DYNAMIC ECONOMETRIC MODELS Vol. 8 Nicolaus Copernicus University Toruń Elżbieta Szulc Nicolaus Copernicus University in Toruń

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1 DYNAMIC ECONOMETRIC MODELS Vol. 8 Nicolaus Copernicus Universiy Toruń Nicolaus Copernicus Universiy in Toruń Modeling of he Dependence Beween he Space-Time Processes 1. Inroducion The main hesis of he paper is a saemenha he basis of an appropriae modeling of he dependence beween he space-ime processes is o consider heir inernal srucures. The space-ime processes are characerized by he ses of double indexed variables he so-called random fields. The models of such random fields, X u i discussed in secions and of he paper, have go an essenial significance for he specificaion of he dependence beween he space-ime processes. The models, discussed in secion 4ake ino accoun he principle of he ime, space and space-ime dynamics which manifess iself by he specificaion of he appropriae ime lags, space shifs, and also simulaneous spaceime shifs in he modeling of he dependence and he principle of congruency as well, which is an exension of he principle of he congruency, used in economerics for he linear dynamic modeling of he dependence of he sochasic processes. The advanages for he modeling of he dependence of economic space-ime processes resuling from such an approach are poined ou. The heoreical consideraions are illusraed in secion 5 by an empirical example, which refers o he dependence beween unemploymen rae and average monhly gross wages and salaries in enerprise secor in Poland. In secion 6 he conclusions are formulaed and he direcions for furher invesigaions are announced.

2 86. Modeling of he Trend-Seasonal Srucure Heerogeneous/non-saionary, wih regard o he mean values, space-ime economic processes may be modeled using he polynomial space-ime rend funcions and he seasonal componen models. Le X u i denoe he space-ime process, observed in he spaial unis i, wih he co-ordinaes of he locaion u i = ( u1 i, ui ), in he ime. The expression of he form: f ( u1 i ui ; ) = r r r r r u i u i 1,, 1 + r + r r, γ (1) presens he spaio-emporal rend of degree r. The model of he space-ime process wih he rend and seasonaliy akes he form: m r r X = γ r, r, r u1 i ui + d kq 1 k + η + r + r r k = 1, () where: Q k seasonal dummies, η homogeneous/saionary space-ime residual process. u i,. Modeling of he Auoregressive Srucure The consrucion of he auoregressive space-ime models is based on he saemenha he values of a phenomenon observed a he esablished poins in ime and space are dependen on he previous observaions of he phenomenon a he oher poins in space. Connecions among he variables in differen unis in space depend in a sysemaical way on he spaial disance, likewise he dependence in ime depends on he ime disance. The dependence among he neighbours of differen orders is considered. For expressing he connecions of he observaions of he variable in one place wih he observaions of he same variable in oher places, i is convenienly o refer o he idea of he spaial lag, which in pracice is named he spaial shif operaor. The spaial shif operaor differs from he emporal shif operaor, because he las one causes shifs of he variable by one or more periods backwards, whereas he spaial operaor acs ino differen direcions, wih regard o he facha he direcion of he shifs in space may be various. The definiion of he spaial shif operaor depends on he spaial daa arrangemen and on wha is known in advance abou he invesigaed phenomenon, e.g. wheher here is well founded he assumpionha he influence of he variable locaed in he given place on such he variable locaed in anoher place

3 Modeling of he Dependence Beween he Space-Time Processes 87 depends mainly on he disance beween he locaions and does no depend on he direcion or wheher i should be assumed o depend on he direcion as well 1. The saring poin o specify he spaial shif operaor is he idenificaion of he neighbours of each place i on he laice D, wih regard o he well-defined crierion of specificaion (i.e. a common border for he so-called neares neighbours). The neighbours of he firs, second ec. order are idenified. The appropriae ses of he neighbours are denoed by: N 1 (i), N (i),... (generally, N s (i), where s denoes he order of he neighbourhood). When he ses of neighbours are fixed for each place ihe spaial shif operaor of he order s (i.e. L (s) ) may be defined as follows: ( s) ( s) L X i = j w N () i ij X. () s From he consideraions above i appearsha he spaial shif operaor is he operaor of he lags disribued in space raher hen he shif operaor in he given direcion. I is assumedha he weighs in () saisfy he following condiions: ( s) 1) w ij 0, ( s) ) w ii = 0, ( s) ) j N = 1 () i w ij. s Usuallyhe spaial weighs are esablished a priori by he researcher. They may reflec he lengh of common borders, number of roads, railways, geographical or economic disance beween he regions. Using he concep of he spaial operaor L (s) he auoregressive space-ime model of order l in space and q in ime, marked by STAR(l, q), i.e.: l q ( s) X i = α sτ L X i, τ + ε i,. (4) s= 0 τ = 1 may be defined. In model (4) he so-called pure spaial auodependence, i.e. he dependence among spaial unis in he same ime is no considered. Generally i is jusified. Since i is possible o agree wih he argumenaionha he evens in dif- 1 Someimes he spaial shif operaor is presened as he so-called srucure of he spaial shifs in a model. Some differen spaial shifs srucures are possible. On heir imporance for defining he so-called STARMA models (space-ime auoregressivemoving-average models) and for he properies of he models, see e.g. Hopper and Hewings (1981). The definiion of he spaial laice may be find in Cressie (199). See, e.g. Giacomini, Granger (004).

4 88 feren poins in space do no influence he evens in oher locaions a once, because he realizaion of he resuls of he influence needs some ime lag. However, an aenion should be paidha he assumpion of he lack of insananeous spaial dependence is imporan, provided he ime disance beween he observaions is smaller han he real ime lag of reacion. If he mechanism generaing he course of he phenomenon creaes i wih frequency greaer han he frequency wih which he daa are observedhen spuriously insananeous influences may appear. Thushe problem wheher he space-ime auoregressive model should include he clear spaial componen depends on he scale of ime realizaion and measuremen of he phenomenon. Furhermore, while he insananeous causal dependence may be doubfulhe spaial correlaion in he same ime (he so-called spaial auocorrelaion) is obviously possible. In he paper he consideraions are limied o he space-ime auoregressive models of he form (4). Thus, i is assumed ha he spaial dependence reqres a leas one ime lag before is effecs are occurred. 4. Modeling of he Dependence Beween Processes The imporan idea of he modeling of he dependence beween space-ime processesaking ino accoun he srucure of he connecions in ime and space is he congruen modeling of random fields. The concep proceeds wih he economeric congruen modeling, which refers o he sochasic processes. The auhor of he paper has already underaken some aemps o consruc he congruen models for random fields and o invesigae heir properies on he ground of he heoreical consideraions and on he basis of he generaed daa 4 as well. In he paper he empirical example of he modeling of he dependence of wo space-ime processes is presened 5. In his case he procedure of he congruen model consrucion is following : 1) The models wih spaio-emporal rend and seasonaliy are idenified: X Y m r r u = i, r, r u 1i ui + ( x ) + r + r r k = 1 k γ d Q + η m r r =, r, r u 1i u i + ( y ) + r + r r k = 1 ( y) k δ d Q + η ( y) k k ( y), (5). (6) ) The space-ime processes η u, η i are idenified as auoregressive ones, and modeled as: 4 See Szulc (1998, 00). 5 See also Szulc (007).

5 Modeling of he Dependence Beween he Space-Time Processes 89 η η l q = u ( s) α s, τ L η τ + ε, (7) i s= 0 τ = 1 h p ( y ) ( s) ( y ) ( y ) = β s, τ L η τ + ε. (8) s= 0 τ = 1 ) The equaion of he dependence for he whie noise space-ime processes ( y) ε ε is consruced:, ( y) = ρε ε, (9) ε + where: ε u i whie noise independen of ε. 4) The congruen model for real processes is obained by aking he residual processes from (5) and (6) and by subsing hem ino (7) and (8) respecively and finallyhe ransformed (7) and (8) ino (9). As he resul he following model is obained: r r r m 1 Yu = θ, r, r u 1i ui + d kqk + ρx i + r + r r k = 1 (10) h p l q ( s) ( s) + βs, τ L Yu α s, L X, i τ + τ u ε i τ + s= 0 τ = 1 s= 0 τ = 1 ( y) where: r = max{ r, r }, α s, τ = ρα s, τ. Taking ino accoun he rend-seasonal componen in he model (10) he heerogeneous/non-saionary mean value is removed from he processes X u i, Y herefore he parameers: α s, τ, β s, τ, ρ measure he dependence beween homogeneous/saionary componens of hese processes. Apar from he curren dependence beween he processes X u i, Y, measured by he parameer ρ, in he model (10) he dependence which is lagged in ime and space is aken ino consideraion. The influence of he explanaory phenomenon observed a he same poins in ime and space a which he explained phenomenon is observed is separaed from he influence of he phenomenon, observed somewhere else and some oher ime. These influences are measured by ρ and α s,τ respecively. The parameers β s, τ reflec he connecions in ime among he magniudes of he explained phenomenon, observed in he neighbouring spaial unis. Thanks o explicie separaion of he variables ( s L ) X u i τ hey will no conain he so-called indirec influences on u i, Y u i.

6 90 Specificaion of he model (10) resuls from he invesigaion of he inernal srucure of individual processes. I is he iniial model, which afer esimaion of he parameers reqres he insignifican componens o be reduced. 5. Empirical Example The empirical example refers o he dependence beween unemploymen rae and average monhly gross wages and salaries (in PLN) in he enerprise secor in Poland by voivodships in he period: January 1999 December 006. The daa are from Saisical Bulleins of voivodships from he appropriae periods and from he inerne sources: hp:// The saisic sample consiss of wo daa ses, including every 96 ime observaions for each of 16 spaial unis, i.e. 156 observaions ogeher. The colleced daa as well on unemploymen as on wages and salaries demonsrae rend and seasonal changes. The suggesion ha he daa may be spaial correlaed is reasonable as well. A. Invesigaing he rend and seasonaliy The models of polynomial funcions of he spaio-emporal rend wih seasonaliy were considered. For wages and salaries he model of he form 6 xi, = i j ( ) ( 18.19) ( 18.19) (.11171) i j ij ( ) ( ) ( ) ( 0.185) i j 78.70i j i ( ) ( ) ( ) ( ) ij i ij i ( ) ( ) ( ) ( ) (11) j j j ( ) ( 0.000) ( ) ( ) Q Q Q Q4 ( ) ( ) ( 15.91) ( ) 4.648Q5 1.88Q Q7.5158Q8 ( 15.64) ( ) ( ) ( 15.64) Q Q Q11 +, ( ) ( 15.91) ( ) was chosen. The model (11) presens he spaio-emporal rend of he rd degree and seasonaliy. The mos of he parameers of he model are significan. The model fi in his case is no high (R =0.4566). 6 For he furher noaion o be simplifiedhe index u i was changed by (i, j).

7 Modeling of he Dependence Beween he Space-Time Processes 91 The analysis of he rend and seasonal changes in he space-ime process of unemploymen allowed o fi he model (1). The mos of is parameers are significan. The coefficien R for he model equals y i, j = i.7187 j i ( ) ( ) ( ) ( 0.057) ( ) 18. j ij i ( ) ( ) ( 0.559) ( ) j i j i ij ( ) ( ) ( ) ( ) i ij i.7988 j ( ) ( ) ( ) ( ) j j Q1 ( ) ( ) ( ) ( ) Q Q Q Q5 ( ) ( ) ( ) ( ) Q Q Q8 ( ) ( ) ( ) ( y) Q Q Q11 +,. ( ) ( ) ( ) (1) B. Invesigaing he auoregressive srucure The purpose of he analysis of he auoregressive srucure of he invesigaed processes was o idenify significance of he larges ime lags and spaial shifs. For boh he processes he ime lags of he 1 h order are significan, whereas he spaial shifs are significan only of he 1 s order. Thus STAR(1, 1) models were esimaed. E.g. for wages and salaries he model ook he form:, = , , , ( 1.748) ( ) ( ) ( ) , , , , 7 ( ) ( ) ( 0.011) ( 0.01) , , , , 11 ( 0.015) ( 0.017) ( ) ( ) , L, L, ( ) ( ) ( ) (1) L, L, L, 5 ( ) ( ) ( 0.041) L, L, L, 8 ( 0.061) ( 0.041) ( ) L, L, L, 11 ( ) ( ) ( ) L, 1 + ei,. ( ) The auoregressive models referring o boh wages and unemploymen included insignifican componens. However, a his sage of he analysis he re-

8 9 ducion of insignifican componens was no carried ou. The reducion was carried ou only wih regard o he iniial congruen model. C. Empirical congruen model Using he procedure presened in secion 4 he congruen model, describing he dependence beween he wages and unemploymen was obained. The model reduced o he significan componens ook he following form: y i, = i j ( ) ( 0.561) ( ) ( ) i j ij ( ) ( ) ( ) ( ) ij i ij i ( ) ( ) ( ) ( ) i j i j i ( ) ( ) ( ) ( 0.000) j j j ( ) ( ) ( ) ( ) Q Q Q 0.87Q ( ) ( ) ( ) ( 0.069) 0.441Q Q Q Q ( ) ( ) ( ) ( ) Q Q Q yi, ( ) ( ) ( ) ( ) yi, yi, yi, 1 ( ) ( ) ( ) L yi, L yi, L y ( ) ( ) ( ) L yi, xi, L xi, ( ) ( ) ( ) L xi, 10 + ei, ; ( ) i, (14) R = The model (14) was obained from he iniial (sufficienly general) congruen model, using he mehod of a poseriori selecion. The model conains: rend and seasonaliy, unemploymen rae in he given voivodship and in he neighbouring voivodships wih ime lags of he 1 s, nd, 11 h and 1 h orders (in monhs), and wages in he given voivodship wih ime lag of he 4 h order. Furhermore, he wages in he neighbouring voivodships wih ime lags of he 1 s and 10 h orders have significan influence on he rae of unemploymen in he given voivodship. The model (14) does no conain he curren wages and salaries. One should noiceha he curren wages and salaries would be presen in he unemploymen rae model if in he model of he dependence beween he

9 Modeling of he Dependence Beween he Space-Time Processes 9 considered processes he rend-seasonal-auoregressive srucures had no been aken ino accoun. Then he model would ake he following form: y i,, = xi, +, j (15) ( ) ( ) and would be characerized by auodependence in he residuals. The model fi would be very low (R = ). Taking ino consideraion only he rend-seasonal srucure of he invesigaed processes causesha in he unemploymen rae model among he explanaory componens he curren wages and salaries in he enerprise secor are presen. In his case he esimaed model is following: yi, = i j ( ) ( ) ( 1.481) ( 0.011) i j ij ( ) ( ) ( ) ( 0.57) i j i j i ( ) ( ) ( ) ( ) ij i ij i ( ) ( ) ( ) ( ).699 j j j (16) ( ) ( ) ( ) ( ) Q Q Q ( ) ( ) ( ) Q Q Q6 ( ) ( ) ( ) Q Q Q9 ( 0.169) ( ) ( ) Q Q xi, +, ( ) ( ) ( ) The coefficien R for he model (16) equals The residuals show he spaial and space-ime auocorrelaion. In differen models he parameers of curren wages and salaries differ from one anoher no only wih regard o significance bu also he value. Moreover, he parameers of he influence of he wages on he unemploymen in various ime disances differ from one anoher wih regard o significance, value and sign. This fac should be conneced, among oher hings, wih he influence of he wages and salaries on he labour demand on he one hand and on he labour aciviy on he oher hand. I is seemed ha hese influences can manifes hemselves wih differen srengh in differen ime. Reurning o he model (14) he residual process analysis was done. The model STAR (1, 1) ook he following form: ( 1) e ˆi, = ei, L ei, 1. (17) ( ) ( ) ( )

10 94 The insignificance of he parameers of he model (17) confirms he lack of he emporal and space-ime auocorrelaion of he 1 s order. The models STAR of higher orders in ime domain were considered as well. Howeverhe parameers in hese models were no significan. 6. Conclusion Taking ino accoun he inernal srucure of he space-ime processes is an imporan elemen of modeling of he dependence beween hese processes. The congruen model, which had he appropriae properies of he residuals, high degree of fi and inerpreabiliy of parameers was obained. In he presened analysis he simplified assumpions were aken. As regards he auoregressive srucure of he invesigaed processes in he spaial dimension, he aenion was limied o he auodependence of he 1 s order, which meansha only he so-called neares neighbours were idenified. In he models he so-called pure spaial auodependence was no aken ino consideraion. These quesions should be considered in furher invesigaions. References Cressie, N. A. (199), Saisics for Spaial Daa, John Wiley & Sons, New York. Giacomini, R., Granger, C. W. J. (004), Aggregaion of Space-Time Processes, Journal of Economerics, 118/1, 7 6. Hopper, P. M., Hewings, G. J. D. (1981), Some Properies of Space-Time Processes, Geographical Analysis, 1, 0. Szulc, E. (1998), On Conformable Economeric Modeling of Space-Time Series, in: Dynamic Economeric Models,, UMK, Toruń, Szulc, E. (00), Idenyfikacja odsępów czasowych realizacji zależności w przesrzenno-czasowych modelach ekonomerycznych, in: A. Zeliaś (red.), Przesrzennoczasowe modelowanie i prognozowanie zjawisk gospodarczych, AE, Kraków (Idenificaion of he Time Lags of Realisaion of Dependence in Spaioemporal Economeric Models, in: A. Zeliaś (ed.) Spaial-ime Modeling and Forecasing of he Economic Phenomena, Academy of Economics, Cracow), Szulc, E. (007), Ekonomeryczna analiza wielowymiarowych procesów gospodarczych, UMK, Toruń (Economeric Analysis of Mulidimensional Economic Processes).