An illustration: Newfoundland and Labrador and Ontario

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Before showing the overall results, this section illustrates a comparison of low income for two provinces—Newfoundland and Labrador and Ontario—using the technique described above. Comparisons for other pairs of provinces are examined in a similar fashion. In Figure 3, we draw the two cumulative distribution functions (CDFs) for Newfoundland and Labrador and Ontario, respectively, where income is needs/region-price adjusted. The low-income cutoff (LICO) headcount rates can be obtained by drawing a standardized LICO line ($15,352) on the x axis. The corresponding y values confirm the information in Figure 1 that the headcount low-income rate is 13.2% in Newfoundland and Labrador and 10.8% in Ontario. However, more importantly, Figure 3 immediately reveals the drawback of the LICO because the answer to the question "Where is low income greater?" crucially depends on where the low-income line is drawn. Indeed, rank order, in this comparison, may lead to a complete reverse outcome when the low-income line is set to below $10,000 of equivalent income.

Tests of stochastic dominance over the full domain of low-income distribution

The task here is therefore to draw statistical inferences to test whether low income differs significantly between two regions. Tests of stochastic dominance are first conducted covering the full spectrum of the lower-end income distribution (i.e., for all possible values of low-income lines between 0+ and $20,000). It should be emphasized that t statistics are calculated at each value of x observed in the sample. For illustration purposes, we only show a grid of 20 low-income lines, which lie from $1,000 to $20,000 at intervals of $1,000, in Table 1. The estimated headcount ratios—along with their asymptotic standard errors—for both provinces as well as the t statistics of the difference for each of these 20 points are presented.¹³ At 5% significance level, Table 1 shows that Newfoundland and Labrador has a lower headcount ratio for all x less than $8,000 (the estimated threshold is $8,416); while Ontario dominates Newfoundland and Labrador for low-income lines above $13,000 (the estimated threshold is $12,366).¹⁴ Since the two distributions crossed within the range of interest and the minimum t statistics show that both provinces dominate each other over some areas of x at significance level, a first-order stochastic dominance cannot therefore be concluded.

Given that there is no clear conclusion for a first-order test, we search for a second-order dominance, which focuses on the low-income gap measure. Figure 4 shows the low-income deficit curves where—for any given poverty line x—the y value represents the mean proportionate low-income gap (as defined by the percentage of x). That is, for instance, a point (y, x)=(0.06, 20,000) in a deficit curve indicates that the mean distance below the low-income line is $1,200 (20000*.06) for the population if the low-income line is set at $20,000. Similar to incidence curves, the two deficit curves still intersected at x around $15,000. The t statistics based on the minimum t ratio at the 5% level show that Newfoundland and Labrador's second order dominates Ontario's for x below $11,424, while Ontario dominates Newfoundland and Labrador for x > $18,470. As a result, second-order dominance is not obtained.

In a search for third-order dominance, we plot low-income severity curves in Figure 5, where the y value is the mean proportionate squared low-income gap. This measure takes into account inequality among the poor by giving unequal weights to the poor population, where the weights are the proportionate low-income gaps themselves. By squaring the gap, low-income comparison between these two provinces has become more clear as the Newfoundland and Labrador curve now lies below that of Ontario for all x < $14,684 at the 5% level, and no reversal is found for all other x values < $20,000. This is not surprising, because this measure gives more weight to the poorest of the poor, and Ontario appears to have a higher proportion of poor people at the bottom of the distribution. As a result, we conclude that Newfoundland and Labrador has less low income than does Ontario, as Newfoundland and Labrador's third-order dominates Ontario over the domain ($0+, $14,684).

The exercise also reveals the sensitivity of low-income measures used. In fact, it shows that no poverty-measure ordering (see Zheng 2000) can be found when LICO is used. That is, under LICO, Ontario is considered to have less low income than Newfoundland and Labrador in terms of headcount rates at the 5% level; the ordering then becomes ambiguous in the low-income gap measure and a reverse ordering is obtained in the squared low-income gap measure.

Tests of stochastic dominance over restricted domain

Notice that dominance relations from above may not hold if the range of interest is redefined over a restricted domain (z_min, z_max), rather than over the full range of the lower-end income distribution (0, z_max). Recall that the two distributions crossed at around $10,000. At the 5% level, Newfoundland and Labrador dominates Ontario for x less than $8,416 and Ontario dominates Newfoundland and Labrador for x greater than $12,366. This implies that one can actually obtain a very different conclusion that Ontario's first order dominates Newfoundland and Labrador's for all x > $12,366, if the lower limit of interest is set to over $8,416.

The challenge, however, is to pick up a reasonable lower limit, where t statistics are to be calculated. In this paper we consider an arbitrary choice of $5,000. Even though this value does not really make sense for a 'minimal survival poverty line,' we choose it because the value is small enough to make our comparisons more robust, but it is large enough to avoid problems of small observations and measurement errors that usually prevail in the lower tail of the distribution. Given the restricted domain of interest, the testing result (the minimum t statistic at the 5% level) still concludes that Newfoundland and Labrador dominates Ontario stochastically at the third order condition over ($5,000, $14,684) with a reported lower threshold censored at $5,000.

The simple illustration above demonstrates that the rankings of low income based on commonly used LICO indicators are not robust, because such comparisons only rank the headcount at one low-income line, and a contradictory result may occur when different low-income lines are chosen. Using the stochastic-dominance approach, this example shows that the two distributions of income can be ranked over a wide range of possible low-income lines. We also show that ranking of distributions may alter when different domains of interest are assumed. In fact, since the t statistics are calculated at each point in the sample, the lower/upper thresholds for dominance can be obtained at a certain level of significance using the minimum t-statistic approach. It allows us to check until we reach which minimum/maximum values of the low-income line we can go to in order to rank low income across two provinces.

Does the choice of cost-of-living deflator matter?

In Figures 6, 7, and 8 we repeat the same exercise as above, but now we use the market-basket measure (MBM)-based cost-of-living index as scaling factors for equivalent income.¹⁵ The LICO equivalence scale is still used to adjust for family composition. Contrary to previous findings, a dominance relation cannot be established between Newfoundland and Labrador and Ontario when the range of interest covers all possible values between zero and $20,000. It is clear from graphs that the two density curves crossed for all first-, second- and third-order conditions and both provinces' low incomes dominate each other at different intervals of the distribution. Low-income severity curves, for instance, show that Ontario's third order dominates Newfoundland and Labrador's for low-income lines above $14,358 at the 5% level, while also showing a complete reverse outcome for low-income lines below $9,091. This reveals that the rankings of low income exhibit sensitivity to the choice of scaling factors relating to cost of living. It is, however, noteworthy that the two low-income incidence curves cross at a much lower value of x when the MBM cost-of-living index is used. Indeed, a reverse outcome will not happen at the 5% level until x < $6,000. Therefore, one may obtain restricted first-order dominance of Ontario over Newfoundland and Labrador for low-income lines between $8,430 and $20,000 if the range of interest is set to above $6,000.

Does the choice of equivalence scale matter?

A similar robustness argument can be applied to the choice of equivalence scale. To examine this, we re-compute equivalent income using two other equivalence scales: the 'squared root of family size' and 'modified OECD scale,' respectively.¹⁶ At the significance of the 5% level, the results (not shown) greatly resemble those in the base case. It concludes third-order dominance of Newfoundland and Labrador over Ontario if test statistics are calculated over the full domain. For restricted dominance, it concludes that Ontario's first-order dominates Newfoundland and Labrador's for low-income lines above $10,471 (compared with $12,366 in the base-case model). This suggests that the rankings of low income are generally less sensitive—at least in this illustration—to the choice of equivalence scales.

Relative concept of low income

How does switching from an absolute to a relative low-income concept affect the rankings of low income? To answer this question, we normalize equivalent income for each individual by dividing respective provincial median income. Income is still-needs adjusted and prices adjusted, using LICO factors. The maximum possible low-income line is set at 70% of the provincial median income. For a restricted case, t statistics are computed for a range of low-income lines from 15% to 70% of the estimated median income, on the basis that they are considered the reasonable lower and upper limits to the low-income lines. Figure 9 reveals that the two CDF lines exhibit quite similar patterns, except for the lower portion of the distribution. The minimum t statistic from Table 4 indicates that Newfoundland and Labrador's first order dominates Ontario's for low-income lines below 34.2% of respective median income. There is no need to look for higher order conditions because no reverse outcome is found in the range of interest at significance level.

Compared with the base-case results, this exercise shows sensitivity to the choice of absolute or relative low-income lines. In the latter case, Ontario never stochastically dominates Newfoundland and Labrador for any range of low-income lines. It is also worth noting that low-income rankings, based on relative concepts, are less affected by the choice of cost-of-living deflator (results not shown) because individuals are now compared with the standard (% of median) in the province of residence. Thus, inter-provincial price differences are irrelevant and only differences in intra-provincial prices matter.