# Kuznets's inverted-U hypothesis: comment.

I. Introduction

In a recent article appearing in this journal, Ram |5~ presents evidence from the United States contradicting the Kuznets inverted-U hypothesis associated with income inequality, which states that inequality would initially increase with economic development but then would fall eventually |3; 4~. Ram conjectures than the advanced nature of the U.S. economy is such that it would probably lie on the second leg of Kuznets's inverted U. That is, he expected postwar U.S. data to reveal a decreasing relationship between inequality and income levels. However, his empirical results failed to support his expectations. Ram reports |5, 1113~:

The main conclusion is that the data are not consistent with the Kuznets-type quadratic relation between income and distribution; nor do they support the prediction of a monotonic decline in income inequality with growth even at such a high level of economic development. On the contrary, the regression estimates fit well an uninverted U-curve, indicating that inequality first declines with economic growth and then rises after reaching a bottom.

While Ram presents both time-series and across-states results, the time-series evidence appears to provide a stronger case against the Kuznets hypothesis. Furthermore, the time-series results represents the more novel evidence.

The present note finds the Ram's time-series results are attributable, in part, to his use of inequality measures based on family incomes with no control for family composition. Second, the level of development (income) variable employed as a regressor is so strongly correlated with the square of itself that Ram's results on the Kuznets-type quadratic relation are not very meaningful statistically. Third, the error term in Ram's specification is found to be highly autocorrelated,(1) and maximum likelihood estimation that corrects for this potential statistical problem reveals different results in some cases. The main finding here for the postwar U.S., contrary to Ram's empirical results, is that there is a decreasing relationship between inequality and the level of income. The present evidence, therefore, supports Ram's initial conjecture that the relatively advanced U.S. economy may be on the second leg of an inverted U-curve, consistent with the Kuznets hypothesis.

II. Evidence

As Ram |5~ points out, virtually all the tests of the Kuznets hypothesis have involved cross-sectional, usually inter-country data, which are fraught with the usual structural problems, including the probable lack of comparability of the income measures.(2) Ram's use of time-series, as well as across-states, data from the U.S., then, represents an important novelty for all the reasons cited by Ram |5, 1113~. His basic conjecture is that "since the post-War U.S. probably lies to the right of the 'turning point' on the inverted-U," one should expect income inequality to decrease with economic growth.

To test the above hypothesis with time-series data, Ram employs two measures of income inequality: the Gini and Bourguignon's L. Since he obtains similar results with both measures, we shall base the present analysis on the more popular measures, the Gini. Using 1947-1988 data for the U.S., Ram reports the following results (Table I, p. 1116; absolute t ratios in parentheses):

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

where INEQA is the Gini measure of inequality based on family incomes, and ln Y is the natural logarithm of real GNP per capital. As Ram concludes, the above results do not support the hypothesis of a declining inequality with respect to income during the postwar period, contrary to the author's conjecture. Furthermore, as additionally noted by Ram, the results actually suggest a U-shaped inequality-income relationship, in direct contradiction to the inverted-U Kuznets hypothesis.

Employing identically defined variables and exact data sources as Ram's,(3) we first report in Table I as equation (A.1) and (A.2) Ordinarily Least Squares (OLS) results based on the above specifications. It is apparent that these results are, respectively, nearly identical to Ram's as reported above. Unfortunately, we also note that the error term exhibits high autocorrelation as indicated by the very low values of the Durbin-Watson statistic (DW).(4) Using the Maximum Likelihood (ML) estimation to correct for this potential statistical problem reverses the sign of the coefficient of ln Y in the case of the simple regression (equation (B.1) compared to equation (A.1)). The coefficient in equation (B.1) is, moreover, statistically significant, though the adjusted coefficient of determination is still rather low. Hence, it appears that the positive, albeit weakly significant, coefficient on lnY obtained by Ram in equation (1) could be attributable to possible problems of serial correlation in the error term.

Correcting for possible autocorrelation of the error term in the multiple regression where LNYSQ is additionally specified yields estimated regression coefficients not appreciably different from those of the OLS, however (compare equations (B.2) and (A.2)). Hence, these ML results appear to confirm those of Ram's OLS quadratic results. In the case of the quadratic specification, therefore, Ram's basic finding contradicting the Kuznets hypothesis cannot be attributed to problems of serial correlation in the error term.

We note, however, that LNY and LNYSQ are highly positively correlated with a first-order correlation coefficient of .999. This translates to a variance inflation factor (VIF) of 500,(5) and TABULAR DATA OMITTED to a very correlation of -.999 between the estimated coefficients of LNY and LNYSQ. Hence, the estimated coefficients of LNY and LNYSQ are subject to a great deal of instability between them, notwithstanding their statistically significant t ratios. That is, the very high VIF suggests that the estimates are unreliable, while the extremely large negative correlation between the estimates is an indication of a potentially high tradeoff between the estimates. The results of the quadratic specification cannot, therefore, be relied on as providing evidence in favor of a U-shaped, rather than Kuznet's inverted-U, relationship between inequality and earnings.

Nevertheless, that the quadratic equations provide higher goodness of fit than their simple regression counterparts suggests that there could be an important variable omitted from the simple regression model. It is possibly this phenomenon that is being reflected by the better fit of the quadratic specification.

The Role of Family Composition

As noted above, the measure of income inequality used by Ram and in our results of Table I is based on family income. Unfortunately, family composition has been changing over time in the U.S.,(6) and this structural change may have important implications for income inequality. For example, the proportion of families with married couples, despite its earlier near-constancy at about 87 percent, appears to have declined appreciably since the late 1960s, reaching a low of 79 percent in 1988. The decline in the importance of this family type appears to coincide roughly with a rise in family income inequality. A fall in the fraction of the heretofore dominant two-parent families, with a concurrent increase in the proportion of the relatively low-earning single-parent families, should raise income inequality among families. Unfortunately, Kuznets |3; 4~ did not incorporate these structural changes in family composition into his analysis. Thus, where inequality is based on family incomes, evidence bearing on the Kuznets hypothesis must appropriately control for family composition.

A family composition variable, the proportion of families with married couples MCFAM, is now included in the regression, along with LNY.(7) The results are reported as equations (A.3) and (B.3) for the OLS and ML, respectively. As both sets of results indicate, and consistent with expectations, a higher MCFAM tends to lower inequality. Hence, the rise in inequality as of the late 1960s, depicted in Figure 2, may be explained, at least in part, by the decrease in MCFAM over roughly the same period. Moreover, the coefficient of LNY is now strongly negative and the goodness of fit, as measured by the adjusted coefficient of determination, appears to be substantially greater than that for the specifications containing LNY alone or both LNY and LNYSQ. Meanwhile, the first-order correlation coefficient for LNY versus MCFAM is relatively small. At -.841, it yields a VIF of only 3.4, which is well within the generally acceptable range,(8) in contrast with a VIF of 500.0 for the equation containing LNY and LNYSQ. Thus, family composition appears to be an important variable in the inequality equation. Meanwhile, the extremely high collinearity between LNY and LNYSQ renders the results based on the quadratic specification rather unreliable.

III. Conclusion

In summary, we conclude that once family composition is accounted for, family income inequality decreased with economic growth during the postwar period in the U.S. This finding, contrary to Ram's empirical results, is consistent with the advanced-economy hypothesis offered by Ram, which is in concert with the Kuznets hypothesis.

1. This is indicated by the very low values of the Durbin-Watson statistic. Also, see footnote 4 below.

2. For a review of these problems see, for example, Saith |6~.

3. All variables and data sources are defined in Table I.

4. Alternatively, these may reflect possible misspecifications of the inequality equation |1~.

5. The VIF is computed as 1/(1 - |r.sup.2~), where r is the correlation coefficient between LNY and LNYSQ. While there are no generally accepted criteria for judging when a VIF is definitely deleterious, the rule of thumb is that for standardized data, a VIF greater than 10 indicates harmful collinearity |2, 153~.

6. See Figure 1.

7. Source: See Table 1.

8. See footnote 5 above.

References

1. Chaudhuri, M., "Autocorrelated Disturbances in the Light of Specification Analysis." Journal of Econometrics, 1977, 301-13.

2. Kennedy, Peter. A Guide to Economics. Cambridge, Mass.: MIT Press, 1985.

3. Kuznets, Simon, "Economic Growth and Income Inequality." American Economic Review, March 1955, 1-28.

4. ------, "Quantitative Aspects of the Economic Growth of Nations: VIII. Distribution of Income by Size." Economic Development and Cultural Change, January 1963, Part II, 1-8.

5. Ram, Rati, "Kuznets's Inverted-U Hypothesis: Evidence from a Highly Developed Country." Southern Economic Journal, April 1991, 1112-23.

6. Saith, Ashwani, "Development and Distribution: A Critique of the Cross-Country U Hypothesis." Journal of Development Economics, December 1983, 367-82.

7. U.S. Bureau of the Census. Money Income of Households, Families, and Persons in the United States: 1987. Current Population Reports, Series P-60, No. 162, Washington, D.C.: U.S. Government Printing Office, 1989.

8. -------. Money Income and Poverty Status in the United States: 1988. Current Population Reports, Series P-60. No. 166, Washington, D.C.: U.S. Government Printing Office, 1989.

9. ------. Household and Family Characteristics: March 1990 and March 1989. Current Population reports, Series P-20, No. 447, Washington D.C.: U.S. Government Printing Office, 1990.

10. U.S. Government. Economic Report of the President. Washington, D.C.: U.S. Government Printing Office, 1990.

In a recent article appearing in this journal, Ram |5~ presents evidence from the United States contradicting the Kuznets inverted-U hypothesis associated with income inequality, which states that inequality would initially increase with economic development but then would fall eventually |3; 4~. Ram conjectures than the advanced nature of the U.S. economy is such that it would probably lie on the second leg of Kuznets's inverted U. That is, he expected postwar U.S. data to reveal a decreasing relationship between inequality and income levels. However, his empirical results failed to support his expectations. Ram reports |5, 1113~:

The main conclusion is that the data are not consistent with the Kuznets-type quadratic relation between income and distribution; nor do they support the prediction of a monotonic decline in income inequality with growth even at such a high level of economic development. On the contrary, the regression estimates fit well an uninverted U-curve, indicating that inequality first declines with economic growth and then rises after reaching a bottom.

While Ram presents both time-series and across-states results, the time-series evidence appears to provide a stronger case against the Kuznets hypothesis. Furthermore, the time-series results represents the more novel evidence.

The present note finds the Ram's time-series results are attributable, in part, to his use of inequality measures based on family incomes with no control for family composition. Second, the level of development (income) variable employed as a regressor is so strongly correlated with the square of itself that Ram's results on the Kuznets-type quadratic relation are not very meaningful statistically. Third, the error term in Ram's specification is found to be highly autocorrelated,(1) and maximum likelihood estimation that corrects for this potential statistical problem reveals different results in some cases. The main finding here for the postwar U.S., contrary to Ram's empirical results, is that there is a decreasing relationship between inequality and the level of income. The present evidence, therefore, supports Ram's initial conjecture that the relatively advanced U.S. economy may be on the second leg of an inverted U-curve, consistent with the Kuznets hypothesis.

II. Evidence

As Ram |5~ points out, virtually all the tests of the Kuznets hypothesis have involved cross-sectional, usually inter-country data, which are fraught with the usual structural problems, including the probable lack of comparability of the income measures.(2) Ram's use of time-series, as well as across-states, data from the U.S., then, represents an important novelty for all the reasons cited by Ram |5, 1113~. His basic conjecture is that "since the post-War U.S. probably lies to the right of the 'turning point' on the inverted-U," one should expect income inequality to decrease with economic growth.

To test the above hypothesis with time-series data, Ram employs two measures of income inequality: the Gini and Bourguignon's L. Since he obtains similar results with both measures, we shall base the present analysis on the more popular measures, the Gini. Using 1947-1988 data for the U.S., Ram reports the following results (Table I, p. 1116; absolute t ratios in parentheses):

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

where INEQA is the Gini measure of inequality based on family incomes, and ln Y is the natural logarithm of real GNP per capital. As Ram concludes, the above results do not support the hypothesis of a declining inequality with respect to income during the postwar period, contrary to the author's conjecture. Furthermore, as additionally noted by Ram, the results actually suggest a U-shaped inequality-income relationship, in direct contradiction to the inverted-U Kuznets hypothesis.

Employing identically defined variables and exact data sources as Ram's,(3) we first report in Table I as equation (A.1) and (A.2) Ordinarily Least Squares (OLS) results based on the above specifications. It is apparent that these results are, respectively, nearly identical to Ram's as reported above. Unfortunately, we also note that the error term exhibits high autocorrelation as indicated by the very low values of the Durbin-Watson statistic (DW).(4) Using the Maximum Likelihood (ML) estimation to correct for this potential statistical problem reverses the sign of the coefficient of ln Y in the case of the simple regression (equation (B.1) compared to equation (A.1)). The coefficient in equation (B.1) is, moreover, statistically significant, though the adjusted coefficient of determination is still rather low. Hence, it appears that the positive, albeit weakly significant, coefficient on lnY obtained by Ram in equation (1) could be attributable to possible problems of serial correlation in the error term.

Correcting for possible autocorrelation of the error term in the multiple regression where LNYSQ is additionally specified yields estimated regression coefficients not appreciably different from those of the OLS, however (compare equations (B.2) and (A.2)). Hence, these ML results appear to confirm those of Ram's OLS quadratic results. In the case of the quadratic specification, therefore, Ram's basic finding contradicting the Kuznets hypothesis cannot be attributed to problems of serial correlation in the error term.

We note, however, that LNY and LNYSQ are highly positively correlated with a first-order correlation coefficient of .999. This translates to a variance inflation factor (VIF) of 500,(5) and TABULAR DATA OMITTED to a very correlation of -.999 between the estimated coefficients of LNY and LNYSQ. Hence, the estimated coefficients of LNY and LNYSQ are subject to a great deal of instability between them, notwithstanding their statistically significant t ratios. That is, the very high VIF suggests that the estimates are unreliable, while the extremely large negative correlation between the estimates is an indication of a potentially high tradeoff between the estimates. The results of the quadratic specification cannot, therefore, be relied on as providing evidence in favor of a U-shaped, rather than Kuznet's inverted-U, relationship between inequality and earnings.

Nevertheless, that the quadratic equations provide higher goodness of fit than their simple regression counterparts suggests that there could be an important variable omitted from the simple regression model. It is possibly this phenomenon that is being reflected by the better fit of the quadratic specification.

The Role of Family Composition

As noted above, the measure of income inequality used by Ram and in our results of Table I is based on family income. Unfortunately, family composition has been changing over time in the U.S.,(6) and this structural change may have important implications for income inequality. For example, the proportion of families with married couples, despite its earlier near-constancy at about 87 percent, appears to have declined appreciably since the late 1960s, reaching a low of 79 percent in 1988. The decline in the importance of this family type appears to coincide roughly with a rise in family income inequality. A fall in the fraction of the heretofore dominant two-parent families, with a concurrent increase in the proportion of the relatively low-earning single-parent families, should raise income inequality among families. Unfortunately, Kuznets |3; 4~ did not incorporate these structural changes in family composition into his analysis. Thus, where inequality is based on family incomes, evidence bearing on the Kuznets hypothesis must appropriately control for family composition.

A family composition variable, the proportion of families with married couples MCFAM, is now included in the regression, along with LNY.(7) The results are reported as equations (A.3) and (B.3) for the OLS and ML, respectively. As both sets of results indicate, and consistent with expectations, a higher MCFAM tends to lower inequality. Hence, the rise in inequality as of the late 1960s, depicted in Figure 2, may be explained, at least in part, by the decrease in MCFAM over roughly the same period. Moreover, the coefficient of LNY is now strongly negative and the goodness of fit, as measured by the adjusted coefficient of determination, appears to be substantially greater than that for the specifications containing LNY alone or both LNY and LNYSQ. Meanwhile, the first-order correlation coefficient for LNY versus MCFAM is relatively small. At -.841, it yields a VIF of only 3.4, which is well within the generally acceptable range,(8) in contrast with a VIF of 500.0 for the equation containing LNY and LNYSQ. Thus, family composition appears to be an important variable in the inequality equation. Meanwhile, the extremely high collinearity between LNY and LNYSQ renders the results based on the quadratic specification rather unreliable.

III. Conclusion

In summary, we conclude that once family composition is accounted for, family income inequality decreased with economic growth during the postwar period in the U.S. This finding, contrary to Ram's empirical results, is consistent with the advanced-economy hypothesis offered by Ram, which is in concert with the Kuznets hypothesis.

1. This is indicated by the very low values of the Durbin-Watson statistic. Also, see footnote 4 below.

2. For a review of these problems see, for example, Saith |6~.

3. All variables and data sources are defined in Table I.

4. Alternatively, these may reflect possible misspecifications of the inequality equation |1~.

5. The VIF is computed as 1/(1 - |r.sup.2~), where r is the correlation coefficient between LNY and LNYSQ. While there are no generally accepted criteria for judging when a VIF is definitely deleterious, the rule of thumb is that for standardized data, a VIF greater than 10 indicates harmful collinearity |2, 153~.

6. See Figure 1.

7. Source: See Table 1.

8. See footnote 5 above.

References

1. Chaudhuri, M., "Autocorrelated Disturbances in the Light of Specification Analysis." Journal of Econometrics, 1977, 301-13.

2. Kennedy, Peter. A Guide to Economics. Cambridge, Mass.: MIT Press, 1985.

3. Kuznets, Simon, "Economic Growth and Income Inequality." American Economic Review, March 1955, 1-28.

4. ------, "Quantitative Aspects of the Economic Growth of Nations: VIII. Distribution of Income by Size." Economic Development and Cultural Change, January 1963, Part II, 1-8.

5. Ram, Rati, "Kuznets's Inverted-U Hypothesis: Evidence from a Highly Developed Country." Southern Economic Journal, April 1991, 1112-23.

6. Saith, Ashwani, "Development and Distribution: A Critique of the Cross-Country U Hypothesis." Journal of Development Economics, December 1983, 367-82.

7. U.S. Bureau of the Census. Money Income of Households, Families, and Persons in the United States: 1987. Current Population Reports, Series P-60, No. 162, Washington, D.C.: U.S. Government Printing Office, 1989.

8. -------. Money Income and Poverty Status in the United States: 1988. Current Population Reports, Series P-60. No. 166, Washington, D.C.: U.S. Government Printing Office, 1989.

9. ------. Household and Family Characteristics: March 1990 and March 1989. Current Population reports, Series P-20, No. 447, Washington D.C.: U.S. Government Printing Office, 1990.

10. U.S. Government. Economic Report of the President. Washington, D.C.: U.S. Government Printing Office, 1990.

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Title Annotation: | comment on Rati Ram, Southern Economic Journal, p. 1112, April 1991 |
---|---|

Author: | Fosu, Augustin Kwasi |

Publication: | Southern Economic Journal |

Date: | Jan 1, 1993 |

Words: | 1769 |

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