A brief theoretical perspective

According to ^{Sen (1992)}, equity implies equality in one particular aspect, but not in all aspects. Hence, to become an equitable society, what variable or aspect must be equalized? To put it briefly, it can be said that there are three main approaches to answer this question, depending on whether equality occurs in: i) well-being, if one considers that the goal is to maximize the total sum of a society’s utility; ii) assets or resources-for ^{Rawls (1971)} there must be equality in terms of primary goods (such as rights, freedoms and opportunities, as well as income and wealth), while for ^{Dworkin (1981)}, equality must be achieved for resources, i.e., each individual should be satisfied with their endowment (or basket) of goods; or iii) opportunities, where the capacity-centered approach posited by Sen and Roemer’s “level the playing field" (or “smooth the way") approach (^{Formichella, 2011}) are considered, among others.

According to Sen's capacity focus, the aspect that should be equalized is the ability or freedom that people have to choose their own way of life, whereas in Roemer's approach, there must be equal opportunities. Here the results an individual obtains are considered to be a consequence of actions that they carry out, and are determined by two types of circumstances: those that the individual can control, and those that are beyond their control (^{Formichella, 2011}). In this way, a distinction is made between inequality caused by individual responsibility or factors over which one has control-what ^{Roemer (1998)} calls “effort"-and inequality caused by predetermined circumstances over which the individual has no control, such as race, gender, family circumstances, cultural capital, inheritance, and family income, among others. Thus, unequal opportunities are related to conditions that are beyond a person's control (^{Paes et al., 2008}).

Given its importance in an individual’s development, a certain level of education is necessary to establish equal opportunities between individuals-but how is this level or equality of opportunities in education measured? According to ^{López et al. (2017)}, the literature identifies two ways to measure educational inequality: unequal access to education (e.g. years of schooling), and academic achievement (e.g. student performance). While some empirical works study one of these aspects, others study both, since according to ^{Gamboa and Waltenberg (2015)}, both aspects are important and interrelated, and as such they should be measured simultaneously.

Empirical literature

Some international researchers point out that predetermined circumstances play an important role in unequal educational and economic opportunities. ^{Ferreira and Gignoux (2014)} found that unequal opportunities due to predetermined circumstances explain up to 35% of differences seen in educational achievement and that gender, mother’s education and father's occupation are variables that show robust results in explaining educational inequality. Along the same lines, ^{Paes et al. (2008)} found that mother's education and father's occupation are important causes of both educational inequality and income inequality for adults. Meanwhile ^{Gamboa and Waltenberg (2012)} consider that the education of parents and the type of school attended by 15-year-olds in six Latin American countries are important factors of inequality of opportunity, not gender.

Other international work addresses educational inequality conditioned by family history specifically (which are circumstances over which an individual has no control), and finds that family income is an important determinant of educational opportunities. ^{Haveman and Wolfe (1995)} indicate that the level of education a person reaches is determined by family income and the investment the family makes in education; they also show that the dropout rate of young people between 15 and 19 years of age is determined mainly by financial problems.

^{Papay et al. (2015)} find large gaps in educational achievement between low and high-income students in Massachusetts, United States, and posit that these gaps are wider at higher levels of education, including higher secondary education (HSE) graduation rates and university enrollment. ^{Reardon (2011)} and ^{Bailey and Dynarski (2011)} also demonstrate that there are gaps in academic achievement between children and youth who grow up in low-income families, as opposed to those who grow up in higher-income families in the United States. ^{Acemoglu and Pischke (2001)} argue that family income has a substantial effect on university enrollment, specifically that an increase in family income of 10% is associated with an increase of 1.4 percentage points in the probability of attending university for four years.

^{Gasparini (2002)} finds that higher disparities in household income are transformed into higher disparities in school elections in Argentina, while ^{González et al. (2014)} point out that low household income reduces the probability of demanding upper secondary or higher education in Colombia, and that young people who come from high-income households have a positive and increasing probability in terms of time in school.

Furthermore, ^{Ibáñez et al. (2020)} show that in Argentina, the older adolescents get, their being economically active and living in a home with a lower educational climate lowers the probability they will attend school. Meanwhile, the probability of attendance is higher among women, those who have health coverage and those whose parents work in the formal labor market.

Studies have been carried out in Mexico that analyze the problem of educational inequality using data from censuses (^{Martínez Rizo, 2002}; ^{Solís, 2010}; ^{Navarro and Favila, 2013}), or from the National Household Income and Expense Survey (^{De la Luz and Díaz, 2010}; ^{Martínez et al., 2011}; ^{Rangel and Ramírez, 2011}). Some of these works show an association between an individual’s socioeconomic characteristics and the years of schooling completed (^{Martínez Rizzo, 2003}; ^{Solís 2010}; ^{Solís et al., 2013}; ^{Navarro and Favila, 2013}), or between family income and school enrollment (De la Luz and Díaz, 2010). Specifically, ^{Solís (2010)} found that in Mexico, most of the variation seen in years of schooling can be explained by class origins (their family of origin’s economic and cultural capital). De la Luz and Díaz (2010) indicate that income level is a determinant of the educational decisions taken by the population group of 15- to 23-year-olds, and that a 10% increase in family income increases the probability of school enrollment by 2.6% (in 2000), 2.5% (in 2002) and 2.4% (in 2004).

Similarly, ^{Solís et al. (2013)} use the Federal District’s Transition to Higher Secondary Education Survey to demonstrate that young people in the lower quartile of the Social Origin Index have a probability of 0.55 of continuing to study at the HSE level, compared to a probability of 0.85 for youth in the top quartile. Moreover, in the National Survey of Higher Secondary Education Dropouts [Spanish acronym ENDEMS] (^{SEP, 2012}), it was found that among the main causes for leaving HSE are economic factors (lack of money in the household for supplies, transportation or enrollment)-in Mexico, household income does seem to be a determinant of whether or not a young person continues to study. For their part, ^{Vargas and Valadez (2016)} use data from the 2010 National Youth Survey and from Cox regression models, and find that the risk of dropping out of school is indirectly yet significantly associated with quality of education and the economic status of the young person.

Although the literature on the effect of family income on school attendance in Mexico is abundant, it appears that this relationship has not been studied using a one-year longitudinal panel-information that is more robust in dealing with probable simultaneity problems. In addition to this contribution, the present study reinforces the idea of an intra-generational gap based on family income, providing specific information that can help provide a focus for public policies in favor of education and the reduction of educational inequality.

Calculation methodology: probit regression model

Statistically modeling school attendance presents the specificity that the response variable (attendance) is not continuous, but rather discrete; that is, one attends or does not attend. For this reason, it is not convenient to use traditional linear regression models given that they assume continuity in the dependent variable.

Therefore, regression methods are used for binary dependent variables and in this case the probit model was selected. The approach consists of modeling the probability that a young person between 14 and 18 years old who attended school during period t will continue to attend school one year later, that is, at t + 1, as a function of household income over period t and other control variables. This is performed using a normal cumulative probability distribution whose argument is a linear combination of the response variable determinants (^{Wooldridge, 2016}).

In the context of a probit model, this can be represented as:

Where Pr denotes Probability, E _it (E _it + 1) is a binary indicator that takes the value of 1 if young person i attends school during period t (t + 1), Φ denotes the normal cumulative distribution, and X _it is a vector of specific characteristics of the young person and their family during period t, such that:

The change in the probability that a young person attends school when independent variable x _j changes -which is called the marginal response-is equal to:

Where 𝜙 is the density of the normal distribution. The coefficient calculated β _j and the marginal response are represented by the same symbol, but are not the same. In this sense, the coefficients do not have a direct interpretation, so it is important to evaluate the impact of the regressors through the marginal responses and determine whether they are significantly different from zero.

Two additional comments on the model: the dependent variable and regressors have a lag, and school attendance refers to period t + 1, while the regressors are dated to period t. This is convenient to break any possible simultaneity of the variables, especially as regards family income. By using the lagged family income-that of period t -it is already given at t + 1 such that it cannot be affected by school attendance at t + 1.

Finally, unlike cross-sectional studies, there may be changes over time in factors common to all individuals (e.g. an economic crisis or natural disaster) that affect the response variable. For this reason, it is pertinent to include dichotomous variables for the time periods to capture this type of fixed effect.

Variables

Although the goal here is to calculate the effect of family income on school attendance, other variables were also included that may be important in explaining young people’s school attendance, such as the young person’s characteristics, their family, their environment and the temporality of observations in the model. In this way, biases due to variable omissions are avoided and the goodness-of-fit is improved. The independent variables used in the model are described below.

Variables for the young person’s characteristics: dichotomous variables were coded that take the value of 1 if the cited characteristic is fulfilled and 0 if it is not fulfilled; these variables are Male, Enters late (goes to school for one year less than they should), Secondary school, Work, Married and Migrant. Age in completed years was included.

Family variables: the following dichotomous variables were tested-Extended household (an additional relative lives in the household), Has older siblings, Has younger siblings, Has both parents, Mother works. The mother’s education (years of study), the number of siblings and the per capita family income were included, the latter consisting of the income of all household members except the young person, expressed in 2016 pesos and in per capita terms.

Environment and time variables: 31 dichotomous variables were included by state and one for rural areas; 10 dummy variables for years and 3 dummy variables for quarters were also incorporated.

Data

This work uses data from the ENOE, from the first quarter of 2005 to the third quarter of 2016. The ENOE has a rotating panel structure in which interviews are conducted with the same households and individuals for five consecutive quarters, resetting each quarter at a fifth of the total sample (^{INEGI, 2016b}).

International studies use data panels with an individual's history, including their school career and personal and family characteristics, to measure the effect of family income on children's school attendance. However, in Mexico there is no panel data of this type, so the ENOE-with its rotating panel structure-becomes the best available instrument, providing us with microdata disaggregated by age, state, gender, rural-urban conditions, individual characteristics, characteristics of the home, etc., for the same individual throughout a year of their life (five consecutive quarters). Only young people between 14 and 18 years old who were living with their parents at the time of the first interview are considered here. Thus, the sample includes 252,077 observations in total.

The database was organized as follows: individuals who are between 14 and 18 years old and live with their parents, who are in school and are participating in the interview for the first time are selected each quarter, and their individual and home characteristics are recorded; using a personal identifier, information from the first and fifth interviews of each selected young person are combined. Thus, the same observation includes the characteristics of the young person and their home, as reported in the initial interview (period t)-such as their age, gender, number of members of their home or their family income-as well as the young person’s school attendance at the time of the final interview (period t + 1). Each quarter includes a different group of young people with information from their initial interview and with their attendance status as recorded in their final interview.

Table 2 shows the trends of young people’s school attendance between their first and last interview. Of the total sample, 69.7% (175,798 youth) reported attending school at the time of the last interview. Of the total number of young people who reported being in school during their first interview, 86.5% continued studying one year later, while among those who reported not being in school in their first interview, 13.7% returned to school one year later.

Table 3 shows the averages for the total sample, for the individuals who continued attending school one year later and for those who did not continue. It is worth reiterating that the characteristics of the young person and their household at the time of the initial interview are used in this model. Given that this study is limited to those young people who were attending school during the initial interview, the sample size is reduced to 193,947 young people.

Although all simple comparisons of the averages for those who attended and did not attend school are significantly different from 0 to 1%, the differences are more notable for certain variables. In particular-compared to those who attended school in the final interview-those who were not see a higher incidence among those who are lagging in their school career, those who work, those who reside in rural areas, those with less educated mothers and those living in households with lower per capita real income. The differences are smaller, although appreciable, between males, those who are in secondary school, those who live in extended households and those who do not live with both parents.

At the time of the initial interview, the household per capita income for the homes of the young people in the sample was MXN $2,123 (in 2016 pesos) per month. For the group of young people who reported not attending school in the final interview, the household per capita income was on average MXN $1,745 per month, i.e., MXN $468 less per person. Table 4 shows that the family per capita income of the sample average is between decile VI and VII, while the family per capita income for young people who stopped attending school falls between decile V and VI.

Table 5 breaks down the percentage of young people who reported attending school in the final interview-given that they also attended school in the initial interview-according to different characteristics. For example, of the males who attended school in the initial interview, 85% reported continuing to attend during the final interview, while this is the case for 87% of females.

Retention-the proportion of young people who attended school in the final interview and as well as in the first interview-tends to be higher among those under the age of 16, compared to those who are 17 or 18 years old (this difference is consistent with a very common result in the literature where dropping out increases with age); those who do not work; those who are single; those who live in a nuclear household (compared to an extended one); and those with mothers who have a high school education or higher.

It is also observed that the percentage of young people who attended school in the first interview increases across the years, rising from 72% in 2005 to 80% in 2015. Meanwhile, the percentage of young people who attended school in the second interview, as well as attending in the first interview, also increased-although to a lesser extent, going from 84 to 87% during the same period. Additionally, a breakdown of attendance by state indicates that Coahuila and Michoacán presented the lowest percentage of attendance during the first interview of young people between 14 and 18 years old, with 64% attendance on average. Mexico City reports the highest attendance rate with 85%. Regarding the retention percentage, that is, young people who attended school in both interviews, Baja California Norte and Mexico City reported the highest rates, 90 and 89%, respectively, while Guanajuato has the lowest rate at 80%.