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Kuznets's inverted-U hypothesis: reply.

The comment by Fosu |6~ makes three points relative to estimate from the U.S. time-series data reported in my paper |8~. These are (a) stochastic error terms in the regressions are autocorrelated, (b) there is high collinearity between the (logarithmic) income (LY) and income-square (LYSQ) terms, and (c) if a "family composition" variable is added, the results are different from what I reported, and support the view that the post-War U.S. lies on the "declining" segment of Kuznets's |7~ inverted-U.

My overall response is that the points mentioned in the comment are inconsequential, do not alter any of the conclusions stated in my paper, and the evidence does not support the view proposed in the comment. I shall deal with each of the three points very briefly and mentioned a few related aspects.

I. Autocorrelation of the Error Terms

Table I reports Gini-regression estimates based on ordinary least-squares (OLS) and an iterative generalized least-squares procedure premised on the postulate of a first-order autoregressive error process (AR1). The estimated models include the complete Kuznets-type quadratic in (logarithm of) real income (GNP per capita) and the shorter version that excludes the quadratic term, both with and without the family-composition variable PCMC which is the percentage of families that contain married couples. Data for PCMC are taken from U.S. Bureau of the Census |10, 200-203~; all other data are the same as used in my original paper |8~.

It is obvious that the correction for autocorrelation makes no significant difference in any case. In the short version that has neither the quadratic income term nor PCMC, OLS estimates show an extremely poor fit, and the income terms has a positive coefficient that is not significant at the usual 5% level; ARI estimates also show a very poor fit, and the income term now has a negative coefficient estimate with a t-statistic of -0.54. The conclusion is the same in both cases, namely, the model explains very little and is clearly misspecified. Even if t-statistic for AR1 estimate of the LY coefficient were larger, as reported in the comment, the fit remains very poor and the model is still misspecified.(1) In other models also, OLS and AR1 estimates lead to the same conclusions.


II. Collinearity between LY and LYSQ Terms

It is well known that high covariance between regressors can be a problem if precision of the estimates falls to such an extent that proper inferences about the parameters are hindered. Typically, such a situation is marked by high |R.sup.2~s but low t-statistics for individual parameter estimates, and one cannot clearly conclude about the separate effects of some of the variables. Collinearity diagnostics are designed primarily for situations of that kind.

The position in this case is entirely different. The quadratic-form estimates in my original paper |8, 1116~ and in Table I show clearly that parameters of both LY and LYSQ are estimated with a high degree of precision, as reflected in very high t-statistics, and there is no problem at all in conducting tests of hypotheses about the individual coefficients. Therefore, collinearity diagnostics are irrelevant to the situation.

Far from being "unreliable" or "unstable", coefficient estimates for LY and LYSQ are sturdy and robust under a variety of circumstances, including autoregressive-error correction. Given the vastly superior fit of the full quadratic and high statistical significance of the individual estimates, this is clearly the model of choice, and much of the discussion in the comment about the "problem" of collinearity in this version is irrelevant and even misleading.

It is also important to remember that even if there were a serious collinearity problem, one cannot simply drop the quadratic term. Besides the fact that term has high statistical significance and fit of the full quadratic model is vastly superior to that without the LYSQ term, a quadratic structure is an essential component of Kuznets's hypothesis. It should be obvious from the extensive literature on the subject that one cannot meaningfully conduct a test of Kuznets's hypothesis without a quadratic income term in the model. Moreover, the scatter plot reported in my page |8, 1117~ indicates that postulating a monotonic relation between inequality and real income (or time) would be a gross misspecification.

III. Addition of a Family-Composition Variable

It is conceivable that changes in family composition might have had some impact on measured interfamily inequality over the period, especially during the 1970s and the 1980s.(2) The most obvious way to take account of that possibility is to add the family-composition variable PCMC to the Kuznets-type quadratic model. Last two rows in Table I contain estimates of such an augmented model. It is clear that the basic structure of the estimates is the same as before although PCMC has a statistically significant coefficient estimate. The estimated parameters of LY and LYSQ show that even holding family composition constant, the period is marked by an uninverted U-Curve structure in which inequality first declines and then increases with income.

It may also be noted that the quadratic model with PCMC is obviously preferable to the shorter version that excludes LYSQ. Besides its conformity to the basic Kuznets-type structure, it is statistically superior to the version without the quadratic income term. It is surprising that the comment does not even mention the quadratic-form estimates after adding PCMC, and instead chooses to focus on the abridged version without LYSQ, which is clearly misspecified when one considers the significance of the LYSQ term and compares |R.sup.2~s and regression standard errors in the two models. It is also interesting to note that having worried about collinearity to the extent of suggesting exclusion of LYSQ, the comment works with a model that could be deemed to have severe collinearity problems on the basis of the well known diagnostics suggested by Belsley, Kuh and Welsch |1~.(3)

Besides the statistical results reported in Table I, there are several other considerations which suggest that changes in family composition cannot account for the increase in income inequality observed since 1970. For example, family-composition changes of the kind mentioned in the comment were rather dramatic during the 1970s and were relatively mild over the 1980s; the proportion of female-headed (non married-couple) families, which are particularly characterized by low incomes, increased by about 35 percent from 1970 to 1980 while the increase from 1980 to 1988 was only about 11 percent.(4) On the contrary, interfamily inequality (Gini) increased only moderately by about 3.1 percent during the 10-year period from 1970 to 1980, and increased much more sharply by about 8.2 percent during the 8-year period 1980-88.(5)

Moreover, while cross-section evidence on intrastate inequality may not be "novel," it is very important. The full-model estimates for 1949, 1959, 1969 and 1979 cross-sections reported in my paper |8, 1118~ reveal the same uninverted-U structure that the time-series data show. As explained in the paper |8, 1119-21~, the cross-section results illustrate rather dramatically how misleading the regression estimates can be if the quadratic income terms is dropped.

In addition to my work, many scholars have reported strong evidence that shows an uninverted-U structure for the U.S. or brings out increasing inequality over the 1970s and/or the 1980s. For example, Braun |3, 520~ recently concluded, "The Kuznets hypothesis that higher wealth will be predictive of lower income inequality is reversed among all 3,136 U.S. counties. Instead, multiple regression shows household income inequality sharply increasing as mean household income grows." Similarly, on the basis of Lorenz-dominance criterion applied to the period 1967-1986, Bishop, Formby and Smith |2, 134~ reported "The tests reveal a sharp rise in U.S. inequality between 1978 and 1982 as well as a shift toward greater inequality over the entire period." Studies by Coughlin and Mandelbaum |4; 5~ and Ray and Rittenoure |9~ have documented and analyzed increasing interstate (regional) inequality which is a component of the total U.S. inequality.

I conclude by saying that there is compelling evidence to suggest that the post-War United States does not conform to the Kuznets-pattern, that inequality is not observed to decline monotonically even at such a high level of economic development, and that an univerted U-curve is observed in which inequality first declines and then rises with increasing income. Nothing stated in the comment alters that scenario; in particular, addition of a family-composition variable to the proper quadratic-form model leaves the univerted-U structure unchanged. As my original paper |8, 1121~ indicated, it is conceivable that the observed increase in inequality is a "short run" departure from the Kuznets-pattern, and a "long run" picture might be in greater harmony with the hypothesis. It is, however, too early to make a judgement on that possibility.

1. The OLS estimates in Table I are based on REG procedure of Statistical Analysis System (SAS). The AR1 estimates are derived from AUTOREG procedure, and the estimation method is iterated "Yule-Walker (YW ITER) which is a feasible generalized least-squares method (GLS). Maximum-likelihood method (ML) yielded estimates for the simple model (with constant and LY terms) that are closer to what the comment reports, but convergence was not achieved. It is interesting that GLS and ML estimates differ only for this misspecified simple model; in all other cases, both methods yield almost identical results.

2. As an aside, care is needed in assessing the effect of changes in family composition on inequality. A proper measure of interfamily inequality should be based on similar family size. Two married-couple families each of which has an income of $20,000, but one of which has 3 members and the other 6, are not "equal". A change in family composition that transforms these into three families whose incomes are $20,000, $10,000 and $10,000, and each of which has 3 members, would generate a better measure of inequality. U.S. Bureau of the Census |10, 3~ shows that average family size in married-couple, male-headed, and female-headed families in 1990 was 3.25, 3.04 and 3.10 respectively. More important, relative to the average size of married-couple families, average size of female-headed families increased from 1970 to 1990.

3. One high "condition number" of 179 is associated with high variance-decomposition factors of 0.999, 0.990 and 0.809. Belsley, Kuh and Welsch |1, 92-93, 106-14~ also explain some weaknesses of the criteria based on VIFs.

4. The proportions are derived from U.S. Bureau of the Census |10, 200-201~, and are approximately 0.108, 0.146 and 0.163 for 1970, 1980 and 1988 respectively. The corresponding Gini indices are 0.354, 0.365 and 0.395.

5. This two-period pattern suggests that proportionate decline in (the percentage of) married-couple families is inversely related with the proportionate increase in interfamily inequality.


1. Belsley, David A., Edwin Kuh, and Roy E. Welsch. Regression Diagnostics. New York: John Wiley & Sons, 1980.

2. Bishop, John A., John P. Formby, and W. James Smith, "Lorenz Dominance and Welfare: Changes in the U.S. Distribution of Income, 1967-1986," Review of Economics and Statistics, February 1991, 134-139.

3. Braun, Denny, "Income Inequality and Economic Development: Geographic Divergence." Social Science Quarterly, September 1991, 520-36.

4. Coughlin, Cletus C. and Thomas B. Mandelbaum, "Why Have State Per Capita Incomes Diverged Recently?" Federal Reserve Bank of St. Louis Review, September/October 1988, 34-36.

5. ------ and ------, "Have Federal Spending and Taxation Contributed to the Divergence of State Per Capita Incomes in the 1980s?" Federal Reserve Bank of St. Louis Review, July/August 1989, 29-42.

6. Fosu, Augustin Kwasi, "Kuznets's Inverted-U Hypothesis: Comment." Southern Economic Journal, January 1993.

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

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

9. Ray, Cadwell L. and R. Lynn Rittenoure, "Recent Regional Growth Patterns: More Inequality." Economic Development Quarterly, 1987, 240-48.

10. U.S. Bureau of the Census, Current Population Reports, Series P-20, No. 447, Household and Family Characteristics: March 1990 and 1989. Washington, DC: U.S. Government Printing Office, 1990.
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Title Annotation:response to article by Augustin Kwasi Fosu in this issue, p. 523
Author:Ram, Rati
Publication:Southern Economic Journal
Date:Jan 1, 1993
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