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Income Research Paper Series – Research Paper
Low-income Dynamics and Determinants under Different Thresholds: New Findings for Canada in 2000 and Beyond
Theoretical considerations and empirical strategy
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Conceptual questions and notations
Identifying low income
Transitory and persistent low income
Analysing low-income dynamics
Analysing low-income persistence
Analysing incidence of low income and probability of being in low income
Duration aversion and comprehensive measure of low-income durations
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Conceptual questions and notations
We would like to discuss the following conceptual questions that are relevant to our empirical analysis. What are the possible time-invariant and time-varying low-income thresholds? What are the issues arising in low-income identification? How should one analyse low income over time and across the population? What are the differences between transitory and persistent low income? How should one measure transitions into and out of low income? How should one analyse low-income incidence, duration and transition in terms of key characteristics of the members of the population?
To facilitate the discussion, we use U to denote the target population, in which there are N individuals. We study these individuals over T periods. Let y_{it} be the income of individual i in period t, where i = 1, 2, ….N and t =1,2,…T. Let y_{.t} = [y_{1t}, y_{2t}, ….,y_{Nt}] be the income vector of the total population in period t and y_{i}. = [y_{i1}, y_{i2}, ….,y_{iT}] be the income vector of individual i over T periods. Let w_{it}be the vector containing socioeconomic and demographic information, such as gender, age, educational attainment, activity limitation, immigration status, minority status, family size, family composition and area of residence for individual i in period t.
Identifying low income
Low income is identified by comparing family income with the low-income thresholds the family faces. The three low-income thresholds are LICO, LIM and MBM, which are established and regularly updated by the Canadian government and widely employed by researchers.^{1}LICO is established using data from the Family Expenditure Survey, now known as the Survey of Household Spending. When a family has to spend 20 percentage points more of its income on necessities (food, shelter and clothing) than the average family, it is classified as a low-income family. Separate thresholds are defined for seven sizes of families—from unattached individuals to families of seven or more people—and for five community sizes—from rural areas to large population centers with more than 500,000 people. To determine whether a person (or a family of which the person is a member) is in low income, an appropriate LICO (given the family size and community size) is applied to the income of the person's economic family. In this study, we use after-tax family incomes and after-tax LICOs.^{2} If the economic family income is below the cut-off, all individuals in that family are considered to be in low income.
LIM is a low-income threshold that is defined as a fixed percentage, 50%, of the median 'adjusted family income'^{3}—family income adjusted for size using the equivalent scale, which takes account of the economies of scale in consumption. By design, LIM is not adjusted for differences in community size: it is, however, automatically adjusted each year for changes in family income distribution. As with low-income identification under LICO, if the income of an economic family is below its LIM threshold, all individuals in that family are considered to be in low income under LIM. Our analysis follows the convention of Statistics Canada and the literature to use after-tax income and the after-tax LIM.^{4}
Fixing LIM at 50% of the median adjusted family income can be somewhat arbitrary.^{5} For example, the European Union has chosen 60% for a conceptually similar low-income threshold and the OECD, 70%. Statistics Canada proposed modifications to the LIM by replacing the economic family-based income with household-based income, by replacing the current LIM equivalence scale with the square root of household size, and by taking household size into consideration in determining the low-income thresholds.^{6}
MBM is based on the costs of a basket of goods and services such as food, shelter, clothing, transportation and other essentials. Statistics Canada, on behalf of HRSDC, collects price data on the goods and services in the basket to calculate thresholds for 19 specific communities and 29 community sizes in the 10 Canadian provinces. The MBM thresholds are calculated for a reference family of two adults aged 25 to 49 with two children, a boy aged 13 and a girl aged 9. The costs for all other household configurations are then calculated using the LIM equivalence scale.
The income compared with the MBM threshold is different from the after-tax income, as the income relevant to MBM further excludes from total income other non-discretionary expenses such as support payments, work-related child care costs, transportation costs and employee contributions to pension plans and to Employment Insurance.^{7}^{,}^{8} If the economic family's disposable income defined as such is below the established MBM threshold, all individuals in that family are considered to be in low income.
Although LICO, LIM and MBM are in the context of family or household income, it would be much clearer if we convert these incomes and cut-offs into individual equivalent incomes and cut-offs.^{9} Generally, if the income of individual i at time t, y_{it}, is lower than a suitably chosen low-income threshold, the individual with that income is considered a low-income individual. For any low-income threshold, z_{t}, we may use the indicator function to simplify the discussion.^{10} Let A_{it}refer to the event y_{it} < z_{t} for all i and t. If the indicator function I(A_{it}) = 1 holds, individual i in period t is identified as in low income. If I(A_{it}) = 0, then individual i in period t is identified as not in low income.
With the above low-income identification, in period t the target population U which is the same for all T periods, is classified into two subpopulations: individuals whose incomes are below the low-income threshold in period t, S_{t} = {i: y_{it} < z_{t}}, and individuals whose incomes are greater than or equal to the low-income threshold in period t,
. The total target population is . In our study, U contains the same N individuals for all T periods within a panel.Transitory and persistent low income
In this paper, we distinguish transitory (or transient) low income from persistent (or chronic) low income. While the distinction between the two is probably well understood, there are many possible interpretations at the operational level. For example, Borrooah and Greedy (2002) consider one year in poverty as temporary poverty and two years in poverty as permanent poverty. Hulme et al. (2001, 2003) use more refined grades of poverty duration. According to their definitions, a 'chronically poor' person is an individual whose income is lower than the low-income threshold in each of the T periods or in most of the T periods, e.g., five or four out of five years; a 'transitorily poor' person is an individual whose income fluctuates around the low-income threshold over time, or whose income falls below the low-income threshold in one of the T periods; a 'non-poor' person is an individual whose income is always greater than the low-income threshold during the study period. These definitions are consistent with the classification of the low-income individuals into transitory and persistent low-income groups in this paper.
Clearly, the period of an individual's lifespan covered by the survey dictates how persistent low income can be best measured. If we follow a cohort only for six years, then the maximum duration in low income would be limited to six years. Censoring and truncation will inevitably occur.^{11} There are two types of censoring. When low income starts before the first survey year, we call this 'left censoring.' When low income persists beyond the last survey year, we call this 'right censoring.' Truncation occurs when low income is so brief that the annual survey does not detect the spell of low income. The critical difference between censoring and truncation is that the former is detectable, while the latter is undetectable with annual data. Annual surveys are inherently incapable of capturing brief low-income spells within a year.^{12} To minimize challenges from censoring and truncation, we will use the longest panel data possible and ignore truncation in our analysis.
Analysing low-income dynamics
To measure low-income dynamics, let the low-income indicator be d_{it} = I(A_{it}).d_{it} = 1 if individual i in period t is in the low-income state; d_{it} = 0 otherwise. Let d._{t} be the vector of low-income indicators for the population in period t and d_{i}. be the vector of low-income indicators for individual i for all T periods.
When we follow a cohort in and out of low income with its probabilities (or proportions) in and out of that low-income state for all T periods, which is the maximum number of years, we call this cohort a closed Markov system.^{13} Let q_{t} be the number of the low-income individuals in the target population. Then the proportion^{14} of individuals in low income (denoted by state 1) in period t is π_{1t} = q_{t} /N and the proportion of individuals not in low income (denoted by state 2) in period t is π_{2t} = 1 - q_{t} /N. By definition, π_{1t} + π_{2t} = 1. Let π_{t} = [π_{1t} , π_{2t}]. The history of proportions in low income and not in low income over T periods is therefore given by π_{1}, ...,πt-1, π_{t}, π_{t}+1,…,π_{T}.
We can also examine low-income dynamics by using the history of the proportions of individuals in low income and out of low income in period t, conditional on their previous low-income state in period t-1 over periods t = 2,…,T.
To make sense of the history of conditional proportions, {π_{nt} | π_{mt-1}}, we need the transition probabilities. The transition probability from an income (low-income or not-low-income) state m in period t-1 to an income state n in period t is denoted by p_{mn}(t). In our analysis, we have a transition probability matrix in period t
for t =1, 2,…,T. Here, p_{11}(t) + p_{12}(t) = 1 and p_{21}(t) + p_{22}(t) = 1. We can identify interesting patterns in the mobility of the low-income states by analysing the estimated transition probabilities. With the transition probabilities, we can calculate π_{nt} | π_{mt-1}=π_{mt-1} p_{mn}(t) and π_{nt}=π_{mt-1}p_{mn}(t) + π_{nt-1}p_{nn}(t) for period t =2,…,T and state m, n = 1, 2 which can be written compactly as π_{t}=π_{t-1}p(t) for all t.
In particular, we can use the long-run average of the diagonal elements of P(t) to measure the immobility in various low-income states. More specifically,
can be used to evaluate the overall immobility regardless of being in, or not in, low income. Because the state 1 (2) is the low (or non-low) income state, we can use to evaluate the overall immobility of being (or not being) in low income. Similarly, we can use to evaluate the mobility from state m to state n, where m, n = 1, 2.^{15}
Analysing low-income persistence
Given income data for N individuals over T periods, we can use the set framework to describe transitory and persistent low income.^{16} In our study, individuals may maintain or change their low-income states over time. As mentioned before, in the population U, we use S_{t} and
to identify those who are in or out of low income in period t. In order to analyse low-income persistence, we can examine S_{t} (or ) over time t. A range of possible configurations is available. One extreme is that the individuals are in low income in all T periods: this group is denoted by ∩^{T}_{t=1} S_{t} . The other extreme is that individuals are not in low income in all T periods: this group is denoted by_{}. Then the set U - _{}represents the individuals who are in low income for at least one year; the set U - ∩^{T}_{t=1} S_{t}represents those who are not in low income for at least one year. There are many intermediate configurations between these two extremes.To make the above framework operational, let D_{iT} be the number of the years in low income for individual i over T periods, which is given by
We can analyse the distribution of normalized low-income durations. The normalized low-income duration is defined as
Here, δ_{T} = [δ_{1T}, δ_{2T},…, δ_{NT}] describes low-income durations as fractions of T periods of the target population. Obviously, 0 ≤ δ_{iT} ≤ 1. The longer (or briefer) the low-income duration is, the higher (or lower) value δ_{iT} has. As usual, the mean and variance of δ_{T}provide useful information about the distribution. Sometimes, we are interested in the distribution of non-zero δ_{iT}. In this case, we can denote these normalized low-income durations by the vector δ_{p}. Similarly, the mean and variance of δ_{p }provide important information about the distribution of δ_{p}.
We can analyse the proportion of the population who are in low income for k out of T periods:
where #(D_{iT} = k) is a count function for the number of individuals whose low-income periods equal k, with k = 0, 1,…T. We can also examine the proportion of the population who are in low income for at least k periods,
Clearly, the relationship between π^{T}_{t=1}(D ≥ 1) and π^{T}_{t=1}(D = 0) is given by
As to what constitutes transitory or persistent low income, researchers must make a reasonable choice. For our panel data of six years, it appears reasonable to view one to three years in low income (1 ≤ D ≤ 3) as transitory and four or more years in low income (4 ≤ D ≤ 6) as persistent, although other configurations are possible.
Analysing incidence of low income and probability of being in low income
We will now analyse incidence in low income and probability of being in low income with respect to the available panels of data. First, the incidence in low income and probability of being in low income can be captured by binary dependent variables. To learn what might increase the probability of being in low income and/or cause low income, we can study the low-income incidence and/or the probability of being in low income that are related to covariates such as gender, age, education, minority status, language, student status, activity limitation status and family composition in the regression framework of the form
,
where d_{i}represents low-income state for individual i at the six-year aggregate level, wi represents the vector of covariates associated with individual i, and u_{i}represents the error term, i = 1, 2, …, N. Therefore, when the inner product of the coefficient vector and w_{i} vector enters f( ), the coefficient estimate for each explanatory variable in the covariates vector wi is the marginal effect^{17} attributing to the probability of being in low income. Modeling the probability of being in low income at the panel-aggregate level has the benefit that the binary variables of the low-income state can be created by the degree of low-income persistence over the panel period, e.g., at least one year in low income, at least four years in low income, and all six years in low income. In doing so, we are able to analyse factors that may contribute to higher probabilities in transitory low income as well as those in persistent low income.
Duration aversion and comprehensive measure of low-income durations
In addition to the above discussion, following Osberg and Xu (2000b), we consider an index for low-income duration over T periods. This is particularly useful for comparing low-income persistence between the populations in two different (possibly overlapped) panels.
We can establish the following seven axioms for a low-income-duration index:
- Focus axiom for low-income duration: The low-income-duration index should be independent of the subpopulation that does not experience any low income.
- Weak monotonicity axiom for low-income duration: A reduction in a person's low-income duration, holding other low-income durations constant, must decrease the low-income-duration index.
- Impartiality axiom for low-income duration: The low-income duration index may be defined over ordered low-income durations without loss of generality.
- Weak transfer axiom for low-income duration: The low-income-duration index should increase if an individual's shorter low-income duration is further reduced at the expense of a similar or greater increase in a long initial low-income duration of another person and the set of low-income people does not change.
- Strong upward transfer axiom for low-income duration: The low-income duration index should increase if an individual's shorter low-income duration is further reduced at the expense of a similar or greater increase in a long initial low-income duration of another individual.
- Continuity axiom for low-income duration: The low-income duration index varies continuously with low-income durations.
- Replication invariance axiom for low-income duration: The low-income duration index does not change if it is computed based on a distribution of low-income durations that is generated by the k-fold replication of the original distribution of low-income durations.
Under the above axioms, we can construct the Sen-Shorrocks-Thon (SST) index of low-income duration over T periods as
where G(δ_{t}) is the Gini coefficient of the normalized low-income durations (in non-decreasing order) of the population. The SST index of low-income duration over T periods is the product of the proportion of the population that is ever in low income, average normalized low-income duration of the subpopulation who are ever in low income, and the inequality measure of normalized low-income durations. The higher (or lower) the SST index of low-income duration is, the lower (or higher) the well-being the target population has.
Notes
- Zhang (2010) offers a comprehensive assessment on the technical details of the existing low income measures in Canada.
- There are after-tax and before-tax LICOs produced by Statistics Canada. The former is the benchmark used for after-tax incomes while the latter is for before-tax incomes. We work on after-tax LICOs and after-tax income.
- We can also discuss the matter on the basis of equivalent individual incomes. The adjusted family income is the sum of equivalent individual incomes of the family members.
- LIMs are calculated three times—using market income, before-tax income and after-tax income. Similar to LICO, we only use the after-tax LIM.
- However, the existence of arbitrary elements is not unique to LIM. Other low-income thresholds contain their own arbitrary elements. For example, under LICO, an arbitrary 20% factor was employed.
- Interested readers may refer to Murphy et al. (2010). Owing to time constraint, we have not considered these modifications in the current study.
- MBM is more sensitive than LICO or LIM to the significant geographical variations (both among and within provinces) in the cost (especially for shelter and transportation) of many typical items of expenditure.
- The conceptual framework of MBM was developed and adopted by HRSDC in 2000: therefore, MBM is not directly available for 1999 or earlier. To enable our analysis on low-income persistence across different choices of low-income thresholds, we impute MBM for 1999 by converting the MBM in 2000 (using the MBM 2007 basket) with the Consumer Price Index. Therefore, we will use caution when interpreting 1999 results under MBM.
- We can explain the simplest conversion here. Let Y(s) be the income of a family of size s. The required family income for a family of size s due to economies of scale is increasing and concave in s. This concept is also applicable to the family low-income cut-off or threshold, Z(s), which is increasing and concave in s. If we use as the adjusted family size, we can convert both family income and low-income threshold into the individual (or per capita) equivalent income and low-income threshold as and The comparison between Y(s) and Z(s) is identical to the comparison between y and z. The individual absolute low-income gap would be if y < z. The individual relative low-income gap would be x = (z – y)/z = [Z(s) – Y(s)]/Z(s) if y < z. Apparently, the relative low-income gap x is scale-free, in the sense that it is the same for an individual and for the family of which the individual is a member.
- Here we adopt the low-income criteria with the income strictly 'less than' the low-income threshold, consistent with most of the relevant literature and data processing conventions. Some authors—e.g., Borrooah and Creedy (2002)— use 'less than or equal to' rather than 'less than.'
- Osberg and Xu (2000) have addressed censoring and truncation in the context of monthly incomes when discussing theoretical issues in poverty measurement and poverty duration.
- The truncation at the annual data level will occur if an annual income is higher than a suitably chosen annual low-income threshold but some monthly incomes are actually below the monthly low-income threshold that corresponds proportionally to the annual low-income threshold.
- See Bartholomew (1982) for more information.
- We use 'proportion' instead of 'probability' although the latter may be more precise in discussing the stochastic process.
- Here we follow the spirit of Section 2.3 in Bartholomew (1982).
- Although Borrooah and Greedy (2002) adopt a similar framework, they only consider two-year poverty status. That is, they define temporary poverty as one year in poverty and permanent poverty as two years in poverty. In our context, we call one to three years in low income transitory and four or more years in low income persistent—over a six-year period.
- Marginal effect is also known as partial or net effect in other contexts. It reflects the impact on the dependent variable caused only by the explanatory variable in question, with everything else held constant.
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