REGIONAL INCOME DISPARITIES IN CENTRAL-EASTERN EUROPE A MARKOV MODEL APPROACH

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1 REGIONAL INCOME DISPARITIES IN CENTRAL-EASTERN EUROPE A MARKOV MODEL APPROACH Aa Decewicz Abstract The paper cocers aalysis of regioal icome disparities ad their dyamics by use of Markov models icludig spatial effects. The mai objective is to examie spatial iteractios betwee regios by meas of a o-homogeeous Markov model with trasitio probabilities coditioed by variables reflectig regio s positio i its eighbourhood. The attetio focuses o disparities of GDP per capita related to Europea average to iclude differeces i regioal ecoomies value ad their levels of well-beig. Data referrig to GDP per capita o NUTS3 i are take from Eurostat regioal database ad atioal statistic offices. Aalysis starts with studyig cross-sectioal distributio of regioal GDP per capita by meas of parametric ad o-parametric estimatio. Regioal spatial effects are measured by Mora spatial statistics, usig wage matrices of k-earest eighbours ad maximal distace. Several Markov models are estimated to examie dyamics of GDP ad idetify patters of icome distributio evolutio ad to explai how eighbourhood affects regio s curret ad future positio. Key words: regioal icome, Markov chai, spatial effects JEL Code: C23, R11 Itroductio The aim of this paper is to aalyze disparities betwee regios of ew members of Europea Uio by use of Markov models. Markov models are preset i literature (Figleto, 1007, Magrii, 1999, Le Gallo, 2004) as a alterative to ecoometric modellig of covergece givig the possibility of predictig future icome distributio, limit distributio ad the speed of covergece (or perhaps divergece). The mai purpose of the paper is a attempt to icorporate spatial effects betwee regios directly ito a Markov model. The attetio is focused o disparities ad covergece of Gross Domestic Product per capita to iclude differeces i regioal ecoomies value ad their levels of well-beig. Data take from Eurostat regioal database ad atioal statistic offices iclude aual GDP (as a 299

2 percetage of UE27 ad percetage of up-regioal average) at NUTS3 level for period for 211 regios located i Bulgaria, Czech Republic, Estoia, Hugary, Latvia, Lithuaia, Polad, Romaia, Sloveia ad Slovakia. The aalysis begis with studyig cross-sectioal distributio of regioal GDP. I order to study the dyamics of GDP distributio a o-homogeeous Markov model icorporatig spatial depedece betwee regios is costructed. I the Markov model trasitio probabilities deped o variables describig regio s curret positio i its geographic eighbourhood (i.e. the regioal GDP is related to waged average of its earest eighbours or up-regioal average). This is a modificatio of previous approach called spatial Markov chai (Rey, 2001, Le Gallo, 2001) i which spatial effects ad other factors impactig trasitio probabilities were take ito accout by estimatig several trasitio matrices for differet groups of regios distiguished by the iitial class to which their eighbours belog. 1 Modellig covergece by Markov models 1.1. Discrete Markov model By defiitio (Bhat, 1984) Markov chai with state-space S 1, 2,..., r process 0 X with memory-less property: P 0 i, j, i 1,..., i S (1) ( X 1 is a stochastic 1 j X i, X 1 i1,..., X 0 i0) P( X j X i) pij( ), meaig that trasitio probabilities deped oly o the last state observed. Homogeeous Markov chai is a process with time-stable trasitio probabilities, usually oted i trasitio matrix P p ij. With kow trasitio matrix it is possible to predict future distributio, accordig to formula d1 dp, (2) where d... d d deotes distributio at time, d P( X i). A Markov chai 1 r is called ergodic if its limit (also called ergodic or statioary) distributio exists ad does ot deped o the iitial distributio. Applyig Markov chais i modellig icome covergece cosists i defiig a fiite umber of states referrig to a set of classes distiguished basig o icome distributio i 300

3 amog regios. Trasitio probabilities of a homogeeous Markov chai are estimated from pael data with formula ij pˆ ij, (3) with ij deotig observed umber of trasitios i i j ad i umber of visits i state i. Covergece is evidet whe the probability mass i ergodic distributio is cocetrated aroud oe state, otherwise divergece or club covergece may be cosidered. Modificatios of this simple model icludig spatial effects (Rey 2001, Le Gallo, 2004) cosists i decompositio of trasitio matrix i a way eablig to extract trasitio probabilities coditioed by a class to which regio s eighbours belog. The usual assumptio i modellig covergece by meas of Markov chai is models homogeeity, however it seems obvious that trasitio mechaism may be time ad crosssectioal varyig. Heterogeeity may be icluded by meas of treatig trasitio probabilities as fuctios of some explaatory variables or studyig parameters stability. I order to iclude the impact of explaatory variables o trasitio probabilities ito a discrete model each row of a trasitio matrix should be estimated by a ordered logit model with r classes correspodig to a chai s states. 1.2 Cotiuous Markov model estimatio from pael data Cotiuous Markov process describes trasitio mechaism i cotiuous time. A movemet from ay state to aother may take place at ay momet, opposite to a discrete model i which trasitios occur oly i fixed time poits, usually idetical with momets of observatio. The followig estimatio method assumes that movemets take place ay time but the process is observed i fixed time momets. Markov property i cotiuous case takes form 0 i, j, i 1,..., i S, t 0 t1... t t1 (4) P( X t j X t i, X t i,..., X t i0 ) P( X t j X t i) pij ( t 1 t The process is called homogeeous if trasitio probabilities withi period of the legth t are costat ad it is usually described by a trasitio itesities matrix q1 q12 q1r q 21 q2 q21 Q, qr1 qr2 qr ). 301

4 ' ij p ij with trasitio itesities q (0), q p (0) q i ' i ij js. Relatio betwee trasitio itesity matrix ad trasitio probability matrix i time period from 0 to t takes form Qt P( t) e. Memory-less assumptio is equivalet to statig that sojour time i state i has expoetial distributio with parameter q i. Estimatig parameters of Markov process from pael data (Kalbfleisch&Lawless, 1985) cosists i maximizig likelihood subject to momet L( Q) p k, l qij, with st st k s k, k, 1 t st t k, 1 tk k, k, 1,, (5) p deotig probability that a idividual k observed at the t, i state s, moves to state t k s t k, 1 at the momet t k, 1. I order to idetify impact of exogeous factors o trasitio itesities a Markov model with covariates might be applied. Trasitio itesities are defied as fuctios of variables vector (0) T q β q ij k, ij exp ij zk, z. (6) The hazard ratio exp( β ij ) is the iterpreted as approximate rate of icreasig trasitio z k, itesity from state to state respect to variable z,. k 1.3 Spatial effects Estimatio method described i sectio 1.2 eables to aalyze dyamics of icome distributio by a Markov model icludig spatial depedecies betwee regios by meas of variables reflectig the impact of eighbourhood o regio s movemet from state to state. To measure spatial effects the eighbourhood has to be established first. The distace matrix is costructed basig o physical distace betwee regios geographical cetres ad a k-ear eighbours ad maximal distace wage matrices are applied. The spatial autocorrelatio is tested by Mora global statistics 1 I w l, m lm w l, m lm 1 x xx x x x with deotig umber of regios, l l l m W wage matrix. w lm 2, (7) Positive result of spatial autocorrelatio tests proves that eighbour regios are more similar to each other the more distaced oes, i.e. poor regios have poor eighbours ad 302

5 rich regios have also rich eighbours. Negative autocorrelatio cosists i eighbourhood of regios which are ot similar to each other. It is coveiet to illustrate spatial autocorrelatio o Mora scatterplot showig value of icome i a regio versus waged value i its eighbourhood. Regios lyig below regressio lie o Mora scatterplot have bigger value of icome the their eighbours, regios above the lie are surrouded by richer eighbours. 2 Regioal GDP per capita distributio Figure 1 plots desity fuctio for GDP per capita i 2000, 2005 ad The GDP per capita (i euro) distributio seems to be 3-modal i 2000 with the mai mode aroud 18% of Europea average. The other modes seem to be vaishig i later years ad are hardly see i Table 1 gives the results of best fitted (accordig to BIC criteria) mixture of ormal distributio for 2000, 2005 ad Fig. 1: GDP per capita i 2000, 2005, 2010 Source: Author s computatio, R CRAN Tab. 1: Mixture of ormal distributio, GDP per capita (euro) Compoet 1 Compoet 2 Compoet 3 Mixig proportio Average Variace Mixig proportio Average Variace Mixig proportio Average Variace Source: Author s computatio, with spded package of R CRAN 303

6 To study spatial depedecies betwee regios wage matrix has bee computed basig o k-ear eighbours (k=5). Distace is measured betwee geographical cetres of regios. Mora statistics for years 2000, 2005 ad 2010 are preseted i Table 2. Each case proves sigificat positive autocorrelatio, however it seems to be decreasig i proceedig years. The results obtaied for maximal distace (250 km) matrix show similar tedecy. Tab. 2: Global Mora statistics - GDP per capita relative to EU average GDP per capita (euro) GDP per capita (pps) Mora I statistic stadard deviatio = , Mora I statistic stadard deviate = , 2000 p-value < 2.2e-16, alterative hypothesis: greater p-value < 2.2e-16, alterative hypothesis: greater Mora I statistic Expectatio Variace Mora I statistic Expectatio Variace Mora I statistic stadard deviate = , Mora I statistic stadard deviate = , 2005 p-value < 2.2e-16 p-value < 2.2e-16 Mora I statistic Expectatio Variace Mora I statistic Expectatio Variace Mora I statistic stadard deviate = , Mora I statistic stadard deviate = 6.969, 2010 p-value < 2.2e-16 p-value = 1.596e-12 Mora I statistic Expectatio Variace Mora I statistic Expectatio Variace Source: Author s computatio, with spded package of R CRAN Mora scatterplot preseted o Figure 2 shows the most outstadig poits o the plot lyig below the regressio lie referrig to regios which are sigificatly richer comparig to their eighbours: SI021 Osredjesloveska (Sloveia), CZ010 Hlaví mesto Praha (Czech Republic), PL127 Miasto Warszawa (Polad), PL415 Miasto Pozań (Polad), HU101 Budapest (Hugary). 304

7 Fig. 2: Mora scatterplot GDP per capita (euro) Source: Author s computatio, with spded package of R CRAN 3 Distributio dyamics estimatio of Markov models To aalyze dyamics of regioal icome distributio by meas of Markov chai oe has to remember that defiitio of states, i.e. the method of trasformig a cotiuous variable to a discrete oe, may ifluece the results. Distiguishig more classes should better reflect icome distributio but results i icreasig umber of model s parameters ad possible problems with quality of estimates. Seve states of a Markov model have bee distiguished basig o 15 th, 30 th, 45 th, 60 th, 75 th ad 90 th percetiles of regioal icome distributio. As available data form a pael the aalysis is coducted with methods described i sectio 1.2. The itesity matrix for homogeeous model has bee estimated ad oe-year trasitio probability matrix, limit distributio ad sojour times i each state have bee calculated (Table 3 ad 4). Probabilities to stay i the same state i the ext year are very high, particularly for state 1 ad 7 (the poorest ad the richest class). Expected time to ext movemet (up or dow) is betwee 2.86 years for regios i state 4 (middle class) to years i state 7. For states 2, 5 ad 6 movemets dow withi oe year are more likely tha ups, for states 3 ad 4 the opposite. Process is ergodic ad its limit distributios gives o sigs of GDP covergece. 305

8 Tab. 3: Trasitio itesities, sojour time ad ergodic distributio for homogeous model to from (0.0237) (0.0237) (0.0224) ( ) (0.0171) (0.0176) (0.0269) (0.0203) (0.0227) (0.0385) (0.0308) (0.0254) (0.0298) (0.0155) (0.0174) (0.0229) (0.0148) (0.0207) (0.0207) sojour time (0.50) (1.44) (0.65) (0.31) (0.44) (0.97) 11.13(2.56) ergodic distributio Notes: Stadard errors i parethesis Source: Author s computatio, with msm package of R CRAN Tab. 4: Oe-year trasitio probabilities for homogeeous model to from Notes: Stadard errors i parethesis Source: Author s computatio, with msm package of R CRAN Next part of aalysis is a attempt to iclude spatial relatios betwee regios directly ito a model. The first ohomogeeous Markov model (Model 1) with covariates is the oe with variable N defied as waged average of GDP per capita level i regio s eighbourhood defied by k-ear eighbours matrix W computed before. Such defiitio is supposed to regard impact of eighbours curret positio o trasitio itesities ad trasitio probabilities. I the secod model (Model 2) covariate U refers to regios positio i its eighbourhood measured by percetage of NUTS2 level average of the area to which it belogs. The third model (Model 3) has time covariate. Table 5 cotais hazard ratios exp( β ij ) for each pair of states with o-zero trasitio itesities for three ohomogeeous 306

9 models. The likelihood ratio test idicates models 1 ad 2 to perform better tha the homogeeous model, model 3 has bee rejected. Tab. 5: Hazard ratio, ohomogeeous models Model 1 Model 2 Model 3 from to HR L U HR L U HR L U Notes: L ad U are lower ad upper limit of 95% cofidece iterval. Source: Author s computatio, with msm package of R CRAN To see how eighbourhood impacts limit distributio several itesities matrices for the whole rage of possible values of covariates resultig from their distributios have bee calculated for Model 1 ad Model 2. Limit distributio of GDP per capita i regio surrouded by poor eighbours cocetrates i state 1, for regios surrouded by rich eighbours mass of probability moves to states 4, 6 ad 7. Similarly, the mass of probability i limit distributio moves to higher states as regio s positio compared to up-regioal average icreases (Figure 3). Fig. 3: Limit distributios for models with covariates Model state N=10 N=20 N=30 N=40 N=50 N=60 N=70 N= Model state 6 7 U=30 U=50 U=70 U=90 U=110 U=130 U=150 U=170 U=190 Source: Author s computatio, with msm package of R CRAN 307

10 Coclusios The purpose of this paper was to show how Markov models with covariate ca be applied to aalyze icome distributio dyamics. This approach is a modificatio of classic applicatio of Markov chais i modellig covergece givig possibility to extract impact of particular factors (of time or spatial ature) o trasitio probabilities. Markov models applied for GDP per capita distributio dyamics i regios of ew member coutries of UE from 2000 to 2010 show strog tedecy to stay i the same class of GDP level i succeedig years. Models with spatial effects prove that regio is more probable to be i higher states (referrig to higher level of GDP per capita) if it has rich eighbours. Refereces Aseli L. (1988). Spatial Ecoometrics: Methods ad Models. Kluwer, Dordrecht. Bhat, N. (1984). Elemets of applied stochastic processes. Wiley&Sos. Figleto, B. (1997). Specificatio ad testig of Markov chai models: A applicatio to covergece i the Europea Uio. Oxford Bulleti of Ecoomics ad Statistics, 59(3), doi: / Kalbfleisch, J., & Lawless, J. (1985). The aalysis of pael data uder a Markov assumptio. Joural of the America Statistical Associatio, 80(392), doi: / Kopczewska, K. (2006). Ekoometria i statystyka przestrzea z wykorzystaiem programu R CRAN. CeDeWu. Le Gallo, J. (2004). Space-time aalysis GDP disparities cross Europea regios: A Markov chai approach. Iteratioal Regioal Sciece Review 27, doi: / Magrii, S. (1999). The evolutio of icome disparities amog the regios of the Europea Uio. Regioal Sciece ad Urba Ecoomics, 29(2), doi: /S (98) Quah, D.T. (1996). Empirics for Ecoomic Growth ad Covergece. Europea Ecoomic Review, 40(6), doi: / (95) Quah, D.T. (1996). Regioal Covergece Clusters across Europe. Europea Ecoomic Review 40 (3-5), doi: / (95) Rey, S.J. & Motouri, B.D. (1999). US Regioal Icome Covergece: A Spatial Ecoometric Perspective. Regioal Studies 33(2),

11 Rey, S.J. (2001). Spatial empirics for regioal ecoomic growth ad covergece. Geographical Aalysis 33(3), Cotact Aa Decewicz Istitute of Ecoometrics, Warsaw School of Ecoomics Al. Niepodległości 162, Warszawa 309