A Nonparametrc and Semparametrc Analyss on the Inequaly-Development Relatonshp Ku-Wa L a, * and Xanbo Zhou b a Cy Unversy of Hong Kong and Unversy of Geneva b Lngnan College, Sun Yat-sen Unversy, Chna Abstract: Ths paper studes the ncome nequaly and economc development relatonshp by usng unbalanced panel data on OECD and non-oecd countres for the perod 1962-2003. Nonparametrc estmaton results show that ncome nequaly n OECD countres are almost on the backsde of the nverted-u, whle non-oecd countres are approxmately on the foresde, except that the relatonshp n both country groups shows an upturn at a hgh level of development. Development has an ndrect effect on nequaly through control varables, but the modes are dfferent n the two groups. Model specfcaton tests show that the relatonshp s not necessarly captured by the conventonal quadratc functon. Cubc and fourth-degree polynomals, respectvely, f the OECD and non-oecd country groups better. Our fndng s robust regardless whether the specfcaton uses control varables. Development plays a domnant role n mgatng nequaly. Keywords: Kuznets nverted-u; Nonparametrc and semparametrc models; Unbalanced panel data JEL classfcaton: C14; C33; O11. * Correspondng author: Cy Unversy of Hong Kong, Tel.: 852 3442 8805, E-mal: efkwl@cyu.edu.hk, and Unversy of Geneva, Tel.: +41 (0) 22 379 9596. 1
I Introducton There are dfferent forms of nequaly n human hstory, ncludng arstocratc, racal, sexual, relgous, polcal, socal and terroral nequales. Some nequales are rrevocable. Whle the Gn coeffcent shows an nter-personal comparson and provdes a statc snapshot measure of ncome nequaly, mprovement n ncome nequaly can often be made ntra-personally, as a person s ncome mproves through experence, skll, job dversy and personal endowment (L, 2002). Indeed, gven that modern socetes tran and educate people for employment wh dfferent rewards, ncome nequaly s nevable (Letwn, 1983). The relatonshp between ncome nequaly and economc development has been characterzed by the Kuznets nverted-u curve (Kuznets, 1955) whch argued that ncome nequaly tends to ncrease at an nal stage of development and then decrease as the economy develops, mplyng that ncome nequaly wll eventually fall as ncome contnues to rse n developng countres. Studes conducted along the lne of the nverted-u relatonshp nclude Sen (1991, 1992 and 1993) who dscussed nequaly through ndvdual capably and functonng. Some studes concentrate on the causes of ncome nequaly whch nclude human capal, technologcal advancement, job dversy and polcal stably, whle other studes examne the long run ncome nequaly convergence (Galor and Zere, 1993; Galor and Moav, 2000; Gould et al. 2001; Acemoglu, 2001; Desa et al. 2005; Bẻnabou, 1996; Ravallon, 2003). The Kuznets nverted-u relatonshp between nequaly and economc development has attracted both supporters and crcs. In partcular, whether the relatonshp s consdered as a law or can be mproved through approprate economc polces (Kanbur, 2000). Nevertheless, the Kuznets nverted-u relatonshp has not been fully confrmed 2
and valdated n studes wh parametrc quadratc models (L et al. 1998, Barro, 2000, Bulíř, 2001, Iradan, 2005). The mportance of the relatonshp and possble msspecfcaton of parametrc quadratc models have led to the use of nonparametrc models wh cross-secton data. For example, by usng nonparametrc estmaton based on a sample of cross-secton country data, Mushnsk (2001) showed that the quadratc parametrc form of the relatonshp between Gn coeffcent and real ncome per capa s msspecfed. Huang (2004) presented a flexble nonlnear framework for a cross-secton data of 75 countres and showed evdence of nonlneary n the nverted-u relaton between the Gn and per capa GDP. Ln et al. (2006) confrmed the valdy of the nverted-u relatonshp by presentng a semparametrc partally lnear nvestgaton wh some control varables usng the dataset n Huang (2004). In studyng the relatonshp between nequaly and development, the choce s whether or not some control varables can be ncluded n the regresson model. Some studes have compled wh the orgnal work of Kuznets and examned the total effect, nstead of the drect effect, of development on nequaly by usng uncondonal models (Mushnsk, 2001; Wan, 2002). In other words, only one regressor of development s used n the regresson model. Such a specfcaton consders the nequaly-development relatonshp as a law and mnmzes the mpact of economc polcy. Other studes consder the determnants of nequaly and examne the mpact of polcy n affectng nequaly, besdes development. That s, the regressors n the regresson model nclude other polcy varables and/or economc ndcators, besdes development (L et al. 1998; Bulíř, 2001; Wang, 2006; Huang et al. 2009). The emprcal results from these condonal models do reflect both the drect and ndrect (va control varables) effects of development on nequaly. 3
Ths paper presents a nonparametrc (whout control varables) and semparametrc (wh control varables) nvestgaton on the nequaly-development relatonshp by usng unbalanced panel data from the developed OECD (Organzaton for Economc Co-operaton and Development) and the developng non-oecd countres. The unbalanced panel data set provde observatons over several perods of tme. Such an analyss can ncorporate heterogeney across countres. In the panel data model specfcaton, we allow country-specfc effects to be fxed effects that are dependent on the regressors. Ths can help to obtan consstent estmators n the nonparametrc regresson functon when nequaly s regressed on the development varable and/or other control varables. We modfy the methodology n Henderson et al. (2008) to cater for the nonparametrc and semparametrc estmatons wh unbalanced panel data, and conduct data-drven specfcaton tests for the selected models. The emprcal results from uncondonal and condonal models show that the channel effects of development on nequaly va the control varables n both OECD and non-oecd countres are dfferent, and are dependent on the level of development n each country groupng. There s, however, much resemblance n the shapes of the nonparametrc functons from both nonparametrc and semparametrc estmatons n each country group, mplng that the control varables as a whole do not change the dynamc mode but the degree of nequaly. Development stll plays a domnant role n mgatng nequaly. For the Kuznets nverted-u hypothess, our fndngs support the cubc and fourth-degree, nstead of quadratc polynomals, for OECD and non-oecd countres, respectvely, n capturng the nonlneary suggested by the nonparametrc and semparametrc regressons. Secton 2 dscusses the data and model specfcaton and presents a parametrc study 4
on the nverted-u relatonshp. Secton 3 brefly generalzes the methodology to su the unbalanced panel data. Secton 4 conducts the nonparametrc and semparametrc estmatons and tests, whle Secton 5 concludes the paper. II Data and Model Specfcaton The Gn coeffcent for each country n the sample s used as the nequaly proxy, and the dataset s obtaned from All the Gns Database under the World Bank project Inequaly around the World. 1 Ths dataset represents a complaton and adaptaton of three datasets: the Dennger-Squre dataset that covers the perod 1960-1996, the WIDER dataset that covers the perod 1950-1998 and the World Income Dstrbuton dataset that covers the perod 1985-2000. The key varable chosen from the database s Gnall that gves the values of Gn coeffcents from household-based surveys for 1,067 country/years. The observatons are comparable n prncple. Ths study chooses the Gn coeffcents sample of country/years wh D =1, where the dummy varable D refers to the welfare concept of the Gn coeffcent ndcated eher by ncome (=1) or by consumpton (=0). To su the nonparametrc estmaton, we have selected the countres wh at least two years data. The fnal dataset used n ths study contans 401 observatons on 30 OECD countres and 303 observatons on 45 non-oecd countres for the perod from 1962 to 2003 (summarzed n the Appendx). We use real GDP per capa as the proxy for the level of economc development. Besdes studyng the total relatonshp between nequaly and development, we also study the drect effect of development on nequaly by controllng some other varables. 1 The web reference for the dataset s: http://web.worldbank.org/projects/nequaly. The book reference s Mlanovc (2005). Detals for the dataset descrpton of the December 2006 verson are provded n the web. Some recent years data are also gven. 5
We consder two knds of controls. The polcy control s ndcated by the varables of openness (ndcated by the percentage trade share of GDP n 2005 constant prces), urbanzaton (ndcated by the percentage of urban populaton n total populaton), and nvestment (ndcated by the percentage of nvestment share n real GDP per capa), denoted as openk, urbanze and k, respectvely. The other control varables of GDP growth and nflaton (ndcated by the annual percentage of GDP deflator) reflect the economc characterstcs of the sample country. These data are obtaned from the Penn World Table and World Development Indcators. Table 1 reports the basc statstcs of these varables for both OECD and non-oecd countres. One observaton s that non-oecd countres on average have a larger nequaly and varaton than OECD countres, whle OECD countres have hgher level of development wh more varatons than non-oecd countres. Table 1 Basc Statstcs Gn gdppc growth openk urbanze k deflator OECD mnmum 17.80 2,028.78-19.29 5.29 2.90 10.41-2.00 maxmum 58.00 63,419.40 12.49 264.14 67.60 48.83 208.00 mean 34.34 18,658.44 2.53 45.59 28.84 26.85 9.35 Standard devaton 7.70 8,009.19 3.80 33.88 11.70 6.07 14.53 Non-OECD mnmum 20.69 561.52-21.60 10.32 0.00 2.11-8.00 maxmum 63.66 33,401.80 16.47 399.22 94.94 56.14 4,107.00 mean 44.45 7,167.09 1.73 74.08 46.53 22.78 83.19 Standard devaton 10.45 5,723.07 5.55 61.44 23.53 9.59 376.19 Table 2 shows the correlaton statstcs between the varables. In OECD countres, the Gn has a negatve correlaton of -0.35 wh the logarhm of GDP per capa, whch s the hghest among s correlatons wh the other varables. In non-oecd countres, such correlaton s smaller but posve (0.13). However, ths only provdes the correlaton 6
between nequaly and development for lnear parametrc regresson models, but s nvald n the nonlnear and nonparametrc relatonshps. Inequaly and development are also moderately correlated wh other varables. For example, the correlatons of log GDP per capa wh openness and urbanzaton are larger when compared to the other varables. Both nequaly and level of development are moderately correlated wh openness n OECD countres and wh nvestment n non-oecd countres. These recprocal relatons mply that, n addon to the drect effect on nequaly, the level of development may have an ndrect effect on nequaly through other channels. 2 Table 2 Correlatons between Varables Gn log(gdppc) growth(-1) openk urbanze k nflaton OECD Gn 1.0000-0.3506 0.1994-0.3245-0.0789-0.1309 0.0394 log(gdppc) -0.3506 1.0000-0.0465 0.3689-0.4740 0.1870-0.3698 growth(-1) 0.1994-0.0465 1.0000-0.0565 0.1285 0.3530-0.2705 openk -0.3245 0.3689-0.0565 1.0000-0.1844 0.0376-0.0196 urbanze -0.0789-0.4740 0.1285-0.1844 1.0000 0.3676 0.1828 k -0.1309 0.1870 0.3530 0.0376 0.3676 1.0000-0.2246 nflaton 0.0394-0.3698-0.2705-0.0196 0.1828-0.2246 1.0000 Non-OECD Gn 1.0000 0.1266 0.0255-0.0144-0.0835-0.2156 0.1248 Log(gdppc) 0.1266 1.0000-0.0807 0.4232-0.7964 0.2183 0.0532 growth(-1) 0.0255-0.0807 1.0000 0.0810 0.1103 0.2938-0.1586 openk -0.0144 0.4232 0.0810 1.0000-0.3339 0.4074-0.0652 urbanze -0.0835-0.7964 0.1103-0.3339 1.0000-0.1201-0.1577 k -0.2156 0.2183 0.2938 0.4074-0.1201 1.0000-0.1393 nflaton 0.1248 0.0532-0.1586-0.0652-0.1577-0.1393 1.0000 To study the relatonshp between nequaly and economc development, we frst specfy the followng nonparametrc (uncondonal) panel data model wh fxed effects 2 The ndrect effects va channels can also be found n growth studes (Barro, 2000; Frankel and Rose, 2002). 7
whout control varables: gn g( lgdp ) u v, t 1,2,, m ; 1,2,, n, (1) where the functonal form of g() s not specfed and lgdp s the natural logarhm of real GDP per capa. For country, there are m observatons from year 1 to m. Country s ndvdual effects u are the fxed-effects that are correlated wh country s economc development wh an unknown correlaton structure. On the contrary, the ndvdual effects u are random effects when they are uncorrelated wh lgdp. For consstent estmaton of g(), we use a nonparametrc estmaton wh fxed effects model. The error term v s assumed to be..d. wh a zero mean and a fne varance, and s mean-ndependent of lgdp, namely E( v lgdp ) 0. Note that n Model (1) there s no control varable n uncoverng the relatonshp between nequaly and development. Ths s consstent wh the orgnal dea n Kuznets nverted-u relatonshp that provdes a general framework to explan nequaly uncondonal on other varables other than the level of economc development. However, recent studes usng econometrc models begn to consder determnants of nequaly wh control varables to study the Kuznets nverted-u relatonshp, as that can provde ceters parbus an analyss on the causaly from economc development to nequaly. In ths study we use both uncondonal and condonal models. The semparametrc (condonal) counterpart of Model (1) wh control varables can be shown as: gn g lgdp x u v t m n (2) ' ( ), 1,2,, ; 1,2,,, where x s the vector of the control varables. We adopt the assumptons n Model (1) and that v s also mean-ndependent of x. In our models, the control varable growth s used n s lagged form snce growth may potentally be endogenous n the 8
nequaly model (Huang et al. 2009). In Model (2) the ndrect effect of development on nequaly s controlled by the term x ', hence g( ) reflects the nequaly from development drectly. The mechansm n Models (1) and (2) and ther relatonshps are ntuvely llustrated n Fgure 1. The g(z) n nonparametrc Model (1) gves the gross contrbuton of development to nequaly, whle the g(z) n semparametrc Model (2) gves the net contrbuton of development to nequaly, gven x. The dfference between the two g(z) s the ndrect contrbuton of development to nequaly va control varables x. g(z) n nonparametrc model (1) (no control varables) Inequaly Development g(z) n semparametrc model (2): gvenx Inequaly Development x ' b g(z) n nonparametrc model = g(z) n semparametrc model + x ' b Fg. 1 The Mechansm n the Nonparametrc and Semparametrc Models When g() s specfed as a parametrc quadratc, cubc or fourth-degree polynomal functon of lgdp, Model (1) and Model (2) become parametrc unbalanced panel data models wh fxed effects, whch can be estmated by the conventonal method 9
(Baltag, 2008). However, n order to keep the approach comparable to the nonparametrc counterpart, we use the dfference of y - y1 nstead of the transformaton of y - or the dfference of y - y, t - 1 n removng the fxed effects. Table 3 contans the parametrc estmaton results for the two samples of OECD and non-oecd countres. The conventonal quadratc specfcaton s used to test the Kuznets hypothess, and the coeffcents on the lnear and quadratc terms are expected to be posve and negatve, respectvely. The estmates for the non-oecd countres have the expected sgns and are hghly sgnfcant, whle those for the OECD countres do not have the expected sgns, regardless whether control varables are added nto the model. We estmated models wh hgher-degree polynomals of the logarhm of GDP per capa, as shown by the cubc and 4-th degree columns n Table 3. For OECD, the cubc specfcaton presents sgnfcant estmates of the coeffcents n both the condonal and uncondonal models, whle the 4-th degree polynomal specfcaton does not provde sgnfcant estmates. For non-oecd, the estmates for the models whout controls are all deal whle the estmate n the 4-th degree specfcatons wh controls s perfect, although the quadratc estmate s also deal as an explanaton of the nverted-u relatonshp. These parametrc estmaton results show that the quadratc specfcaton does not gve a best f n both samples of OECD and non-oecd countres, thereby castng doubts on the conventonal quadratc specfcaton n descrbng the nequaly-development relatonshp. y 10
quadratc cubc 4-th degree OECD lgdp -1.372 388.180* -87.381 (5.071) (72.324) (875.06) lgdp 2-0.098-42.309* 34.887 (0.267) (7.823) (141.78) lgdp 3 1.517* -4.029 (0.281) (10.175) lgdp 4 0.1489 (0.273) Table 3 Parametrc Estmaton Results Parametrc model quadratc cubc 4-th degree -17.801* (7.053) 0.661** (0.351) growth(-1) 0.117* (0.023) openk 0.037* (0.007) urbanze 0.009 (0.037) k 0.220* (0.028) nflaton -0.025* (0.007) Non-OECD lgdp 24.775* 147.99* -1797.5* 32.343* (4.297) (42.84) (423.62) (4.287) 262.500* (69.337) -29.430* (7.414) 1.074* (0.264) 0.113* (0.023) 0.035* (0.007) 0.031 (0.038) 0.1973* (0.028) -0.029* (0.007) 799.739 (803.449) -116.847 (130.456) 7.366 (9.380) -0.169 (0.252) 0.114* (0.023) 0.035* (0.007) 0.028 (0.038) 0.200* (0.028) -0.029* (0.007) Semparametrc model 0.121** (0.069) -0.001 (0.017) -0.171** (0.095) 0.231* (0.083) -0.030 (0.020) 68.221 (43.601) -1358.56* (410.51) lgdp 2-1.272* (0.256) -16.05* (5.118) 336.356* (76.508) -1.710* (0.258) -6.037 (5.240) 253.14* (74.33) lgdp 3 0.583* (0.202) -27.540* (6.095) 0.171 (0.207) -20.576* (5.939) lgdp 4 0.834* (0.181) 0.617* (0.177) growth(-1) 0.024 (0.029) 0.025 (0.029) 0.038 (0.030) 0.023 (0.065) openk 0.034* (0.006) 0.033* (0.006) 0.032* (0.006) 0.021** (0.013) urbanze 0.037 (0.029) 0.027 (0.031) 0.007 (0.032) -0.004 (0.060) k -0.180* (0.021) -0.177* (0.022) -0.166* (0.022) -0.148* (0.048) nflaton 0.001* (0.000) 0.001* (0.000) 0.001* (0.000) 0.001* (0.000) Notes: The dependent varable s Gn. The numbers n the parentheses are standard errors of the coeffcent estmates. Estmates of the ntercepts n parametrc models are not reported. * = 5% sgnfcance and ** = 10% sgnfcance. 11
III Nonparametrc Estmaton and Testng Method wh Unbalanced Panel Data We use the same notaton as those n Henderson et al. (2008) to llustrate our model estmaton n the unbalanced panel data case. For smplcy, we denote y z gn and lgdp. Models (1) and (2) can be estmated by the eratve procedures modfed slghtly from Henderson et al. (2008) to cater for the unbalanced panel data. To remove the fxed effects n Model (1), we wre y y y gz ( ) gz ( ) vv gz ( ) gz ( ) v. 1t 1 1 1 Denote y ( y 2,, y )', v ( v 2,, v )', and g ( g 2,, g )', where g g( z ). m m m The varance-covarance matrx of v and s nverse are calculated, respectvely, as ( ) and 2 ' v Im 1 em 1em1 ( I e e / m ), where I 1 s an 1 2 ' v m1 m1 m1 m denty matrx of dmenson m 1 and e 1 s a ( m 1) 1 vector of uny. The creron functon s gven by 1 1 ( g, 1) ( 1 1)' ( 1 1), 1,2,, g y g ge m y g ge m n. 2 Denote the frst dervatves of ( g, g 1) wh respect to g as g, ( g, g 1), t 1, 2, m. Then ( g, g ) e ( y g g e ), ' 1,1g 1 m1 1 m1 ' 1 g, 1, 1 1 m 1 ( g, g ) c ( y g g e ), t 2, m where c, 1 s an ( m 1) 1 matrx wh ( t 1) th element/other elements beng 1/0. Denote gz dgz dz (, )' ( ), ( )/ '. It can be estmated by solvng the frst order 0 1 condons of the above creron functon through eraton: 1 m n m K ˆ ˆ h( z z) G g, [ 1] ( 1),, ( 0, 1 )',, [ 1] ( ) 0 g l z G g l zm, 1 t1 12
where the argument g, s gˆ [ l 1] ( zs) for s t and G ( 0, 1)' when s t, and gˆ [ l 1] ( zs) s the ( l 1) th eratve estmates of ( 0, 1)'. Here G 1, ( z z) / h' and () 1 k ( / ) h v h k v h, k( ) s the kernel functon. The next eratve estmator of ( 0, 1)' 1 s equal to gˆ ˆ [] l ( z), g[] l ( z ) ' = D1 ( D2 D3), where 1 D e e K z z G G c c K z z G G n m ' 1 ' ' 1 ' 1 m 1 1 ( 1 ) 1 1, 1, 1 ( ), m h t t h 1 m t2 n m 1 ' 1 ' 1 ˆ ˆ 2 m1 m1 h 1 1 [ l1] 1, t1, t1 h [ l1] 1 m t2 D e e K ( z z) G g ( z ) c c K ( z z) G g ( z ), n m 1 ' 1 ' 1 D3 Kh( z 1z) G 1em 1,[ 1] ( ), 1,[ 1], H l Kh z z Gc t H l 1 m t2 and,[ l 1] H s an ( m 1) 1 vector wh elements y ( gˆ ( z ) gˆ ( z )), t 2,, m. [ l1] [ l1] 1 The seres method s used to obtan the nal estmator for g( ). The convergence creron for the eraton s set to be Further, the varance m m 1 1 g ( z ) g ( z ) / g ( z ) 0.01. n 2 n 2 ˆ ˆ ˆ [] l [ l 1] [ l1] 1 m t2 1 m t2 2 v s estmated by 2 ˆv 1 1 2n 1 n m 2 ( y ˆ ˆ y 1( gz ( ) gz ( 1))). 1 m t2 The varance of the eratve estmator gz ˆ( ) s calculated as ( nh ˆ ( z)) 1, where k 2 () v dv, and ˆ 1 m 1 ( z) K ( z z)/ n n m 2 ˆ h v. 1 m t2 We use the seres method to obtan an nal estmator for ( ) and then conduct the eraton process. The convergence creron for the eraton s set to be 13
Further, the varance m m 1 1 g ( z ) g ( z ) / g ( z ) 0.01. n 2 n 2 ˆ ˆ ˆ [] l [ l 1] [ l1] 1 m t2 1 m t2 2 v s estmated by 2 ˆv 1 1 2n 1 n m 2 ( y ˆ ˆ y 1( gz ( ) gz ( 1))). 1 m t2 The varance of the eratve estmator gz ˆ( ) s calculated as ( nh ˆ ( z)) 1, where k 2 () v dv, and ˆ 1 m 1 ( z) K ( z z)/ n n m 2 ˆ h v. 1 m t2 For the estmaton of semparametrc Model (2), we denote the nonparametrc estmator of the regresson functons of the dependent varable y and the control varables x, respectvely, as gˆ y() and gˆ () = ( gˆ x,1(),, gˆ xd, ())', where d s the x number of controls. Then b s estmated by ˆb n -1 æ ö æ n ö ' 1 ' 1 = - - x* S x* / m x* S y* / m çå è ø çå è, ø = 1 = 1 where y * and x * are, respectvely, ( m 1) 1 and ( m 1) d matrces wh the t -th row element beng y* = y -( gˆ y( z) -gˆ y( z1)) and x* = x -( gˆ x( z )-gˆ x( z 1)). The nonparametrc functon g( ) s estmated by the same method shown above, except that y s replaced by y - x b whenever occurs. ' ˆ For the selected model to ncorporate a data-drven procedure, we further modfy the specfcaton tests to an unbalanced panel data case. Regardless whether the models have control varables as regressors, we perform the followng two specfcaton tests. The frst specfcaton test s to choose n Model (1) between parametrc and nonparametrc models whout control varables. The null hypothess H 0 s parametrc model wh gz ( ) g0( z, ). For example, g z z z 2 0(, ) 0 1 2. The alternatve H 1 s that gz ( ) s nonparametrc. The test statstc for testng ths null s 14
1 1 I g z g z n m (1) 2 ( ˆ ˆ n 0(, ) ( )) n 1 m t1, where ˆ s a consstent estmator of the parametrc panel data model wh fxed effects; gˆ() s the eratve consstent estmator of Model (1). The second specfcaton test s to choose n Model (2) between parametrc and semparametrc models wh control varables. The null hypothess H 0 s parametrc model wh gz ( ) g0( z, ). The alternatve s that gz ( ) s nonparametrc n Model (2). The test statstc for testng ths null s 1 1 I g z x g z x ˆ, n m (2) ' ' 2 ( ˆ n 0(, ) ( ) ) n 1 m t1 where and are consstent estmators n the parametrc panel data model wh fxed effects; gˆ() and ˆ are the eratve consstent estmator of model (2). In the followng emprcal study, we apply bootstrap procedures n Henderson et al. (2008) to approxmate the fne sample null dstrbuton of test statstcs and obtan the bootstrap probably values for the test statstcs. IV Emprcal Results The kernel n both the estmaton and the testng s the Gaussan functon and the bandwdth s chosen accordng to the rule of thumb 3 n - : h 1.06s ( m ) 1/5 = å, where s z s the sample standard devaton of { z }. All the bootstrap replcatons are set to be 400. The last column n Table 3 reports the coeffcent estmaton for the control varables n the parametrc part of semparametrc Model (2). For the OECD countres, wh the excepton of openk and urbanze, the coeffcent estmates of all other control z = 1 3 We also slghtly change the constant nstead of 1.06, and fnd that the estmaton and test results are not sgnfcantly affected. 15
varables have the same sgns and smlar values n both parametrc and semparametrc models. For countres n the non-oecd sample, wh the excepton of the urbanze varable, the coeffcent estmates of all other control varables are hghly smlar n both parametrc and semparametrc models. The nconsstency n the coeffcent estmates of such varables as urbanze and openk casts doubts on model specfcaton once agan, but ths wll be tested n the fnal stage of our analyss. An nterestng fndng s the dfferent sgns n the coeffcents of nvestment and nflaton between the two samples. Investment share has a posve effect on nequaly n OECD, but a negatve effect n non-oecd, mplyng that nvestment aggravates nequaly n OECD countres, whle allevates nequaly n non-oecd countres. The effect of nflaton s exactly the oppose to that of nvestment between the OECD and non-oecd countres. In Table 4, the nonparametrc functon g( ) s estmated at some quantle ponts of the logarhm of GDP per capa by usng nonparametrc Model (1) and semparametrc Model (2). For the OECD countres, when ther development level s at the 2.5 percent quantle from the bottom, the estmate of g( ) from semparametrc Model (2) s larger than that from nonparametrc Model (1). The dfference s the total contrbuton by control varables to nequaly. But when the development level s one of the other quantles, the estmate of g( ) from nonparametrc Model (1) becomes larger, whch mples that the ntegrated contrbuton by control varables to nequaly becomes posve. In short, polcy varables and economc characterstcs can ndeed play a role n affectng nequaly n the hgher stage of development. 16
Table 4 Nonparametrc Estmaton of g( ) at Dfferent Ponts of lgdp Quantle of lgdp Nonparametrc model (1) Semparametrc model (2) z m(z) std. err. m(z) std. err. OECD 2.5% 8.5937 40.7812 5.4563 42.5567 5.2224 25% 9.4911 35.4923 1.9810 34.5420 1.8960 50% 9.8375 33.4746 1.5039 31.6731 1.4394 75% 10.0647 32.3819 1.5291 30.3893 1.4635 95% 10.3257 31.4750 2.1687 28.6795 2.0757 97.5% 10.3968 30.3402 2.5887 27.2607 2.4777 at sample mean of z mean 9.7299 34.2492 1.5917 32.5366 1.5234 Non-OECD 2.5% 7.0233 40.5527 3.2093 42.6521 3.0549 25% 8.0689 44.9614 2.1619 47.1526 2.0579 50% 8.6832 46.5913 1.5724 48.3060 1.4968 75% 9.0522 45.2422 1.6618 47.0502 1.5819 95% 9.8949 42.3625 3.2022 43.5062 3.0481 97.5% 9.9989 42.9649 3.4717 43.8056 3.3047 at sample mean of z mean 8.5836 46.6044 1.6252 48.3800 1.5470 However, the estmaton results n OECD countres are oppose to those n non-oecd countres. Table 4 shows that for the non-oecd countres all the estmates of g() at each quantle from semparametrc Model (2) are larger than that at the same quantle from nonparametrc Model (1). The ntegrated contrbuton of control varables to nequaly s therefore negatve, namely, they totally decrease nequaly. We next dscuss and compare the results from the OECD and non-oecd countres. OECD Countres Fgures 2 and 3 llustrate the nonparametrc and semparametrc estmatons of g() n Models (1) and (2), respectvely, for the OECD countres, and show also the lower and upper bounds of 95 percent confdence ntervals for the estmates. The nonparametrc estmates are reasonable, though the estmaton has some boundary effects. The two 17
curves of g( ) n Fgures 2 and 3 look very smlar n shape, whch mples that the control varables, though havng an effect on nequaly, have played ltle role n changng the nonlneary shapes of g( ). 4 gn 70 60 50 40 30 20 10 7 8 9 10 11 log(gdppc) low er bound g(z) upper bound Fg. 2 Nonparametrc Estmaton n Model 1: OECD 70 60 50 gn 40 30 20 10 7 8 9 10 11 log(gdppc) low er bound g(z) upper bound Fg. 3 Semparametrc Estmaton n Model 2: OECD Our fndng shows that the shapes of g( ), whether estmated from semparametrc Model (2) or nonparametrc Model (1), are very smlar to that n Mushnsk (2001) who used a nonparametrc estmaton model wh cross-secton data. The results reflect an 4 Huang (2004) also shows ths fndng n hs cross-secton analyss. 18
ncreasng, albe short, porton at lower levels of development, a turnng pont (where the effect of development on nequaly changes from posve to negatve) at 8.2 (about $3,640 n 2005 dollars), followed by a longer decreasng porton of the curve. Whle the results n Mushnsk (2001) reflected the predomnance of mddle-ncome countres n hs dataset, the result from our dataset usng nonparametrc estmaton for the OECD countres (representng manly hgh-ncome or upper-mddle-ncome countres) reflect the predomnance of the hgh-ncome and upper-mddle-ncome group. Also, the curves hnt an upturn at a hgher ncome level (around 10.8, about $49,020), whch accords wh the hgh-level upturn pont from a hghly developed economy (Ram, 1991). The contrbuton of development to the reducton of nequaly va control varables can be seen from the vertcal dfference between the two nonparametrc and semparametc curves n Fgure 4. The ntegrated effect of development on nequaly va the control varables s negatve at lower ncome levels (lower than 9.2, about $9,900) snce the curve from nonparametrc estmaton s below that from semparametrc estmaton. Control varables can contrbute to reduce nequaly at or below ths level of ncome. When the development level s above $9,900, ths ndrect ntegrated effect becomes posve, mplyng that control varables as a channel tend to ncrease nequaly at ths hgher stage of ncome level. In the orgnal verson of the Kuznets hypothess, the channel effect of development on nequaly va other determnants s gnored. Here we nclude the control varables n the semparametrc model to show the ndrect effect of development on nequaly. The evdence shows that whether the channel effects of development are posve or negatve n OECD countres depend on the development level. 19
60 50 gn 40 30 20 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 log(gdppc) nonparametrc semparametrc para-cubc Fg. 4 Comparng Non- and Sem-parametrc Estmaton: OECD The analyss so far shows that nonparametrc and semparametrc estmatons can provde addonal nformaton on the mechansm of economc development on nequaly. However, whch specfcaton s most sued to the sample requres further hypothess tests. Table 5 presents varous test results. In the case whout control varables, the p-values for the quadratc and cubc parametrc specfcatons aganst nonparametrc specfcaton are, respectvely, 0.07 and 0.06. At the 10 percent sgnfcant level, the nulls of quadratc and cubc parametrc specfcatons are rejected and the alternatve, nonparametrc specfcaton s accepted. However, at the 5 percent level, the nulls are accepted. In the case wh control varables, the p-values are less than 5 percent. So we reject the nulls of parametrc specfcaton at the 5 percent level and accept the semparametrc specfcaton. The test results shown n Table 5 provde further support to our analyss. 20
Table 5 Nonparametrc and Semparametrc Model Specfcaton Tests: OECD Model Hypotheses I n Model selected ( p-value) whout H 0 : Quadratc parametrc Parametrc or 4.006 control H 1 : Nonparametrc varables Nonparametrc? (0.070) wh control varables Parametrc or Semparametrc? H 0 : Cubc parametrc H 1 : Nonparametrc H 0 : Quadratc parametrc H 1 : Semparametrc H 0 : Cubc parametrc H 1 : Semparametrc 3.964 (0.060) 10.205 (0.013) 11.823 (0.008) Nonparametrc (10%) Parametrc (5%) Nonparametrc (10%) Parametrc (5%) Semparametrc Semparametrc Snce the cubc parametrc model whout control varables s accepted at the conventonal 5 percent sgnfcant level (note that the quadratc curve s not consdered here because s estmate s not sgnfcant, see Table 3: OECD), we nclude the curve of the estmated cubc functon n Fgure 4. Ths cubc functon mples a turnng pont whch s almost the same as those n the curves from the nonparametrc and semparametrc estmatons, and hence can capture some of the non-concaves suggested by the nonparametrc and semparametrc regressons. The F test for the parametrc specfcatons n Table 6 shows that the cubc functon of the logarhm of GDP per capa provdes a f better than the quadratc functon and cannot be rejected aganst the alternatve fourth-degree specfcaton. Table 3 (OECD) also shows that all terms n the cubc polynomal are sgnfcant, and they are better than the estmates n quadratc and fourth-degree polynomals. Hence the tests support the estmaton of a cubc functon of development for the OECD sample rather than a quadratc or fourth-degree functon n the case of no control varables. 21
Table 6 Parametrc Model Tests for Incluson of Polynomal Terms Whout control varables Wh control varables Degree of polynomal F statstc: OECD / Non-OECD F statstc: OECD / Non-OECD H 0 :Second vs H 1 :Thrd 11.2043* / 5.3481* 0.3217 / 1.0814 H 0 :Thrd vs H 1 :Fourth 1.5146 / 9.2195* 2.8767 / 5.5398* H 0 :Fourth vs H 1 :Ffth 0.2849 / 1.0849 2.9167 / 0.0058 Note: * = 5% sgnfcance. In the case wh control varables, the tests n Table 6 present no obvous evdences to show whch parametrc specfcaton s best. One can conclude from Table 3 (OECD) that the cubc form s preferred snce all the estmated coeffcents of the cubc polynomal statstcally preval over those of the quadratc and fourth-degree counterparts. However, as the test wh control varables n Table 5 shows, parametrc cubc specfcaton s rejected and semparametrc model s accepted at the 5 percent sgnfcant level. Hence, the nsgnfcance of the F tests n the case of control varables for OECD countres s expected snce the F tests based on the estmaton of parametrc model may not be vald for semparametrc models. Non-OECD Countres Fgures 5 and 6 respectvely present the nonparametrc estmaton of g() n Models (1) and (2) for the non-oecd sample countres. The estmates provde a result stronger than those n Fgures 1 and 2 snce the boundary effect for nonparametrc estmaton s less sgnfcant. There s much resemblance between the shapes of nonlneary of g() n Fgures 5 and 6. The results reflect a rapd ncreasng, albe short, porton at lower levels of development and a frst turnng pont at 7 (about $1,100), then another ncreasng, albe long and flat, porton at the mddle ncome level of 22
development, then followed by a slowly decreasng porton of the curve wh the second turnng pont at 8.7 (about $6,000). Fnally, the curves also hnt at an upturn at a hgher ncome level (around 10, about $22,026), smlar to the processes shown n OECD countres and the fndngs n Ram (1991) and Mushnsk (2001). The second turnng pont s hgher than the frst one and the fnal upturn occurs at even hgher nequaly level. Ths process presents a roller coaster mode, albe flat and long n the mddle of the process. 5 In the non-oecd countres, f the fnal upturn s not accounted for, the process approxmately accords wh the nverted-u hypothess. 60 50 gn 40 30 20 10 6 7 8 9 10 11 log(gdppc) low er bound g(z) upper bound Fg. 5 Nonparametrc Estmaton n Model 1: Non-OECD 5 Ths roller coaster mode also appeared n the study by Keane and Prasad (2002) that emprcally examned Poland s nequaly-development relatonshp and generalzed the mode by usng the sample of transonal economes. 23
60 50 gn 40 30 20 10 6 7 8 9 10 11 log(gdppc) lower bound g(z) upper bound Fg. 6 Semparametrc Estmaton n Model 2: Non-OECD Fgure 7 contans both the nonparametrc and semparametc curves estmated for the non-oecd countres. The vertcal dfference reflects the contrbuton of control varables to nequaly. The ntegrated effect of development on nequaly va control varables s always negatve, whch s dfferent from that n the OECD countres. In short, control varables generally mgate nequaly n non-oecd countres, except when the logarhm of ncome level s very large (greater than 10.2). Ths evdence shows that the channel effect of development on nequaly va the control varables as a whole s negatve n non-oecd countres. Table 7 presents the test results for choosng between nonparametrc or semparametrc and parametrc specfcatons n the non-oecd countres. The quadratc and cubc parametrc specfcatons cannot be rejected at any usual sgnfcant level whether or not the model ncludes control varables as regressors. Among the parametrc models, the F tests for parametrc specfcatons n Table 6 show, when compared to the thrd- and ffth-degree polynomal specfcatons, that the fourth-degree polynomal of the logarhm of GDP per capa gves a suffcently accurate descrpton of the non-oecd 24
countres, whether or not the control varables are added nto the models. Recall n Table 3 (non-oecd) that all coeffcent estmates n the fourth-degree polynomals are sgnfcant at the 5 percent level. Hence the fourth-degree functon specfcaton gves a better descrpton of the data than the quadratc and cubc polynomals. The tests for the non-oecd countres support the estmaton of a fourth-degree functon of development rather than a quadratc or cubc form. 6 Table 7 Model Specfcaton Tests: Non-OECD Model Hypotheses I n ( p-value) whout H 0 : Quadratc parametrc Parametrc or 6.748 control H 1 : Nonparametrc varables Nonparametrc? (0.415) wh control varables Parametrc or Semparametrc? H 0 : Cubc parametrc H 1 : Nonparametrc H 0 : Quadratc parametrc H 1 : Semparametrc H 0 : Cubc parametrc H 1 : Semparametrc 5.212 (0.375) 6.796 (0.438) 5.942 (0.465) Model selected Quadratc parametrc Cubc parametrc Quadratc parametrc Cubc parametrc Fgure 7 contans also the curve of the estmated fourth-degree polynomal. It shows that the fourth-degree functon can capture some of the non-concaves suggested by nonparametrc and semparametrc regressons. Although the shape resembles the nverted-u relastonshp, wh the excepton when the development reaches a very hgh level, the conventonal quadratc form used to estmate the nequaly-development relatonshp mght be msspecfed for the non-oecd dataset. 6 Such a support on the use of a fourth-degree polynomal rather than a quadratc form can also be seen n Mushnsk (2001). 25
60 50 gn 40 30 20 6 7 8 9 10 11 log(gdppc) semparametrc nonparametrc 4th-degree polynomal Fg. 7 Comparng Non- and Sem-parametrc Estmaton: Non-OECD Comparng OECD and non-oecd Countres Addonal dfferences between the estmates of the nequaly-development relatonshp for the OECD and non-oecd countres can be seen from the nonparametrc approach. The nonparametrc and semparametrc estmates of the relatonshp between nequaly and development n OECD countres have only one turnng pont at a low ncome level of about $3,640 (see Fgure 4), whle the estmates n non-oecd countres have two turnng ponts, wh the frst beng at a rather low ncome level (about $1,100) and the other beng at a mddle ncome level (about $6,000) (see Fgure 7). Generally, when the level of economc development s not very hgh, nequaly n OECD/non-OCED countres wll decrease/ncrease wh development. However, both curves make an upturn at an advanced level of development, though they act wh a dfferent manner: nequaly n OECD countres shows an upturn at a lower nequaly level than non-oecd countres. If ths upturn were not accounted for, the evdence from OECD countres would roughly suggest that the nverted-u relatonshp between nequaly and development s located beyond the turnng pont and on the backsde of the 26
nverted-u, whle the non-oecd evdence shows that the relatonshp roughly accords wh a full nverted-u relatonshp. Fgure 8 presents an ntuve comparson of the nonparametrc (left) and semparametrc (rght) estmates of the nonparametrc functon g(.) between OECD and non-oecd countres. On the same porton of development (lgdpî[7.6,10.4]), the two curves for OECD and non-oecd countres ntersected at one pont at about 8.4, namely at GDP per capa = $4,447. It can be seen that OECD countres has hgher nequaly than non-oecd countres when GDP per capa s less than $4,447, but has lower nequaly when development has exceeded ths level. The dynamcs n Fgure 8 mples that non-oecd countres seem to face hgher nequaly at mddle or hgh ncome levels. The dfference of the ntersecton of OECD and non-oecd countres between nonparametrc and semparametrc estmates reflects the role of control varables. The effects of polcy varables on nequaly change the mode of the ntersecton, but have ltle effect on the relatve locaton of the two curves from nonparametrc and semparametrc estmatons. 55 55 45 45 gn 35 gn 35 25 25 15 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 log(gdppc) 15 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 log(gdppc) OECD Non-OECD OECD Non-OECD Fg. 8 Nonparametrc (Left) and Semparametrc (Rght) Estmaton of g(.) 27
The estmaton and test results from nonparametrc model whout control varables mply that the conventonal quadratc concave functon may not necessarly capture the relatonshp between nequaly and development n both OECD and non-oecd countres. The tests show that the cubc polynomal of development levels can capture the relatonshp more accurately than quadratc and fourth-degree polynomals n OECD countres, whle the fourth-degree polynomal can gve a better descrpton of the data than other polynomals n non-oecd countres, whether or not the control varables are added as regressors n the models. In both OECD and non-oecd countres, analyss based on a quadratc specfcaton for the relatonshp between nequaly and development s msleadng. The estmaton and test results from the semparametrc model wh control varables show that the data-drven model selecton for OECD countres requres a semparametrc specfcaton whle the non-oecd countres requre a fourth-degree polynomal parametrc specfcaton. Gven the ntegrated contrbuton by control varables to nequaly shown above, we next study the effects of the control varables on nequaly by comparng the estmates n the parametrc part of the semparametrc model for the OECD countres (shown n the last column n Table 3) and the 4-th degree parametrc model for the non-oecd countres (shown n the 4-th degree column n Table 3). The mplcatons of the effects of the control varables are dfferent n the two sample country groups, except the varable growth (-1) whch has a posve effect on nequaly. Specfcally, the effect of openness on nequaly n OECD countres s negatve, albe nsgnfcant both economcally and statstcally, whle openness has a posve and sgnfcant effect on nequaly n non-oecd countres. Openness generally wll aggravate ncome nequaly n non-oecd countres, but has an emollent effect on 28
nequaly n OECD countres. Although ntegraton to the world market s expected to help non-oecd countres to promote prospery, ncreasng opportunes to trade are also lkely to affect ncome dstrbuton. Whether or not ncreasng openness to trade s accompaned by a reducton or an ncrease nequaly has strongly been debated (Julen, 2007; Wood, 1997). The effect of urbanzaton on nequaly s negatve (-0.171) and sgnfcant for OECD countres, but posve (0.007) and nsgnfcant for non-oecd countres. Urbanzaton helps to mgate nequaly n OECD countres, but ncreases nequaly n non-oecd countres, albe nsgnfcantly. Accordng to Anand (1993), the urban-rural dfference generally results n larger nequaly n total ncome dstrbuton due to urbanzaton. Hence, n the process of urbanzaton, ncome nequaly wll frst ncrease and then decrease wh urbanzaton or the mgraton of rural populaton to ces. In our case, OECD countres have much hgher urbanzaton than non-oecd countres. Hence the negatve effect of urbanzaton on nequaly n OECD and the posve effect n non-oecd accord wh ths general urbanzaton-nequaly relatonshp. The fndng that nvestment share aggravates nequaly n OECD countres but reduces nequaly n non-oecd countres contrasts wh the result n Barro (1999) that showed ltle overall relatonshp between ncome nequaly and nvestment. One explanaton s that nvestment may have potental endogeney n nequaly models. Inflaton has a negatve albe nsgnfcant effect on nequaly n OECD countres but has a posve and sgnfcant effect on nequaly n non-oecd countres. Generally, cross-country evdence on nflaton and ncome nequaly suggests that they are posvely related. For example, Albanes (2007) argued that the correlaton between nflaton and ncome nequaly s the outcome of a dstrbutonal conflct underlyng the 29
determnaton of government polces, and nflaton s posvely related to the degree of nequaly as low ncome households are more vulnerable to nflaton. Snce non-oecd countres have a hgh average nflaton, ther monetary authores should reduce nflaton to allevate ncome nequaly. However, the mpact of nflaton on ncome dstrbuton may be nonlnear (Bulíř, 2001), and the posve and sgnfcant effect of nflaton on nequaly would need to be explaned wh cauton. V Concluson Ths paper provdes evdences on the relatonshp between nequaly and development from estmatons and tests of nonparametrc and semparametrc panel data models wh fxed effects. Based on an unbalanced panel dataset, ths study contrbutes to the lerature by presentng new evdences about the nequaly-development relatonshp n OECD and non-oecd countres and provdes addonal nformaton on the mechancs of the effect of development on nequaly. For the OECD countres, nequaly generally decreases wh development, wh the excepton of an upturn at a hgher ncome level. The control varables wll help reduce ncome nequaly at lower ncome levels (below about $9,900), but they tend to ncrease nequaly when development exceeds that level. For the non-oecd countres, the nequaly-development relatonshp appears n a roller coaster mode wh two turnng ponts, and one upturn appears at a very hgh ncome level. When compared to the performance n OECD countres, the effect of development on nequaly va the control varables s always negatve n non-oecd countres, except after an upturn at a hgh ncome level. Non-OECD countres seem to face serous nequaly at the mddle or hgh ncome level. 30
Nonparametrc estmatons whout control varables suggest that a polynomal of hgher-degree mght gve a suffcently accurate descrpton of both OECD and non-oecd countres. Our tests support estmatng a cubc functon of development for the OECD sample and a fourth-degree functon of development for the non-oecd sample. Both the parametrc estmates capture some of the non-concaves suggested by nonparametrc and semparametrc regressons. Semparametrc estmatons and tests wh control varables present some mplcatons on polcy. In OECD countres, growth and nvestment share n GDP aggravate nequaly, but openness, urbanzaton and nflaton would reduce nequaly. In non-oecd countres, growth, openness, urbanzaton, and nflaton generally ncrease nequaly, but nvestment share helps mgate nequaly. It would be approprate to argue that nvestment n non-oecd countres should be geared more to mprove the condons of the low ncome groups, as that shall reduce nequaly overtme. 31
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