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 Regonal convergence of German labour markets represents a poltcally mportant queston. Dfferent studes have examned convergence processes n Germany. We derve equatons to estmate the speed of convergence on the bass of an extended Solow model. The technque of geographcally weghted regresson permts a detaled analyss of convergence processes, whch has not been conducted for Germany so far yet. It allows to estmate a separate speed of convergence for every regon resultng from the local coeffcents of the regresson equatons. The applcaton of ths technque to German labour market regons shows regons movng wth a dfferent speeds towards ther steady states. The half-lve tmes n the model of condtonal convergence dsperse less than the same coeffcents n the absolute convergence model. Moreover, the speed of convergence s substantally slower n the manufacturng sector than n the servce sector. JEL C1, R11, R58 Keywords: Regonal Convergence, Spatal Econometrcs, Geographcally Weghted Regresson * We are grateful to the referees for ther helpful comments. Prof. Dr. Hans-Fredrch Eckey, Economcs Department, Unversty of Kassel, Nora-Platel-Str. 4, D Kassel, Germany. Telephone: /3045, Telefax: /3045, e-mal: eckey@wrtschaft.un-kassel.de. Prof. Dr. Renhold Kosfeld, Economcs Department, Unversty of Kassel, Nora-Platel-Str. 4, D Kassel, Germany. Telephone: /3084, telefax: /3045, e-mal: rkosfeld@wrtschaft.un-kassel.de. Matthas Türck, Economcs Department, Unversty of Kassel, Nora-Platel-Str. 4, D Kassel, Germany. Telephone: /3044, telefax: /3045, e-mal: tuerck@wrtschaft.un-kassel.de.

3 Introducton 1. Introducton 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, passed a law n 004 (33. Rahmenplan der Gemenschaftsaufgabe "Verbesserung der regonalen Wrtschaftsstruktur") about regonal polcy. In that law regonal equalzaton s explctly mentoned (cf. 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 have dealt wth the ssue of convergence and dvergence. On the one hand, researchers provde surveys on the 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 the extent to whch the poor 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, such as 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 has been 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 and West Germany. Before unfcaton there were dfferent ways of mgraton. Whle n the GDR there were tendences towards urbansaton, n West Germany there was deconcentraton. The 1990s are represented by a convergent development towards a suburbansaton n both parts of Germany. At the end of the decade there were further tendences towards dvergence n the nternal mgraton. 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) does not 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 & Blen (004) supply an accurate analyss of the dstrbuton of employment. The shft share regresson of West German admnstratve dstrcts yelds a negatve effect of regonal wages on employment growth. Employment growth dffers, especally suburban regons gan jobs from the core ctes. The study of Görzg & Gorng & Werwatz (005) shows that the catchng-up process of East German

4 Introducton 3 wages has slowed down durng the last ten years. Bayer & Juessen (004) conduct a tme seres approach to fnd out, whether the 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 & 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 as mportant factors. Kosfeld & Eckey & Dreger (006) estmate a convergence rate of 7.6 % across German regonal labour markets. On the bass of an extended Solow model they predct 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 & 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 regonal dvergence n the long run. Some researchers use regresson models wth ncome or GDP per capta as the dependent varable. Herz & Röger (1995) fnd a half-lfe tme of 16 years for 75 West German "Raumordnungsregonen" usng the perod Bohl (1998) dentfes tendences towards regonal dvergence, because the null hypothess of the unt root test for panel data (Levn Ln test) s not rejected. But the result s lmted by the fact that there can be found statonarty n some federal states. Kosfeld & 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 & Nebuhr (005a) use regresson models n order to explan economc growth from West Germany's plannng regons. Because the estmatons do not ft well, they provde a kernel approach to cluster the regons. They fnd three convergence clubs, whch have dfferent growth equlbrums. Funke & Nebuhr (005b) provde nsghts of β-convergence n West Germany's plannng regons. For the perod of 1976 to 1996 a slow rate of convergence s detected. Kosfeld & Eckey & Dreger (006) study 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 β-convergence Barro & Sala--Martn (1991, pp. 11) ntroduced 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 & Dreger (006) fnd dmnshng varances of ncome per capta and labour productvty n German labour regons, so the hypothess of σ-convergence s confrmed.

5 Growth model 4 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 the technque of geographcally weghted regresson, whch s developed by Brunsdon, Charlton and Fotherngham (see for example Brunsdon & Fotherngham & Charlton, 1998, p. 957; Fotherngham & Brunsdon & Charlton, 000 and Fotherngham & Brunsdon & Charlton, 1998). Only one convergence study uses ths model. Bvand & Brunstad (005) estmate a geographcally weghted regresson of Western Europe. Ther coeffcents have changng sgns. They fnd convergence of some regons and dvergence of others. However, a model, whch uses dfferent regresson coeffcents for German regons, has not been estmated untl now. In addton most models of condtonal β-convergence, whch use the approach of Mankw & Romer & 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. In secton we present a neoclasscal growth model, whch augments the Solow model by human captal. Secton 3 outlnes the geographcally weghted regresson. In partcular, 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.. Growth model We use a model that has been suggested by Mankw & Romer & Wel (199). They add 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 the Solow model. Human captal s mportant, because ths producton factor explans tendences towards convergence n East Germany at the begnnng of the 1990s (Barrel & Velde, 000). The producton functon of type Cob-Douglas n perod t s gven by: β t α 1 α β (1) ( ) Y t = K t H At Lt, 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). Dvdng the producton functon (1) by At Lt yelds the equaton: α β () y t = kt h t, where the lower cases stand for quanttes per effectve unt of labour,. e. yt = Yt ( At Lt ), kt = Kt ( At Lt ) and ht = Ht ( At Lt ). The steady state of output per effectve can derved from ths producton functon (s. Romer, 1996, p. 133 and Mankw & Romer & Wel, 199, pp. 416): β (3) = s 1 α y * ln k h 1 α n + g + δ α wth k s as savng rate of captal, n as growth rate of labour L, g as rate of technologcal progress and δ as deprecaton rate. Barro & Sala--Martn (004, p. 61) have shown

6 Growth model 5 that approxmatng equatons () and (3) around the steady state yelds, f one takes a Taylor seres expanson: λt λt α (4) ln yt ln y0 = (1 e ) ln y0 + (1 e ) lnsk 1 α β + (1 e λt ) 1 β α β lns h (1 e λt ) 1 α + β α β ln (n + g + δ). 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 (4) studes condtonal convergence. Ths s a convergence process, where poorer regons grow faster than rcher regons after controllng for relevant varables. Then the regresson coeffcent for the startng level of output wll be sgnfcantly 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 formula (4): λt λt α (5) ln y ln y = (1 e ) ln y + (1 e ) lnsk t 0 + (1 e λt ) 1 0 β α ln h * (1 e 1 α λt ) 1 α α ln (n + g + δ). 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 (6) ln yt ln y0 = (1 e ) ln y0. In emprcal analyses quanttes per capta and not per effectve unt of labour are usually used, because the level of technology A s unknown. The equatons (5) and (6) are gven n unts per capta as follows (s. Hemmer & Lorenz, 004, p. 49 and Temple, 1999, p. 1): Y Y Y (7) 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 (8) 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 (5) of absolute equal coeffcents for ln sk and ln (n + g + δ) leads to the equaton:

7 Growth model 6 (9) Y Y Y ln t ln 0 t t (1 e ) ln 0 α λ λ = + (1 e ) k L t L 0 L 0 1 α * λt β H λt + ( 1 e ) ln + (1 e ) ln A0 + gt. 1 α L [ lns ln (n + g + δ) ] However, the nvestments n human captal are dffcult to measure, because the foregone labour earnngs can hardly be fgured out. Mankw & Romer & 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 & Thompson (1999, pp ). They argue that the ndcator from Mankw & Romer & Wel (199) wll underestmate human captal n poor countres and overestmate t n rch countres. Dnopoulus & Thompson (1999) pont out further reasons for the nadequacy 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 practce. 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 & Wel (199, p. 411) gt s a constant, because the perod t s fxed [s. formula (9)]. 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: (10) ln A 0 = α + ε. Ther assumpton of noncorrelaton between ths error term and the ndependent varables seem not to be convncng (Islam, 1995, p.1134 and Klenow & Rodrguez- Clare, 1997). The technologcal growth depends rather on nsttutonal characterstcs and endowments, whch dffer across regons (Gundlach, 005, pp. 553). So we measure gt and A 0 by locally specfc constants. However, there s emprcal evdence for threshold values of regonal convergence or dfferent regonal parameters of β- convergence (s. for example Bvand & Brunstad, 005; Funke & Nebuhr, 005a; Juessen, 005; Huang, 005). Thus we allow also regonally dfferent values for all other parameters to prevent nconsstent estmators (Temple, 1999, p. 16). In addton we do not use the unconvncng assumpton of the same growth rate of technologcal progress and rate of deprecaton n all regons (see Temple, 1999, p. 1, Kosfeld & Eckey & Dreger 006). All regresson coeffcents, especally the rate of convergence, are estmated separately for all regons. Ths model of condtonal convergence s gven by:

8 Regresson model and data sources 7 (11) 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 (1) 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 3.1 Geographcally weghted regresson The nfluence between a dependent varable Y and some ndependent varables X k dffers often across regons (spatal nonstatonarty). Therefore our regresson models consst of locally dfferent parameters [cf. formulas (11) and (1)]. We use a geographcally weghted regresson (GWR) whch has been developed by Brunsdon, Charlton and Fotherngham n the past ten years (see for example Brunsdon & Fotherngham & Charlton, 1998, p. 957; Fotherngham & Brunsdon & Charlton, 000 and Fotherngham & Brunsdon & Charlton, 1998). The global regresson model wthout takng nto consderaton a spatal dependence s wrtten by the form (13) 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 (13) s gven by m (14) y = β + β x + u 0 k k= 1 k wth y as vector of the dependent varable, x k as vector of the kth ndependent varable and u as vector of the error term. In geographcally weghted regresson the global regresson coeffcents n (13) are replaced by local parameters: (15) y = β0 + βk xk + u, m 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 (13) or (14) respectvely holds (cf. Brunsdon & Fotherngham & Charlton, 1996, pp. 8 and Fotherngham & Charlton & Brunsdon, 1997, pp. 6). In model (1) 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 (11) dffers only regardng X from model (1). The matrx X contans two further columns, frst [ ln sk ln (n + g + δ) ], second ln ( H L).

9 Regresson model and data sources 8 Thus the global model s a specal case of the GWR functon. For every regon separate parameters are estmated, whch s an advantage over the spatal error and the spatal lag model (cf. 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 (15) 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 two regons d j to calculate the weghts (Gaussan weghtng functon): (16) 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 W for regon : ˆ 1. ˆβ s the GWR estmator for the th regon: (17) β = ( X W X) X W y (18) ˆ ( ˆ ˆ ˆ ) β = β0 β1 β m and W a n by n dagonal matrx, whch s denoted by the weghts w j, j = 1,,, n: w1 0 0 (19) 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 squares n comparson to the OLS estmaton. The sum of squared resduals from the OLS model can be expressed as: (0) u ˆ 0 u ˆ 0 = y R0 y wth (1) R0 = ( I S0 ) ( I S0 ) and 1 0. S () = X ( X X) X S 0 s called OLS smoothng operator, because t transfers or "smooths" the observed values y to the expected values ŷ : (3) yˆ = S 0 y.

10 Regresson model and data sources 9 The th row of the GWR smoothng operator S 1 s gven by 1 (4) s x = ( X W X) X W. Lettng R 1 be a quadratc matrx computed wth the GWR smoothng operator: (5) R1 = ( I S1) ( I S1), the GWR resduals may be wrtten usng the quadratc form of ths matrx (6) u ˆ 1 u ˆ1 = y R1 y. If we assume y has a normal dstrbuton, the rato ( uˆ ˆ ˆ ˆ ) v (7) 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 use a GWR approach. 3. Sources of data We estmate an absolute and a condtonal convergence model for Germany [cf. formulas (11) and (1)]. As spatal unts we do not use 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 180 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, Kosfeld & Eckey & Dreger, 006; Eckey & Kosfeld & Türck, 005 and Eckey & Kosfeld, 005). The offcal data on the bass of 440 admnstratve unts (Krese) can be aggregated to labour market regons.

11 Regresson model and data sources 10 Rostock Rostock Hamburg Hamburg Bremen Bremen Hanover Berln Hanover Berln Lepzg Lepzg Cologne Cologne Saarbrücken Frankfurt/Man Nuremberg <= 0.15 <= 0.0 <= 0.5 <= 0.35 <= 0.94 Saarbrücken Frankfurt/Man Nuremberg <= <= <= 0.01 <= <= 0.03 Stuttgart Stuttgart Munch Munch a) Investment rate n physcal captal b) Rate of technologcal progress g Fgure 1. Investment rate and technologcal progress per annum ( ) Offcal data are used 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 labour productvty (Y/L) whch s measured by gross value added per employee. 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 added [s. Fgure 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 1 The regonal captal stock s not denoted by the offcal statstcs. We use therefore an estmaton, whch s descrbed n the appendx.

12 Emprcal evdence on convergence 11 Table 1. Varable Descrptve statstcs Mean Standard devaton Mnmum Maxmum Gross value (Y) Gross value (Y) Labour force (L) Labour force (L) Human captal (H) Physcal captal (K) Physcal captal (K) Investment rate n physcal captal ( s k ) per annum ( ) Growth rate of labour force (n) per annum ( ) Growth rate of technologcal progress (g) per annum ( ) Rate of deprecaton (δ) per annum ( ) In many studes a constant rate of technologcal progress s used for all regons (see for example Islam, 1995, p. 1139; Barro, 1999, p. 1 and Kosfeld & Eckey & Dreger, 006, pp. 759), whch s not realstc (Gundlach, 005, pp. 553). We estmate g wth a panel GWR approach of the producton functon: (8) f ( dummy West / East,dummyfor g,l,k,h) Y =. 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. Fgure 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 Germany and low n the ndustrally shaped Ruhr dstrct and Saarland. The average growth rate for Germany corresponds wth the estmaton of Grömlng (004) Emprcal evdence on convergence 4.1 Absolute convergence At frst we estmate an absolute convergence model of the neoclasscal growth theory for two reasons. On the one hand, ths model, whch was developed by Barro and Sala-Martn (199 and 1991, pp. 11), s now a standard model. It s used by many 3 In addton we estmated the rate of technologcal progress wth the Solow resdual (Barro, 1999 and 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 great deal, because the coeffcent g s negatve n about 10 % of the regons. Our estmaton averages out 1.1 %.

13 Emprcal evdence on convergence 1 researchers (see for example Kosfeld & Eckey & Dreger, 006; Setz, 1995; Km, 003; Cuadrado-Roura, 001; Martn, 001; Fngleton, 003 and Gundlach, 003) but wth the excepton of Bvand & Brunstad (005) only for statonary regresson coeffcents. Ths assumpton of the same convergence rate of every regon s not realstc (Temple, 1999, pp. 16). On the other hand, the absolute convergence model permts a sectoral analyss of the convergence process, because sectoral data are not avalable for the control varables. The absolute convergence s estmated wth labour productvty, 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 (1) and (15)]: ln y,00 ln y,1995 (9) = β0 + β1 ln y, u. 7 The results of the calculatons are lsted n Table. The coeffcent of determnaton R (global OLS estmaton of the equlbrum ln y,00 ln y,1995 (30) = β0 + β1 ln y, u ) 7 yelds a value of 33.7 %. 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 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 [cf. formulas (9) and (30)]: 7λ (1 e ) (31) β1 =. 7 The speed of convergence n the global OLS model, 1 1 (3) λ = ln(1 + β1 7) = ln[1 + ( 0.03) 7] = 0.036[ = ˆ 3.6 %], 7 7 shows qute a fast declne n regonal dspartes. A 3.6 % convergence rate mples about a: ln(1/ ) ln(1/ ) (33) t = = = 19 λ 0,036 year half-lfe tme of the convergence process.

14 Emprcal evdence on convergence 13 Table. Coeffcent Absolute convergence of the labour productvty Mnmum Lower Quartle Medan Upper Quartle Maxmum Global OLS β 0 or β ** β 1 or β ** R or R ** AIC = ; Bandwdth = 1.419; Global test of nonstatonarty: F = 7.646** 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 (7)] 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 well nterpreted. 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 0.7 % and 5.5 %. Hamburg Rostock Hamburg Rostock Bremen Bremen Hanover Berln Hanover Berln Cologne Lepzg Cologne Lepzg Frankfurt/Man Saarbrücken Nuremberg <= <= <= 0.00 <= <= Frankfurt/Man Saarbrücken Nuremberg <= 0 <= 5 <= 35 <= 50 <= 96 Stuttgart Stuttgart Munch Munch a) Labour productvty growth b) Half-lfe tme Fgure. Average growth of labour productvty and half-lfe tme of the convergence process Fgure a) shows the dstrbuton of the average labour productvty growth n the perod from 1995 to 00 and the half-lfe tme across German labour regons. Especally regons n the former GDR and n Bavara have comparably hgh growth 4 In addton the nonstatonarty of the two regresson coeffcents s checked by Monte Carlo smulaton (the p-values are smaller than 0.01). These tests confrm the result of the global test of nonstatonarty.

15 Emprcal evdence on convergence 14 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 tme of the convergence process vares n German labour regons [s. Fgure b)]. Its value ncreases from the north to the south. Regons n south Bavara and Baden-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. Rostock Rostock Hamburg Hamburg Bremen Bremen Hanover Berln Hanover Berln Lepzg Lepzg Cologne Frankfurt/Man Labour productvty >45 Cologne Frankfurt/Man Average growth >0.0 Saarbrücken Stuttgart Nuremberg <=45 <=30 >30 Half-lfe tme Saarbrücken Stuttgart Nuremberg <=0.0 <=30 >30 Half-lfe tme Munch Munch a) Half-lfe tme and labour productvty b) Half-lfe tme and labour productvty 00 growth Fgure 3. Half-lfe tme of the convergence process Fgures 3 a) and 3 b) provde a vsual mpresson of the spatal structure of the half-lfe tme 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 tme. Ther steady state of the labour productvty wll probably not reach the value of 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 tme. They wll be the most prosperous regons of Germany on a long-term bass.

16 Emprcal evdence on convergence 15 We also estmate an absolute convergence model of the manufacturng sector and the servce sector. For both models the global test of nonstatonarty s sgnfcant. The explaned varance n the approach of the servce sector s much hgher than the same value of the servce sector model (cf. Table 3 and Table 4). Table 3. Coeffcent Absolute convergence of the labour productvty (manufacturng sector) Mnmum Lower Quartle Medan Upper Quartle Maxmum Global OLS β 0 or β ** β 1 or β ** R or R ** AIC = -5.0; Bandwdth = 1.567; Global test of nonstatonarty: F = 4.346** 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 Table 4. Coeffcent Absolute convergence of the labour productvty (servce sector) Mnmum Lower Quartle Medan Upper Quartle Maxmum Global OLS β 0 or β ** β 1 or β ** R or R ** AIC = ; Bandwdth = 1.51; Global test of nonstatonarty: F = 4.999** 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 The half-lfe tme of the manufacturng sector exceeds the correspondng value of the servce sector (s. Fgure 4). The reason s that most regons have smlar basc servces (Corrado & Martn & Weeks, 005, p. C145). Note that the spatal pattern of both sectors s dfferent, too. Many regons, whch have a long half-lfe tme n one sector, wll converge qute fast n the other sector. On an aggregated level ths dfference wll compensate each other.

17 Emprcal evdence on convergence 16 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 <= 8 <= 10 <= 1 <= 18 <= 6 Stuttgart Stuttgart Munch Munch a) Manufacturng sector b) Servce sector Fgure 4. Half-lfe tme of the convergence process (dfferent sectors) 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 & 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 [cf. formulas (11) and (15)]: ln y,00 ln y,1995 (34) = β + β ln y + β [ 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. 5 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 5 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.

18 Emprcal evdence on convergence 17 hgher than n the model of the absolute convergence, although we use substantally more varables. Table 5. Coeffcent Mnmum Condtonal convergence of the labour productvty 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 ** AIC = ; Bandwdth =.49; Global test of nonstatonarty: F = 6.090** 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 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 that all regons are convergng. The parameters ndcate a speed of convergence, whch dsperse less than the coeffcents n the model of absolute convergence. Rostock Rostock Hamburg Hamburg Bremen Bremen Hanover Berln Hanover Berln Cologne Frankfurt/Man Saarbrücken Stuttgart Nuremberg Lepzg <= <= 4 <= 8 <= 35 <= 55 Cologne Frankfurt/Man Saarbrücken Stuttgart Nuremberg Lepzg Labour productvty >45 <=45 <=7 >7 Half-lfe tme Munch Munch a) Half-lfe tme b) Half-lfe tme and labour productvty 00 Fgure 5. Half-lfe tme of the convergence process Fgure 5 a) shows the spatal structure of half-lfe tme, whch s calculated usng the speed of convergence. The half-lfe tme ncreases from northwest to southeast. 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. Fgure 5 b) gves a vsual mpresson of the half-lfe tme n combnaton wth the labour

19 Concluson 18 productvty n 00. The whte shaped regons have a small labour productvty and a short half-lfe tme. 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 tme. Most regons n Bavara and n Baden-Württemberg have above-average values of labour productvty and half-lfe tme. 5. 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 regons. That wll cause locally dfferent parameters, whch s gnored 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 technque of geographcally weghted regresson 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 that Bavaran regons have a long and north German dstrcts a short half-lfe tme. 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. The substantally varyng coeffcents show that a global convergence model, whch was estmated by many researchers (see for example Kosfeld & Laurdson, 004; Funke & Nebuhr, 005a; Funke & Nebuhr, 005b; Kosfeld & Eckey & Dreger, 006) mght be mproved 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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