UNIKassel VERSITÄT. Regional Convergence in Germany. A Geographically Weighted Regression Approach

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1 UNIKassel VERSITÄT Fachberech Wrtschaftswssenschaften Regonal Convergence n Germany A Geographcally Weghted Regresson Approach von Hans-Fredrch Eckey Renhold Kosfeld Matthas Türck Nr 76/05 Volkswrtschaftlche Dskussonsbeträge

2 Regonal Convergence n Germany A Geographcally Weghted Regresson Approach Hans-Fredrch-Eckey *, Renhold Kosfeld, Matthas Türck Abstract The convergence of German regons represents a poltcally explosve queston Dfferent studes examned convergence processes of Germany We derve equatons to estmate the convergence speed on bass of a Solow model, to whch human captal s added The geographcally weghted regresson permts a detaled analyss of convergence processes, whch has not been conducted for Germany so far yet We estmate a separate speed of convergence for every regon, whch results from the local coeffcents of the regresson equatons The applcaton of ths procedure to German labour market regons shows that regons move wth a dfferent speed towards ther steady state level The half lves are substantally longer n the manufacturng sector than n the servce sector One model even yelds tendences towards dvergence of some perpheral regons Zusammenfassung De Konvergenz von deutschen Regonen stellt ene poltsch brsante Fragestellung dar Verschedene Studen haben Konvergenzprozesse n Deutschland untersucht Auf Bass enes um Humankaptal erweterten Solow-Modells werden de emprsch zu untersuchenden Glechungen entwckelt De räumlch-gewchtete Regresson erlaubt ene dfferenzerte Analyse von Konvergenzprozessen, we se für Deutschland bsher noch ncht vorgenommen wurde Wr schätzen für jede Regon ene separate Konvergenzgeschwndgket, de sch aus den lokalen Koeffzenten der Regressonsglechungen ergbt De Anwendung deses Verfahrens auf deutsche Arbetsmarktregonen zegt, dass sch Regonen mt vollkommen unterschedlcher Geschwndgket auf hren Steady-State zubewegen De Halbwertszeten snd dabe m produzerenden Gewerbe wesentlch länger als m Denstletungssektor In enem Modell bestehen sogar dvergente Tendenzen für enge perphere Regonen * Prof Dr Hans-Fredrch Eckey, Economcs Department, Unversty of Kassel, Nora-Platel-Str 4, D Kassel, Germany Telephone: /3045, Telefax: /3045, e-mal: eckey@wrtschaftun-kasselde Prof Dr Renhold Kosfeld, Economcs Department, Unversty of Kassel, Nora-Platel-Str 4, D Kassel, Germany Telephone: /3084, telefax: /3045, e-mal: rkosfeld@wrtschaftun-kasselde Matthas Türck, Economcs Department, Unversty of Kassel, Nora-Platel-Str 4, D Kassel, Germany Telephone: /3044, telefax: /3045, e-mal: tuerck@wrtschaftun-kasselde

3 JEL C1, R11, R58 Keywords: Regonal Convergence, Spatal Econometrcs, Geographcally Weghted Regresson Schlüsselwörter: Räumlche Konvergenz, räumlche Ökonometre, räumlch-gewchtete Regresson

4 Introducton 3 1 Introducton The regonal convergence s a poltcally mportant queston In the European Communty contract the adjustment of the lvng condtons s explctly mentoned (s Lammers 1998, p 197) The Basc Consttutonal Law of the Federal Republc of Germany (Grundgesetz) mentons unform lvng condtons (Art 7 GG) From ths the necessty for an economc polcy algned to convergence can be justfed The Plannng Commttee for Regonal Economc Structure, to whch mportant federal mnsters are members, has passed a law n 004 (33 Rahmenplan der Gemenschaftsaufgabe "Verbesserung der regnalen Wrtschaftsstruktur") about the regonal polcy In that law the regonal equalzaton s explctly mentoned (prnted n Ebersten/Karl 1996 cont, D II, pp 7, see also Irmen/Strubelt 1998, p ) So many fscal programmes am at a reducton of regonal dfferences Because of the hgh poltcal mportance, many studes are dealng on the ssue of convergence and dvergence On the one hand researchers provde surveys on European level (see for example Neven 1995, Thomas 1995, Engel/Rogers 1996, Thomas 1996, Hellwell 1998, Ntsch 000, Martn 001, Nebuhr 00, Fngleton 003, Bottazz/Per 003, Arba/Paelnck 003, Greunz 003, López-Bazo/Vayá/Arts 004 and Eckey/Kosfeld/Türck 005 a) On the other hand there are many elaborate analyses of the convergence process n Germany Smolny (003) examnes, to what extent the bad economc development n East German regons can be explaned by a growth model But he fnds no convncng explanaton for the low productvty He assumes that East Germany wll not catch up wth West Germany soon The same concluson s conducted by Ragntz (000), who gves several structural reasons for the productvty lag lke dfferent sectoral patterns, a low captal ntensty of producton and a weak market poston of frms Klodt (000) concludes that the fadng out of the catchng-up process snce the md-1990s s caused by an napproprate desgn of ndustral polcy, whch s concentrated on the subsdy of physcal captal Kemper (004) analyses the nternal mgraton n East und West Germany Before unfcaton there were dfferent ways of mgraton Whle n the GDR had been tendences towards urbansaton, n West Germany a deconcentraton takes place The 1990s are represented by a convergent development towards a suburbansaton At the end of the decade there are tendences towards dvergence n the mgraton patterns Other aspects of convergence are the process of specalsaton of ndustres as well as the employment The paper of Nebuhr (000) s based on two economc growth equlbrums Frst, the development of productvty s explaned by the level of producton Second, a regresson model analyses the relatonshp between an ndcator of agglomeraton and the growth of employment Both approaches show a convergent development n West Germany's plannng regons Suedekum (004) doesn't fnd a trend towards spatal specalsaton or concentraton Only n some regons the specalsaton s ncreasng These areas beneft from t by an above-average employment growth Suedekum and Blen (004) supply an accurate analyss of the dstrbuton of employment The shft share regresson of West German admnstratve dstrcts yelds out a negatve effect of the regonal wages on the employment growths The employment growths dffer, especally suburban regons gan jobs from the core ctes Bayer and Juessen (004) conduct a tme seres approach to fnd out, f the

5 Introducton 4 unemployment rate n the West German Länder wll converge The used data cover the perod from 1960 to 00 Some tests suggest a convergence process and others not An mportant ndcator s also labour productvty Barrel and Velde (000) provde emprcal evdence for a catchng-up process of East Germany compared to West Germany They estmate unbalanced panel models and dentfy the emergence of West German frms besdes exogenous and endogenous techncal processes, lke the absorpton of deas and the techncal progress, as mportant factors Kosfeld, Eckey and Dreger (005) estmate a convergence rate of 7,6 % n German plannng regons Ther Solow's growth model predcts an ncrease of relatve labour productvty from 74 % to 88 % n East Germany n comparson to West Germany n the decade from 000 to 010 Researchers often use the GDP per capta to examne convergence processes Funke and Strulk (000) develop a two regon endogenous growth model to study the regonal development of the output The speed of convergence n unfed Germany depends on the expanson of the nfrastructure They ntroduce several scenaros, whch all suggest a qute fast convergence process of both parts of Germany In the most optmstc scenaro East Germany wll reach 80 per cent of West Germany's GDP per capta after 0 years Juessen (005) fnds out usng descrptve statstcs of the GDP per capta that poor regons are catchng up A nonparametrc kernel approach provdes evdence for a regonal dvergence n the long run Some researchers use regresson models wth the ncome or the GDP per capta as the dependent varable Bohl (1998) dentfes tendences towards regonal dvergence, because the null hypothess of the unt root test for panel data s not rejected But the result s lmted by the fact that there can be found statonarty n some federal states Kosfeld and Laurdson (004) estmate an error-correcton mechansm to cover tendences towards convergence n German labour market regons The adjustment coeffcent s not sgnfcant, so they conclude: "At the end of the 0s century only weak local adjustment processes ( ) towards a global equlbrum can be establshed" (Kosfeld/Laurdson 004, p 70) Funke and Nebuhr (005 a) use regresson models n order to explan the economc growth from West Germany's plannng regons Because the estmatons don't ft well, they provde a kernel approach to cluster the regons They fnd three convergence clubs, whch have dfferent growth equlbrums Funke and Nebuhr (005 b) provde nsghts of the β-convergence n West Germany's plannng regons For the perod of 1976 to 1996 a slow rate of convergence s detected Kosfeld, Eckey and Dreger (005) study the β-convergence n German functonal regons for the perod from 199 to 000 In an absolute convergence model the speed of convergence amounts to 6,5 % The convergence rate n a condtonal grows model decreases to 4 % Besde of the β-convergence Barro und Sala--Martn (1991, pp 11) ntroduced the σ- convergence Ths concept measures the changng n earnng dfferences If the gap s closng, there s a tendency towards convergence Only a few papers use the concept of σ-convergence Bode (1998) analyses ths approach by usng Markov chan models He concludes that West German regons are convergng snce the 1970s Kosfeld, Eckey and Dreger (005) fnd dmnshng varances of the ncome per capta and the labour productvty n German labour regons, so the hypothess of σ-convergence s confrmed A new aspect s the calculaton of locally dfferent parameters of β-convergence, because the varaton of parameters can lead to nconsstent estmators (Temple 1999, pp 16) Locally dfferent parameters can be calculated by a geographcally weghted

6 The growth model 5 regresson, whch s developed by Brunsdon, Charlton and Fotherngham (Brunsdon/Fotherngham/Charlton 1998, p 957) Only one convergence study uses ths model Bvand and Brunstad (005) estmate a geographcally weghted regresson of Western Europe Ther coeffcents have changng sgns They fnd a convergence of some regons and a dvergence of others However a model, whch uses dfferent regresson coeffcents for German regons, s not estmated untl now In addton most models of condtonal β-convergence, whch use the approach of Mankw, Romer and Wel (199), assume the same growth rate of technologcal progress and rate of deprecaton n all regons or neglect the term (see for example Islam 1995, Huang 005) The am of ths paper s to estmate a convergence model wth locally dfferent parameters of German regons takng nto consderaton the problems specfed above The paper s organzed as follows Frst we derve a neoclasscal growth model, whch augments the Solow-model by human captal Secton 3 outlnes the geographcally weghted regresson Especally, we show how ths approach s estmated and tested In addton we explan the used data The estmated models are presented n Secton 4 Secton 5 concludes the results The growth model We use a model that has been suggested by Mankw, Romer and Wel (199) They add the human captal, whch s stressed n the endogenous-growth theory (s for example Lucas 1988, Grossmann/Helpman 1989 and as an overvew Frenkel/Hemmer 1999, pp 00), to a Solow model The producton functon of type Cob-Douglas n perod t s gven by: β t α 1 α β (1) ( ) Y = K H A L, t t t t where Y s the output, K represents the stock of physcal captal, H the stock of human captal, A the level of technology and L the labour (Mankw/Romer/Wel 199, p 416) The producton elastctes of physcal captal α and human captal β are presumed to be added up smaller than one: () α + β < 1 Formula () mples decreasng returns to scale Dvde the producton functon (1) by A yelds the equaton: t L t α β (3) y t = kt h t, where the lower cases stand for the quanttes per effectve unt of labour, e yt = Yt ( At Lt ), kt = Kt ( At Lt ) and ht = Ht ( At Lt ) We presume a closed economy wth no publc spendng Then the savngs take the same value as the gross nvestments: (4) It = sk Yt The growth rate of physcal captal can be expressed by the net nvestments, whch are defned by the gross nvestments mnus the deprecaton: (5) K t = It δ Kt = sk Yt δ Kt, where δ denotes the deprecaton rate The growth of physcal captal per effectve unt s gven by:

7 The growth model 6 (6) k t k t t K t L t A = = t k t k t K t Lt At Substtutng equaton (5) nto formula (6) gves the growth of physcal captal per effectve labour unt: k (7) t sk Yt δ K Y = t n g = s t k δ n g, k t K t K t where n stands for the growth rate of labour L and g for the rate of technologcal progress A: nt (8) Lt = L0 e and gt (9) At = A0 e If we multply both sdes of (7) wth k t, we get: Y (10) t K k t t = sk ( δ + n + g) k t = sk yt ( δ + n + g) k t K t At L t The evoluton of the economy for the human captal can be derved n the same way as equaton (10): h t t (11) h t = = sh yt ( n + g + δ) h t The model s based on three assumptons Frst, the output can be transformed costless n captal Second, human and physcal captal have the same deprecaton rate Thrd, there are decreasng returns to scale, whch s expressed by formula () The decreasng returns to scale mply that there s a steady state level of convergence The steady state corresponds to a stuaton, where the changes n physcal and human captal are zero: (1) s k y * ( n + g + δ) k* = 0, (13) s h y * ( n + g + δ) h* = 0 We solve equatons (1) und (13) for (14) k * h * = ( n + g + δ) k * s k α (15) k * h * = ( n + g + δ) h * β s k y t and h yt, s takng nto account formula (3): α β s h Takng natural logs of these equatons yelds (s Romer 1996, p 133): (16) lns k α ln k * +β ln h* = ln( n + g + δ) + ln k * +, (17) lns h α ln k * +β ln h* = ln( n + g + δ) + ln h * + The values of physcal und human captal n the steady state can be calculated on the bass of (16) and (17): ( α β) 1 β β 1 1 (18) s = k s k * h, n + g + δ

8 The growth model 7 ( ) α α 1 1 α β (19) s = k s h * h n + g1 + δ Thus the steady state levels of physcal and human captal depends n a postve way on the proportons of physcal and human captal on the ncome A hgher growth of the labour force n, of the technologcal change or the deprecaton rate lead to lower steady state levels The steady state of output per effectve unt s derved from substtutng (18) and (19) n the producton functon (3) (s Romer 1996, p 133): α β α + β (0) ln y* = ln sk + lnsh ln( n + g + δ) 1 α β 1 α β 1 α β Usng the human captal stock n the steady state, we get the relatonshp: α β (1) = s 1 α y * ln k h 1 α n + g + δ Barro and Sala--Martn (004, p 61) show that approxmatng equatons (18) and (19) around the steady state yelds, f one takes a Taylor seres extenson: y () t = λ ( ln y* ln yt ), yt where the convergence rate λ s gven by: (3) λ = ( n + g + δ) ( n α β) The convergence rate λ s a measure, how fast regons attan ther long run equlbrum path From Equaton () follows: λt λt (4) ln yt ln y0 = (1 e ) ln y* (1 e ) ln y0, whch yelds after substtutng (0): λt λt α (5) ln yt ln y0 = (1 e ) ln y0 + (1 e ) ln sk 1 α β λt β λt α + β + (1 e ) lnsh (1 e ) ln (n + g + δ) 1 α β 1 α β The growth of the output s postvely determned by the proporton of output, whch s nvested n physcal and human captal, and negatvely affected by the ntal level of output as well as the growth rate of labour force, technologcal progress and deprecaton (Mankw/Romer/Wel 199, p 43) Equaton (5) studes condtonal convergence Ths s a convergence process, where poorer regons grow faster than rcher regons and all other varables are hold fxed Then the regresson coeffcent for the startng level of output wll be sgnfcant negatve The condtonal convergence model can be also expressed wth the human captal n the steady state nstead of the nvested share n human captal The relatonshp follows from formulas (5) and (19): λt λt α (6) ln yt ln y0 = (1 e ) ln y0 + (1 e ) ln sk 1 α λt β λt α + (1 e ) ln h * (1 e ) ln (n + g + δ) 1 α 1 α

9 The growth model 8 A sgnfcant condtonal convergence does not necessarly mean that an absolute convergence process takes place Absolute convergence apples a negatve relatonshp between the ntal output and growth of output wthout usng control varables: λt (7) ln yt ln y0 = (1 e ) ln y0 In emprcal analyses are usually used quanttes per capta and not per effectve unt of labour, because the level of technology A s unknown The equatons (6) and (7) are gven n unts per captal as follows (s Hemmer/Lorenz 004, p 49; Temple 1999, p 1): Y Y Y (8) ln t ln 0 t (1 e ) ln 0 λ λt = + (1 e ) ln A0 + gt L t L 0 L 0 and Y t (9) t Y0 t Y0 α ln ln λ (1 e ) ln λ = + (1 e ) ln sk L t L 0 L 0 1 α * λt β H λt α + (1 e ) ln (1 e ) ln (n + g + δ) 1 α L 1 α λt + ( 1 e ) ln A0 + gt Note that the restrcton of model (6) of absolut equal coeffcents for ln sk and ln (n + g + δ) leads to the equaton: Y L t Y L 0 Y α k L 0 1 α * λt β H λt + ( 1 e ) ln + (1 e ) ln A0 + gt 1 α L (30) ln t ln 0 λt (1 e ) ln 0 λt = + (1 e ) [ lns ln (n + g + δ) ] However, the nvestments n human captal are dffcult to measure, because the forgone labour earnngs can hardly be numeralzed Mankw, Romer and Wel (199) use the percentage of workng-age-populaton that attends secondary school They assume that ther nput based ndcator s proportonal to the nvestments So the estmators wll not be based Ths nput based ndcator s crtcsed by Dnopoulus and Thompson (1999 pp ) They argue that the ndcator from Mankw, Romer and Wel (199) wll underestmate human captal n poor countres and overestmate t n rch countres Dnopoulus and Thompson (1999) pont out further reasons for the nadequateness of ths ndcator Frst, human captal nvolves tertary educaton and tranng on the job too Second, the attendance at school does not necessarly mply, that human captal s rsng There are dfferences n the ablty of learnng In addton some educated sklls are dffcult to use n praxs However, because the human captal stock n the steady state s unknown and there are not avalable data for the nvestments n human captal, the human captal stock should be used nstead (Hemmer/Lorenz 004, p 158) Note that n the condtonal convergence model of Mankw, Romer and Wel (199, p 411) gt s a constant, because the perod t s fxed [s formula (30)] Mankw, Romer and Wel argue that the dfferences n the technologcal level e clmatc and nsttutonal crcumstances can be measured by a constant α and a country-specfc error term ε: (31) ln = α + ε A 0

10 Regresson model and data sources 9 Ther assumpton of noncorrelaton between ths error term and the ndependent varables seem not to be convncng (Islam 1995, p1134) So we measure gt and A 0 by locally specfc constants However, there s emprcal evdence for threshold values of regonal convergence or regonal dfferent parameters of β-convergence (s for example Bvand/Brunstad 005, Funke/Nebuhr 005 a, Juessen 005, Huang 005) Thus we allow also regonal dfferent values for all other parameters to prevent nconsstent estmators (Temple 1999, p 16) In addton we do not use the not convncng assumpton of the same growth rate of technologcal progress and rate of deprecaton n all regons (s Temple 1999, p 1, Kosfeld/Eckey/Dreger 005) All regresson coeffcents, especally the rate of convergence, are estmated separately for all regons Ths model of condtonal convergence s gven by: (3) Yt Y0 t Y (1 e ) ln 0 ln ln λ L = t L 0 L 0 [ lns ln(n + g + δ )] ( λ t α + 1 e ) 1 α k λ H ( 1 e t β ) λ t + ln + (1 e ) ln A0 gt 1 α L + The analogcal model of absolute convergence can be expressed as follows: Y Y Y (33) ln t ln 0 t (1 e ) ln 0 λ λ t (1 e = + ) ln A0 gt L t L 0 L Regresson model and data sources 31 Geographcally weghted regresson The nfluence between a dependent varable Y and some ndependent varables X k dffers often across the regons (spatal nonstatonarty) Therefore our regresson models consst of locally dfferent parameters [s formulas (3) and (33)] We use a geographcally weghted regresson (GWR), whch has been developed by Brunsdon, Charlton and Fotherngham n the past ten years (Brunsdon/Fotherngham/Charlton 1998, p 957) The global regresson model wthout takng nto consderaton a spatal dependence s wrtten by the form (34) y = β0 + βk xk + u, m k= 1 where y, = 1,,, n, are the observaton of the dependent varable Y, β k (k = 0, 1,,, m) represent the regresson coeffcents, x k s the th value of X k and u are the error terms In matrx notaton (34) s gven by m (35) y = β + β x + u 0 k k= 1 k wth y as vector of the dependent varable, k x as vector of the kth ndependent varable and u as vector of the error term In geographcally weghted regresson the global regresson coeffcents n (34) are replaced by local parameters:

11 Regresson model and data sources 10 (36) y = β + β xk + u 0 m k k= 1 where β k (k = 0, 1,,, m) s the regresson coeffcent, whch expresses the nfluence of x k on y If the β k are constant for all = 1,, 3,, n, the global model of equaton (34) or (35) respectvely holds (Brunsdon/Fotherngham/Charlton 1996, pp 8; Fotherngham/Charlton/Brunsdon 1997, pp 6) In model (33) the dependent varable y s ln ( Y ) ( ) t Lt ln Y 0 L0 and X assembles the ndependent varables n a n -matrx It contans a column of 1s to estmate the nfluence of ln A0 + gt on Y In the second column stand the values of ln ( Y 0 L 0 ) The condtonal convergence model (3) dffers only regardng X from model (33) The matrx X contans two further columns, frst [ lnsk ln (n + g + δ) ], second ln ( H L) Thus the global model s a specal case of the GWR-functon For every regon are estmated separate parameters Here s the advantage over the spatal-error- and the spatal-lag-model (s Anseln 1988) A spatal dependence n the error term can be caused by a mssng spatal varyng relatonshp (Brunsdon/Fotherngham/Charlton 1999, pp 497) How can the GWR-parameters be estmated, because there are more unknowns n (36) than degrees of freedom? In the calbraton observatons are weghted n accordance wth ts proxmty to regon As the dstance between two regons becomes smaller, the weght wll be greater We use the Eucldean dstance between to regons d j to calculate the weghts (Gaussan weghtng functon): (37) 0,5 ( d ) j bandwdth w j = e The bandwdth ndcates the extent to whch the dstances are weghted Wth a greater bandwdth the smoothng ncreases Then regons and j get a smaller (greater) weght w j, f they are far from (close to) each other The bandwdth s computed by crossvaldaton or mnmsng the Akake nformaton crteron (Fotherngham/Brunsdon/Charlton 000, pp 56; Fotherngham/Charlton/ Brunsdon 1998, p 1910) The regresson coeffcents are estmated by weghted least squares (WLS) The values of the ndependent varables from regons, whch are nearer to regon, have a greater nfluence, because they are multpled wth the weght matrx for regon W : ˆ 1 = (38) β ( X W X) X W y ˆβ s the GWR-estmator for the th regon: (39) β ˆ ( βˆ βˆ βˆ ) = 0 1 m and W a n by n dagonal matrx, whch s denoted by the weghts w j, j = 1,,, n: w1 0 0 (40) 0 w 0 W = 0 0 wn However, one should test, f the GWR-model s approprate The global test of nonstatonarty compares a regresson of y on X wth sum of squared resduals to a geographcally weghted regresson The extra complexty of varyng regresson coeffcents s worthwhle only, f the GWR-model supples a smaller resdual sum of

12 Regresson model and data sources 11 squares n comparson to the OLS-estmaton The sum of squared resduals from the OLS-model can be expressed as: (41) u ˆ 0 u ˆ 0 = y R0 y wth (4) R0 = ( I S0 ) ( I S0 ) and 1 0 S = (43) X ( X X) X S 0 s called OLS-smoothng operator, because t transfers or "smoothes" the observed values y to the expected values ŷ : (44) yˆ = S 0 y The th row of the GWR-smoothng operator S 1 s gven by 1 (45) r = ( X W X) X W y Lettng R 1 be a quadratc matrx computed wth the GWR-smoothng operator: (46) R1 = ( I S1) ( I S1), the GWR-resduals may be wrtten usng the quadratc form of ths matrx (47) u ˆ 1 u ˆ1 = y R1 y If we assume y has a normal dstrbuton, the rato ( uˆ ˆ ˆ ˆ ) v (48) F 0 u0 u1 u = 1, ( uˆ 1 uˆ 1) w where v denotes the trace of R1 R0 and w the trace of R 1, s approxmatve F- dstrbuted (Brunsdon/Fotherngham/Charlton 1999, pp 501) If the null hypothess of statonarty s rejected, the GWR-model s approprate Besde the nonstatonarty of all regresson coeffcents one can check, f one parameter s nonstatonary The test s based on a Monte Carlo smulaton (for detals see Fotherngham/Brunsdon/Charlton 000, pp 56) If the null hypothess of statonarty s rejected for some but not all parameters, a mxed GWR-model could be approprate (Fotherngham/Brunsdon/Charlton 000, pp 65; Me/He/Fang 004) If the global test of nonstatonarty suggests usng a geographcally weghted regresson model and the Monte Carlo smulaton s not sgnfcant for all coeffcents, one should also calculate a GWR-approach 3 Sources of data We estmate an absolute and a condtonal convergence model for Germany [s formulas (3) and (33)] As spatal unts we do not use the admnstratve unts (Krese) A regresson analyss wth admnstratve unts can provoke spatal autocorrelaton (Kelbach 000, pp 10 and Dörng 005, p 100), whch s strengthened by suburbanzaton tendences (Kühn 001, Kaltenbrunner 003, Motzkus 001, pp 196 and Schönert 003) Ths spatal autocorrelaton would cause an neffcency of the geographcally weghted regresson Instead our analyss s based on labour market regons, whch Eckey defned by commuter flows (Eckey/Horn/Klemmer 1990; Eckey 001) Ths demarcaton worked satsfactorly n dfferent studes (s for example Kosfeld/Laurdsen 004, Kos-

13 Regresson model and data sources 1 feld/eckey/dreger 005, Eckey/Kosfeld/Türck 005 and Eckey/Kosfeld 005) The offcal data on the bass of admnstratve unts (Krese) can be aggregated to labour market regons We use the offcal data to estmate the convergence models, whch cover the perod between 1995 and 00 (Statstsche Ämter des Bundes und der Länder 003 and 004) We focus on the labour productvty (Y/L), whch s measured by gross value added per employee, and the gross value added per capta (Y/N) The condtonal model contans addtonal varables The nvestment rate n physcal captal s k s gven by gross nvestments n physcal captal dvded by gross value (s Fg 1 a) Human captal covers the labour force wth a degree of an upper school provdng vocatonal educaton (tertary educaton), a unversty of appled scences or a unversty We use the ntal values to prevent an endogenety bas (s Temple 1999, pp 18) The growth of labour force n s gven by the offcal statstcs The deprecaton rate can be computed usng the gross nvestments and the physcal captal stock 1 In many studes a constant rate of technologcal progress s used for all regons (see for example Islam 1995, p 1139 and Kosfeld/Eckey/Dreger 005, pp 198) We estmate g wth a panel GWR-approach of the producton functon: (49) f ( dummy West / Ost,dummyfor g,l,k,h) Y = Fg 1: Investment rate and technologcal progress Rostock Rostock Hamburg Hamburg Bremen Bremen Hanover Berln Hanover Berln Lepzg Lepzg Cologne Cologne Saarbrücken Frankfurt/Man Nuremberg <= 015 <= 00 <= 05 <= 035 <= 094 Saarbrücken Frankfurt/Man Nuremberg <= 0007 <= 0010 <= 001 <= 0014 <= 003 Stuttgart Stuttgart Munch Munch a) nvestment rate n physcal captal b) rate of technologcal progress g The dummy varable for estmatng g s 1, f the values of 00 are used The regresson coeffcent belongng to ths dummy yelds the growth rate over the whole perod Fg 1 b provdes a vsual mpresson of the spatal structure The rate of technologcal progress s hgh n some regons around Munch and Stuttgart as well as n East 1 The regonal captal stock s not denoted by the offcal statstcs We use therefore an estmaton, whch s descrbed n the appendx

14 Emprcal evdence on convergence 13 Germany and low n the ndustrally shaped Ruhr dstrct and Saarland The average growth rate for Germany corresponds wth the estmaton of Grömlng (004) 3 Table 1: Descrptve statstcs Varable Mean Standard devaton Mnmum Maxmum Gross value (Y) Gross value (Y) Populaton (N) Populaton (N) Labour force (L) Labour force (L) Human captal (H) Physcal captal (K) Physcal captal (K) Investment rate n physcal captal ( s k ) Growth rate of labour force (n) Growth rate of technologcal progress (g) Rate of deprecaton (δ) Emprcal evdence on convergence 41 Absolute convergence At frst we estmate an absolute convergence model of the neoclasscal growth theory On the one hand the annual labour productvty s used, whch s measured by gross value added per employee The average growth of labour productvty s explaned by the ntal labour productvty level The GWR-equlbrum of ths model can be expressed as [s formulas (33) and (36)]: ln y,00 ln y,1995 (50) = β0 + β1 ln y, u 7 The results of the calculatons are lsted n Table The coeffcent of determnaton (global OLS-estmaton of the equlbrum ln y,00 ln y,1995 (51) = β0 + β1 ln y, u ), 7 R 3 In addton we estmated the rate of technologcal progress wth the Solow resdual (Barro/Sala-- Martn 004, pp 434) The regonal results of ths approach, whch s usually used (see for example Grömlng 001 and Grömlng 004), do not convey a bg deal, because the coeffcent g s negatve n about 10 % of the regons Our estmaton averages out 11 %

15 Emprcal evdence on convergence 14 yelds a value of 337 % Ths proporton of explaned varaton s sgnfcant The regresson coeffcents have the expected sgn We obtan a level of technology n the base perod 1995, whch s expressed by the ntercept, of 0137 The negatve coeffcent of the ntal labour productvty level confrms a convergence of German regons Regons, whch have a low labour productvty, grow faster than regons wth a hgh labour productvty The parameter β 1 s lnked to the speed of convergence λ by the followng relatonshp (s Barro/Sala--Martn 004, p 46): λ (5) β 1 = (1 e ) The speed of convergence n the global OLS-model, (53) λ = ln( 1 β1 ) = ln[ 1 ( 003) ] = ln(103) = 0031[ = ˆ 31 %], shows a qute fast declne n regonal dspartes A 31 % convergence rate mples about a: ln(1/ ) ln(1/ ) (54) t = = = λ 0,031 year half lfe of the convergence process Table : Absolute convergence of the labour productvty Coeffcent Mnmum Lower Quartle Medan Upper Quartle Maxmum Global OLS β 0 or β ** β 1 or β ** R or R ** Bandwdth = 1419; Global test of nonstatonarty: F = 7646** Notes: R : coeffcent of determnaton; R : local coeffcent of determnaton; F: emprcal F-value; **: sgnfcant at the 1 % level; *: sgnfcant at the 5 % level; ( * ) : sgnfcant at the 10 % level Because the null hypothess of the global test of nonstatonarty [s formula (48)] s rejected, we estmate a GWR-model, too 4 The regresson coeffcents vary remarkable, but the sgns are all the same Thus the results can be nterpreted well The ntercept s always postve and t shows the dfferent extent of usng technology The slope has a negatve sgn, so German labour regons are convergng concernng the labour productvty The convergence speed covers the range between 07 % and 45 % Fg a shows the dstrbuton of the average labour productvty growth n the perod from 1995 to 00 and the half lfe across German labour regons Especally regons n the former GDR and n Bavara have comparably hgh growth rates The values ncrease from the west to the east The subsdes n the former GDR favoured nvestments n captal ntensve branches (Quehenberger 000, pp 1-13) Ths process caused a labour-savng technologcal progress and a hgh growth n labour productvty The half lfe of the convergence process vares n German labour regons (s Fg b) Its value ncreases from the north to the south Regons n south Bavara and Baden- 4 In addton the nonstatonarty of the two regresson coeffcents s checked by Monte Carlo smulaton (the p-values are smaller than 001) These tests confrm the result of the global test of nonstatonarty

16 Emprcal evdence on convergence 15 Württemberg as well as n Saarland need more than ffty years to acheve half of the rse n labour productvty to ther steady-state, whle ths value les n Northern Germany at less than 0 years Fg : Average growth of labour productvty and half lfe of the convergence process Rostock Rostock Hamburg Hamburg Bremen Bremen Hanover Berln Hanover Berln Lepzg Lepzg Cologne Cologne Saarbrücken Frankfurt/Man Nuremberg <= 0010 <= 0015 <= 000 <= 0030 <= 0058 Saarbrücken Frankfurt/Man Nuremberg <= 0 <= 5 <= 35 <= 50 <= 99 Stuttgart Stuttgart Munch Munch a) Labour productvty growth b) Half lfe Fgs 3 a and 3 b provde a vsual mpresson of the spatal structure of the half lfe n combnaton wth the labour productvty and the labour productvty growth The regons n the former GDR have a low labour productvty and a short half lfe Ther steady-state of the labour productvty wll probably not reach the value from most regons n West Germany, because ther relatve hgh growth n the md 1990s s declnng Some regons n the south of Bavara and near Stuttgart have a hgh fnal labour productvty, an above average growth of ths varable and a long half lfe They wll be the most prosperous regons of Germany on a long-term bass

17 Emprcal evdence on convergence 16 Fg 3: Half lfe of the convergence process Rostock Rostock Hamburg Hamburg Bremen Bremen Hanover Berln Hanover Berln Lepzg Lepzg Cologne Frankfurt/Man Labour productvty >45 Cologne Frankfurt/Man Average growth >00 Saarbrücken Stuttgart Nuremberg <=45 <=30 >30 Half lfe Saarbrücken Stuttgart Nuremberg <=00 <=30 >30 Half lfe Munch Munch a) Half lfe and labour productvty 00 b) Half lfe and labour productvty growth In addton we estmate an absolute convergence model of the labour productvty usng the manufacturng and the servce sector The estmaton results are reported n the appendx The half lfe of the manufacturng sector exceeds the correspondng value of the servce sector (s Fg ) Note that the spatal pattern of both sectors s dfferent, too Many regons, whch have a long half lfe n one sector, wll converge qute fast n the other sector On an aggregated level ths dfference wll compensate each other Fg 4: Half lfe of the convergence process (dfferent sectors) Rostock Rostock Hamburg Hamburg Bremen Bremen Hanover Berln Hanover Berln Lepzg Lepzg Cologne Cologne Saarbrücken Frankfurt/Man Nuremberg <= 0 <= 5 <= 35 <= 50 <= 435 Saarbrücken Frankfurt/Man Nuremberg <= 1 <= 14 <= 15 <= 18 <= 9 Stuttgart Stuttgart Munch Munch a) Manufacturng sector b) Servce sector

18 Emprcal evdence on convergence 17 Besde the labour productvty the gross value added (GVA) per capta s an often used ndcator to measure absolute convergence In equaton (50) y now stands for ths varable The results of the calculatons are shown n Table 3 The global OLSestmaton yelds a R-square of only 0088 Thus the ft of the model s not partcularly good However, the F-test ndcates a sgnfcant proporton of explaned varance Both regresson coeffcents have the expected sgn The speed of convergence s wth a value of 1 % much slower than n the absolute convergence model wth the labour productvty Ths result corresponds to studes of Spansh (Tortosa-Ausna et al 005) and German (Kosfeld/Eckey/Dreger 005) regons Table 3: Absolute convergence of the gross value added per capta Coeffcent Mnmum Lower Quartle Medan Upper Quartle Maxmum Global OLS β 0 or β ** β 1 or β ** R or R ** Bandwdth = 1640; Global test of nonstatonarty: F = 4680** Notes: R : coeffcent of determnaton; R : local coeffcent of determnaton; F: emprcal F-value; **: sgnfcant at the 1 % level; *: sgnfcant at the 5 % level; ( * ) : sgnfcant at the 10 % level In addton to the global OLS-model a GWR-approach s calculated, because a global nonstatonarty s proved (s last row n Table 3) 5 The ntercept exceeds zero n all regons, but there s a changng sgn of the slope Ths result s also reported n the analyss of Bwang and Brunstad (005, p 19), who use EU-regons The regons, whch have a postve slope, do not reach ther steady-state, but they dverge Fg 5: Average growth of GVA per capta and half lfe of the convergence process Hamburg Rostock Hamburg Rostock Bremen Bremen Hanover Berln Hanover Berln Cologne Lepzg Cologne Lepzg Frankfurt/Man Saarbrücken Nuremberg <= 0010 <= 0015 <= 000 <= 0030 <= 0057 Frankfurt/Man Saarbrücken Nuremberg <0 < 40 < 60 < 150 <5550 Stuttgart Stuttgart Munch Munch a) Growth of GVA per capta b) Half lfe 5 Monte Carlo smulatons provde evdence for nonstatonarty of both coeffcents, too

19 Emprcal evdence on convergence 18 The dvergng regons have a negatve convergence speed They are located at the Western border (s Fg 5 b) These regons are surrounded by dstrcts, whch are convergng slowly The comparson wth Fg b shows that the dvergng regons have n the other model partly the hghest half lfe value But there are smulartes between the two models The half lfe of the convergence process ncreases from southwest to northeast 4 Condtonal convergence The condtonal model dffers from the model of the absolute convergence by the fact that control varables are ncluded We use a model, whch was conducted by Mankw, Romer and Wel (199) They added the human captal, whch s stressed n the endogenous-growth theory (s for example Lucas 1988, Grossmann/Helpman 1989 and as an overvew Frenkel/Hemmer 1999, pp 00), to a Solow model The equaton of the labour productvty growth model wth locally dfferent regresson coeffcents s gven by [s formulas (3) and (36)]: ln y,00 ln y,1995 (55) = β + β ln y + β ln[ lns ln(n + g + δ )] 0 1, k + β 3 ln h + u, where y, 00 represents the labour productvty 00 n regon and y, 1995 the same quantty n 1995 and all other varables are denoted as before The global test of nonstatonarty suggests usng a geographcally weghted regresson model 6 The nfluence of the control varables s qute small In the global OLS-estmaton the coeffcent of human captal s not sgnfcant at the 10 % level In the GWR a sgnfcance test of the local parameters s not computed, but the coeffcents le all n the proxmty of zero The regresson coeffcent of the nvestment rate and the growth rate of labour force and technologcal progress as well as the rate of deprecaton rate s sgnfcant at the 5 % level So the local coeffcents of determnaton are only slghtly hgher than n the model of the absolute convergence, although we use substantally more varables Table 4: Condtonal convergence of the labour productvty Coeffcent Mnmum Lower Quartle Medan Upper Quartle Maxmum Global OLS β 0 or β ** β 1 or β ** β or β * β 3 or 3 R or β R ** Bandwdth = 49; Global test of nonstatonarty: F = 6090** Notes: R : coeffcent of determnaton; R : local coeffcent of determnaton; F: emprcal F-value; **: sgnfcant at the 1 % level; *: sgnfcant at the 5 % level; ( * ) : sgnfcant at the 10 % level 6 The Monte Carlo smulaton does not reject the null hypothess of statonarty for all regresson coeffcents n both condtonal convergence models However, the local determnaton coeffcents are hgher n the GWR-model, so ths model s more approprate

20 Emprcal evdence on convergence 19 The GWR-parameters of the ntal labour productvty le n the range between and The negatve sgns confrm the result of the absolute convergence model wth the labour productvty as dependent varable that all regons are convergng The parameters ndcate a speed of convergence, whch dsperse less than the coeffcents n the model of absolute convergence Fg 6 a shows the spatal structure of half lfe, whch s calculated usng the speed of convergence The half lfe ncreases from northeast to southwest Some regons at the east border of Saxony and Bavara wll need more than 35 years to acheve half of the rse n labour productvty to ther steady-state value Fg 6 b gves a vsual mpresson of the half lfe n combnaton wth the labour productvty n 00 The whte shaped regons have a small labour productvty and a short half lfe They are located perpherally n the Harz, n the north of the former GDR and between Cologne and Saarbrücken In contrast to the models of absolute convergence many regons of East Germany exhbt an above average half lfe Most regons n Bavara and n Baden- Württemberg have above average values of labour productvty and half lfe Fg 6: Half lfe of the convergence process (condtonal model of labour productvty) Rostock Rostock Hamburg Hamburg Bremen Bremen Hanover Berln Hanover Berln Lepzg Lepzg Cologne Frankfurt/Man Saarbrücken Stuttgart Nuremberg <= 4 <= 6 <= 8 <= 35 <= 55 Cologne Frankfurt/Man Saarbrücken Stuttgart Nuremberg Labour productvty >45 <=45 <=7 >7 Half lfe Munch Munch a) Half lfe b) Half lfe and labour productvty 00 In addton we estmate the model of condtonal convergence of the gross value added per capta The global coeffcent of determnaton s sgnfcant but qute small Only 10 % of the varance of the gross value added per capta s explaned by the three ndependent varables The nfluence of the control varables s agan small The regresson coeffcent of human captal yelds no sgnfcant nfluence

21 Dscusson and concluson 0 Table 5: Condtonal convergence of the gross value added per capta Coeffcent Mnmum Lower Quartle Medan Upper Quartle Maxmum Global OLS β 0 or β ** β 1 or β ** β or β β 3 or 3 R or β ( * ) R ** Bandwdth = 117; Global test of nonstatonarty: F = 408** Notes: R : coeffcent of determnaton; R : local coeffcent of determnaton; F: emprcal F-value; **: sgnfcant at the 1 % level; *: sgnfcant at the 5 % level; ( * ) : sgnfcant at the 10 % level In contrast to the absolute convergence model (s Table 3) all local regresson coeffcent of the ntal gross value added per capta have a negatve sgn and ndcate a convergence A short half lfe characterzes the regons n the northern parts of Germany The dvergng regons n the absolute convergence model (s Fg 5 b) have the hghest half lfe value (s Fg 7 a) The comparson between Fg 6 and Fg 7 shows that n the frst mentoned model most regons n East Germany have a half lfe above average, whle t s pronounced below average n the other model Fg 7: Half lfe of the convergence process (condtonal model of gross value added per capta) Rostock Rostock Hamburg Hamburg Bremen Bremen Hanover Berln Hanover Berln Lepzg Lepzg Cologne Frankfurt/Man Saarbrücken Stuttgart Nuremberg <40 < 45 < 50 < 65 <737 Cologne Frankfurt/Man Saarbrücken Stuttgart Nuremberg GVA per capta >0 <=0 <=50 >50 Half lfe Munch Munch a) Half lfe b) Half lfe and GVA per capta 00 5 Dscusson and concluson The assumpton of statonarty cannot be founded theoretcally for most research questons The behavour and atttudes of people as well as the nfrastructure vary across

22 Dscusson and concluson 1 regons That wll cause locally dfferent parameters, whch s neglected by a global approach In addton a global estmaton may lead to a bas and provoke autocorrelaton To that extent the geographcally weghted regresson represents an mportant extenson of spatal econometrcs The geographcally weghted regresson procedure s appled to a convergence model of German labour market regons The estmaton yelds dfferent speeds of convergence of the regons In partcular t showed up that Bavaran regons have a long and north German dstrcts a short half lfe The approach provdes evdence that the south German regons wth a hgh labour productvty and a small unemployment rate wll be the most prosperous regons n Germany On the bass of the economc development n the long-run there wll be a gap between north and south Germany One model even shows dvergence tendences of some perpheral regons The substantally varyng coeffcents show that a global convergence model, whch was estmated by many researchers (see for example Kosfeld/Laurdson 004, Funke/Nebuhr 005 a, Funke/Nebuhr 005 b, Kosfeld/Eckey/Dreger 005), mght be nproved by a geographcally weghted regresson approach Our paper represents the frst step of a local analyss of convergence processes n Germany

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