Tests for Richness and Poorness: A Stochastic Dominance Analysis of Income Distributions in Hong Kong

Transcription

1 Tests for Richness and Poorness: A Stochastic Dominance Analysis of Income Distributions in Hong Kong Sheung-Chi Chow 1, Francis T. Lui 2, Ma. Rebecca Valenzuela 3 and Wing-Keung Wong 1 1 Department of Economics, Hong Kong Baptist University 2 Department of Economics, The Hong Kong University of Science and Technology 3 Department of Economics, Monash University. June 8, 2015 Abstract In this paper, we propose the use of new stochastic dominance tests to achieve a more robust analysis of relative welfare levels in the study of income distributions. In particular, we applied the theory of descending stochastic dominance to enrich results that are obtained using the standard stochastic dominance techniques. The feasibility of the proposed approach and the new tests of richness and poorness developed for it are demonstrated using distributions of incomes in Hong Kong. Empirically, we find that between the years 2001, 2006 and 2011, the year 2006 had the most favorable income distribution in the sense of having the highest welfare level in the income distributions. Across age groups, our new test of poorness reveal that, at the lowest income levels, the younger age groups have less proportions of poor income units compared to their older counterparts; while our test for richness show that, at the upper end of the income scale, the 65+ age group has the higher proportions of very rich individuals compared to all the other groups in the distribution. These help explain the high level of inequality in Hong Kong. JEL Codes: C19, C44, I30 - Corresponding author. address: Rebecca.Valenzuela@monash.edu 1

2 1 Introduction Singular measures of income and consumption inequality provide useful though limited views of actual disparities in welfare and living standards in the population. Such indices are instructive in providing complete orderings of distributions as they appear comprehensible and unambiguous in many applications. At the same time, however, singular measures are known to suffer from the lack of universal acceptance of the value judgments of the underlying welfare functions. Contradicting conclusions can thus emerge and present problems for policy analysis and decision making. An alternative procedure for analyzing distributions is to use the stochastic dominance approach. Stochastic dominance (SD) is often used in investment and decision theory to refer to situations where one outcome can be ranked to be superior to another. While the SD theory is very well developed in the finance literature, the same cannot be said in the study of welfare in income distributions. The SD approach has however become more popular in recent years mainly due to the greater availability of micro-unit records. Some of the more recent ones include: Maasoumi & Heshmati (2008) which studies inequality among households in the United States, Sarkar (2012) which investigates inequality in India, Maasoumi, Su and Heshmati (2013) provides empirical results for China, and Valenzuela, Lean and Athanasoupolous (2014) which analyses inequality and welfare among households in Australia. To analyze income distributions, the SD approach uses a systematic method for a pairwise comparison of points in two distinct distributions which rank welfare situations over very wide classes of welfare functions. SD results thus presents a fuller and more accurate picture of existing inequality levels by virtue of its use of more detailed information (at all possible points of comparison). In other words, SD analysis considers the whole distribution of outcomes and is therefore better able to reveal crucial details and potentially important distinctions between different parts of the distribution. As such, it avoids the overly narrow cardinalisation of welfare functions as is done by indices. 2

3 The use of SD techniques to analyze income distributions are important especially when alternative inequality indices fail to provide unambiguous rankings of distributions and also it is common that different indices give different rankings. SD analysis is thus often preferred over the scalar index approach for many reasons. For example, SD tests are able to rank welfare situations over very wide class of welfare functions and so are ethically robust, unlike scalar measures which are known to suffer from the lack of universal acceptance of the value judgments of the underlying welfare functions and from which contradicting conclusions can arise. Stochastic dominance relations therefore have the potential to offer powerful inferences with regards to the welfare rankings of different distributions. If dominance relations are inferred, one can distinguish the better off group from its counterparts and therefore have a clear vision of the future policies, strategies and programs for helping the targeted groups. On the other hand the inability to infer a dominance relation is equally valuable and informative indicating that any welfare ordering based on a particular index is highly subjective and does not apply to all segments of populations. Conclusions drawn from SD analysis are thus more robust and more useful in the wider policy sense. Currently, inequality analyses using the stochastic dominance techniques measure relative welfare from the bottom of the distribution; That is, to infer inequality from a pairwise comparison which emphasizes the poor end of the income distribution curve. Such approach is certainly useful for drawing inference about the degree of relative poverty in the distribution, but misses to account for the contribution of the relatively rich in the overall inequality. Thus, in this paper, we wish to correct this omission and propose a new way of thinking about inequality analysis by incorporating analysis from the upper end of the income scale, and accounting for the contribution of the rich end in total inequality. We believe that this approach improves on the current ones, and provides more robust results of welfare comparisons by combining SD results from measuring both relative poorness as well as relative richness in a population. To do so, we propose to incorporate the theory of descending stochastic dominance theory (DSD) of income inequality to the traditional SD theory, which we will hereon refer to as the ascending stochastic dominance (ASD) in this paper. ASD thus looks into the effect of the 3

4 poorness while DSD could measure the effect of richness for income distribution. The theories of both ASD and DSD are well developed, but, as far as we know, only the theory of ASD has been used for income inequality. Thus, the main contribution of this paper to existing literature is the introduction of DSD theory to income distribution and the recommendation to use both ASD and DSD to study the joint welfare effects of the poorness and richness of income distribution. The theory of DSD has been well-developed in the finance area, see, for example, Hammond (1974), Meyer (1977), and Stoyan (1983). Readers may refer to Levy (2015), Wong (2007), and Sriboonchitta, et al. (2009) for more review on the theory of DSD. For this paper, we propose to employ both ASD and DSD test statistics to test the significance of the poorness and richness of income distribution. We also propose to improve the testing procedure recommended by Bai et al. (2015). This testing procedure could allow us to have a more reasonable assumption that the income series (or consumption series) under comparison could be dependent. Such testing procedure with this assumption is important since according to Bai et al. (2015) if the series under comparison are in fact dependent, the test statistic of the stochastic dominance test could be totally different and the result could be misleading. In addition, the test from Bai et al. (2015) could also allow us to compute the test statistics for a pre-described set of grids so that the tests could be used to measure different degrees of the poorness and richness, say, for example, the top 10 per cent of the richest income units as well as the bottom 20 per cent of the poorest income units. The use of the stochastic dominance approach is thus an important feature of this paper. The addition of this new approach from the top end is very relevant in this day and age, as economies grow richer and high inequality in many advanced countries is often perceived to be caused by significant wealth or incomes accumulating on the top-end. (e.g. occupy wall street movement makes this point). The approach is also relevant for many economies with ageing populations where a growing number of the equal and above 65+ population are seen to have accumulated wealth over their lifetime. To demonstrate the feasibility of this approach, we will apply the proposed technique to analyze the distribution of incomes in Hong Kong and study the poorness and richness in Hong Kong over time and across demographic groups. Conclusions drawn will be compared to those derived from the implementation of singular measure which we also implement here for comparison purposes. 4

5 This paper has the following structure. After a brief introduction and overview in Section 1, Section 2 presents the theory and introduces the proposed tests of stochastic dominance, their distributional characteristics and the bootstrap technique used. Section 3 derives and presents the new tests of stochastic dominance for richness and poorness. Section 4 sets the scene with an overview of the Hong Kong economy. Section 5 presents the data, results and the discussion- analysis. Section 6 concludes. 2 Conceptual Framework To apply SD technique to income distribution analysis, we let F be a distribution function and define a social welfare function (SWF) of the form W( F) u x df x (1) where the utility function u is increasing in its argument, which are household or person income units or units for short. In general, we assume W to be symmetric and increasing in all its arguments so that various ethical criteria of desirable, well-behaved SWFs can be used. It is well known that different SWFs of the form in (1) give the same order of ranking as that of SD if one stochastically dominates the other of the first-order. If we impose an additional restriction that the second derivative of u is negative, then all SWFs in this restricted class likewise give a unanimous ranking of two distributions if one dominates the other at second order. To implement the approach, we consider incomes or wealth X and Y defined over the real number space ab, with probability distribution functions F and G, respectively. We note that a should be non-negative for income but in this paper we do not include this restriction in our definition because we use X and Y for both income and wealth and one could have negative wealth. For any x, we define the k-order cumulative distribution functions A Fk and A Gk of X and Y to be 5

6 x a A x a F A k (x) = F k 1 (x) dx and G A k (x) = G k 1 (x) dx, (2) A for any k 1 and F A 0 = f and G A 0 = g are the probability density functions of X and Y. We call the above SD ascending stochastic dominance (ASD) in this paper to distinguish it from descending stochastic dominance (DSD) that we will discuss later in this paper. We adopt the following definition of ASD for income inequality: A Definition 1: X is said to first (second)-order ASD dominate Y, denoted by X 1 Y or A A F 1 G ( 2 A A A A A X Y or F 2 G ) if and only if F1 ( x) G1 ( x) ( F2 ( x) G2( x) ); and third- A A A A order ASD dominate Y, denoted by X 3 Y or F 3 G if and only if F3 ( x) G3 ( x) and X. Y We use FASD, SASD, and TASD to denote the first-, second-, and third-order ascending stochastic dominance, respectively. The k-order ASD can be defined similarly for any k>3. A A We also call F k and G k in (2) the k -order ASD integrals of X and Y because these integrals are integrated from bottom to top. In the context of income distributions, if X FASD Y; that is, X 1 A Y, one can claim that X has less proportion of poor units compared to Y. More specifically, if all individuals with A incomes equal to or below a specified value of x are considered poor, then X 1 Y means that distribution F will always have less proportion of poor income units than distribution G A for any specified value of x. Further, if X 1 Y, then the expected level of welfare from the F distribution is at least as great as that of G distribution for all increasing welfare functions. On the other hand, if income distribution F SASD dominates an income distribution G; that is, X 2 A Y, this means that the integral of the cumulative probability of X is smaller than that of Y. The intuitive implication of SASD is different from FASD. That is, the income distribution of the units in X does not have less proportion of poor units compared to that in Y for any income level but has less proportion of poor units compared to that in Y for all the relative lower income levels. If one finds that X dominates Y by SASD, one can infer that 6

7 there are less very poor people in X than Y. We can also infer that distribution F has a higher expected welfare level than distribution G if the social welfare function is increasing and concave (Levy, 2015). Because of the above properties, we also call the test that could detect ASD relation as test for poorness in the next section. We now discuss the relationship between ascending stochastic dominance and social welfare. In particular, suppose U denotes the class of all monotonic, symmetric additively separable social welfare functions of the form W(F) = u(x)df(x) where u: R + R is any continuous function. It is well known (Based on Bawa, 1975; Foster and Shorrocks, 1988) that F A k G if and only if W(F) > W(G) for all W U A k for k = 1, 2, and 3 in which U A 1 U is defined by including the condition u > 0; U A A 2 U 1 is defined by including the additional condition u < 0 ; and U A A 3 U 2 is defined by including another additional condition u > 0. This result is also applied in inequality analysis by Atkinson (1970). Second-order stochastic dominance has been shown to be equivalent to the Lorenz dominance in Atkinson (1970). We describe briefly in the following. According to Atkinson (1970), we define the Lorenz curve to be Or, equivalently, x where p = F(x) = f(q)dq 0 by parts, we get x 0 L(p) = μ 1 qf(q)dq, (3) p 0 L(p) = μ 1 F 1 (t)dt, (4) and μ is the mean of the distribution. Integrating the function x μl(p) = qf(q)dx 0 x 0 = xf(x) F(q)dq. (5) Compare the Lorenz curves for two distributions with same mean; that is, it could be shown that L(F{x}) L (F {x}) = 0, 7

8 x [F(q) F (q)]dq 0 Using the fact that = μ[l(f{x}) L (F {x})] x[f(x) F (x)]. (6) x * F * * * 1 * L F x F x t dt 0 F x 0 F x * 1 * * 1 * x * F x * F Fx * * 1 *. F x t dt F x t dt L F x F x t dt (7) Then, Atkinson (1970) gets x [F(q) F (q)]dq 0 (8) = μ[l(f{x}) L F{x} (F{x})] + [ F 1 (t)dt F {x} x(f{x} F {x})]. Let x be the argument in F, that is, F 1 (F {x}). Using the first mean value theorem for the integrals, the second term on the right hand side of Equation (8) could be rewritten as [F 1 (c 1 ) F 1 (F {x})](f{x} F {x}) to show that it will always be positive. Then, if f(t) Lorenz dominates f * t such that L(F{x}) L (F {x}) 0 for all F, then the second- x order stochastic dominance condition [F(q) F (q)]dq 0 0 will be satisfied. Comparing the Lorenz curves for two distributions at a point p = F(x) = F (x ), we have * x x * * L p L p x x p F q dq F q dq 0 0 * * x x 0 x * * F q F q dq F q dq x x F x. (9) Similarly, applying the first mean value theorem for integrals, the second term in the first line of Equation (9) can be expressed as [F(c 2 ) F(x)](x x) and therefore it will always be positive. Then, if f (t) second-order stochastically dominates f (t) such that [F(q) x 0 8

9 F (q)]dq 0, then the Lorenz dominant condition L(F{x}) L (F {x}) 0 for all F will be satisfied. The results imply that the second-order stochastic dominance condition is equivalent to the Lorenz dominance condition. In this paper, we also introduce the descending stochastic dominance theory (DSD) to income distribution and inequality analysis. The theory of DSD has been well-developed, see, for example, Hammond (1974). Readers may refer to Levy (2015), Wong (2007), and Sriboonchitta, et al. (2009) for more review on the theory of DSD. However, as far as we know, there is no paper introduce the DSD theory to income distribution and inequality analysis. Thus, in this paper we bridge the gap in the literature to introduce DSD to income distribution and inequality analysis. To do so, we first define the k-order reverse cumulative distribution functions X and Y to be D k b b D D D and F x F x dx x k 1 k x k 1 D Fk and D Gk of G x G x dx (10) for any k 1 and F D 0 = f and G D 0 = g be the probability density functions of X and Y. We are ready to state the following definition of descending stochastic dominance (DSD) for income inequality: D Definition 2: X is said to first(second)-order DSD dominate Y, denoted by X 1 Y or D D F 1 G ( 2 X Y or F 2 D G ) if and only if F 1 D (x) G 1 D (x) ( F 2 D (x) G 2 D (x)); and third- D order DSD dominate Y, denoted by X 3 Y or F 3 D G if and only if F 2 D (x) G 2 D (x) and X. Y We use FDSD, SDSD and TDSD to denote the first-, second-, and third-order descending stochastic dominance, respectively. The k-order DSD can be defined similarly for any k>3. We also call D Fk and from top to bottom. D Gk in (10) the k -order DSD integrals of X and Y because they integrate 9

10 If X with distribution function F dominates Y with distribution function G in the sense of FDSD; that is X 1 D Y, this implies that X has a higher proportion of richer units than Y for any income level. More specifically, if all individuals with incomes equal to or above a D specified value of x are considered to be rich, then X 1 Y implies that the reverse cumulative distribution of X will always have same or higher percentage of rich individuals than that of Y for any income level. This further implies that the expected level of welfare for X is at least as great as that for Y for all increasing welfare function. On the other hand, if income distribution F SDSD dominates income distribution G, that is, X D 2 Y, this implies that the integral of the reverse cumulative probability of X is always on top of that of Y. The intuitive implication of SDSD is different from FDSD. That is, X does have higher proportion of rich individuals compared to Y for all relative higher income levels but this statement is not true at least not for some very low income levels. If we get SDSD of X over Y, we can infer that there are more very rich people in X than Y. Because of the above properties, we also call the test that could detect DSD relation as test for richness. There is a relationship between descending stochastic dominance and social welfare. Suppose again U denotes the class of all monotonic, symmetric additively separable social welfare functions of the form W(F) = u(x)df(x), where u: R + R is any continuous function. Let U D 1 U be defined by the condition u > 0; let U D D 2 U 1 be defined by u > 0; and let U D D 3 U 2 be defined by u > 0. Under these setting, we could apply the results on stochastic dominance (Wong and Li, 1999), to obtain the following rule: F D D k G if and only if W(F) > W(G) for all W U k for k = 1, 2, and 3 Thus, a distribution F has a higher welfare level than distribution G if the utility is increasing and convex. This is an important result that has not been considered in the previous literature of income distribution analysis. The important book of Sen (1997) states that it is remarkable for Atkinson (1970) to prove that Lorenz ranking (second ordered ASD ranking) could allow one to directly compare the social welfare level. The result from Atkinson (1970) and Foster and Shorrocks (1988) is remarkable but it is only true when the monotonic, symmetric additively separable social welfare function is a concave function. 10

11 However, a symmetric additively separable social welfare function may not necessarily be a concave function. In the extreme case, this is possible when the set of people in society P s each have an increasing marginal utility in income, and at the same time, the social welfare function for this group, W s (x), is positive definite 1. But since 2 W s (x) x i 2 > 0 and 2 W s (x) x i x j = 0 for all i, j P s and i j, the social welfare function associated with this group is clearly a convex function. In general, it is not difficult to imagine a social welfare function is positive definite once people in the economy are assumed to have an increasing marginal utility in income. If this is the case, the DSD approach to analysis is the more appropriate one to use. In practice, the true form of the social welfare function is unknown, which leaves the choice between ASD and DSD approaches indeterminate. As in this paper, we recommend the use the ASD and DSD approaches simultaneously, with clear advise for caution in interpretation of results. If income distribution F does not stochastic dominate income distribution G in both first-order ASD and first-order DSD, then we could only say that social welfare in income distribution F is higher than social welfare in income distribution G when the following two conditions are satisfied: (i) F dominates G in the sense of ASD (DSD) and (ii) when F dominates or equals to income distribution G in the sense of DSD (ASD). For example, if income distribution F SASD income distribution G and income distribution F SDSD G, then we can safely conclude that the social welfare in F is higher than the social welfare in G. However, we are unable to state that the same conclusion if F SASD G at the same time that G SDSD F. Levy and Levy (2002), Chan et al. (2012), Levy (2015), and others have developed some properties to study the relationships between different orders of ASD and DSD. In this paper, we discuss some of their relationships for income distributions as follows. First, hierarchy exists in both ASD and DSD for income distribution. That is to say, if the income distribution X is found to be k-order ASD dominates that of Y, then X will be k+1-order ASD dominates that of Y. Similarly, if the income distribution X is found to be k-order DSD dominates that of 1 For more details about the assumption of increasing marginal utility in income see Friedman and Savage (1948). 11

12 Y, then X will be k+1-order DSD dominates that of Y. Thus, one could only report the lowest dominance order of ASD and DSD being found. If X is found to FASD dominate Y, then X will FDSD dominate Y. That is to say, if X has less proportion of poor units compared to Y for any level of income, then we will know that X has higher proportion of rich individuals than Y for any level of income. On the other hand, Levy and Levy (2002) show that if X and Y have the same mean, then X dominates Y in SASD while Y dominates X in SDSD. One could easily relax the condition of equal means to be nearly equal such that if the means of X and Y are about the same, then X dominates Y in the sense of SASD if and only if Y dominates X in the sense of SDSD. In this situation, we can infer that incomes are more evenly distributed in X than Y such that there are less percentages of both extreme poor and extreme rich individuals in X, and Y is less evenly distributed that there are more proportions of extreme poor and extreme rich in Y. This implication is very useful because if the income in one country, say X, is found to SASD dominate that in another country, say, Y and the means incomes of both X and Y are about the same, this result could imply that the income distribution in X is more even than Y and have both less super rich and super poor than Y. From the above discussion, we notice that X and Y have the same direction of domination for both ASD and DSD if the dominance is of the first order. However, if their means are about the same, then the directions of ASD and DSD dominance are reverse. How about the third order? If X and Y have the third order ASD and DSD, will the dominance in the same direction or different direction? Chan et al. (2012) show that under some conditions, the directions of the dominance of TASD and TDSD are the same but under other situations, the directions are reverse. This finding is useful when one compare different income distributions. 3. Improved Stochastic Dominance Tests for Richness and Poorness In this section, we will discuss the improved stochastic dominance tests for richness and poorness. The stochastic dominance (SD) test (we call it the ascending stochastic dominance 12

13 (ASD) test to distinguish it from descending stochastic dominance (DSD) test in this paper) has been well developed. Commonly used SD tests include the tests developed by Davidson and Duclos (2000), Barrett and Donald (2003) and Linton et al. (2005). In this paper, we recommend to use the SD test proposed by Davidson and Duclos (2000) and modified by Bai, et al. (2015) to measure income distribution for the poorness because it has been found to be one of the most powerful, but yet less conservative in size (Wei and Zhang, 2003; Tse and Zhang, 2004 and Lean et al., 2008). It is found to be robust to non-i.i.d. data, including heteroscedastic data (Lean, et al., 2008). It can also be used to compare any portion of different distributions. Test for Poorness To test for poorness, we recommend to employ the ASD test. We assume {f i }(i = 1,2, N f ) and {g i }(i = 1,2, N g )are observations drawn from the income distributions X and Y, with distribution functions F and G, respectively, and with their integrals F A j (x) and G A j (x) defined in (2) for j = 1,2,3. For a grid of pre-selected points x 1, x 2,..., x k, the j-order ASD A test statistic, T j, proposed by Davidson and Dulcos (2000) and modified by Bai, et al. (2015) is: T j A (x) = F ja (x) G ja (x) V ja (x) (11) where V ja (x) = V Fj A (x) + V Gj A (x) 2V FGj A (x); N h 1 H ja (x) = N h (j 1)! (x z i) j 1 +, i=1 13

14 N h V Hj A (x) = 1 1 [ N h N h ((j 1)!) 2 (x z 2(j 1) i) + H ja (x) 2 ], H = F, G; h = f, s; V FGj i=1 N h A (x) = 1 1 [ N h N h ((j 1)!) 2 (x f i) j 1 + (x s i ) j 1 + F ja (x)g ja (x)]. i=1 It is not possible to test empirically the null hypothesis for the full support of the distributions. Thus, Bishop et al. (1992) propose to test the null hypothesis for a pre-designed finite numbers of values x. Specifically, for all i = 1,2,..., k; the following hypotheses to test for poorness: H 0 : F j A (x i ) = G j A (x i ), for all x i ; H A : F j A (x i ) G j A (x i ), for some x i ; H A1 : F j A (x i ) G j A (x i ), for all x i, F j A (x i ) < G j A (x i )for some x i ; H A2 : F j A (x i ) G j A (x i ), for all x i, F j A (x i ) > G j A (x i )for some x i. We note that in the above hypotheses, H A is set to be exclusive of both H A1 and H A2. This means that if the test does not reject H A1 or H A2, it will not be classified as H A. Therefore, Bai et al. (2015) modify the decision rules to be: max T j A (x k ) < M j α, accept H 0 : X = j Y 1 k K max T j A (x k ) > M j α and min T j A (x k ) < M j α, accept H A : X j Y 1 k K 1 k K max T j A (x k ) < M j α and min T j A (x k ) < M j α, accept H A1 : X j Y 1 k K 1 k K max T j A (x k ) > M j α and min T j A (x k ) > M j α, accept H A2 : Y j X 1 k K 1 k K where M α j is the bootstrapped critical value of the j-order ASD statistic. The test statistic is compared with M α j at each point of the combined sample. 2 However, it is empirically difficult to do so when the sample size is very large. In order to make the computation easy, we specify K equal-distance grid points {x k, k = 1,2,..., K} which cover the common support of random samples {X i } and {Y i }. Simulations show that the performance of the modified DD statistics is not sensitive to the number of grid points for some reasonably large number, say, for example K=100. Thus, in practice, we follow Fong et al. (2005), Gasbarro et al. (2007), 2 Refer to Bai et al. (2015) for the construction of the bootstrapped critical value M α j. 14

15 and others to choose K = 100. We note that Bai et al. (2015) improve the ASD test by deriving the limiting process of the ASD statistic T j A (x) so that the SD test can be performed by using max T j A (x) to take care of the dependency of the partitions. We follow their x recommendation in our analysis. In this paper, we improve on the ASD test in the sense that we use both limited number of grids and max x T j A (x) comparison. Fong et al. (2005), Valenzuela, et al. (2014), and others use the former while Bai et al. (2015) adopt the latter but they do not use both while we do. In addition, Fong et al. (2005) and others use the Studentized Maximum Modulus distribution (Richmond, 1982) to compute the critical value M α j that relies on the assumption that the grids are independent. However, Bai et al. (2015) have proved that the grids are dependent and thus they recommend to obtain the critical value by using simulation. In this paper, we follow their recommendation to obtain the simulated critical value M α j in our analysis. Test for Richness To test for richness, we recommend to use the DSD test. We assume {f i }(i = 1,2, N f ) and {g i }(i = 1,2, N g ) are observationsdrawn from the income distributions X and Y, with distribution functions F and G, respectively, and their reverse integrals F j D (x) and G j D (x) defined in (10) for j = 1,2,3. For a grid of pre-selected points x 1, x 2,..., x k, the j-order DSD test statistic, T j D, developed by Bai, et al. (2015) is: T j D (x) = F jd (x) G jd (x) V jd (x) (12) where V jd (x) = V Fj D (x) + V Gj D (x) 2V FGj D (x); 1 N h H jd (x) = (z N h (j 1)! i x) j 1 i=1 +, 15

16 N h V Hj D (x) = 1 1 [ N h N h ((j 1)!) 2 (z 2(j 1) i x) + H jd (x) 2 ], H = F, G; h = f, s; V FGj i=1 N h D (x) = 1 1 [ N h N h ((j 1)!) 2 (f i x) j 1 + (s i x) j 1 + F jd (x)g jd (x)]. i=1 For i = 1,2,..., k, the following hypotheses are tested for richness: H 0 : F D j (x i ) = G D j (x i ), for all x i ; H D : F D j (x i ) G D j (x i ), for some x i ; H D1 : F D j (x i ) G D j (x i ), for all x i, F D j (x i ) > G D j (x i ) for some x i ; H D2 : F D j (x i ) G D j (x i ), for all x i, F D j (x i ) < G D j (x i ) for some x i. Similarly to the situation in testing the ASD test, we follow Fong et al. (2005, 2008) and Valenzuela, et al. (2014) to make 10 major partitions with 10 minor partitions within any two consecutive major partitions in each comparison, and we follow the approach recommended by Bai, et al. (2015) to use max T j D (x) to test for the preference of risk seekers on the x income distributions. Not rejecting either H 0 or H A or H D implies the non-existence of any SD relationship between X and Y, and that neither of these distributions is preferred to the other. If H A1 (H A2 ) of order one is accepted, X(Y) stochastically dominates Y(X) at first order, while if H D1 (H D2 ) of order one is accepted, distribution X(Y) stochastically dominates Y(X) at first order. If H A1 (H A2 ) [H D1 (H D2 )] is accepted at order two (three), a particular distribution stochastically dominates the other at second- (third-) order. We recommend to use a limited number of grids for ASD and DSD comparison to accommodate the effect of almost SD. 3 We follow the ideas from Leshno and Levy (2002), Guo et al. (2014, 2014a), and others to address almost ASD and DSD in our findings. We note that Gasbarro et al. (2007) and others use a conservative 5% cut-off point in checking the 3 Almost SD allows a small area violation computed from the compared distributions to reveal a preference for most decision makers but not for all of them. Readers may refer to Leshno and Levy (2002) and Guo et al. (2014, 2014a) for more information. 16

17 proportion of test statistics for statistical inference. Using a 5% cut-off point, they conclude that one prospect dominates another prospect only if we find that at least 5% of the statistics are significant. For example, if X 10% SASD Y but Y 1% SASD X, Gasbarro et al. (2007) call this X SASD Y. On the other hand, if X 15% TDSD Y but Y 2% TDSD X, Qiao, et al. (2012, 2013) call this X TDSD Y. In this paper, we suggest to call the first example X 1%-almost SASD Y and call the second example, X 2%-almost TDSD Y. In the first example, it means that X SASD Y except 1% in the grids while in the second example, it means that X TDSD Y except 2% in the grids. We note that this is a preliminary test for the almost ASD and DSD. We note that the definition of almost ASD and DSD are different from what we are using, one may refer to Guo et al. (2014a) and others for the proper definition of almost ASD and DSD and one may refer to Quo et al (2015) for the proper almost ASD test. We note that the almost ASD and DSD tests are for the preliminary study of almost ASD and DSD, 4 indicating the ASD and DSD relationship except how many percentages in the grids. We will indicate the location of this almost areas because this provides a very important information to the income distribution. For example, we will tell readers where 1% of the claim that Y SASD X does not hold. is this in the top 1% richest, in the bottom 1% poorest, or 1% middle income group, this makes a big difference in the income distribution comparison of X and Y. Another advantage of using limited number of grids for comparison is that it enables one to investigate the dominance between two income distributions up to any interesting point from the poorness as well as from the richness to be compared. For example, one may be interested in know whether bottom 20% in country X dominates that in country Y and whether top 10% in country X dominates that in country Y. Using k=100, to compare ASD test for the first 20 grids will obtain the comparison for the former and to compare DSD test for the top 10 grids will obtain the comparison for the latter. Bai et al. (2015) have mentioned the advantages of this approach that it allows two distributions to be dependent and the bootstrap method to decide critical point so that our critical value will be closer to the true critical values. In analysis income distribution and 4 We note that the almost ASD test used in this paper is the preliminary test proposed by Guo, et al. (2015). They propose a proper almost ASD tests but it is still under construction. 17

18 inequality, this approach provides a number of additional advantages: first, it allows a more accurate analysis on social welfare (or living standard). Second, the approach does not require any specific poverty line. Since the second order ascending stochastic dominance is equivalence to Lorenz dominance as discussed in Section 2, our test for relative poorness (using our test in mean divided income series) in second order could also be considered as an improved test for Lorenz dominance. Third, our approach can be used to measure and test for both richness and poorness. Fourth, our approach accommodates the effect of almost SD. Fifth, our approaches allow analyists to measure the dominance between two income distributions up to any interesting point from the poorness as well as from the richness. Overall, the approach allows for a more in depth analysis on income distribution. 4. The Empirical Setting: An Overview of Hong Kong and its Economy Hong Kong, officially known as Hong Kong Special Administrative Region of the People's Republic of China, is a city on the southern coast of China. Hong Kong is well known for its expansive skyline, deep natural harbour and extreme population density (some seven million inhabitants over a land mass of 1,108km 2 (427.8 sq mi) in 2014). The current population of Hong Kong comprises 93.6% ethnic Chinese, most of whom are descendants of skilled labourers from the neighbouring Guangdong province. Except for a brief occupation of Japan during the second world war, Hong Kong was a British colony from 1842 until As a result of the negotiations between China and Britain, Hong Kong was transferred to the People's Republic of China under the 1984 Sino-British joint declaration. The city became China's first special administrative region on 1 July 1997 under the principle of "one country, two systems." Over the past 4 decades, Hong Kong has experienced dramatic economic growth. The economy took off in the 1970s and has successfully transformed itself into a regional hub of business services in the 1990s, with an annual GDP growth rate of 5.8 per cent from 1971 to For the more recent data, average nominal GDP growth is 4.17 per cent, average real 18

19 GDP growth is 3.99 per cent and 2.3 per cent (using the GDP deflator and CPI to deflate prices, respectively) from 2001 to Towards the late 1970s, Hong Kong became established as a major entrepôt between the world and China. The city has developed into a major global trade hub and financial centre, ranking fifth on the 2014 Global Cities Index after New York City, London, Tokyo, and Paris. The service economy is characterised by low taxation and free trade and has been regarded as one of the world's most laissez-faire economic policies, with the local currency, the Hong Kong dollar, ranking as the 13th most traded currency in the world. Hong Kong has one of the highest per capita incomes in the world. However, it also has the reputation of the one with the most severe income inequality among the advanced economies. In terms of inequality, Hong Kong s have always stood at the higher end early estimates of the Gini coefficient put it to be 0.43 in 1971, rising to in 1986, and further up to in 1991 (Chow and Papanek, 1981). 6 The most recent government statistics indicate Hong Kong Gini coefficient to be in 2001, in 2006, in placing the city among the most income-unequal societies in the world, alongside the poorest African and Latin American countries. So, by international standards, Hong Kong s income inequality has continuously stood at a relatively high level in recent decades. On the other hand, there are some studies argue that the income inequality in Hong Kong is not high. For example, in an earlier study, Hsia and Chau (1978) argue that rapid industrialization in Hong Kong had significantly contributed to the rise in the standard of living and to the decrease in the dispersion of the size distribution of household income. Chow and Papanek (1981) and Fields (1984) show that despite high rates of both economic 5 Using CPI to deflate prices is meaningful because CPI captures more relevant information on people's living standards in HK than GDP deflator, which captures the trading sector, which is almost 4 times of HK's GDP. 6 The Gini coefficients of 1986 and 1991 for all domestic households come from Census and Statistic Department, Summary Results of the 1996 Population By-census 7 For easier comparison, we use the Gini coefficient for all domestic households based on pre-tax, pre-benefit household income rather than the Gini coefficient based on post-tax, post-benefit per capita income since that number, for the best we known, doesn t available for year before The Gini coefficient based on post-tax, post-benefit per capita income is in 1996, in 2001, in 2006 and in These Gini coefficients come from Census and Statistic Department, Hong Kong SAR Thematic Report Household Income Distribution in Hong Kong, 2011 Population Census 19

20 and population growth, Hong Kong s income distribution did not clearly worsen. Turner et al. (1991) suggest that there was significant narrowing of income differentials between different classes of employees during the period from the mid-1970s to the mid-1980s. Chau (1994) suggests that there has been extensive upward mobility of low-income households in Hong Kong especially since the late 1970s. Wu (2009) finds that compared to China, the Hong Kong population maintain a higher degree of tolerance of income inequality and a higher degree of perceived fairness of income distribution; apparently, public discontent is not as pronounced as it has been in mainland China although high inequality started to be used in the mid 2000s by pro-democracy movements to generate interest and mobilize populations into political action. Whether Hong Kong s income inequality is high or not high? In this paper it is not our intention to test whether the income inequality is high or not high because we just compare the inequality in Hong Kong over time for different age groups. Howeer, our finding could be used to draw inference on whether the inequality is wider or narrower over time for different age group. 5 Data, Results and Discussion In our analysis we use the 5 per cent sample data set obtained from 2001, 2006 and 2011 Population Census conducted by Hong Kong Census and Statistics Department. Three particular series of data are used in the sample data set including Age, Total Personal Income from all employment (Einc) and Other Cash Income (Oinc). Einc pertains the amount a person has earned monthly from employment including salary or wage, bonus, commission, overtime, housing allowance, tips and other cash allowances. 3F8 Oinc, on the other hand, refers to total recurrent cash incomes received by a person each month. This includes a cash received which are not remuneration for work e.g. rent income, interest, dividend, education grants (excluding loan), regular/monthly pensions, social security payment, old age allowance, disability allowance, comprehensive social security assistance, scholarships, regular contribution from persons outside the household, contribution from charities. In addition, the series Total Personal Income (Total) is generated by adding income from all 8 New Year bonus and double pay are excluded. 20

21 employment and Other Cash income together. That is, Total = Einc + Oinc. All data used in the paper are adjusted by inflation rate. 9 Table 1 presents some descriptive statistics from our sample population. From the table, we find that the population share of age group has steadily shrunk between 2001 and 2011, in favour of a rise in the share of the age group. The share of the under 35s decreased by 12.6 percentage points from 2001 to 2011, which is a 1.3 per cent per annum reduction rate. In contrast, the population share of the group grew by 0.8 per cent per year, while the population share of the 65+ group remains small at less than 2 per cent and is virtually unchanged between 2001 and We present some singular measures of poverty and inequality that we calculated from the data in Tables 2 to 5: The Sen-Shorrocks-Thon (SST) indices (Indices A and B) proposed by Shorrocks (1995) based on the pioneering work of Sen (1976) that used to examine the degree of poverty,5f10 and the Gini index proposed by Gini (1912) that used to examine the degree of inequality. From the last row in Table 2, we find that poverty increased from 10 to 12 per cent from 2001 to 2011, and there is no discernible age group that bore the burden of this poverty increase. From Table 3, we see that poverty rate for each census year is generally higher for the older age group (age groups above 65). Examining the changes over time, we find that between 2001 and 2011, poverty levels increase more than doubled for the youngest age group (increases from 6.9 per cent to 15.3 per cent) and the oldest age group as well (increases from 19.3 per cent to 41.5 per cent). This is a trend that can be cause for concern, but more so for the younger set as they take a significant proportion of the whole population, compared to the 85+ which comprise less than 0.5 per cent of the population.f The observations in relative poverty levels over time and across age groups can also be seen in the columns for Earned Income, Einc in Table 2 and 3. But the zeroes in the Oinc clearly stand out. This only highlights the disadvantage of setting a poverty line using one particular way, in our case, half the median income: because the median for Oinc is mostly zero, we are 9 Some modifications have also been applied to the data set: 1) All the data that is label as N.A in any of the two series are deleted. 2) All the data that is label as 0 in their corresponding age column are deleted. 10 We set the poverty line as that income that is half the median of the equivalent income. 21

22 not able to get much information for the SST and many other singular index type poverty measurement tools. Additionally, if we use other methods to set the poverty line, then the corresponding singular indices we calculate will not be comparable. Tables 4 and 5 present the estimated Gini coefficients for various age groups. We can see from here that there was a steady increase in inequality between 2001 and 2011 and that the Gin indices appear to rise with age. Between earned and other income, we also find that there seems so much more inequality in Oinc, compared to Einc, where these differences are up to three times higher. Further, we find that inequality based on earned income is positively related with age, while inequality based on other income is inversely related with age. We also note here the apparent relationship between the calculated values of the Gini indices and the SST indices, that is, the higher the Gini index is, the higher is the poverty is, given that other things being hold constant. Tables 6 to 8 present the results of our stochastic dominance analysis of Hong Kong incomes. In these tables, Einc t t t m, Oinc m and Total m refer to personal income, other cash income, and total income, respectively, in period t for age group m, with t = 01, 06, 11 representing 2001, 2006 and 2011 and m = 15~34, 35~64, 65 representing the age groups of 15-34, and all age that are larger or equal to 65, respectively. Einc t, Oinc t and Total t are the sum of Einc t t t m, Oinc m and Total m for all age groups. Using the decision rule state in Section 3, the SD tests results for comparing the poorness and richness in Hong Kong for t different groups and different time periods are summarized in Tables 6, 7 and 8 for Einc m t Oinc m and Total t m, respectively. We first start our analysis for the employment income, Einc, in Table 6. We find that the distribution of employment income in 2006 is the most favourable one because it TADS distribution in 2001 and TDSD distribution in For the age group of 15 to 34, most distributions of employment income for this age group are descending dominated by distributions of the age group of 35 to 64 and the age group of 65 and above but ascending dominate the age group of 65 and above. The comparison of this age group between different years shows that income distribution in 2001 is the best one. 22

23 For the age group of 35 to 64, all distributions of employment income for this age group are first-order stochastic dominate the age group of 65 and above. The comparison of this age group between different years only allow us to state that income distribution in 2006 is better than 2001.It is because distribution in 2001 TDSD distribution in 2011 but distribution in 2011 SASD distribution in 2001 and distribution in 2006 SDSD distribution in 2011 but distribution in 2011 SASD distribution in The comparison between different years of the age group of 65 and above, shows that 2006 is better than 2001.It is because distribution in 2001 TDSD distribution in 2011 but distribution in 2011 SASD distribution in 2001 and distribution in 2006 SDSD distribution in 2011 but distribution in 2011 TASD distribution in In Table 7, present results for other cash income, Oinc, which could be viewed as a proxy for wealth or non-human capital. We find that the distribution of other cash income in 2001 is first-order ascending and descending stochastic dominated by distribution of other cash income in The distribution of other cash in 2001 is also second-order ascending dominated by distribution of other cash income in On the other hand, distribution of other cash income in 2006 is dominates the distribution of other cash income in 2011, in both the SASD and SDSD tests. For the age group of 15 to 34, the distribution of other cash income for this age group first- order ascending and descending stochastic dominated by nearly all other income distributions. The comparison of this age group between different years show that income distribution of other cash income in 2006 should be the most favourable one since it FASD and FDSD the distributions of 2001 and For the age group of 35 to 64, the distribution of other cash income for this age group firstorder ascending and descending stochastic dominate nearly all other income distribution of age group 15 to 34 but FASD and FDSD almost all other income distribution of age group age that are larger or equal to 65. The comparisons between different years show that the 2006 s income distribution is the best under our stochastic dominance criteria. Over all, the distributions of other cash income of age group for all age that are larger or equal to 65 seem to be the most favourable one. 23

24 The SD tests results for comparing distributions of total personal income, Total, are summarized in Table 8. Here, we find that the distribution of total personal income in 2001 SASD the distribution of total personal income in 2006 but is second-sorder descending stochastic dominated by the distribution of total personal income in We find no stochastic dominance relationship when we compare the distributions of 2001 and On the other hand, we find that the distribution of total personal income in 2006 dominates the distribution of other cash income in 2011 using the TDSD test criteria. For the age group of 15 to 34, the distribution of total personal income for this age group second- order ascending and descending stochastic dominated by nearly all other income distributions of age group 35 to 64. It is interesting that, the distribution of total personal income for 15 to 34, SASD almost all distributions of total personal income for 65 and above, but descending stochastic dominated by almost all distributions of total personal income for 65 and above. These imply that age group of 15 to 34 has less proportion of poor units compared to age group of 65 and above for all the lower income levels but age group of 65 has more very rich people than the age of 15 to 34 in terms of total income. For the age group of 35 to 64, the distribution of other cash income for this age group FASD and FDSD almost all other income distribution of age group for all age that are larger or equal to 65. Our final table, Table 9, shows the results of stochastic dominance tests of relative richness and poorness in the income distributions of Hong Kong. We are using the same approach as above but derive the results using all data by its corresponding mean. As we have mentioned in section 2, the second order ascending stochastic dominance implication is equivalent to Lorenz dominance. In the first panel, it can be seen that for employment incomes, the earnings distribution of 2001 exhibited no ascending dominance over the earnings distribution of The descending tests however show that the 2006 dominated the 2001 distribution in the secondorder. Between 2001 and 2011, we find that the 2001 distribution also exhibited no ascending dominance over the distribution of 2011, but that the latter distribution was however found to descending dominate the former in the third order. And between 2006 and 2011, the