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The Kuznets hypothesis: An indirect test
economics
letters Economics Letters 54 (1997) 81-85
ELSEVIER
The Kuznets hypothesis: An indirect test Zaki Eusufzai Department of Economics, Loyola Marymount University, 7900 Loyola Boulevard, Los Angeles, CA 90045.8410, USA Received 25 March 1996; final version received 21) February 1997
Abstract
The Quandt Log Likelihood Ratio test, used on cross country data, shows that there is a break in the relationship between per capita GNP and income inequality; the breakpoint occurs at a per capita between $696 ancl $773.
Keywords: Kuznets hypothesis; Quandt log-likelihood ratio test; Income inequality JEL t'lassification: 04; O11
1. Introduction
In 1955 Simon Kuznets proposed the hypothesis that during the course of economic growth a country would initially experience worsening income inequality. As economic growth proceeded further, the income inequality would reach a peak and then begin to improve (Kuznets, 1955). While Kuznets did not empirically test his hypothesis, he did illustrate the process with a numerical example. The Kuznets hypothesis, if true, has important policy implications for developing countries. If true, developing countries following policies to achieve economic growth should expect to see the income inequality in their countries rise before it eventually falls. As such, developing countries are keenly interested in the level of per capita GNP at which income inequality can be expected to start improving, i.e. the 'turning point'. Because of its policy implications, many attempts have been made to (a) empirically test the hypothesis, and (b) calculate the turning point based on the empirical estimates. However, no research has conclusively shown the hypothesis to be true or false or has come up with a stable dollar value for the turning point. In fact, there has been a wide variation in the calculation of the turning point. In the existing literature the turning points have been mathematically calculated following estimation of the functional relationship. The variations possible in the procedure have given 0165-1765/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved Pll S 0 1 6 5 - 1 7 6 5 ( 9 6 ) 0 0 9 0 0 - 7
Z. Eusufzai / Economics Letters 54 (1997) 81-85
82
rise to n u m e r o u s s t u d i e s . 1 A different measure of inequality, or a different functional form, or a different data set have been used to extend the literature. What I do in this paper is to attack the problem from a different point of view. I argue that if the relationship between per capita GNP and income inequality is indeed inverted Ushaped, then econometric tests that test for the stability of regression equations should be able to detect a break in the relationship, and that the per capita GNP at which the break occurs will be the turning point. I use such a test to show that (a) there is indeed a break in the relationship between per capita GNP and income distribution, (b) that there exists a positive (negative) relationship between per capita GNP and income distribution before (after) the breakpoint, (c) the breakpoint occurs at a per capita GNP of between $696 and $773 (1970 US dollars), and (d) these results hold true for a variety of income inequality measures.
2. Logic behind the empirical test If, as hypothesized by Kuznets, the relationship between per capita GNP and income inequality is indeed U-shaped, then the relationship can be thought of as being decomposable into two parts: the positive relation between the two variables up to an initial level of per capita GNP and then a negative relationship for higher levels ~f per capita GNE Looked at from this point of view, a suitable econometric test should be able to detect this break in the relationship as well as pinpoint the per capita GNP at which this break takes place. Such a test is the Quandt log-likelihood ratio test (QLRT). The test is laid out in a series of papers, including Goldfeld and Quandt (1972, 1976) and Brown et al. (1975). This technique is appropriate when it is believed that the regression relationship may have changed abruptly at an unknown point t = r from one constant relationship specified by /3~.~, o'2~ to another constant relationship specified b y / 3 ~ , ~r.,:. For each arbitrary r, we compute and plot A, - ~r. log ~" + ½ ( T - r) log ~r~ - ~Tiogl w~" ,
(1)
where tt i - (residual sum of squares for the first ro~,,)/r, o'~ = (residual sum of squares for T - r o ~ ) l T - r, o " = (residual sum of squares for To~)IT. The value of r at which A, attains its minimum value is the estimate of the point at which the switch from one relationship to another has occurred. Since the distribution of Ar on the null Ilypothesis is unknown, no test has been devised for minimum A,. A graph of A, against r indicates any switch in the relationship as well as the precise point (per capita GNP) at which the switch takes place.
* See Ram (1995) and Hsit:~: and $myth (1994) for a partial list of such studies.
Z. Eusufzai / Economics Letters 54 (1997) 81-85
83
3. Data All the data used in this paper are obtained from A n a n d and K a n b u r (1993). The per capita G N P is measured in 1970 US dollars. The six inequality indices considered are (i) Theil's entropy index, T; (ii) Theil's second measure L, which is the log of the ratio of the arithmetic to the geometric mean income of the distribution; (iii) the squared coefficient of variation, $2; (iv) a decomposable transform of the Atkinson inequality i n d e x , / ( 2 ) ; (v) the Gini coefficient, G ; and (vi) the variance of log-income, tr 2. In each case a higher value for the inequality measure reflects a higher degree of inequality. D a t a for these variables are available for a total of 54 countries, 41 of them developing countries ?
4. Empirical results For our purpose, we had r take on a value of 10 to 44, thus generating 35 sets of regressions for each of the six inequality measures, each set determining a value of A, according to Eq. (1). 3 The results of the Q L R T are presented in Fig. l(a) and (b). We can examine the figures to determine the minimum ;t for each income inequality measure. 4 From Fig. l(a), we see that for three of the inequality measures - T, S 2, and G - there is a clear minimum at r = 35, r = 37 and r = 35, respectively (for r = 35, the corresponding per capita G N P is $696.1, f o r r - - 3 7 it is $773.4). For L (in Fig. l(b)), the minimum value is reached at r = 10. If the value for r = 10 is ignored, then the second lowest value occurs at r = 33. s F o r / ( 2 ) and tr 2, beyond the corner values, there is no 'natural' minimum. For all that, the results are striking. Four out of the six inequality measures show a break in their relationship with per capita G N P and all around a similar per capita G N P level! A curious aspect of the figures is that for most of the graphs there is not a constant, uniform progress towards a minimum and a smooth ascent thereafter. Rather, even with a global minimum, they seem to be characterized by multiple local minima. While a definitive reason for this cannot be given, two things may be pointed out: (a) there is no requirement in the test for such an uniform progress towards the minimum, 6 and (b) in as much as the multiple local minima reflect a 'tendency' for regime change, it could be occurring because of the cross
2 Further details can be found in Anand and Kanbur (1993). 3The range for r was chosen so as to give a minimum of five degrees of freedom. The dependent variable in the regression was an inequality measure and the explanatory variables were a constant term and the log of per capita ONE 4 As noted in Goldfeld and Quandt (1976, p. 8), and Brown et al. (1975, p. 157), the usual test statistic -2 log A does not have the 'appropriate' asymptotic chi-squar~ distribution. Rather, for certain special cases (see Feder, 1975a,b), the likelihood ratio test statistic is distributed as the maximum of a number of currelated chi-square variables. Hence, it should be recognized that there is an implicit confidence interval around the switch point. 5 Since r = 10 represents a 'corner value' and is an outlier~ ignoring it may be justified. A minimum at a corner value is hard to interpret without more data points. 6 In Brown et al. (1975) the procedure was used on a data set to illustrate the method. The resulting graph of lambda, as shown in their Fig. 7 (p. 161), exhibits the same kind of multiple minima that Fig. l(a) and (b) do.
84
Z. Eusufzai / Economics Letters 54 (1997) 81-85 r
i ....
i
r
, ~ i , i ,
(b)
(a)
i -2
~ qi
L-,.-G_ ! ,4
,10
• 12 1 -14
~
"
• 16 i -14 ~"
• 18 "- l e ,+.
Fig. I.
country nature of the data. Each data point is being generated by a different country and empirical tests of the Kuznets hypothesis, in using a cross country data set to test the hypothesis, implicitly assume that the data set can be thought to have been generated by one country across time. In reality, since the underlying structure of the income-inequality relationship may be slightly different across countries, this may be generating the multiple local minima. The QLRT only indicates some kind of a break in the relationship, but we cannot be sure, by this test alone, that the break reflects the Kuznets hypothesis. To be sure of that, we undertake a further test, calculating the correlation coefficient between the inequality measures and the per capita GNP, prior to and subsequent to the breakpoint. If the break is reflecting the Kuznets hypothesis, we should expect to find a positive relation between the two variables prior to the break and a negative relationship subsequent to the break. The results are displayed in Table I, Again the results are striking. For every one of the four inequality mea-:ures, the Pearson coff¢latioa coefficient is positive prior to the breakpoint indicated by the QLRT and negative subsequently. Tests of equality of the correlation coefficients between the two periods using the Fisher r to z transform (see Snedecor and Cochran, 1980) indicate that ~he null hypothesis of equality of the correlation coefficients can be rejected for all four of the ii~equality measures at the 5% level of significance.
Z. Eusufzai / Economics Letters ,¢4 (1997) 81-85
85
Table 1 Correlation coefficients Correlation coefficient Variable
Before break
After break
T L S" G
0.25 0.36 0.21 (I.26
-0.49 -0.44 -0.54 -0.46
5. Conclusions We cannot hope for a decisive conclusion on the validity of the Kuznets hypothesis on the basis of a single test. Still, ti~ere seems to be some promise in using regression stability tests to further investigate the issue.
Acknowledgements I am grateful to J. Earley and an anonymous referee for helpful comments and suggestions.
References Anand, S. and R. Kanbur, 1903, The Kuznets process and the inequality-development relationship, Journal of Developmer.~t Economics 40, 25-52. Brown, R., J. Durbin and J. Evans, 1975, Techniques for testing the constancy of regression relationships over time, Journal of the Royal Statistical Society Series B 37, 149-192. Feder, El., 1975a, On asymptotic distribution theory in segmented regression problems-identified case, The Annals of Statistics 3, 49-.-83. Feder, P.I., 1975b, The log-likelihood ratio in segmented regression, The Annals of Statistics 3, 84-97. Goldfeld, S.M. and R.E. Quandt, 1972, Non-linear methods in econometrics (North-Holland, Amsterdam). Goldfeld, S.M. and R.E. Quandt, 1976, Studies in non-linear estimation (Ballinger Publishing Company, Cambridge). Hsing, Y. and D. Smyth, 1994, Kuznets's inverted-U hypothesis revisited, Applied Economics Letters 1, 111-113. Kuzncts, S., 1955, Economic growth and income inequality, American Economic Review 45, I--28. Ram, R., 1995, Economic development and income inequality: An overlooked regression constraint, Economic Development and Cultural Change 43, 425-434. Snedecor, G.W. and W.G. Cochran, 198(I, Statistical methods, 7th edn. (Iowa State University Press. Ames).
letters Economics Letters 54 (1997) 81-85
ELSEVIER
The Kuznets hypothesis: An indirect test Zaki Eusufzai Department of Economics, Loyola Marymount University, 7900 Loyola Boulevard, Los Angeles, CA 90045.8410, USA Received 25 March 1996; final version received 21) February 1997
Abstract
The Quandt Log Likelihood Ratio test, used on cross country data, shows that there is a break in the relationship between per capita GNP and income inequality; the breakpoint occurs at a per capita between $696 ancl $773.
Keywords: Kuznets hypothesis; Quandt log-likelihood ratio test; Income inequality JEL t'lassification: 04; O11
1. Introduction
In 1955 Simon Kuznets proposed the hypothesis that during the course of economic growth a country would initially experience worsening income inequality. As economic growth proceeded further, the income inequality would reach a peak and then begin to improve (Kuznets, 1955). While Kuznets did not empirically test his hypothesis, he did illustrate the process with a numerical example. The Kuznets hypothesis, if true, has important policy implications for developing countries. If true, developing countries following policies to achieve economic growth should expect to see the income inequality in their countries rise before it eventually falls. As such, developing countries are keenly interested in the level of per capita GNP at which income inequality can be expected to start improving, i.e. the 'turning point'. Because of its policy implications, many attempts have been made to (a) empirically test the hypothesis, and (b) calculate the turning point based on the empirical estimates. However, no research has conclusively shown the hypothesis to be true or false or has come up with a stable dollar value for the turning point. In fact, there has been a wide variation in the calculation of the turning point. In the existing literature the turning points have been mathematically calculated following estimation of the functional relationship. The variations possible in the procedure have given 0165-1765/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved Pll S 0 1 6 5 - 1 7 6 5 ( 9 6 ) 0 0 9 0 0 - 7
Z. Eusufzai / Economics Letters 54 (1997) 81-85
82
rise to n u m e r o u s s t u d i e s . 1 A different measure of inequality, or a different functional form, or a different data set have been used to extend the literature. What I do in this paper is to attack the problem from a different point of view. I argue that if the relationship between per capita GNP and income inequality is indeed inverted Ushaped, then econometric tests that test for the stability of regression equations should be able to detect a break in the relationship, and that the per capita GNP at which the break occurs will be the turning point. I use such a test to show that (a) there is indeed a break in the relationship between per capita GNP and income distribution, (b) that there exists a positive (negative) relationship between per capita GNP and income distribution before (after) the breakpoint, (c) the breakpoint occurs at a per capita GNP of between $696 and $773 (1970 US dollars), and (d) these results hold true for a variety of income inequality measures.
2. Logic behind the empirical test If, as hypothesized by Kuznets, the relationship between per capita GNP and income inequality is indeed U-shaped, then the relationship can be thought of as being decomposable into two parts: the positive relation between the two variables up to an initial level of per capita GNP and then a negative relationship for higher levels ~f per capita GNE Looked at from this point of view, a suitable econometric test should be able to detect this break in the relationship as well as pinpoint the per capita GNP at which this break takes place. Such a test is the Quandt log-likelihood ratio test (QLRT). The test is laid out in a series of papers, including Goldfeld and Quandt (1972, 1976) and Brown et al. (1975). This technique is appropriate when it is believed that the regression relationship may have changed abruptly at an unknown point t = r from one constant relationship specified by /3~.~, o'2~ to another constant relationship specified b y / 3 ~ , ~r.,:. For each arbitrary r, we compute and plot A, - ~r. log ~" + ½ ( T - r) log ~r~ - ~Tiogl w~" ,
(1)
where tt i - (residual sum of squares for the first ro~,,)/r, o'~ = (residual sum of squares for T - r o ~ ) l T - r, o " = (residual sum of squares for To~)IT. The value of r at which A, attains its minimum value is the estimate of the point at which the switch from one relationship to another has occurred. Since the distribution of Ar on the null Ilypothesis is unknown, no test has been devised for minimum A,. A graph of A, against r indicates any switch in the relationship as well as the precise point (per capita GNP) at which the switch takes place.
* See Ram (1995) and Hsit:~: and $myth (1994) for a partial list of such studies.
Z. Eusufzai / Economics Letters 54 (1997) 81-85
83
3. Data All the data used in this paper are obtained from A n a n d and K a n b u r (1993). The per capita G N P is measured in 1970 US dollars. The six inequality indices considered are (i) Theil's entropy index, T; (ii) Theil's second measure L, which is the log of the ratio of the arithmetic to the geometric mean income of the distribution; (iii) the squared coefficient of variation, $2; (iv) a decomposable transform of the Atkinson inequality i n d e x , / ( 2 ) ; (v) the Gini coefficient, G ; and (vi) the variance of log-income, tr 2. In each case a higher value for the inequality measure reflects a higher degree of inequality. D a t a for these variables are available for a total of 54 countries, 41 of them developing countries ?
4. Empirical results For our purpose, we had r take on a value of 10 to 44, thus generating 35 sets of regressions for each of the six inequality measures, each set determining a value of A, according to Eq. (1). 3 The results of the Q L R T are presented in Fig. l(a) and (b). We can examine the figures to determine the minimum ;t for each income inequality measure. 4 From Fig. l(a), we see that for three of the inequality measures - T, S 2, and G - there is a clear minimum at r = 35, r = 37 and r = 35, respectively (for r = 35, the corresponding per capita G N P is $696.1, f o r r - - 3 7 it is $773.4). For L (in Fig. l(b)), the minimum value is reached at r = 10. If the value for r = 10 is ignored, then the second lowest value occurs at r = 33. s F o r / ( 2 ) and tr 2, beyond the corner values, there is no 'natural' minimum. For all that, the results are striking. Four out of the six inequality measures show a break in their relationship with per capita G N P and all around a similar per capita G N P level! A curious aspect of the figures is that for most of the graphs there is not a constant, uniform progress towards a minimum and a smooth ascent thereafter. Rather, even with a global minimum, they seem to be characterized by multiple local minima. While a definitive reason for this cannot be given, two things may be pointed out: (a) there is no requirement in the test for such an uniform progress towards the minimum, 6 and (b) in as much as the multiple local minima reflect a 'tendency' for regime change, it could be occurring because of the cross
2 Further details can be found in Anand and Kanbur (1993). 3The range for r was chosen so as to give a minimum of five degrees of freedom. The dependent variable in the regression was an inequality measure and the explanatory variables were a constant term and the log of per capita ONE 4 As noted in Goldfeld and Quandt (1976, p. 8), and Brown et al. (1975, p. 157), the usual test statistic -2 log A does not have the 'appropriate' asymptotic chi-squar~ distribution. Rather, for certain special cases (see Feder, 1975a,b), the likelihood ratio test statistic is distributed as the maximum of a number of currelated chi-square variables. Hence, it should be recognized that there is an implicit confidence interval around the switch point. 5 Since r = 10 represents a 'corner value' and is an outlier~ ignoring it may be justified. A minimum at a corner value is hard to interpret without more data points. 6 In Brown et al. (1975) the procedure was used on a data set to illustrate the method. The resulting graph of lambda, as shown in their Fig. 7 (p. 161), exhibits the same kind of multiple minima that Fig. l(a) and (b) do.
84
Z. Eusufzai / Economics Letters 54 (1997) 81-85 r
i ....
i
r
, ~ i , i ,
(b)
(a)
i -2
~ qi
L-,.-G_ ! ,4
,10
• 12 1 -14
~
"
• 16 i -14 ~"
• 18 "- l e ,+.
Fig. I.
country nature of the data. Each data point is being generated by a different country and empirical tests of the Kuznets hypothesis, in using a cross country data set to test the hypothesis, implicitly assume that the data set can be thought to have been generated by one country across time. In reality, since the underlying structure of the income-inequality relationship may be slightly different across countries, this may be generating the multiple local minima. The QLRT only indicates some kind of a break in the relationship, but we cannot be sure, by this test alone, that the break reflects the Kuznets hypothesis. To be sure of that, we undertake a further test, calculating the correlation coefficient between the inequality measures and the per capita GNP, prior to and subsequent to the breakpoint. If the break is reflecting the Kuznets hypothesis, we should expect to find a positive relation between the two variables prior to the break and a negative relationship subsequent to the break. The results are displayed in Table I, Again the results are striking. For every one of the four inequality mea-:ures, the Pearson coff¢latioa coefficient is positive prior to the breakpoint indicated by the QLRT and negative subsequently. Tests of equality of the correlation coefficients between the two periods using the Fisher r to z transform (see Snedecor and Cochran, 1980) indicate that ~he null hypothesis of equality of the correlation coefficients can be rejected for all four of the ii~equality measures at the 5% level of significance.
Z. Eusufzai / Economics Letters ,¢4 (1997) 81-85
85
Table 1 Correlation coefficients Correlation coefficient Variable
Before break
After break
T L S" G
0.25 0.36 0.21 (I.26
-0.49 -0.44 -0.54 -0.46
5. Conclusions We cannot hope for a decisive conclusion on the validity of the Kuznets hypothesis on the basis of a single test. Still, ti~ere seems to be some promise in using regression stability tests to further investigate the issue.
Acknowledgements I am grateful to J. Earley and an anonymous referee for helpful comments and suggestions.
References Anand, S. and R. Kanbur, 1903, The Kuznets process and the inequality-development relationship, Journal of Developmer.~t Economics 40, 25-52. Brown, R., J. Durbin and J. Evans, 1975, Techniques for testing the constancy of regression relationships over time, Journal of the Royal Statistical Society Series B 37, 149-192. Feder, El., 1975a, On asymptotic distribution theory in segmented regression problems-identified case, The Annals of Statistics 3, 49-.-83. Feder, P.I., 1975b, The log-likelihood ratio in segmented regression, The Annals of Statistics 3, 84-97. Goldfeld, S.M. and R.E. Quandt, 1972, Non-linear methods in econometrics (North-Holland, Amsterdam). Goldfeld, S.M. and R.E. Quandt, 1976, Studies in non-linear estimation (Ballinger Publishing Company, Cambridge). Hsing, Y. and D. Smyth, 1994, Kuznets's inverted-U hypothesis revisited, Applied Economics Letters 1, 111-113. Kuzncts, S., 1955, Economic growth and income inequality, American Economic Review 45, I--28. Ram, R., 1995, Economic development and income inequality: An overlooked regression constraint, Economic Development and Cultural Change 43, 425-434. Snedecor, G.W. and W.G. Cochran, 198(I, Statistical methods, 7th edn. (Iowa State University Press. Ames).