Statistics Canada
Symbol of the Government of Canada

Low-income measures and stochastic dominance

Warning View the most recent version.

Archived Content

Information identified as archived is provided for reference, research or recordkeeping purposes. It is not subject to the Government of Canada Web Standards and has not been altered or updated since it was archived. Please "contact us" to request a format other than those available.

By far the most widely used low-income measure is the headcount index, which simply measures the proportion of the population that is counted as living in low income. In this paper, we also include two other measures—the low-income gap and the squared low-income gap—in the analysis. These three measures belong to the Foster-Greer-Thorbecke class of poverty indices (see Foster, Greer and Thorbecke 1984). These measures may be written generally as

formula

formula

where y1 is the value of per-adult equivalent income for the 1-th person, and N represents the total population. Then gi is the income shortfall—the gap between individual income and the low-income threshold—for individual for a given low-income line z, and is a measure of the sensitivity of the index to the income shortfalls themselves. Foster, Greer and Thorbecke (1984) interpret this parameter as an indicator of "aversion to poverty" because it gives greater emphasis to the poorest poor as a becomes larger. When a =0, P0 is simply the headcount ratio; when

a =1, P1 is the poverty or low-income gap index, defined by the mean distance below the low-income line, where the mean is formed over the entire population, with the non-poor counted as having a zero low-income gap; and when =2, P2 (the squared low-income gap) is called the poverty or low-income severity index, because it is sensitive to inequality among the poor. Although, in general, P can be derived for any desired order, it becomes more difficult to interpret for larger . Therefore, we restrict our discussion to the first three measures in this paper.

Note that it is useful to consider the low-income gap (P1) and squared low-income gap (P2) measures in addition to the commonly used headcount index (P0), since the latter is neither monotonic nor distribution-sensitive. For example, a small transfer of income from a rich person to a very poor person may not change the headcount ratios, while this welfare improvement is reflected in a reduction of both the P1 and P2 measures. Also, a transfer of income from a poor person to a poorer person may not alter the P0 and P1, but it lessens inequality among the poor, and it is reflected in a reduction of the P2 measure.4 Policies on low-income reduction may be more appropriately targeted—whether reducing headcount or reducing severity of the poor— when outcomes of all three measures are understood.

Stochastic dominance and statistical inference

To make a robust low-income comparison for two income distributions, it is important to check whether low income in one distribution always dominates the other, no matter what low-income line is used. This requirement can be addressed by drawing on the technique of stochastic dominance, which is based on the comparisons of cumulative distribution functions (CDFs). Consider two distributions of incomes with CDF, FA and FB, respectively. Let

formula, and

formulafor any integer s ≥ 2 .

Distribution B is said to dominate distribution A stochastically at order s if formula for all low-income lines over the domain of interest. The graph of D1(x) is often referred to as the low-income incidence curve because it is traced out as one plots the headcount index on the vertical axis and the low-income line on the horizontal axis, allowing the low-income line to vary from zero to an arbitrarily selected maximum low-income line zmax. The graph of D2(x) is usually

regarded as the low-income deficit curve, and D3(x) as the low-income severity curve.

Since two density curves may be very close to each other, there is a need to assess whether the difference between them is statistically significant. Studies have suggested various hypotheses that could be used in a testing procedure for stochastic dominance.5 In this paper, we employ a null hypothesis of non-dominance of B over A,formula for all x over a domain of interest. If the null is rejected, then it legitimately infers the dominance of B over A. It can be shown that such a hypothesis is asymptotically bounded by the nominal level of a test based on the standard normal distribution. The test is based on the minimum t-statistic approach proposed by Kaur, Prakasa Rao and Singh (1994) for the null against the alternative of dominance. Similar to Kaur, Prakasa Rao and Singh, we calculate the t statistic for each value of x that is observed in the sample. We reject the null of non-dominance and accept the alternative of dominance if the minimum t statistic is significant at the 5% level. This procedure is often interpreted as an intersection-union test because dominance of B over A can only occur if the t statistic for the difference in any ordinate pair is significant.6

In reality, it is often possible that the two distributions of incomes may cross within the range of interest (as in Figure 2).7 In this case, there are two closed intervals observed and two minimum t statistics are obtained with the opposite sign. If both minimum t statistics are significant at a certain level, we can conclude dominance of B over A between formula and also dominance of A over B betweenformula. As a result, dominance relation over the entire domain is uncertain or undetermined. If it occurs, one may resolve the problem by looking for higher order stochastic dominance, which focuses on a measure that places more weights on the poorer persons, to help reach a clear conclusion. In the case of second-order dominance, it is to compare the low-income deficit curve, which can be traced out by calculating the areas under the cumulative distribution function (low-income incidence curve), and plot its value against the low-income line. Similarly, third-order dominance can be employed by comparing the low-income severity curve (the areas under the low-income deficit curve). If this fails to reject the null of non-dominance up to third-order condition, we declare that the two distributions of income are not comparable.

In many circumstances, especially in the discussion of welfare economics, interests of poverty dominance are often restricted to over an arbitrarily defined intervalformula, as suggested in Atkinson 1987. In Figure 2, for example, first-order stochastic dominance of A by B is not found over the whole range of the income distribution, while dominance may be obtained over the restricted domain formula. The comparisons, therefore, refer only to a 'partial,' rather than a complete, ordering of the distributions. Davidson and Duclos (2006) also point out some rationales for focusing on testing of restricted dominance and they emphasize that such focus would avoid comparisons over areas where there is too little information.

In fact, it may be more informative to estimate the thresholds for dominance (or restricted dominance) relations between regions. Since, in this paper, t statistics are calculated at each value of x over the domain of interest, it is possible to find the estimates of lower/upper thresholds in which interval one distribution stochastically dominates the other. To do this, we must first choose a range of low-income lines where t statistics are calculated. Then, the minimum t statistic is used to test the null of non-dominance at significance of 5% level. If there does exist only dominance of B over A for the range of interest, we declare dominance and report the estimates of lower/upper thresholds to which range the distribution B ranks over A. However, if there is a failure to reject the null, either because the minimum t is not significant or because there exists a reverse case (dominance of A over B) at another interval within the range of interest, we declare no dominance and search for higher order tests.

In this paper, t statistics are calculated mainly for two different ranges of interest over the lower part of the income distribution: full domain ($0+, $20,000); and, restricted domain ($5,000, $20,000). In either case, we make an arbitrary choice of maximum possible low-income line zmax =$20,000 of equivalent income (see definition below), while the lower limit is set to $5,000 of equivalent income for the restricted model.