Non-parametric investigation of the Kuznets hypothesis for transitional countries

Transcription

1 Abstract No-parametric ivestigatio of the Kuzets hypothesis for trasitioal coutries Demidova Olga State Uiversity Higher School of Ecoomics The Kuzets hypothesis states that the relatio betwee icome distributio ad ecoomic developmet is characterized by a iverted - U curve. I other words, iequality i the icome distributio first rises but the falls. May researchers have verified this hypothesis, usig the data for developed coutries excludig trasitioal coutries. While testig for the Kuzets iverted-u relatioship, most studies follow the parametric quadratic specificatio by regressig the Gii idex o GDP per capita ad its squared term. But goodess of fittig for such models is usually quite low. The purpose of this paper is to test the Kuzets hypothesis usig the data for 9 trasitioal coutries. We estimate the ukow relatioship betwee the Gii idex ad GDP per capita estimatig the Nadaraya-Watso oparametric regressio with Gaussia kerel. The graphical outcomes show clear evidece of a iverted-u relatioship betwee the Gii idex ad per capita GDP. We use a alterative measure of icome distributio, amely, the ratio of icomes for 0% richest ad 0% poorest people; the ratio of icomes for 0% richest ad 0% poorest people reproduces that estimatio ad results i the same iverted-u shape of the curve. The graphical evidece cofirms the validity of the Kuzets' hypothesis for trasitioal coutries. All the estimated fuctios Y i = f(), i =,,3 (where Y is the Gii idex, Y is the ratio of icomes for 0% richest ad poorest people, Y 3 is the ratio of icomes for 0% richest ad poorest people, is per capita GDP) decrease for GDPs per capita which are greater tha ca. 000 (PPP USD). The GDP per capita for Russia equals 990 (PPP USD) ad we ca expect reductio of the iequality level i this coutry with icreasig GDP per capita.. JEL Classificatio: O, O5, C4 Keywords: Kuzets' hypothesis; Iverted -U curve; Trasitioal coutries. Itroductio The cetral topic of this paper is the relatio betwee the icome distributio iequality ad the mea icome icrease. Whe does the iequality i the distributio of icome decrease as the mea icome icreases? I 955 S.Kusets (Kusets S., 955) suggested that the value of icome iequality rises iitially, ad the drops after reachig a turig poit. This hypothesis is referred to as the Kuzets' hypothesis of the iverted - U curve. The Gii idex is the most commoly used measure of icome iequality. That is why may ivestigators search how the Gii idex or the low icome share depeds o per capita GDP usig cross coutries survey. Barro R. (Barro R., 000), Papaek G. ad Ky O. (Papaek G.986 ) regress the Gii idex o the per capita GDP ad its squared term, D.Mushiski (D. Mushiski, 00) used fourth-degree poliom. Auluwalia M. (Auluwalia M., 976) observed a U-shaped relatioship betwee the low icome share ad per capita GDP. demidova@hse.ru

2 Ho-Chua River Huag (Ho-Chua River Huag, 004) used oparametric approach to test the validity of the Kuzets hypothesis iverted-u shape. Some authors ote that the ature of the relatioship differs accordig to a coutry s level of ecoomic developmet ad divide coutries ito two groups (developed ad less developed) as a prerequisite to testig the Kuzets hypothesis. Savvides A. (Savvides A., 000) used the threshold regressio model. Sukiassya G. (Sukiassya G., 007) remarks that the existig literature o the iequality ad ecoomic developmet has virtually igored trasitio ecoomies ad his paper fills a importat gap o the theme. The author idicates that the effect of iequality o growth is egative for the trasitio ecoomies of Cetral ad Easter Europe ad the Commowealth of Idepedet States. This paper pursues the followig two objectives: ) To test the Kuzets hypothesis o the theoretical level, to determie the coditios o which the iverted-u depedece of the Gii coefficiet o the mea icome might take place, ad to give a ecoomic iterpretatio of the obtaied mathematic al results; ) To test whether the Kuzets hypothesis is valid for coutries with trasitio ecoomy.. Theoretical approach Suppose the populatio of a coutry is divided ito equal groups ad the group s icome icreases with its umber.

3 / j j 3 Let be the icome of the poorest group, be the icome of the richest group, i Z i is the mea icome, pi is the icome share of i-th group, i i=,,. j The ratio of the crosshatched regio area to the area of the triagle OAB multiplied by 00% is called the Gii idex G (Fig.) Figure. The Gii idex j B ( )/ j j / j j A 0 / / Cumulative populatio share Oe ca show that G... 00% () Z Z Z or G p p... p 00% () Thus G liearly depeds o,,- ad iversely o Z. The Gii idex G also liearly depeds o the icome shares p,,p-. The coefficiets of the icome shares p,,p- are egative ad their absolute values decrease as the umber of the icome share (ad the correspodig icome) icreases.

4 4 Remark. For the case of quatile groups ( = 5) from formula () it follows that G 00% (0.8.6 p. p 0.8 p3 0.4 p4 ) Geerally, the Gii idex G is a fuctio of variables:,,, -, Z. Note that G Z G i... 00% 0, (3) Z 00% 0, i,..., (4) Z Hece, the Gii idex icreases as the mea icome icreases (the icomes of - poorest groups beig costat) ad decreases as the icome of ay icome group with umber,,- icreases (other factors beig costat). The graph of the fuctio G is preseted i Fig.. Hereafter we keep oly variables Z ad for simplicity. I geeral case, the graph of the fuctio G coicides with dimesioal maifold G ~.

5 5 Figure : Maifold G Z Suppose γ is a smooth curve o the maifold G ~, γ GZ is the projectio of the curve γ oto the plae GOZ, γ Z is the projectio of the curve γ oto the plae OZ (Fig.3). Let γ GZ be iverted-u curve. It is quite iterestig to determie the form of the curve γ Z i this case. Figure 3: Projectios G (The Gii idex) γ GZ γ Z γz

6 6 Let the curve γ GZ be preseted as G = g(z). Suppose that * g( Z) 0 for 0 Z Z, * g( Z ) 0, ( ) ( 0 * g Z for Z Z ad g Z) 0 (5) The graph of the fuctio g(z) is preseted i Fig. 4. Figure 4: Iverted U form G (The Gii idex) G = g(z) turig poit Z* Z (mea icome)

7 7 Substitutig g(z) for G ad zero for,, - i (), we obtai expressio for the curve γ Z : ( ), where Z ( Z) Z( g( Z)) (6) Differetiatig both sides (6) two times, we obtai ( Z) g( Z ) Z g( Z), (7) ( ) ( Z) (g( Z) Zg ( Z)) (8) ( ) Substitutig Z * * for Z i (7) ad (8) ad ote that g ( Z ) 0, we get * * ( Z ) g( Z ) (9) * * * ( Z ) Z g ( Z ) (0) ( ) Usig (6) for Z = Z *, we get ( Z * * * ) Z g( Z ), hece * * ( Z ) ) * ( Z () Z Takig ito accout (7), (8) ad (5), we obtai * ( Z) 0 ad ( Z) 0 for Z Z () From (0), (), (), we get the followig graph for the fuctio (Z) (Fig.5). Fuctio φ(z) is covex for * Z Z.

8 8 Figure 5: Projectio o the plae OZ = Z = φ(z) O turig poit Z* Z Remark. If g(z) is a polyomial of degree k the φ(z) is also a polyomial with degree k+. E.g. if g(z) is a quadratic fuctio the φ(z) is a cubical fuctio. Remark 3. If the projectio of the curve γ oto the plae GOZ has iverted-u form, the the projectio of this curve oto ay plae joz (j =,,-) has the same form as fuctio (Z) (Fig.5). Remark 4. Suppose G = g(z) where g(z) has iverted-u form. The the relatioship betwee p i, i =,, - ad mea icome Z is U-shaped. It follows from (). Let us state the mai result of this sectio. I order for the Gii idex to begi droppig startig from a certai level of the mea icome Z *, it is essetial for the icome of low-icome groups to icrease with the mea icome growth. I particular, for the Gii coefficiet to decrease quadratically, the icome of the most low-icome group should icrease cubically. 3. Empirical approach Suppose we have a sample for m coutries. Let us deote the set of i i i i i A (,,...,,, ) observatios for i th coutry by i Z G, i =,,m, where i j i is a icome of the j th group i the i th coutry, j =,,-, Z is the mea i icome i the i th coutry, G is the Gii idex for the i th coutry. The poits A,,Am belog to the maifold G ~. This set of poits is a proxy for the curve γ.

9 9 Usig the projectios of the poits A,,Am oto the plaes GOZ ad OZ we estimate the fuctios g(z) ad (Z). First, we try to estimate the parameters of the fuctios g(z) ad (Z) usig quadratic ad cubical specificatio correspodigly. The we use oparametric specificatio. 4. Data The data set used i this study is take from the Huma developmet report (006) of World Bak (Appedix). 9 trasitioal coutries were chose. Oe attractive feature of this group of coutries is that their startig poits were remarkably similar. Yet, they subsequetly have experieced substatial divergece i growth rates ad icome iequality (Sukiassya G., 007). For each of the coutries, three variables are cosidered. Those iclude the Gii idex (deoted by GINI, a measure of iequality), the GDP per capita (PPP USD, deoted by GDP, a proxy for the mea icome), ad the 0% low icome share (deoted by P0 - ). We also create the ew variable 0 the icome of the 0 % low icome share, where P0 GDP. 5. Empirical results 5.. Parametric specificatio The traditioal regressios have bee estimated i the followig specificatio: GINI 0 GDP GDP (3) P 0 GDP GDP (4) GDP GDP 3GDP (5) The results of ruig are preseted i Tables, ad 3. Table. Regressio of the Gii coefficiet o per capita GDP ad its squared term Source SS df MS Number of obs = 9 F(, 6) =.5 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = 5.73 GINI Coef. Std. Err. t P>t [95% Cof. Iterval] GDP GDP -4.00e e e e-08 _cos Table. Regressio of the 0 % low icome share o per capita GDP ad its squared term Source SS df MS Number of obs = 9 F(, 6) =.53 Model Prob > F = Residual R-squared = 0.05 Adj R-squared = Total Root MSE =.538

10 0 P0- Coef. Std. Err. t P>t [95% Cof. Iterval] GDP GDP.5e e e e-08 _cos Table 3. Regressio of the 0 % low icome o per capita GDP, its squared ad cubic terms Source SS df MS Number of obs = 9 F( 3, 5) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = Coef. Std. Err. t P>t [95% Cof. Iterval] GDP GDP GDP 3 8.6e e e-0.e-09 _cos The output idicates that regressios (3) ad (4) are isigificat although all coefficiets have expected sigs. Icreasig the polyomial degree does t chage the situatio. The coefficiets of the oliear powers of per capita GDP i regressio (5) are isigificat. 5.. No - Parametric specificatio The low goodess of fittig ad isigificace of the polyomial regressios coefficiets was a reaso why we estimated the ukow relatioship betwee the Gii idex ad per capita GDP (ad two other relatioships) usig the followig relatioship: GINI m(gdp) (6) P 0 m( GDP) (7) 0 m( GDP (8), ) The coditioal expectatio fuctio, m( ) was estimated usig the Nadaraya - Watso estimator ad the Gaussia kerel. Figures 6-8 cotai kerel regressios (6) (8) results.

11 Fig.6 The estimated coditioal mea of the Gii coefficiet o per capita GDP Kerel regressio, bw = 500, k = Grid poits Fig.7 The estimated coditioal mea of the 0 % low icome share o per capita GDP Kerel regressio, bw = 500, k = Grid poits

12 Fig.8 The estimated coditioal mea of the 0 % low icome o per capita GDP Kerel regressio, bw = 500, k = Grid poits As see from Fig. 6 8, the Kuzets hypothesis is supported for the trasitio coutries. Ideed, the depedece of the Gii idex o per capita GDP takes a iverted-u form, the depedece of the 0 % low icome share o per capita GDP takes a U- form, the depedece of the 0 % low icome o per capita GDP looks alike the graph of the fuctio φ i Fig. 5. Accordig to Fig. 6, the turig poit is ca. 000 (PPP USD). The GDP per capita for Russia equals 990 (PPP USD). All coutries with greater GDP per capita have the Gii idex smaller tha Russia. That is why we ca expect reductio of the iequality level i this coutry with icreasig GDP per capita. Some deviatio of the practical results from the theoretical oes is observed oly at the edges that is typical for kerel regressio. This problem ca be solved, say, by usig splies.

13 3 Remark 5.We use a alterative measure of icome distributio, amely, the ratio of icomes for 0% richest ad 0% poorest people; the ratio of icomes for 0% richest ad 0% poorest people with the same result. Remark 6.The depedece of the 0 % low icome share o per capita GDP also has a U-form. The depedece of the 0% low icome o per capita GDP has the same form as the oe for 0% low icome. 6. Coclusio This article is devoted to the verificatio of the Kuzets hypothesis about the relatio betwee icome distributio ad the mea icome. It has bee show that the Gii idex is a fuctio of the mea icome ad the icomes of all icome groups except the richest group. The suggestio about a iverted-u shape of the depedece of the Gii idex o the mea icome was formalized usig the coditios for the first ad secod derivatives of certai fuctios. Whe meetig these coditios, the form of the lowicome groups icome depedece o the mea icome was determied. The drop i the Gii idex after reachig the turig poit is possible oly with the low-icome groups icome growth beig ahead. The shape of the low-icome groups icome depedece o the mea icome after reachig the turig poit must be covex. The data o 9 coutries cofirm the validity of the Kuzets hypothesis for trasitio coutries. For this group of coutries the turig poit of ca. 000 PPP USD was foud. Amog the coutries with lower GDP per capita, Russia is the closest oe to the turig poit. We ca expect reductio of the iequality level i this coutry with icreasig GDP per capita. Refereces Ahluwalia M. (976), Icome distributio ad developmet, America Ecoomic Review 66, pp Barro R. (000), Iequality ad growth i a pael of coutries, Joural of Ecoomic Growth 5, pp Ho-Chua River Huag (004), A flexible oliear iferece to the Kuzets hypothesis, Ecoomics Letters, vol. 84, issue, pp Kusets, S. (955), Ecoomic growth ad icome iequality, America Ecoomic Review 45, pp. 8 Mushiski D.(00), Usig o-parametrics to iform parametric tests of Kuzets hypothesis, Applied Ecoomics Letters, pp Papaek G. ad Ky O. (986), The effect o icome distributio of developmet, the growth rate ad ecoomic strategy, Joural of Developmet Ecoomics 3, pp Savvides A. ad Stegos T. (000), Icome iequality ad ecoomic developmet: evidece from the threshold regressio model, Ecoomics Letters, vol. 69, issue, pp. 07-

14 Sukiassya G. (007), Iequality ad growth: What does the trasitio ecoomy data say?, Joural of comparative ecoomics, vol.35, issue, pp

15 5 Appedix Gii idex GDP per capita. PPP ($) Share of icome or cosumptio. poorest 0% (%) Share of icome or cosumptio. poorest 0% (%) 7 Sloveia Czech Republic Hugary Polad Estoia Lithuaia Slovakia Croatia Latvia Bulgaria Romaia Bosia ad Herzegovia Russia Federatio Macedoia, TFYR Belarus Albaia Ukraie Kazakhsta Armeia Chia Georgia Azerbaija Turkmeista Viet Nam Kyrgyzsta Uzbekista Moldova Tajikista Lao People s Dem. Rep