mone Carlo simulaion from all experimens (A B 2A 2B) referred o he comparison of models TS DS and EC wih regard o: specificaion residual process properies esimaes of parameers by and Δ in models for levels and differences respecively disribuion of he -Suden and Durbin-Wason saisics disribuion of deerminaion coefficien R 2. The deailed resuls wihin he menioned capaciy are in Piłaowska (23).

5 The Effecs of Incorrec Idenificaion of Non-saionariy of Economic Processes 97 Whereas he behavior of forecasing models TS DS and EC provided he differen relaionships for differen frequency componens (experimen B 2B) for samples of 3 is presened below 4. The models reduced o significan variables are used o calculae dynamic forecass wih horizon h = 2 K5 for and h = 2 K for and also o calculae he predicion mean square errors where realizaions y n+ yn+ 2 K yn+ h and x n+ xn+ 2 K xn+ h were obained from appropriae daa g eneraing models of Y and. Forecass of Y were calculaed under assumpion of known values of explanaory process. 3. Performance of Forecasing Models TS DS and EC The esimaes of parameers in reduced models () (2) and (3) disincly differ (see Piłaowska (23)) as a resul of differen relaionship for differen frequency componens (experimen B). Therefore i is expeced ha he performance of model TS (reduced model ()) and models DS and EC (reduced model (2) and (3)) in forecasing will also be differen. Forecass of Y are obained from he model for levels which describes relaionships in he whole frequency band (alhough parameers are averaged wih appropriae weighs) and hence akes ino accoun he relaionships in long and shor run. Whereas forecass of Y are calculaed from he model describing relaionships for high frequencies (i.e. afer eliminaing he low frequencies by difference filer) i.e. from he model referring o relaionships in shor run. The comparison of forecasing properies of models TS DS and EC will be curried ou by he means of he raios of predicion mean square errors (PMSE) calculaed from each forecasing model i.e. PMSE(DS)/PMSE(TS) PMSE(EC)/PMSE(TS) which allow o compare he sraegy always difference wih he sraegy always ake levels. Raios of Predicion Mean Square Errors in Experimen B The PMSE(DS)/PMSE(TS) raios show (fig. 2) ha model TS ouperforms model DS for all parameers values β x significance levels α size of disurbances σ and sample sizes n a he whole forecas horizon because raios PMSE(DS)/PMSE(TS) are greaer han one. As he forecas horizon and parameer values β x increase he dominaion of model TS is greaer a he whole forecas horizon. Hence model DS canno compee wih model TS. From he comparison of he performance of forecasing models TS and EC resuls ha model TS can compee wih model TS. Model EC ouperforms Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House 4 The behavior of forecasing models TS DS and EC provided he same relaionship in he whole frequency band (experimen A 2A) is presened in Piłaowska (23).

6 98 Mariola Piłaowska model TS (raios PMSE(EC)/PMSE(TS) are lower han one) see fig. 3 and 4 for small disurbance (σ = ) in boh sample sizes ( 3) for large disurbance in sample size of 6 and for conservaive sraegy in eliminaing insignifican processes (especially a he % bu also a 5% significance level). In hose cases as a resul of conservaive sraegy he specificaion of model TS is oo parsimonious and does no include lagged processes such x- y - y -2 which play he role of balancing he harmonic srucure of boh sides of model. Therefore such specificaion is no sufficien o describe changes of Y in he case of differen relaionships for low and high frequency componens. 5 5 β x=.9 β x = 3 5 α =.5 σ = α =. σ = α =.5 σ = 3 5 Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House 5 5 α =. σ = horyzon prognozy horyzon prognozy horyzon prognozy Fig. Raios PMSE(DS)/PMSE(TS) in differen sample size ( 3) and disurbance σ = in experimen B β x=.9 β x = horyzon prognozy horyzon prognozy horyzon prognozy Fig. 2. Raios PMSE(DS)/PMSE(TS) in differen sample size ( 3) and disurbance σ = 3 in experimen B Model TS slighly ouperforms model EC (fig. 3 and 4) a almos whole forecas horizon (excep shor horizon) a he 5% and % significance level (liberal sraegy in eliminaing insignifican processes). As he significance level increases he specificaion of model TS includes more frequenly addiional

7 The Effecs of Incorrec Idenificaion of Non-saionariy of Economic Processes 99 elemens such y y 2 x balancing he srucure of model han a he % significance level. For large disurbance (σ = 3) model TS provides forecass wih lower PMSE a all significance levels and in all sample sizes. This dominaion is kep also in small sample () a he 5% and % significance level in spie of parsimonious specificaion of model TS (mos frequenly x and x ). This means ha hese specificaions can well approximae he daa generaing model in he case of differen relaionships for low and high frequency componens. β x =.9 β x = α =.5 σ = 3 5 α =.5 σ = α =. σ = 3 5 Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House α =. σ =.94 horyzon prognozy horyzon prognozy horyzon prognozy Fig. 3. Raios PMSE(EC)/PMSE(TS) in differen sample size ( 3) and disurbance σ = in experimen B β x =.9 β x = horyzon prognozy horyzon prognozy horyzon prognozy Fig. 4. Raios PMSE(EC)/PMSE(TS) in differen sample size ( 3) and disurbance σ = 3 in experimen B Raios of Predicion Mean Square Errors in Experimen 2B In experimen 2B he raios PMSE(DS)/PMSE(TS) indicaes ha model TS ouperforms model DS for all parameer values β significance levels size of disurbance and all sample sizes a he whole forecas horizon because he ra-

8 Mariola Piłaowska ios PMSE(DS)/PMSE(TS) are greaer han one (fig. 5 and 6). As he forecas horizon increases he dominaion of model TS grows a he whole forecas horizon β x=.6 β x =.7 β x = α =.5 σ = α =. σ = α =.5 σ = α =. σ = horyzon prognozy horyzon prognozy horyzon prognozy Fig. 5. Raios PMSE(DS)/PMSE(TS) for differen sample size ( 3) and disurbance σ = in experimen 2B β x =.7 β x = Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House horyzon prognozy horyzon prognozy horyzon prognozy Fig. 6. Raios PMSE(DS)/PMSE(TS) for differen sample size ( 3) and disurbance σ = 3 in experimen 2B Raios PMSE(EC)/PMSE(TS) show he differen performance of forecas- TS and EC depending on he size of disurbance σ. For small disur- ing models bance (σ = ) model EC is in general superior for all sample sizes n significance levels and parameer values β x a he whole forecas horizon (excep β x =.6.8 a he %) because he raios are lower han one (fig. 7 and 8). However for large disurbance (σ = 3) model TS gives lower predicion mean square errors (raios PMSE(EC)/PMSE(TS) are slighly greaer han one).

9 The Effecs of Incorrec Idenificaion of Non-saionariy of Economic Processes α =.5 σ = α =. σ = β x =.7 β x = α =.5 σ = α =. σ =.85 horyzon prognozy horyzon prognozy horyzon prognozy Fig. 7. Raios PMSE(EC)/PMSE(TS) for differen sample size ( 3) and disurbance σ = in experimen 2B β x =.7 β x = Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House horyzon prognozy horyzon prognozy horyzon prognozy Fig. 8. Raios PMSE(EC)/PMSE(TS) for differen sample size ( 3) and disurbance σ = 3 in experimen 2B 4. Conclusions The comparison of he sraegy always ake levels wih he sraegy always difference indicaes ha neiher sraegy has he clear advanage in forecasing. In oher words models for levels (he TS models) can compee wih models for differences (he EC models) even in he case of correc idenificaion of non-saionariy (in mean or in variance). This suggess he usefulness of models for levels as well as models for difference independenly of he ype of non-saionariy however provided ha models saisfy he congruence posulae consising in specifying he model in such a way ha he residual process has whie noise properies. Moreover his may indicae he rule of humb o build models a he same ime for levels and for differences and hen o choose he model having beer saisical properies as well as beer accepance from he economic poin of view. On he oher hand boh models can be used in forecasing o calculae combined forecass.

10 2 Mariola Piłaowska Lieraure Granger C. W. J. (98) Some Properies of Time Series Daa and heir Use in Economeric Model Specificaion Journal of Economerics Kufel T. (22) Posula zgodności w dynamicznych modelach ekonomerycznych (Congruence Posulae in Dynamic Economeric Models) Wydawnicwo UMK Toruń. Kufel T. Piłaowska M. Zieliński Z.(996) Symulacyjna analiza poznawczych własności dynamicznych modeli zgodnych (Simulaion Analysis of Cogniive Properies of Dynamic Congruen Models) in: A. Zeliaś (red.) Przesrzennoczasowe modelowanie i prognozowanie zjawisk gospodarczych (Spaial Time Series Modelling and Forecasing) Wydawnicwo AE Kraków Piłaowska M. (23a) Skuki nadmiernego i niewysarczającego różnicowania procesów ekonomicznych. Analiza symulacyjna (Effecs of Overdifferencing and Underdifferencing. Simulaion Analysis) Przegląd Saysyczny (Saisical Survey) z Piłaowska M. (23b) Modelowanie niesacjonarnych procesów ekonomicznych. Sudium meodologiczne (Modelling Non-saionary Economic Processes. Mehodological Sudy) Wydawnicwo UMK Toruń. Piłaowska M. (24) Realizacja posulau zgodności jako meoda uniknięcia skuków pozornej zależności (Realizaion of Congruence Posulae as a Meod of Avoiding he Effecs of a Spurious Relaionship) Przegląd Saysyczny (Saisical Survey) z Wooldridge J. M. (999) Asympoic Properies of Some Specificaion Tess In Linear Models wih Inegraed Processes w: R. F. Engle H. Whie Coinegraion Causaliy and Forecasing. A Fesschrif in Honour of Clive W. J. Granger Oxford Universiy Press Yule G. U. Kendall M. G. (966) Wsęp do saysyki (Inroducion o Saisics) PWN Warszawa. Zieliński Z. (984) Zmienność w czasie srukuralnych paramerów modelu ekonomerycznego (Time Variabiliy of Srucural Parameers in Economeric Model) Przegląd Saysyczny (Saisical Survey) z. / Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House