Poverty Comparisons with Dependent Samples

Transcription

1 Poverty Comparisons with Dependent Samples Buhong Zheng Department of Economics University of Colorado at Denver April 2001 Abstract Standard inference procedures for poverty comparisons require samples to be independent. For many commonly used income samples, however, this requirement is not fulþlled since samples are rotated. This note introduces an easy-to-use method of correction for sample dependency. We also apply the method to test changes in U.S. poverty in the 1990s and evaluate the marginal effects of public assistance on poverty before and after the recent welfare reform. JEL Classifications: C40, I32 Key Words: Marginal Changes, Poverty Measures, Statistical Inference, Dependent Samples Acknowledgment: I thank John A. Bishop and Brian J. Cushing for helpful conversations and suggestions on the issue of sample dependency. I also thank Laura M. Argys for her comments on an earlier draft of this note. 1

2 Poverty Comparisons with Dependent Samples I. Introduction In 1999 the estimated U.S. poverty rate was 11.8 percent while the rate for the previous year was 12.7 percent. Using the standard inference procedures for sample proportion, the Census Bureau concluded that poverty in the U.S. had decreased from 1998 to 1999 and the drop was statistically signiþcant at the ten percent level. 1 This conclusion, however, can be questioned based on both the poverty measure used and the statistical method applied. The deþciency of using the poverty rate as the indicator of poverty has been widely recognized since Sen s 1976 inßuential work on poverty measurement. A poverty measure, Sen argued, should reßect not only the incidence of poverty but also the intensity of poverty as well as the distribution of income within the poor; the poverty rate reßects only the incidence ofpoverty. Itfollowsthattheexclusiveuseofthepovertyratemaynotaccurately portray changes in poverty and, in order to draw valid conclusions, one should also use other more comprehensive poverty measures such as that proposed by Sen (1976). In the literature, more than a dozen such measures have been developed and some of them have been applied in empirical poverty research. In contrast with the measurement issue, the statistical issue in poverty comparisons has received relatively less attention. Although Kakwani (1993) and others have developed inference procedures for the newly developed poverty measures, the procedures, like that used by the Census Bureau, assume several textbook conditions on the sampling process. One such condition is that samples must be independently drawn from different populations. For many income samples used in poverty comparisons, however, this requirement is not always satisþed. This is the case because samples such as the Current Population Survey (CPS), the Panel Studies of Income Dynamics (PSID), and the Consumer Expenditure Survey (CEX) are rotating waves or panels and may contain information about a cross-section of individuals at two or more points in time. For example, in the CPS sample each household is surveyed for two consecutive years and in each year half of the households are replaced by a new panel. Therefore, the samples from any two adjacent years may have a substantial overlap of sampling units and are not independent. As a consequence, one needs to take sample dependency into account in computing the standard error for the difference between two years poverty estimates. The purpose of this note is to extend the inference procedure of Kakwani to the situation where samples are dependent. SpeciÞcally, we introduce a method of correction for the standard error of the difference between two poverty estimates. Since dependent samples can be either completely dependent (matched pairs) or partially dependent (rotated samples), we Þrst derive the covariance formula for completely dependent samples and then provide an easy-to-use two-step correction procedure for 1 See the tables compiled by the U.S. Census Bureau in its web site. 2

3 partially dependent samples. To illustrate the method of correction for sample dependency, we test the year-to-year changes in U.S. poverty in the 1990s and analyze the marginal effects of public assistance and welfare programs on poverty in both 1990 and These empirical applications suggest that it is necessary to take sample dependency into account in poverty comparisons. II. Poverty Measures, Marginal Changes and Statistical Inference Consider a joint distribution between two variables x (0, ) and y (0, ) with a continuous c.d.f. K(x, y). To simplify the reference, we may interpret x as an individual s income without public assistance and y as the individual s income with public assistance. The marginal distributions of x and y are denoted as F (x) and G(y). For convenience, we also assume that functions F and G are strictly monotone and the Þrst two moments of x and y exist and are Þnite. Throughout the note, z denotes the poverty line which is an exogenously determined positive value. 2 An individual is poor if and only if his income is strictly below the poverty line. II.1. Poverty measures and the marginal change in poverty Following Atkinson (1987) and Kakwani (1993), we consider in this note a popular class of poverty measures additively separable (decomposable) poverty measures. For a given income distribution, say F (x), and a given poverty line z, an additively separable poverty measure is P (F ; z) = Z z 0 p(x, z)df(x) (2.1) where p(x, z) is the poverty-deprivation function which is nonnegative and is continuous in both x and z. This class contains several frequently used poverty measures. For example, the class of poverty measures proposed by Foster et al. (1984) corresponds to p(x, z) =(1 x/z) k with k 2; iftheparameterk were allowed to take values 0 and 1, the Foster et al. class would also contain two well-known measures of poverty the poverty rate and the poverty gap ratio; if p(x, z) =lnz ln x then P (F ; z) is the measure introduced by Watts in The marginal change in poverty refers to the increase or decrease in poverty after the population has experienced a change in income. For example, public assistance and welfare programs in the U.S. help low-income individuals and, thus, reduce the level of poverty. In this case, the marginal change in poverty due to public assistance can be deþned as the difference between the poverty level without public assistance 2 Alternatively, the poverty line could be relatively determined as, say, one half of the mean or median income. Preston (1995) and Zheng (2001) have derived statistical inference for poverty measures with relative poverty lines. The results of this note can also be similarly generalized to the case of relative poverty lines. 3 The poverty gap ratio reßects the intensity of poverty while both the Foster et al. measures with k 2 and the Watts measure are also sensitive to the income distribution of the poor. For a survey of all poverty measures, see, for example, Foster and Sen (1997) and Zheng (1997). 3

4 and the poverty level with public assistance. For a given poverty measure P,this marginal change can be formerly deþned as P (F, G; z) =P (F ; z) P (G; z). (2.2) II.2. Large sample properties of the marginal change in poverty Assume that a paired sample of size n, {(x i,y i )}, is randomly drawn from the population K(x, y). Then for an additively separable poverty measure P, consistent estimates of poverty indices associated with F (x) and G(y) are, respectively, ˆP (F ; z) = 1 nx p(x i,z)i(x i z) and n ˆP (F ; z) = 1 nx p(y i,z)i(y i z) (2.3) i=1 n i=1 where I(x i z) =1for x i <zand 0 otherwise. Consequently, the marginal change in P can be consistently estimated as ˆP (F, G; z) = ˆP (F ; z) ˆP (G; z). (2.4) Using the law of large numbers and through direct calculations, one can easily establish the asymptotic normality of ˆP (F, G; z) and derive the corresponding variance formula. The following proposition summarizes these results, which generalize Kakwani s results to the case of dependent samples. Proposition 1. Under the assumptions that K(x, y), F (x) and G(y) are continuous and the Þrst two moments of x and y exist and are Þnite, the estimate of the marginal change, ˆP (F, G; z), has a limiting normal distribution in that n 1 2 ( ˆP P ) is asymptotically normally distributed with mean zero and variance ε 2 = ε 2 x + ε2 y 2ε xy (2.5) where ε 2 x and ε 2 y are, respectively, the asymptotic variances of n ˆP (F ; z) and n 2 ˆP (G; z) whicharegiveninkakwani(1993) 4,andε xy istheasymptoticcovariancebetween n ˆP (F ; z) and n 2 ˆP (G; z) and is ε xy = Z z Z z 0 0 p(x, z)p(y, z)dk(x, y) P (F ; z)p (G; z). (2.6) II.3. Asymptotic variance of ˆP when samples are partially dependent Assume that two samples of possibly different sizes m and n, {x i } and {y i },are drawn from distribution K(x, y). Since the samples are only partially dependent, we may assume that the Þrst t observations of the two samples are matched pairs and the remaining observations are independently drawn from F (x) and G(y), respectively. That is, (x 1,x 2,..., x t ) and (y 1,y 2,...,y t ) are paired, (x t+1,..., x m ) is independent of 4 Also see Jäntti (1992), Preston (1992), Bishop et al. (1995), and Rongve (1997). 4

5 {y i }, and (y t+1,..., y n ) is independent of {x i }. For many income samples such as CPS, it is also reasonable to assume that (x t+1,..., x m ) is independent of (x 1,x 2,...,x t ), and {y t+1,...,y n } is independent of (y 1,y 2,..., y t ). It follows that the variances of ˆP (F ; z) and ˆP (G; z) can be estimated in the way as Kakwani (1993) described. Denoting α x = 1 P ti=1 p(x m i,z)i(x i z), β x = 1 P mi=t+1 p(x m i,z)i(x i z), α y = 1 P tj=1 p(y n j,z)i(y i z), andβ y = 1 P nj=t+1 p(y n j,z)i(y i z), wecanwritethecovariance term between ˆP (F ; z) and ˆP (G; z) as cov( ˆP (F ), ˆP (G)) = cov(α x + β x, α y + β y ) (2.7) = cov(α x, α y )+cov(α x, β y )+cov(β x, α y )+cov(β x, β y ). Since β x is independent of {y j } (hence α y and β y )andβ y is independent of {y i } (hence α x and β x ), we have cov(α x, β y )=cov(β x, α y )=cov(β x, β y )=0and thus cov( ˆP (F ; z), ˆP (G; z)) = cov(α x, α y ). (2.8) Noting further that α x = 1 P ti=1 p(x m i,z)i(x i z) = t P ti=1 p(x m i,z)i(x i z) i and α y = 1 P h tj=1 p(y n j,z)i(y i z) = t 1 P tj=1 p(y n t j,z)i(y i z) i,wehave cov( ˆP (F ; z), ˆP (G; z)) (2.9) = t m t n cov 1 tx p(x i,z)i(x i z), 1 tx p(y j,z)i(y i z). t i=1 t j=1 Clearly cov( 1 P ti=1 p(x t i,z)i(x i z), 1 P tj=1 p(y t j,z)i(y i z)) can be directly calculated by treating (x 1,..., x t ) and (y 1,..., y t ) as completely dependent samples. Thus, cov( ˆP (F ; z), ˆP (G; z)) can be computed using the following two-step procedure: Þrst calculate the covariance of sample statistics of the matched subsamples as if they were completely dependent samples; then multiply the covariance term by the proportions of the matched portions of the two samples (t/m and t/n). 5 III. Applications: Public Assistance and U.S. Poverty in the 1990s To illustrate the inference procedures developed above, we apply them to the CPS data. We wish to examine the recent trend in U.S. poverty and the marginal effects of public assistance and welfare programs on U.S. poverty. In particular, we are interested in measuring the impacts of the recent welfare reform on poverty of different ethnic groups. To this end, we extract the annual CPS samples from 1990 to 1999 which include 1996 the year in which the welfare reform was initially implemented. The poverty measures we use include the poverty rate, the poverty gap ratio, the Foster et al. measure, and the Watts measure. We include in our samples 5 This correction procedure can also be applied to inequality comparisons with dependent samples (Zheng and Cushing, 2001). 5 h 1 t

6 all primary families, subfamilies and primary individuals but exclude all people living in group quarters; we also drop all entries with zero or negative family incomes so that the Watts poverty measure can be applied. To be consistent with the official statistics, we use the Orshansky equivalence scales. III.1. The U.S. poverty dynamics from 1990 to 1999 Table 1 summarizes pairwise comparisons of U.S. poverty from 1990 to Each row of the table reports changes in poverty between the two years; the four cells correspond to the four measures used (FGT is the Foster et al. measure with k =2and W is the Watts measure). In each cell, there are four numbers: the Þrst number is the increase (positive) or decrease (negative) in poverty between the two years; the next two numbers are the estimated standard errors of the change in poverty without and with correcting for sample dependency, respectively; the last number is the percentage reduction in the standard error if sample dependency is corrected. On the shoulder of each difference of poverty estimate there are two letters which indicate the signiþcance levels of the difference: the Þrst letter indicates the level using the standard error without being corrected for sample dependency and the second letter indicates the level using the standard error corrected for sample dependency. The meanings of these letters are: a notsigniþcant at the ten percent level; b signiþcant at the ten percent level; c signiþcant at the Þve percent level; and d signiþcant at the one percent level. An inspection of the table reveals that correcting for sample dependency matters in drawing conclusions on poverty comparisons. Except for 1994/1995, the correction reduces standard error estimates by between 6.5 percent to 20.5 percent. 6 Adifference of this size will surely alter the p-value of the comparison; it is also possible to change the signiþcance level even if only a few such levels are considered. The effect of correcting for sample dependency is indicated by the two letters on the shoulder of each poverty estimate: if two letters are different then the correction changes the signiþcance level. In the table, there are Þve comparisons affected by this correction: the poverty gap ratio at 1991/92, 1992/93 and 1996/97; the Foster et al. measure at 1991/92; and the Watts measure at 1993/94. It is interesting to note that the magnitudeofcorrectiondoesnothavetobelargetomakeadifference and a large magnitude does not necessarily change the signiþcance level; what matters here is whether or not the test statistic with uncorrected standard error is already close to the critical value for a pre-selected signiþcance level. Finally, it is useful to point out that the different poverty measures involved may not always result in consistent conclusions regarding the directions or the signiþcance levels of poverty changes. For example, in 1997/98 the poverty rate indicates an increase in poverty but all other three measures indicate just the opposite. These changes, however, are not statistically signiþcant with or without correcting for sample dependency at the ten percent level. Although a description of the precise 6 The 1995 sample is based entirely on the 1990 Census design and, therefore, the 1994 and 1995 samples are independent and need not adjust for sample dependency. 6

7 trend in poverty is somewhat measure-dependent, we can generally conclude that U.S. poverty increased in the early 1990s and decreased thereafter. This conclusion is roughly consistent with what the official poverty statistics had described. Table 1. Changes in U.S. Poverty in the 1990s Year P. Rate P. Gap FGT Watts 1990/ d,d d,d d,d (0.0023) (0.0011) (0.0008) [0.0019] {19.7} [0.0010] {17.7} {12.8} 1991/ a,a (0.0023) [0.0020] {18.2} 1992/ d,d (0.0024) [0.0020] {17.3} 1993/ d,d (0.0024) [0.0020] {20.0} 1994/ d,d (0.0024) [0.0024] {0} 1995/ a,a (0.0025) [0.0021] {14.9} 1996/ d,d (0.0024) [0.0021] {15.1} 1997/ a,a (0.0022) [0.0018] {19.5} 1998/ d,d (0.0020) [0.0017] {16.2} b,c (0.0012) [0.0010] {15.7} a,b (0.0012) [0.0010] {17.9} d,d (0.0012) [0.0010] {20.5} d,d (0.0012) [0.0012] {0} a,a (0.0012) [0.0011] {14.4} b,c (0.0012) [0.0011] {14.0} a,a (0.0011) [0.0009] {18.2} d,d (0.0010) [0.0008] {14.4} c,d (0.0008) {11.7} a,a (0.0009) {14.4} d,d (0.0008) {16.1} d,d (0.0008) [0.0008] {0} a,a (0.0009) [0.0008] {11.3} a,a (0.0009) [0.0008] {11.0} a,a (0.0008) {13.6} d,d (0.0007) [0.0006] {11.2} d,d (0.0024) [0.0022] {9.9} c,c (0.0025) [0.0023] {9.5} a,a (0.0026) [0.0023] {12.3} b,c (0.0027) [0.0024] {11.7} c,c (0.0029) [0.0029] {0} a,a (0.0029) [0.0027] {7.0} a,a (0.0029) [0.0027] {7.7} a,a (0.0027) [0.0025] {8.9} a,a (0.0026) [0.0025] {6.5} III.2. Public assistance, welfare reform and U.S. poverty In the U.S., a large amount of the National Gross Product is spent annually on public assistance programs such as food stamps and AFDC (aid to families with dependent children). These programs are designed to help the poor and reduce poverty. Earlier studies conþrmed that such programs had indeed reduced poverty and the poverty-reducing effects varied among different ethnic groups (see Danziger et al., 1981, or Moffitt, 1992, for a review). The recent welfare reform attempted to reduce the number of program participants and shorten the duration of payment. 7 As 7 See Danziger (1999) for a detailed discussion of the reform. Also see Witte et al. (1998) and 7

8 a result, one may expect that the poverty-reducing role of public assistance will be affected and that different ethnic groups may be affected differently. To test these hypotheses, we need to compare the actual poverty level with the hypothetical poverty level if the public assistance were absent. Since the two samples used in this comparison are matched pairs and thus are dependent, standard testing procedures are not applicable. In this section, we use our modiþed inference procedure to test these hypotheses. 8 An important issue in testing these hypotheses is the measurement of the marginal effect of public assistance and welfare payment on poverty. While most studies, including that of the Census Bureau, measure the marginal effect as the change in poverty if the amount of public assistance is subtracted from the total income, researchers generally agreed that such a counterfactual of without public assistance does not consider the labor supply effect of welfare payment. In this sense, this simple measure of marginal effect overstates the poverty-reducing effect of public assistance. Despite this awareness, in the literature, there has been relatively few attempts to deþne a more accurate counterfactual. Part of the difficulty lies in the formidable amount of information required by this exercise. In this note, we follow most studies in the literature and adopt their deþnition of marginal effect. Of course, such a measure can only be interpreted as measuring the upper-bound estimates (Danziger et al., 1981, p. 1007) of poverty-reduction. Table 2. Public Assistance in 1990 and 1999 Year Race Coverage (%) Average Amount ($) 1990 Whites Others Whites Others In 1990 dollars. The time periods we choose to evaluate the impacts of the welfare reform are 1990 and 1999 which has the most recent available CPS data. 9 In doing so, we are assuming that all welfare reform policies initiated in 1996 had been fully incorporated and taken into effect in Table 2 summarizes the percentages of participants Mach (2000). 8 It is useful to point out that the issue of public assistance, welfare reform and poverty is much more complicated than what we describe and test here. The objective of this application is to illustrate the proposed methodology. Thus, one should be cautious in interpreting the results obtained here. 9 The years we choose for the comparison are more-or-less arbitrary. We have also used other years before 1996 as the beginning year and Þnd, qualitatively, the results do not change much. 10 We are also assuming that the economic conditions in 1999 were the same as those in Of course, this assumption may not be very realistic. In our example, however, it is lesser a problem than it appears when we compare the changes in the effects of the two population groups between 1990 and

9 and the average payments of public assistance in 1990 and 1999 for whites and the rest of the population. For either group, both the coverage rate and the average payment had been substantially reduced from 1990 to It is however important to note that while the coverage of the white population was reduced by 2.8 percent, the rate of the group others was reduced by 12.2 percent! In this sense, the welfare reform had affected nonwhites much more greatly than whites. This suggests that the marginal effects of public assistance on poverty might have been reduced more for nonwhites than for whites. Our following investigation conþrms thisconjecture. Table 3 documents the comparisons of the marginal effects of public assistance and welfare programs on poverty in 1990 and 1999 for the two population subgroups (whites and others). The top two-third of the table shows the cross-sectional comparisons of the marginal effects between the white population and the rest of the U.S. population; the bottom one-third of the table reports the cross-time comparisons of the two racial groups and their differences. In each cell, there are three numbers: the Þrst one is the marginal effect(rows1,2,4and5)orthedifference between the two population groups (rows 3 and 6) or between 1990 and 1999 (rows 7, 8 and 9); the second number is the standard error without being corrected for sample dependency; the third number is the standard error corrected for sample dependency. Table 3. Marginal Effects on Poverty in 1990 and 1999 Year Race P. Rate P. Gap FGT W Whites (1) 1990 Others (2) Difference (3) = (2) (1) Whites (4) 1999 Others (5) Difference (6) = (5) (4) Whites (7) = (4) (1) 1990/99 Others (8) = (5) (2) Difference (9) = (8) (7) (0.0020) [0.0004] (0.008) [0.0020] a,d (0.0086) [0.0019] (0.0019) [0.0002] (0.0067) [0.0009] a,d (0.0069) [0.0033] a,d (0.0028) [0.0004] a,d (0.0107) [0.0021] a,d (0.0111) [0.0021] (0.0010) [0.0003] (0.0045) [0.0013] d,d (0.0046) [0.0013] (0.0009) [0.0001] (0.0035) [0.0006] d,d (0.0036) [0.0006] d,d (0.0013) [0.0003] d,d (0.0057) [0.0014] c,d (0.0059) [0.0015] (0.0007) [0.0003] (0.0035) [0.0015] d,d (0.0035) [0.0014] (0.0007) [0.0001] (0.0026) d,d (0.0027) d,d (0.0010) [0.0003] d,d (0.0043) [0.0016] d,d (0.0044) [0.0016] (0.0024) [0.0012] (0.0117) [0.0058] d,d (0.0119) [0.0012] (0.0029) [0.0006] (0.0087) [0.0023] d,d (0.0091) [0.0023] d,d (0.0038) [0.0014] d,d (0.0145) [0.0062] d,d (0.0150) [0.0064] 9

10 By comparing the two standard errors in each cell, one can see that correcting for sample dependency affects standard error estimation substantially. For example, for the poverty rate in 1990, the standard error of the difference between the two racial groups is reduced from to , an almost 80 percent drop! In the table, such reductions move Þve comparisons from being insigniþcant at the ten percent level to being signiþcantattheonepercentlevel(thepovertyrateatrows3,6,7,8, and 9) and move one comparison from being signiþcant at the Þve percent level to the one percent level (the poverty gap ratio at row 9). Use the standard error estimates corrected for sample dependency, all comparisons are signiþcant at the one percent level. With this observation, we can draw the following conclusions on the marginal effects of public assistance on poverty between the two population groups and between the two time periods. First, the marginal effects were greater for the nonwhite population than for the white population in both years. Second, the marginal effects were reduced from 1990 to 1999 for both groups as a result of the 1996 welfare reform. Third, the nonwhite population experienced larger drop in the marginal effect than the white population. In this sense the marginal effects between the two groups converged as a result of the welfare reform. It is useful to point out that these conclusions are not sensitive to which of the four poverty measures is used. It is also interesting to note that, for the poverty rate, none of these differences would be signiþcant even at the ten percent level if the standard errors are not corrected for sample dependency. This implies that, if only the poverty rate is used and sample dependency is also not corrected, one would conclude that public assistance had equal effects on both groups in both years and the 1996 welfare reform had not reduced those effects. IV. Conclusions Many commonly used economic data in poverty studies, such as the CPS and CEX samples, are rotated and, thus, samples from different years may not be independent. Yet, a crucial assumption in standard poverty inference procedures is that samples must be independently drawn. In this note we have shown that failure to correct for sample dependency is likely to overestimate the standard errors of the changes in poverty and, as a result, inferences on poverty comparisons may not be robust. This note has introduced a simple method of taking sample dependency into account in estimating the standard errors of changes in poverty. This is done by Þrst deriving the asymptotic covariance term for paired (completely dependent) samples and then extending it to the case of rotated (partially dependent) samples. The twostep method of correction is easy to apply; all one needs is to calculate the covariance term from the overlapped (paired) subsample and then multiply it by the proportions of the subsample in each sample. To illustrate the method, we applied it to test the changes in U.S. poverty in the 1990s and analyze the marginal effects of public assistance on U.S. poverty. We also evaluated the impacts of the 1996 welfare reform on the marginal effects of public 10

11 assistance on poverty for the white population and the rest of the U.S. population. We found that the nonwhite population had larger marginal effect than the white population in both years but the gap between the two effects was smaller after the welfare reform in 1999 than before the welfare reform in

12 References Atkinson, A. (1987): On the Measurement of Poverty, Econometrica 55, Bishop, J. K. V. Chow and B. Zheng (1995): Statistical Inference and Decomposable Poverty Measures, Bulletin of Economic Research 47, Danziger, S., R. Haveman and R. Plotnick (1981): How Income Transfer Programs Affect Work, Savings, and the Income Distribution: A Critical Review, Journal of Economic Literature 19, Danziger, S., ed. (1999): Economic Conditions and Welfare Reform (W. E. Upjohn Institute for Employment Research, Kalamazoo, Michigan). Foster, J., J. Greer and E. Thorbecke (1984): A Class of Decomposable Poverty Measures, Econometrica 52, Foster, J. and A. Sen (1997): On Economic Inequality after a Quarter Century, in: On Economic Inequality (expanded edition, Clarendon Press, Oxford). Jäntti, M. (1992): Poverty Dominance and Statistical Inference, Research Report, Stockholm University. Kakwani, N. (1993): Statistical Inference in the Measurement of Poverty, Review of Economics and Statistics 75, Mach, T. (2000): Three Essays on Welfare Reform, Ph.D. Dissertation, Ohio State University. Moffitt, R. (1992): Incentive Effects of the U.S. Welfare System: A Review, Journal of Economic Literature XXX, Preston, I. (1992): Large Sample Estimation and Inference for Poverty Measures, Discussion Paper, University College London. Preston, I. (1995): Sampling Distributions of Relative Poverty Statistics, Applied Statistics 44, Rongve, I. (1997): Statistical Inference for Poverty Indices with Fixed Poverty Lines, Applied Economics 29, Sen, A. (1976): Poverty: An Ordinal Approach to Measurement, Econometrica 44, Watts, H. (1968): An Economic DeÞnition of Poverty, In D. P. Moynihan (ed.), On Understanding Poverty (Basic Books, New York). Witte, A., M. Queralt, T. Chipty and H. Griesinger (1998): Unintended Consequences? Welfare Reform and the Working Poor, NBER Working Paper Zheng, B. (1997): Aggregate Poverty Measures, Journal of Economic Surveys 11, Zheng, B. and B. Cushing (2001): Statistical Inference for Testing Inequality Indices with Dependent Samples, Journal of Econometrics 101, Zheng, B. (2001): Statistical Inference for Poverty Measures with Relative Poverty Lines, Journal of Econometrics 101,