Asymptotic inference of progressivity dominance

J. Klavus Journal of Health Economics 20 2001 363–377 365 income inequality and the concentration curve of taxes or other payments. Distribution-free statistical tests are constructed to test for differences in the curves as well as for changes in progressivity over time. Following a description of data and variables in Section 3, a brief overview on the implications of economic changes for the health care system is presented along with the main results concerning progressivity changes and dominance relations. The last section summarises and discusses the findings and the method.

2. Asymptotic inference of progressivity dominance

Let Lp be the Lorenz curve of income, interpreted as the fraction of total income received by the lowest pth fraction of the population arranged in ascending order of income. The concentration curve of taxes, Cp, is given by the proportion of taxes paid by the lowest p th fraction of the population. The statistical inference of progressivity involves testing for differences in the ordinates of Lp and Cp at given percentile points, p. In order to conduct the statistical test the joint sampling distribution of Lp and Cp must be known. Consider a sample of observations Y j for j = 1, . . . , N , ranked in ascending order of income. Let w j be the weight attached to the jth observation in the sample j i implies Y j Y i . With sample weights, the proportion of the population represented by each observation is not necessarily the same and, therefore, weighted ranks R i are defined as the cumulative sum of weighted observations up to the midpoint of each group interval cf. Lerman and Yitzhaki, 1989 R i = i− 1 j = w j + w i 2 1 where w j and w i are the population weights the aggregate number of people represented by the ith and jth individual in the sample and w = 0. A relative rank variable r i is obtained by dividing each rank by the sum of the population weights. To construct income groups the ordered sample is partitioned into K groups of equal size, so that the sum of relative ranks at the group limits equals or is just less than p i , where p i is the abscissae p i = iK + 1 for i = 1, . . . , K, corresponding to the income groups. If, for example, deciles are used, K = 9 and p 1 = 0.1, . . . , p 9 = 0.9, and r i is the closest rank not greater than p i . Further, let ε yi and ε ti denote the level of income and tax payments at r i . To establish statistical inference, the first step is to derive the asymptotic distribution of the conditional expectations of income, denoted as γ i , and taxes denoted as δ i , for i = 1, . . . , K + 1. Consider two jointly distributed random variables Y and T. The expectation of Y conditional on r i being less than or equal to p i is γ i ≡ E[Y |r i ≤ p i ] 2 Similarly the conditional expectation of T at p i is given by δ i ≡ E[T |r i ≤ p i ] 3 The obvious estimators of γ i and δ i in the sample are the weighted cumulative means ˆ γ i and ˆδ i of Y and T, respectively. Note that the conditional means at ˆ γ K+ 1 and ˆδ K+ 1 are equal 366 J. Klavus Journal of Health Economics 20 2001 363–377 to the unconditional weighted sample means ˆ γ and ˆδ. From these quantities, the Lorenz curve ordinates can be estimated as ˆ Φ i = p i ˆ γ i ˆ γ , and analogously as ˆ Γ i = p i ˆδ i ˆδ for the concentration curve ordinates. Let the vector of the conditional means corresponding to the abscissae p i [p i |i = 1, . . . , K] be ˆ γˆ γ ˆ γ ≡ ˆ γ 1 , . . . , ˆ γ K ′ and ˆδˆδˆδ ≡ ˆδ 1 , . . . , ˆδ K ′ . Progressivity is given by the distance between each of the ordinate estimates as ˆ Pˆ Pˆ P = p 1 ˆ γ 1 ˆ γ − ˆδ 1 ˆδ , . . . , p K ˆ γ K ˆ γ − ˆδ K ˆδ 4 The asymptotic distribution of ˆ Pˆ Pˆ P depends on the joint asymptotic distribution of its terms, represented by a 2K + 1-vector ˆ θˆ θˆ θ = p 1 ˆ γ 1 , . . . , p K ˆ γ K , ˆ γ , p 1 ˆδ 1 , . . . , p K ˆδ K , ˆδ ′ . It can be shown that ˆ θˆ θˆ θ is asymptotically normal in that N ˆ θˆ θˆ θ − θθθ has a limiting K + 1-variate normal distribution with mean zero and an asymptotic covariance matrix Ω Ω Ω see for example; Beach and Davidson, 1983 and Davidson and Duclos, 1997, where the covariance of γ i and δ j is given by covp i γ i , p j δ j = p i [EY i T i − γ i δ i + 1 − p j ε yi − γ i ε tj − δ j +ε yi − γ i δ j − δ i ] 5 The first two terms in brackets denote the covariance of Y and T conditional on r i ≤ p i . The asymptotic covariance of γ i and γ j and for δ i and δ j by replacing γ i with δ i , and ε yi with ε ti is covp i γ i , p j γ j = p i [λ 2 i + 1 − p j ε yi − γ i ε yj − γ j + ε yi − γ i γ j − γ i ] 6 where λ 2 i is the cumulative variance of Y or T at i. Setting i = j yields the asymptotic variance of γ i or δ i as varp i γ i = p i ⌊λ 2 i + 1 − p i ε yi − γ i 2 ⌋ 7 Thus, the matrix Ω Ω Ω consists of the variance and covariance terms of each ordinate along the curves Lp and Cp, and the covariance terms of each ordinate across the curves. Hereafter, shorthands of γ i γ j and δ i δ j are used for the former and γ i δ j for the latter terms. In accordance with the theorem of Rao 1973 on the limiting distribution of differentiable functions of random variables, the limiting distribution of ˆ Pˆ Pˆ P is also multivariate normal with mean zero and covariance matrix V V V = J J J Ω Ω ΩJ J J ′ , where Ω Ω Ω is as derived above and J J J is the K × 2K + 1 Jacobian defined as J = δ ˆ P i δ ˆ θ i ˆ θ =θ =    1γ . . . 1γ −p 1 γ 1 γ 2 .. . −p K γ K γ 2 −1δ . . . −1δ p 1 δ 1 δ 2 .. . p K δ K δ 2    8 J. Klavus Journal of Health Economics 20 2001 363–377 367 Since the composition of the matrix V V V is rather complex, only the structure of the diagonal terms is presented here. The diagonal terms are v ii = 1 ¯ γ 2 γ i γ i + 1 ¯δ 2 δ i δ i − 2 1 ¯ γ ¯δ γ i δ i − 2 p i γ i ¯ γ 3 γ i γ K+ 1 − 2 p i δ i ¯δ 3 δ i δ K+ 1 +2 p i γ i ¯ γ 2 ¯δ γ K+ 1 δ j + 2 p i δ i ¯ γ ¯δ 2 γ i δ K+ 1 − 2 p i γ i ¯ γ 2 p i δ i ¯δ 2 γ K+ 1 δ K+ 1 + p i γ i ¯ γ 2 2 γ K+ 1 γ K+ 1 + p i δ i ¯δ 2 2 δ K+ 1 δ K+ 1 9 In order to test for statistical significance of progressivity at an individual ordinate i, the null hypothesis H : ˆ D i = 0, where ˆ D i = ˆ γ i ˆ γ − ˆδ i ˆδ , is tested against the alternative hypothesis H : ˆ D i = 0. Substituting sample estimates for the terms in Eq. 9, the test statistics are Z i = ˆ D i ˆ v ii N 12 10 which is asymptotically distributed as standard normal. To test for changes in progressivity ordinates estimated from two independent samples, the test statistics is Z i = ˆ D A i − ˆ D B i [ ˆv A ii N A + ˆ v B ii N B ] 12 11 where ˆ D A i and ˆ D B i refer to two independent progressivity ordinates. For testing the hypoth- esis concerning the dominance relation of entire curves a simultaneous inference approach based on a set of differences at ordinates i = 1, . . . , K can be used see for example; Beach and Richmond, 1985; Bishop et al., 1994. This involves comparing the largest positive and negative values of the individual Z-statistics Z + , Z − to the critical value in a studentised maximum modulus SMM table. The test statistics are selected from Z + = max{0, Z i }; i ∈ [1, K]; Z − = min{0, Z i }; i ∈ [1, K] 12 In the case of two curves A and B, representing either a Lorenz curve and a concentration curve i.e. progressivity according to Eq. 10, or two separate progressivity distributions i.e. changes in progressivity according to Eq. 11, the test hypotheses are 2 Z + not significant; Z − not significant ⇒ H : the two curves are equal Z + significant; Z − not significant ⇒ H 1 : A the reference dominates B Z + not significant; Z − significant ⇒ H 2 : B dominates A the reference Z + significant; Z − significant ⇒ H 3 : the two curves cross It should be noted that when analysing progressivity or changes in two progressivity distributions the inequality implications of dominance differs from those associated with 2 A progressivity distribution is a curve plotting the differences in Lorenz and concentration curve ordinates at given percentile points. 368 J. Klavus Journal of Health Economics 20 2001 363–377 comparisons concerning two Lorenz curves. In the case of Lorenz curves, the dominant lower curve always yields less income inequality. In comparisons concerning progressiv- ity, if the Lorenz curve concentration curve dominates the concentration curve Lorenz curve, the tax is progressive regressive, implying pro-poor pro-rich redistribution. When comparing two progressivity distributions referring, for example, to two different time peri- ods the interpretation of dominance is as follows: if the reference distribution is progressive, the dominant curve yields less progressivity or more regressivity, whereas if the reference distribution is regressive the dominant distribution gives rise to less regressivity or more progressivity.

3. Data and variables