metadata: authors: 0: author: Emran,M. Shahe
1: author: Shilpi,Forhad J.
count: World,Bangladesh,India
docna: 0: docna: Estimating intergenerational mobility with
            incomplete data : coresidency and truncation bias in
            rank-based relative and absolute mobility measures
docty: Policy Research Working Paper
subtopic: Educational Sciences,Labor & Employment Law,Gender and Development,Inequality,Public Sector Management and Reform
teratopic: Law and Development,Public Sector Development,Education,Poverty Reduction,Gender
abstracts: cdata!: The rank-based measures of
            intergenerational mobility have become increasingly popular
            in economics literature. Recent evidence shows that
            rank-based measures are less affected by measurement error
            and life-cycle bias compared with other standard measures
            such as intergenerational regression coefficient and
            intergenerational correlation. However, most of the
            available household surveys suffer from sample truncation,
            because coresidency is used to define household membership.
            There is no evidence on how sample truncation affects the
            rank-based mobility estimates relative to intergenerational
            regression coefficient and intergenerational correlation.
            This paper provides evidence on this in the context of
            intergenerational schooling persistence, using two
            exceptional surveys from India and Bangladesh that include
            all children irrespective of residency status. The analysis
            shows that the measures of relative mobility (slopes) are
            biased downward in coresident samples, but the average bias
            in rank correlation is less than half of that in
            intergenerational regression coefficient, and comparable to
            that in intergenerational correlation in magnitude. The
            intercept estimates are biased upward, with the largest bias
            found in the intercept of the regression used for
            intergenerational correlation. Truncation bias in rank-based
            absolute mobility estimates is the lowest in most cases. The
            results strengthen the case for rank-based measures of
            intergenerational mobility when working with the standard
            household surveys.
geo_reg_key: 517198,517191,517212
repnme: repnme: Estimating intergenerational mobility with
            incomplete data : coresidency and truncation bias in
            rank-based relative and absolute mobility measures
display_title: Estimating intergenerational
            mobility with incomplete data : coresidency and truncation
            bias in rank-based relative and absolute mobility measures
listing_relative_url: /research/2018/04/29816568/estimating-intergenerational-mobility-incomplete-data-coresidency-truncation-bias-rank-based-relative-absolute-mobility-measures
topicv3: Inequality under Law m1326002 925,Human Resources m1326304 1211,Education m1327048 470,Gender m1327621 615,Household Surveys m1504071 2761
publishtoextweb_dt: 2018-04-18T00:00:00Z
historic_topic: Law and Development,Public Sector Development,Education,Poverty Reduction,Gender
no_of_pages: 35
author_keys: 0: author_key: 84831
model_extractions: WPS8414 Policy Research Working Paper 8414 Estimating Intergenerational Mobility with Incomplete Data Coresidency and Truncation Bias in Rank-Based Relative and Absolute Mobility Measures _M. Shahe Emran Forhad Shilpi_ Development Research Group April 2018
Policy Research Working Paper 8414

# **Abstract**

The rank-based measures of intergenerational mobility have become increasingly popular in economics literature. Recent evidence shows that rank-based measures are less affected by measurement error and life-cycle bias compared with other standard measures such as intergenerational regression coefficient and intergenerational correlation. However, most of the available household surveys suffer from sample truncation, because coresidency is used to define household membership. There is no evidence on how sample truncation affects the rank-based mobility estimates relative to intergenerational regression coefficient and intergenerational correlation. This paper provides evidence on this in the context of intergenerational schooling persistence, using two exceptional surveys from India and Bangladesh that include all children irrespective of residency status. The analysis shows that the measures of relative mobility (slopes) are biased downward in coresident samples, but the average bias in rank correlation is less than half of that in intergenerational regression coefficient, and comparable to that in intergenerational correlation in magnitude. The intercept estimates are biased upward, with the largest bias found in the intercept of the regression used for intergenerational correlation. Truncation bias in rank-based absolute mobility estimates is the lowest in most cases. The results strengthen - the case for rank-based measures of intergenerational mobil ity when working with the standard household surveys.

This paper is a product of the Development Research Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/research. The authors may be contacted at fshilpi@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# **Estimating Intergenerational Mobility with Incomplete Data: Coresidency and Truncation Bias in Rank-Based Relative and Absolute Mobility Measures**

**M. Shahe Emran**<sup>1</sup>

IPD, Columbia University

## **Forhad Shilpi**

DECRG, World Bank

**Key Words:** Intergenerational Mobility, Rank correlation, Rank-based Absolute Mobility, Coresidency, Truncation Bias, Developing Countries, Intergenerational Regression Coefficient (IGRC), Intergenerational Correlation (IGC), Bangladesh, India

**JEL Codes:** O12, I32

> 1 We would like to thank David Zimmer and William Greene for helpful discussions and/or communications. The standard disclaimers apply.

Emails for correspondence: shahe.emran@gmail.com, fshilpi@worldbank.org.
# **Introduction**

The role of family background in determining the economic opportunities of a child has been the focus of a large literature in economics and sociology.<sup>1</sup> There has been a surge in the interest in reliable estimates of intergenerational persistence in economic status, motivated by the evidence that economic inequality has increased in recent decades in many countries (World Bank (2006)). The concern is that the observed increase in inequality reflects worsening deep-seated inequality of opportunities in education and the labor market. There is growing evidence of a Great Gatsby curve, showing that crosssectional inequality and intergenerational mobility are negatively correlated (Corak (2013), Fan et al. (2015), Neidh ̈ofer (2016)).

A major constraint on the research on intergenerational mobility has been data limitations in the standard household surveys. A substantial literature in the context of developed countries shows that the estimates of intergenerational income persistence can be severely biased downward due to measurement error (Solon (1992), Mazumder (2005)). A small but growing literature analyzes how different measures of intergenerational persistence used in the literature are affected by data limitations. In a widely-acclaimed paper, Chetty et al. (2014) show that the standard measure of intergenerational income persistence, called intergenerational income elasticity (IGE) and estimated as the slope of a regression of log children’s income on log parental income, is fragile. The lack of robustness reflects the fact that the conditional expectation function of children’s log income given parent’s log income is highly nonlinear. They show that, in contrast, rank correlation, estimated as the slope of a regression of children’s percentile rank in income distribution on parental percentile rank, is approximately linear, and it is much more robust to measurement error and treatments of zero income.<sup>2</sup> Dahl and Deleire (2008) show that the rank-based measures

> 1For excellent surveys of the economics literature, see Solon (1999), Black and Devereux (2011), Bjorklund and Salvanes (2011), and for the sociological literature, see Erikson and Goldthorpe (2002), and Fox et al. (2016).

> 2The advantages of rank-transformed variables in economic analysis have been appreciated since Hotelling and Pabst (1936). They emphasized the robustness with respect to violation of the normality assumption. For analysis of intergenerational mobility, the rank-rank specification was introduced in

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are robust to mis-specifications, and Mazumder (2015) finds that the attenuation bias in the rank-rank slope estimate is significantly smaller compared to that in intergenerational elasticity (IGE). Nybom and Stuhler (2016) use rich data from Sweden to analyze life-cycle bias and attenuation bias in IGE and rank based measures. They find that the rank-based measures are more robust to both life-cycle bias and attenuation bias when compared to the estimates of IGE.

Although attenuation bias due to measurement error has been a central focus of the literature on intergenerational mobility, a second potentially important source of bias arising from coresidency restrictions common in household surveys has received much less attention. Since standard household surveys, especially in developing countries, do not collect information on nonresident children, the estimates of intergenerational persistence are likely to suffer from significant truncation bias. We are aware of only two papers that analyze the implications of coresidency restrictions in household surveys for estimating intergenerational persistence in economic status. In a recent paper, Emran, Greene and Shilpi (2018) show that intergenerational regression coefficient (IGRC), the most widely used measure of intergenerational mobility in development literature, suffers from strong downward bias due to truncation as a result of coresidency criteria used for household membership in a survey. IGRC is estimated as the slope of a level-level regression, usually for intergenerational schooling persistence. This also suggests that some of the IGE estimates for developed countries may be significantly downward biased even after correction of attenuation bias due to measurement error when the data do not include all the children because of nonresidency at the time of the survey.<sup>3</sup> Using the British Panel Household Survey, Francesconi and Nicoletti (2006) report 12% - 39% downward bias due to coresidency in short panels when estimating persistence in the Hope-Goldthorpe index of occupational prestige. To the best of our knowledge, there is no evidence in the literature on how the rank-based the economic literature by Dahl and Deleire (2008).

> 3Their evidence also shows that truncation bias is much less (less than one-third) in another widely used measure called intergenerational correlation (IGC) which estimates Pearson correlation between economic status of parents and children.

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measures of relative and absolute mobility are affected by truncation because of coresidency restrictions in a survey.

Taking advantage of two exceptionally rich household surveys from India and Bangladesh, we provide evidence on the effects of sample truncation caused by coresidency restrictions on the rank-based measures of relative and absolute mobility. Our analysis focuses on intergenerational schooling persistence, as education has been the main indicator of economic status in most of the recent research on intergenerational mobility in developing countries in the absence of reliable income data (see, for example, Azam and Bhatt (2015), Emran and Shilpi (2015), Hertz et al. (2007), Nimubona, A, and D. Vecatachellum (2007), Behrman et al. (2001)).<sup>4</sup> The evidence reported below shows that truncation, in general, results in _downward_ bias in the estimate of the _slope_ (relative mobility) and _upward_ bias in the estimate of the _intercept_ of widely-used intergenerational persistence equations including the rank-rank regression. When the focus is on relative mobility, the downward bias in rank correlation is small and similar in magnitude to that in IGC (about -10%), but the bias in IGRC is much larger (about -25%). Truncation bias in the intercept estimate is, in general, the largest in the regression specification used to estimate IGC (i.e., Pearson correlation), while the bias in the intercept of rank-rank regression is, in most cases (26 out of 30), the smallest. Children’s expected years of schooling (or expected normalized schooling in the case of the IGC specification) conditional on parental schooling are overestimated in coresident samples, implying that the upward biased estimate of the intercept dominates the downward bias in the slope estimate. When the focus is on absolute mobility, the rank-based measure a la Chetty et al. (2014) outperforms an alternative measure based on

> 4The focus on educational mobility is appropriate for at least two reasons. First, as noted above, data are not available to construct reliable estimates of permanent income, and schooling attainment is likely to be a good proxy for life-time income and economic status in most developing countries. Blanden (2013) in her review of cross-country evidence on intergenerational mobility concludes that “....the results for earnings and education tend to be fairly well correlated; this implies that information on educational mobility is a good proxy for earnings mobility in countries where earnings information is not readily available.” Second, there is a broad consensus among policy makers and economists that education is key to social mobility in an increasingly skill-driven economy (Stiglitz (2012), Rajan (2010), The Economist (2012), World Development Report (2006)).

3
the slope and intercept of the IGRC or IGC equation in most of the cases. The truncation bias in the rank-based absolute mobility estimate is lower in 80 percent of cases in both Bangladesh and India.

The evidence in this paper suggests that when working with standard household surveys it is, in general, better to rely on rank-based measures of relative and absolute mobility. The results can be helpful and reassuring for many researchers who, following the lead of Chetty et al. (2014), have adopted the rank-based measures to understand intergenerational mobility in developing countries (see, for example, Fan et al. (2015), Asher et al. (2017)). The rest of this paper is organized as follows. Section 2 provides a discussion on truncation due to coresidency in household surveys in developing countries. The next section discusses the standard measures of relative and absolute mobility widely used in the literature. Section (4) is devoted to data and variables definitions. The estimates of truncation bias for different samples are reported and discussed in section (5). Section (6) provides a brief discussion on possible intuitions behind the results. The paper concludes with a summary of the findings and the implications for research on intergenerational mobility in developing countries.

# **(2) Coresidency, Household Definition, and Sample Truncation in Household Surveys in Developing Countries**

The availability and quality of household surveys in developing countries have improved substantially over last few decades. For example, starting from early 1980s, the Living Standards Measurement Study (LSMS) has generated high quality household surveys in more than 40 countries, while the Demographic and Health Surveys (DHS) cover more than 90 countries. Although there is no uniformity in the definitions of ‘household’ across different surveys, almost all of the surveys use coresidency criteria to define household membership such as ‘living together’, ‘eating together’, and sometimes ‘pooling of funds’ (United Nations (1989), Deaton (1997)). For example, according to the Tanzania DHS, “a household is defined as a person or group of persons, related or unrelated who live 4
together and share a common source of food”. Most LSMS surveys include an individual as a household member only if she/he lived in the household for more than 3 months in the last 12 months (Glewwe (2000)), which would exclude the children in colleges away from home, as the college students in developing countries do not get 3 months break to go back home.<sup>5</sup> Some surveys such as IFLS in Indonesia and IHDS in India (the second round in 2012) include two randomly chosen non-resident children, which implies that the poorer households are more likely to face truncation, because fertility rate usually declines as household income increases. Most household surveys done by national statistical agencies also use similar coresidency restrictions to define a household.<sup>6</sup> The coresidency restrictions used in the surveys result in truncation of the sample as there is no information on the nonresident children, and truncation is likely to be nonrandom.<sup>7</sup> In developing countries such as India and Bangladesh, many girls move out of parental household when they drop out of school because of marriage, and the sons leave the household for jobs. In addition, the rural households are likely to miss more educated children systematically because most of the villages do not have a college (or even a high school), and children have to migrate to pursue higher education.

Figure 1 provides estimates of conditional probability of non-residence of children at the time of the survey at various schooling levels of children. Both in India and Bangladesh the probability of non-residence is much lower at the middle of the schooling distribution. Truncation thus is more prominent at the tails of the schooling distribution. There are also significant differences across countries and gender. The incidence of non-residency

> 5There are exceptions such as infants less than 3 months old are included as household members. For an excellent discussion on the issues involved in defining a household, please see Glewwe (2000). Some surveys use a much higher cut-off; for example, MxFLS in Mexico considers someone a household member if he/she lived in the household for a year or more.

> 6Some panel surveys carefully track the households over the years, and thus take care of the sample selection arising from attrition. However, most of them still use coresidency criteria to define household membership at the baseline. Thus even with no attrition, the data still suffer from truncation bias.

> 7This implies that it is not possible to implement selection correction using the Heckman procedure, as it is not possible to estimate a selection equation without knowing which households are missing children from the survey. The maximum likelihood procedure developed by Bloom and Killingsworth (1985) can be used, but it relies on strong distributional assumptions.

5
at the time of the survey is lower in India in general compared to Bangladesh, and the non-residency rates are higher for girls (daughters) in both countries. The gender gap in non-residency rate is much larger in the case of India, and in Bangladesh the gender gap is higher at the bottom of the schooling distribution and effectively vanishes at the top of children’s schooling distribution.

# **(3) Measures of Intergenerational Mobility: Relative and Absolute**

The most widely used measure of intergenerational mobility is regression-based relative mobility which is estimated as the slope of a log-log (for income) or level-level (for schooling) OLS regression. The standard regression model for intergenerational persistence in schooling is: where _Si_<sup>_c_and</sup><sup>_S_</sup> _i_<sup>_p_areindicatorsofeducationalattainmentforchild</sup><sup>_i_andhis/herparents</sup> respectively. Following the recent literature on rank-based measures of relative and absolute mobility, we focus on the conditional expectation function of children’s schooling without additional controls such as age and gender to ensure comparability (see, for example, Asher et al. (2017) and Chetty et al. (2014)). The focus is on estimating the parameter _β_ 1 which is called the Intergenerational Regression Coefficient (IGRC).<sup>8</sup> We report the data, the measures of mobility, and the status indicators used in a sample of recent papers on intergenerational mobility in developing countries in Table 1; most of the papers use IGRC as the measure of intergenerational mobility, the most common indicator of economic status is education, and all of the data sets suffer from truncation due to coresidency restrictions.

A second widely used measure of relative mobility is estimated from the following OLS regression: where the indicators of educational attainment such as years of schooling are normalized by

> 8The literature on intergenerational persistence in income uses a log-linear model, and thus the slope provides an estimate of intergenerational income elasticity (IGE).

6
their standard deviation, i.e., _σc_ and _σp_ are standard deviations of children’s and parent’s schooling respectively. The focus in most of the literature is on estimating the parameter _ρ_ 1 which provides an estimate of Pearson correlation between educational attainment of two generations. It is called intergenerational correlation (IGC) in the literature.

An increasingly popular measure of relative mobility is rank correlation, used originally by Dahl and DeLeire (2008), and made salient by the recent work of Chetty et al. (2014). The rank correlation is estimated from the following OLS regression: where _Ri_ is the percentile rank of child _i_ in the distribution of children’s schooling, and _Pi_ is the percentile rank of the parent of child _i_ in the distribution of parental schooling. The parameter of interest is _δ_ 1 which provides an estimate of Spearman rank correlation in schooling across generations.

We compare the bias in the estimates of the slope and intercept terms in the rankrank regression in equation (3) with the bias estimates for the slope and intercept from equations (1) and (2). As noted before, Emran et al. (forthcoming) provide estimates of coresidency bias in IGC and IGRC using the same data sets on India and Bangladesh used here. However, the regression specification used in that paper includes quadratic age controls for both children and parents. Thus their estimates are not comparable to the rank-rank estimates without any controls in this paper; we report estimates of _β_<sup>ˆ</sup> 1 and _ρ_ ˆ1 without any controls for comparability.<sup>9</sup> The different measures of relative mobility discussed above have been the preferred ones in the economics literature for analyzing intergenerational mobility in both developed and developing countries. However, as emphasized by Chetty et al. (2014), relative mobility measures have an important limitation: an improvement in relative mobility may be driven

> 9These estimates provide additional robustness checks for their central conclusion that IGC is preferable to IGRC as a measure of relative mobility in coresident samples. We also note that the main conclusions regarding the bias in rank-based relative and absolute mobility estimates in this paper remain valid if we include age of parents and children in the regressions. However, with quadratic age controls for both children and parents, the standard error of the estimate of the intercept becomes large in many cases.

7
by a worsening of outcomes for children at the upper tail of the distribution rather than an improvement for the children at the lower tail of the distribution. To address this issue they propose measures of absolute mobility based on the estimates of _δ_ 0 and _δ_ 1 in equation (3) above. Chetty et al. (2014) define absolute mobility at percentile _p_ as the expected percentile rank of the children whose parents belong to percentile _p_ in the distribution of parental schooling, i.e., _Pi_ = _p_ . Denote the OLS estimates of the parameters by a hat, then the absolute mobility at the percentile _p_ (denoted as _r_ ̄ _p_ ) is calculated as below: Following Chetty et al. (2014), we focus on absolute mobility at _Pi_ = _p_ = 25. With linear conditional expectation function, _r_ ̄25 = _E_ ( _Ri | Pi ≤_ 50), i.e., the expected rank of children born to parents who fall in the lower half of parental schooling distribution, and thus is a measure of upward mobility. In addition, we also report absolute mobility at the 75th percentile of the parental schooling distribution which shows the expected rank of children born to parents in the upper half of parental schooling distribution.

One can also derive expected years of schooling and normalized schooling for children using the intercept and slope estimates from equations (1) and (2) above. Denote the expected years of schooling for child _i_ as _S_<sup>ˆ</sup> _i_<sup>_c_,thenwehave(correspondingtoequation(1)</sup> above): Using expected years of schooling _S_<sup>ˆ</sup> _i_<sup>_c_from equation (5) above, we calculate expected school-</sup> ing attainment of the children from the lower half of the parental schooling distribution as: To get a measure comparable to the absolute mobility measure defined by Chetty et al. (2014) at the 25th percentile, we calculate the percentile rank of _S_<sup> ̄</sup> 25<sup>_c_intheschooling</sup> distribution of children. This provides the expected rank of the children from the lower half of the parental schooling distribution when expected schooling attainment is determined 8
by equation (6) above. We denote this rank as _R_ 25<sup>_β_toconveythat</sup><sup>_β_ˆ0and</sup><sup>_β_ˆ1wereused</sup> to derive this rank estimate. Note that the expected normalized schooling of children (i.e., normalized by the standard deviation) and expected years of schooling of children are monotonically related to each other, and thus the rank remains the same. This implies that if we calculate the rank _R_ 25<sup>_ρ_,basedontheexpectednormalizedschoolingusingOLS</sup> estimates of parameters of equation (2), i.e., _ρ_ ˆ0 and _ρ_ ˆ1, it will be identical to _R_ 25<sup>_β_,i.e.,</sup> _R_ 25<sup>_β_=</sup><sup>_R_</sup> 25<sup>_ρ_.</sup>

# **(4) Data and Variables: India and Bangladesh**

To estimate the bias caused by truncation arising from the coresidency rules in surveys, we need data sets that include all of the children of the household head (and spouse) irrespective of their residency status, and also need to identify which members of the household were coresident at the time of the survey implementation. We take advantage of two exceptionally rich data sets particularly suited for our analysis. The data on India come from the 1999 Rural Economic and Demographic Survey done by the National Council for Applied Economic Research, and the data on Bangladesh are from the 1996 Matlab Health and Socioeconomic Survey (MHSS).<sup>10</sup> The Bangladesh survey collected information on all children of the household head and spouse (including from past marriages) irrespective of their residency status from 4,538 households in Matlab thana of Chandpur district. The India survey also collected information on all of household head’s children from the current marriage, but did not gather information on non-coresident mothers of children from earlier marriage(s). Both of these surveys focus on rural areas in the respective countries. The bias from censoring due to possible non-completion of younger children may not be as important in rural areas, because the proportion of children who go on to have more than middle school (or high school) education is not likely to be large. The children who go for more than high school

> 10The MHSS 1996 is a collaborative effort of RAND, the Harvard School of Public Health, the University of Pennsylvania, the University of Colorado at Boulder, Brown University, Mitra and Associates and the International Centre for Diarrhoeal Disease Research, Bangladesh (ICDDR,B).

9
education (10 years of schooling in Bangladesh and India) are also the children who leave the village household, because the “colleges” (for grades 11 and 12) and universities (for three-four year undergraduate, and graduate study) are located in the cities and large towns.

Our estimation sample consists of household heads and spouses, and their children, including those from other marriages in the case of Bangladesh. Our main empirical analysis is based on a sample of children aged 13-60 years. To test the sensitivity of our conclusions with respect to the specific age cutoffs, we estimate the rank correlation and absolute mobility at 25th percentile for two alternative age ranges; 16-60 and 13-50 years.

Table 2 reports the summary statistics for both the Bangladesh and India data for our main estimation sample (children in the age range 13-60 years). Several interesting observations and patterns emerge. The average schooling attainment is low in rural areas of both of the countries at the time of the survey years. The mean and median years of schooling for children are 4.97 and 5.00 respectively for Bangladesh, and 6.23 and 7 for India. The relatively lower education attainments in Bangladesh compared with India are also observed for the parent’s generation: median years of father’s schooling was 2 years in Bangladesh compared with 2.50 years in India. The age distribution of children also differs: the median age for the Bangladesh data is 30 years compared with 33 years for India. The gender gap in education between boys and girls is about 1 year in Bangladesh in contrast with 2.42 years in India.

In a study of intergenerational schooling persistence, one can define parental schooling in a number of different ways: some researchers use average schooling of father and mother as the relevant indicator, while others use the maximum of parents education, which in most cases amounts to father’s schooling in many developing countries. Also, many existing studies focus on the sub-sample of sons, and use father’s schooling as the relevant indicator of parental human capital. We use three indicators of parental education: father’s years of schooling, mother’s years of schooling, and the average years of schooling for father and mother. The percentile rank for father is estimated from the distribution of 10
father’s schooling, and similarly for mothers, while the percentile rank for average parental education is calculated using the distribution of average schooling.

# **(5) Estimates of Truncation Bias in Rank-Based Measures of Intergenerational Mobility**

The estimates from three different samples are provided: all children sample (includes both sons and daughters), the father-sons sample, and the mother-daughters sample. In all of the rank-rank regressions, the dependent variable (the indicator of educational attainment) is children’s percentile rank in the distribution of years of schooling for all children. Define the bias in the estimate of parameter _ω ∈ δ_ ˆ0 _,_ ˆ _δ_ 1 _, β_ ˆ0 _, β_ ˆ1 _,_ ˆ _ρ_ 0 _,_ ˆ _ρ_ 1 as _PBω_ (short for [ ] percentage bias in the estimate of _ω_ ): _Estimate of ω f rom Coresident Sample − Estimate of ω f rom Full Sample PBω_ = [ _Estimate of ω from Coresident Sample_ ] _×_ 100 (7) Thus when truncation causes a downward biased estimate, then the estimated bias in definition (7) above is negative, and it is positive when the estimate of a parameter from the coresident sample is biased upward. The advantage of using the estimate from the coresident sample as the base in equation (7) above is that it is directly observable to most of the researchers facing data constraints with access to only coresident sample.

# **(5.1) Truncation Bias in Estimates of Relative Mobility**

# **(5.1.1) Estimates from the All Children Sample**

ˆ The estimates of rank correlation _δ_<sup>ˆ</sup> 1 are reported in Table 3; the estimates of IGC _ρ_ 1 and IGRC _β_<sup>ˆ</sup> 1 are also included for comparison. The top panel contains the estimates for Bangladesh and the lower panel for India.

The point estimate of rank correlation _δ_<sup>ˆ</sup> 1 from the coresident sample is _smaller_ than that from the full sample (including nonresident children) across all three different indicators of parental education for both Bangladesh and India. This is consistent with the _a priori_ expectation that truncation causes downward bias in the estimate of the slope parameter(s) 11
of an OLS regression (Hausman and Wise (1977), Cohen (1991)).<sup>11</sup> More striking is the fact that the differences between the two estimates are very small in magnitude across the board. For example, the estimates for father’s schooling as the indicator of parental education in Bangladesh are 0 _._ 523 (full sample) and 0 _._ 483 (coresident sample). This implies that the downward bias in the rank correlation estimate due to coresidency is only _−_ 0 _._ 04, which amounts to _PBδ_ ˆ1 = _−_ 8 _._ 31% according to formula (7) (see the bias estimates in row 2).<sup>12</sup> To appreciate the order of magnitudes involved, it is instructive to compare this estimate to the corresponding estimate of truncation bias in the most widely used measure of intergenerational persistence in the development literature, i.e., IGRC; the bias in IGRC is almost four times as large at _PBβ_ ˆ1 = _−_ 29 _._ 40% (see row 6). The extent of coresidency bias in the rank correlation estimate varies somewhat with different indicators of parental education, with the bias being highest when the percentile rank of mother’s schooling is the indicator (-11.36%). The pattern of bias in rank correlation estimates across different indicators of parental education is similar in India. The average percentage bias in rank correlation across three indicators of parental education in Bangladesh is -8.31% and larger in India at _−_ 13 _._ 19% _._ In comparison, the average bias in IGRC is _−_ 29 _._ 39% in Bangladesh and _−_ 20 _._ 42% in India.

The results in columns 3-6 of Table 3 show that the IGC estimates suffer less truncation bias compared to IGRC, a conclusion established before in Emran et al. (2018) using a different regression specification. The more interesting evidence in Table 3 is that the extent of truncation bias in the rank correlation is, in general, close to that in IGC. When compared ˆ to the bias in IGC (ˆ _ρ_ 1), the estimated bias is lower in rank correlation _δ_ 1 in three out ( ) of six cases in Table 3. The average bias across six estimates for India and Bangladesh is ˆ ˆ _−_ 10 _._ 75% in rank correlation _δ_ 1 , _−_ 9 _._ 69% in IGC (ˆ _ρ_ 1), and _−_ 24 _._ 91% in IGRC _β_ 1 . The <u>( )</u> ( )

> 11Hausman and Wise (1977) in the appendix provide an explanation for the downward bias in the OLS estimate showing that it is smaller than the MLE estimate in a truncated sample.

> 12The estimates from full and coresident samples are statistically different from each other as the estimated standard errors are very small; the intersection of the 95 percent confidence intervals is a null set. However, the statistical precision and formal rejection of equality of the estimates are not informative in our context, as the differences in magnitudes are very small.

12
estimates thus suggest that rank correlation and IGC (Pearson correlation) are practically in a tie in coresident samples, but the most widely used measure IGRC is clearly the most biased by the truncation due to coresidency criteria.

# **(5.1.2) Estimates of Father-Sons and Mother-Daughters Intergenerational Persistence**

There is evidence that the intergenerational linkages between father-sons and motherdaughters may be stronger than the cross-gender effects. In this subsection, we discuss the truncation bias in the estimates of intergenerational persistence between schooling of fathers and sons, and between mothers and daughters. Table 4 presents the estimates.

Consistent with the evidence from the all children sample in Table 3, the estimates in table 4 show that the bias in _β_<sup>ˆ</sup> 1 (IGRC) is significantly higher compared to the bias in (ˆ _ρ_ 1) (IGC) and _δ_ ˆ1 (rank correlation). A comparison of the estimates of (ˆ _ρ_ 1) and _δ_ ˆ1 , how( ) ( ) ever, shows that the conclusion depends on the gender: for the father-sons intergenerational link, the estimated bias in rank correlation is somewhat smaller than that in Pearson correlation. In contrast, for the mother-daughters link, the bias in estimated rank correlation is larger in magnitude compared to the bias in IGC (Pearson correlation coefficient). The results in Table 4 also show that the estimates of mother-daughter schooling persistence in general suffer more severe downward bias compared to the estimates for father-sons. The larger bias in the daughters sample reflects the fact that coresidency rates are much lower for them both in Bangladesh and India. In the father-son sample the coresidency rate is 79 percent in India, while the corresponding rate is 52 percent in Bangladesh. In the mother-daughter samples, the coresidency rates are much lower: 39 percent in India and 26 percent in Bangladesh, reflecting the fact that women leave the natal family following marriage in both countries. The evidence thus suggests that an analysis of the gender gap in intergenerational mobility based on most of the available household surveys may be seriously misleading, because of significant gender differences in coresidency rates.

13
# **(5.1.3) Estimates from Alternative Age Ranges**

The empirical results discussed so far are based on our “main sample” composed of children in the age range 13-60 years. To check if the conclusions above are specific to this sample, we report estimated bias in the slope parameters of equations (1)-(3) from two alternative samples: 13-50 years, and 16-60 years. Table 5A reports the estimated bias from the 13-50 years sample, and Table 5B from the 16-60 years sample. We omit the underlying estimates from the full and coresident sub-samples which allows us to put the estimated bias from the all children sample, and father-sons, and mother-daughters sub-samples in a single Table. The estimates for Bangladesh are in the first three columns, and those for India are in the last three columns in each table.

Consider the bias estimates from the 13-50 years age sample in Table 5A. The estimates ˆ for the slope confirm the conclusion that the bias in the estimates of _δ_ 1 and (ˆ _ρ_ 1) are ( ) ˆ consistently smaller that that in _β_ 1 and this is true in both Bangladesh and India. A ( ) comparison of the bias estimates for the rank correlation with those for IGC shows that the bias in IGC is smaller in six out of ten cases. However, judged in terms of average bias, IGC and rank correlation are close to each other: the average bias across India and Bangladesh ˆ over different indicators of parental educations is _−_ 9 _._ 4% for (ˆ _ρ_ 1) and _−_ 10 _._ 01% for _δ_ 1 . The ( ) estimates form the 16-60 years sample also lead to the same set of conclusions. Averaging over the 20 estimates for 13-50 years and 16-60 years age ranges, the bias estimates are 8 _._ 48% for rank correlation, 8 _._ 38% for IGC, and 21 _._ 74% for IGRC. Thus the evidence cannot discriminate between IGC and rank correlation, but both are clearly much better than the IGRC as a measure of relative mobility. In the light of the evidence, it is advisable to report both rank correlation and IGC when the focus is on relative mobility.

# **(5.2) Truncation Bias in Absolute Mobility Estimates**

# **(5.2.1) Bias in the Estimates of the Intercept**

All of the studies on intergenerational mobility in developing countries listed in Table 1 rely exclusively on some measure(s) of relative mobility estimated as the slope of an OLS 14
regression, and do not consider or report the estimates of the intercepts in equations (1), (2), and (3) above. However, it is important to understand the biases in the estimated intercepts, as absolute mobility depends on both the slope and intercept of the regression equations. Table 6 reports the estimates of the intercept of equations (1)- (3), i.e., _δ_<sup>ˆ</sup> 0, _β_<sup>ˆ</sup> 0 and _ρ_ ˆ0 for the age range 13-60 years.

Consider the estimates for the all children sample in Table 6. The pattern of the estimates is exactly the opposite of that found in the estimates of the slope parameter in first two columns of Table 3 above; the estimate from the coresident sample is consistently _higher_ when compared to the corresponding estimate from the full sample across all three indicators of parental education, and this holds for both Bangladesh and India. Again, more important for our analysis is the evidence that the estimates from the full and coresident samples differ by small magnitudes in the case of the intercept of the rank-rank regression ˆ _δ_ 0.

For comparison, we turn to the estimates of the bias in the intercept terms from equaˆ tions (1) and (2) above, _β_<sup>ˆ</sup> 0 and _ρ_ 0, i.e., _PBβ_ ˆ0 and _PBρ_ ˆ0. The first thing to notice is that ˆ the bias estimates for the intercept term are positive across the board for both _β_<sup>ˆ</sup> 0 and _ρ_ 0, reinforcing the conclusion from the estimates of _δ_<sup>ˆ</sup> 0 that truncation due to coresidency leads to _upward_ bias in the estimated intercept term. The second important point that comes across clearly from the bias estimates in Table 6 is that the upward bias is significantly higher across the board in _β_<sup>ˆ</sup> 0 and _ρ_ ˆ0 when compared to the bias in _δ_<sup>ˆ</sup> 0, with the extent of bias largest in _ρ_ ˆ0. In Bangladesh, the average bias, estimated across three indicators of ˆ parental education, are 6 _._ 54% for _δ_<sup>ˆ</sup> 0, 15 _._ 91% for _β_<sup>ˆ</sup> 0, and 25 _._ 08% for _ρ_ 0. The corresponding ˆ average biases estimates for India are 8 _._ 78% for _δ_<sup>ˆ</sup> 0 , 16 _._ 37% for _β_<sup>ˆ</sup> 0 , and 18 _._ 05% for _ρ_ 0.

The results on the father-sons and mother-daughters samples in Table 6 are, however, somewhat different, as the conclusion depends on the gender. For mother-daughter the twin conclusions that the estimate is upward biased in the coresident sample and the degree of bias is lowest for the intercept of the rank-rank regression hold. However, these conclusions are not valid for the father-sons estimates. The estimates for alternative age ranges reported 15
in Tables 7A and 7B show that the different results for father-sons is sample-specific; for example, the conclusions regarding the direction and magnitude of the bias noted above remain valid for the father-sons estimates in the 13-50 years age range. The important take away from the results on the bias in the intercept estimates in Tables 6, 7A and 7B is that, in most cases (26 out of 30), the upward bias is the lowest in the intercept of the rank specification.

# **(5.2.2) Truncation Bias in Absolute Mobility: Expected Schooling and Expected Schooling Rank**

Following Chetty et al. (2014), we combine the estimates of the slope and intercept of the rank-rank regression and report estimates of absolute mobility using equation (4) above. Since truncation due to coresidency restrictions causes _downward_ bias in the slope estimate, but, at the same time, leads to an _upward_ biased estimate of the intercept in general, one might conjecture that the bias in the absolute mobility is likely to be smaller than that in the estimates of relative mobility because of offsetting effects. It is, however, important to appreciate that the bias in the intercept may dominate the estimates at the lower tail, while the bias in slope is likely to be more consequential for the estimates at the upper tail, and in general, it is not possible to know the direction of net bias at a given percentile of parental schooling.

# **Expected Years of Schooling (and Expected Normalized Schooling)**

Tables 8A and 8B present the results for the 25th and 75th percentiles of the parental schooling distribution. Table 8A reports the estimated expected years of schooling (using ˆ ˆ _β_ 0 , _β_ 1) and the percentage bias in the coresident samples, while Table 8B reports the corresponding estimates for normalized schooling (using _ρ_ ˆ0 , _ρ_ ˆ1). The estimates provide the average expected years of schooling or the average expected normalized schooling for the subset of children whose parental schooling belongs to a certain percentile.

The first point to notice is that the estimates of expected schooling for the 25th perˆ centile are the same as the estimated intercepts _β_<sup>ˆ</sup> 0 , _ρ_ 0, which reflects the fact that almost 16
50 percent parents have zero schooling in our data sets. Thus the results on truncation bias in the intercepts discussed above in subsection (5.2.1) imply that the expected years of schooling at the 25th percentile are likely to be substantially overestimated in the coresˆ ident samples, for both _β_<sup>ˆ</sup> 0 and _ρ_ 0, and this is true in both Bangladesh and India. Also, the extent of upward bias is the largest if one relies on _ρ_ ˆ0. This evidence suggests that even though IGC (ˆ _ρ_ 1) is a robust measure of relative mobility when working with standard household surveys (Emran, Greene, and Shilpi (2018)), the estimates of expected schooling attainment using equation (2) may be the least reliable at the lower tail of the distribution.

The estimates of expected years of schooling for children show that the estimates from the coresident samples are in general larger than those from the full sample at the 25th percentile, and at the 75th percentile, it is true in 9 out of 10 cases.<sup>13</sup> The evidence thus suggests that the upward bias in the intercept dominates the downward bias in the slope estimate. However, the magnitude of bias does not exhibit any consistent pattern across the 25th and 75th percentiles of parental distribution. In Bangladesh, the truncation bias (absolute value of the bias) at the 75th percentile is smaller than that at the 25th percentile in four out of five cases: the average bias is 7 _._ 94 percent for the 75th percentile, while it is 15 _._ 58 percent for the 25th percentile. In India, in 2 out of 5 cases, the bias is higher at the 75th percentile : the average bias estimates are 11.57 percent for the 75th and 13.36 percent for the 25th percentile.

The estimates of expected normalized schooling (i.e., years of schooling normalized by standard deviation) in Table 8B also tell a similar story. The estimates are consistently upward biased in coresident samples, but the magnitude of bias does not follow a consistent pattern across the 25th and 75th percentiles. A comparison of the biases in the expected years of schooling to that in normalized schooling shows that the average bias is substantially higher in the estimates of normalized schooling when compared to the expected years

> 13The only exception at the 75th percentile is the expected years of schooling for sons conditional on father’s schooling. However, the estimates from coresident sample (6.666) and full sample (6.767) are very close to each other.

17
of schooling in Bangladesh. In the case of India, the pattern holds, i.e., the average bias is lower in the expected years of schooling, but the magnitudes are much closer, at the 25th percentile: 13.36% (years of schooling) and 14.58% (normalized schooling), and at the 75th percentile: 11.57% (years of schooling) and 12 _._ 9% (normalized schooling).

# **Expected Rank of a Child**

Table 9 presents the estimates of absolute mobility using the estimated _δ_<sup>ˆ</sup> 0 , _δ_<sup>ˆ</sup> 1 at the 25th percentile of parental schooling distribution, following the definition of Chetty et al. (2014). We also calculate the rank of expected years of schooling of children whose parental schooling belongs to the 25th percentile to have a comparable measure of absolute mobility (using equation (5)). As noted before, the ranking of a child’s expected schooling in the distribution of children’s schooling does not vary between equations (1) and (2) by construction. Thus we focus on the rank of the expected years of schooling from equation (5) in the distribution of children’s schooling. In addition to absolute mobility at 25th percentile, we also report absolute mobility at the 75th percentile.

The results show that the bias is smaller in the rank-based absolute mobility estimate in 8 out of 10 cases both in Bangladesh and India. At the 25th percentile of parental schooling, the average bias (absolute magnitudes ignoring the signs) in rank-based absolute mobility _r_ ̄25 estimates over different specifications in Bangladesh is 5.67% , and is 13.32% for the alternative measure based on the predicted schooling using equation (5) above, i.e., _R_ 25<sup>_β_.</sup> The corresponding estimates at 75th percentile of parental distribution are 2.44 percent for _r_ ̄25 and 6.66 percent for _R_ 25<sup>_β_.TheevidencefromIndiaissimilar:theaveragebias</sup> at 25th percentile is 3.44% in rank-based absolute mobility estimates, and 10.34% in the IGRC-based absolute mobility estimates. The corresponding average bias estimates at the 75th percentile for India are 2.54% (rank-based) and 8.1% (IGRC-based). The evidence thus is strong that absolute mobility estimates based on rank-rank specification are much more robust to the truncation bias arising in coresident samples.

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# **(6) Discussion**

Emran, Greene, Shilpi (2018) discuss a rationale for the observed lower bias in IGC ˆ (ˆ _ρ_ 1) in coresident samples when compared to the bias in IGRC _β_ 1 . They point out ( ) that truncation not only causes downward bias in the estimate of IGRC, it also results in downward bias in the estimate of variance of children’s schooling (Greene (2012), Cohen ˆ ˆ _σp_ ˆ (1991)). Since _ρ_ 1 = _β_<sup>ˆ</sup> 1 ˆ , a downward biased estimate of _σc_ cancels out some of bias due _σc_ to the downward bias in _β_<sup>ˆ</sup> 1. The rank correlation takes an additional step to purge the effects of changing marginal distributions across generations by focusing on the copula of the bivariate distribution of parents’ and children’s schooling.

An explanation for the lower sensitivity of rank correlation estimates to truncation found in this paper can be developed in terms of the fact that truncation tends to exclude observations from the tails of a distribution more. It has long been understood that rank correlation is less sensitive to outliers, because rank-ordering pulls the outlying observations more towards the center of the distribution (Lehmann (1975), Shevlyakov and Oja (2016), Bishara and Hittner (2012)). Since observations from the tails of the distribution are more likely to be lost because of truncation, the effect on OLS is expected to be strong, but rank correlation is more robust to inclusion (or exclusion) of these observations in (from) the sample.

The finding that the intercept estimate from the IGC specification (equation (2) above),ˆ ˆ _<u>β</u>_ 0 in general, contains the largest upward bias can be explained by observing that _ρ_ 0 = ˆ . It _σc_ is widely known that, in general, truncation biases the estimate of _β_<sup>ˆ</sup> 0 upward in the IGRC regression equation (1) (Hausman and Wise (1977), Cohen (1991)). However, as noted above, truncation also biases the estimate of variance of children’s schooling downward in coresident samples. This implies that the upward bias in the estimate of _ρ_ ˆ0 is higher that that in the estimate of _β_<sup>ˆ</sup> 0.

The growing evidence on the robustness of rank-based measures of mobility, however, raises the question: what we miss when we are unable to have reliable estimates of IGRC 19
(or IGE in the case of income persistence). As discussed above, compared to IGRC (or IGE), IGC and rank correlation progressively purge the effects of changes in marginal distributions across generations. Many authors argue that this is desirable as this allows a researcher to focus on the fundamental structure of dependence between parents and children. For example, Bjorklund and Jantti (2009) note that IGC provides a measure of mobility that is not affected mechanically by changes in inequality across generations. But others such as Mazumder (2015) and Mitnik et al. (2014) point out that IGRC or IGE are valuable precisely because they contain information about the marginal distributions and relate to cross sectional inequality directly.

# **Conclusions**

Following the influential contribution of Chetty et al. (2014), many researchers working on intergenerational mobility, both in developed and developing countries, have adopted the rank-based measures. There is an emerging body of evidence that rank-based measures are more reliable as they are less sensitive to measurement error and life-cycle biases. We focus on the implications of a common data limitation faced by researchers for estimates of rank-based relative and absolute mobility: sample truncation due to the fact that household membership is defined in terms of a set of coresidency criteria.

We utilize two exceptionally rich household surveys from Bangladesh and India where information on non-resident children was collected, and the subset of children coresident at the time of survey implementation was identified. The evidence shows that truncation results in downward bias in the slope estimate and upward bias in the intercept estimate. The truncation bias in rank correlation is relatively low and similar in magnitude to that in intergenerational correlation (IGC), and the downward bias in coresident samples is substantially higher in the estimates of IGRC. The evidence thus suggests that a researcher working with coresident samples should report both rank correlation and IGC to understand relative mobility. Our results also show that the magnitude of bias in rank-based absolute mobility proposed by Chetty et al (2014) is usually small, and in most cases, suffers 20
significantly less from truncation bias compared to alternative measures based on years of schooling. This paper thus strengthens the case for rank-based measures for analyzing intergenerational mobility especially in developing countries where most of the available data sets (e.g. LSMS and DHS) suffer from sample truncation due to coresidency restrictions used to define household membership.

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# **Table 1: Intergenerational Mobility in Developing Countries:**

**Data, Status Indicator and Measures**

||**COUNTRY**|**DATA**||
|---|---|---|---|
|**STUDY**|**Status Indicator**|**SAMPLE**|**MEASURES**|
|Thomas (1996)|South Africa<br>Education|Cross-section 1991/93<br>Coresident Children|IGC|
|Behrman and Wolfe (1987)|Nicaragua<br>Education|Single Cross-section: 1977/78<br>Coresident Children of Sisters|IGRC|
|Jalan and Murgai (2008)|India<br>Education|Cross-sections: 92/93, and 98/99<br>Coresident Children|IGRC|
|Maitra and Sharma (2010)|India<br>Education|Single Cross-Section: 2005<br>Coresident Children|IGRC|
|Azam and Bhatt (2015)|India<br>Education|Single Cross-section: 2005<br>Coresident Sons +Nonresident Sons<br>in College '+ Head's Parents<br>But does not include nonresident sons|IGRC and IGC|
|||due to work migration, and household split||
||||IGC and|
|Emran and Shilpi (2015)|India|Cross-sections: 1993, 2006|Sibling Corr.<br>Transition|
||Education|Coresident Children|Matrices|
|Emran and Sun (2011)|China<br>Education|Cross-sections: 1995, 2002<br>Coresident Children ` + Head's and<br>Spouses Parents ` + nonresident members|IGRC|
||Occupation|with financial links to the household||
|Fan, Yi, Zhang (2015)|China|Cross-section, 2010|IGRC, IGC and|
||Income, Education|Coresident + Nonresident Children|Rank-Rank|
|Hertz et al. (2007)|21 Developing<br>Countries|Cross-section, various years<br>Coresident Children|IGRC and IGC|
||Education|+ Head's and Spouse's Parents||
|Nimubona and|South Africa|Repeated Cross-section (Pseudo-Panel)|IGRC|
|Vencatachellum (2007)|Education|Coresident Children||
|Lillard and Willis (1995)|Malaysia|Cross-Section, 1988<br>One coresident and up to 2 nonresident|IGRC|
||Education|children randomly selected||
|Lam and Schoeni (1993)|Brazil|Cross-section|IGRC|
||Earnings|Household Head's and Spouse's Parents||
||||IGRC,|
|Behrman et al (2001)|Brazil, Colombia,|Cross-section (various years)|Transition|
||Mexico, Peru<br>Education|Coresident Children ' + Head's and<br>Spouse's Parents|Matrices|

Notes: IGRC Stands for Intergenerational Regression Coefficient, IGC for Intergenerational Correlation, Sibling Corr. for Sibling Correlation, and Rank-Rank for Spearman Rank Correlation.
# **Table 2: SUMMARY STATISTICS**

|||**ALL CHIL**|**DREN**|**CO-RES**|**IDENT C**|**HILDREN**|
|---|---|---|---|---|---|---|
||**Mean**|**Median**|**N**|**Mean**<br>|**Median**|**N**|
||(1)|(2)||(4)|(5)||
|||||<br>**BANGLADESH**|||
|Years of Education of||**B**|**oth Son**|**s and Daughters S**|**ample**||
|Children|4.97|5.00|18587|5.52|5.00|5852|
|Father|3.39|2.00|14017|3.74|3.00|5599|
|Mother|1.46|0.00|14527|1.81|0.00|5523|
|Average of Parents|2.33|1.00|18505|2.78|2.00|5806|
|||||**Sons Sample**|||
|Children|5.84|5.00|9056|5.56|5.00|3873|
|Father|3.38|2.00|7126|3.53|2.00|3713|
|Mother|1.45|0.00|7261|1.64|0.00|3648|
|Average of Parents|2.34|1.00|9010|2.59|1.50|3844|
||||**D**|**aughters Sample**|||
|Children|4.14|4.00|9531|5.44|5.00|1979|
|Father|3.41|2.00|6891|4.16|3.00|1886|
|Mother|1.47|0.00|7266|2.14|0.00|1875|
|Average of Parents|2.33|0.50|9495|3.14|2.50|1962|
|||||**INDIA**|||
|Years of Education of<br>||**B**<br>|**oth Son**<br>|**s and Daughters S**<br>|**ample**<br>||
|Children|6.23|7.00|14877|6.97|8.00|9132|
|Father|4.37|2.50|14877|4.74|5.00|9132|
|Mother|1.83|0.00|14877|2.12|0.00|9132|
|Average of Parents|3.10|2.50|14877|3.43|2.50|9132|
|||||**Sons Sample**|||
|Sons|7.29|8.00|8341|7.54|8.00|6561|
|Father|4.31|2.50|8341|4.59|5.00|6561|
|Mother|1.82|0.00|8341|1.99|0.00|6561|
|Average of Parents|3.06|2.50|8341|3.29|2.50|6561|
||||**D**|**aughters Sample**|||
|Daughters|4.87|5.00|6536|5.54|6.00|2571|
|Father|4.46|2.50|6536|5.14|5.00|2571|
|Mother|1.84|0.00|6536|2.45|0.00|2571|
|Average of Parents|3.15|2.50|6536|3.79|3.25|2571|

Notes: Data Sources: India: Rural Economic and Demographic Survey (REDS) 1999; Bangladesh: Matlab Health and Socioeconomic Survey 1996.
**Table 3: Truncation Bias in Relative Mobility: All Children Sample**

|||**Rank**|**Correlation**|**I**|**GC**|**IGRC**|
|---|---|---|---|---|---|---|
|||**Full**|**Coresident**|**Full**|**Coresident**|**Full**<br>**Coresident**|
||||**BA**<br>|**NGLADE**<br>|**SH **<br>|<br>|
|**Father's Schooling**||0.523|0.483|0.506|0.459|0.546<br>0.422|
||**Bias**|**-8**|**.31%**|**-1**|**0.4%**|**-29.40%**|
|**Mother's Schooling**||0.548|0.492|0.465|0.421|0.842<br>0.616|
||**Bias**|**-1**|**1.36%**|**-1**|**0.3%**|**-36.67%**|
|**Average Schooling**|**Bias**|0.531<br>**-5**|0.504<br>**.26%**|0.50<br>**-5**|0.47<br>**.15%**|0.71<br>0.58<br>**-22.09%**|
|||||**INDIA**|||
|**Father's Schooling**||0.448|0.409|0.439|0.397|0.483<br>0.419|
||**Bias**|**-9**|**.42%**|**-1**|**0.8%**|**-15.36%**|
|**Mother's Schooling**||0.453|0.382|0.367|0.318|0.569<br>0.454|
||**Bias**|**-1**|**8.77%**|**-1**|**5.5%**|**-25.52%**|
|**Average Schooling**||0.467|0.420|0.456|0.403|0.654<br>0.543|
||**Bias**|**-1**|**1.39%**|**-1**|**3.1%**|**-20.37%**|

Notes: (1) The sample consists of daughters and sons of 13-60 years age. (2) IGC stands for Intergenerational correlation, IGRC stands for intergenerational regression coefficient.

(3) Bias is Percentage Bias = [(Coresident estimate - Full estimate)/Coresident estimate] * 100

**Table 4: Bias in Relative Mobility for Father-Sons** **<u>and Mother-Daughters</u>**

|||**RankC**|**orrelation **|**IGC**||**IGRC**|
|---|---|---|---|---|---|---|
|||**Full**|**Coresident**|**Full**<br>**Coresident**|**Full**|**Coresident**|
||||**BA**<br>|**NGLADESH **<br> <br>|||
|**Father-Sons**||0.516|0.499|0.488<br>0.449|0.565|0.444|
||**Bias**|**-5**|**.19%**|**-8.61%**||**-27.03%**|
|**Mother-Daughter**||0.577|0.505|0.529<br>0.490|0.845|0.597|
||**Bias**|**-14**|**.27%**|**-8.10%**||**-41.60%**|
|||||**INDIA**|||
|**Father-Sons**||0.426|0.421|0.420<br>0.409|0.457|0.427|
||**Bias**|**-1**|**.26%**|**-2.65%**||**-7.22%**|
|**Mother-Daughters**||0.549|0.481|0.466<br>0.428|0.688|0.571|
||**Bias**|**-14**|**.18%**|**-8.70%**||**-20.44%**|

Notes: (1) The samples consist of daughters only (mother-daughters) and sons only (father-sons) for the 13-60 years age range. (2) IGC stands for Intergenerational correlation, IGRC stands for intergenerational regression coefficient. (3) Bias is percentage bias.
**Table 5A: Bias in Relative Mobility: Estimates from 13-50 Years Age Sample**

||**BAN**|**GLADESH**|||**INDIA**||
|---|---|---|---|---|---|---|
||**Rank Corr.**|**IGC**|**IGRC**|**Rank Corr.**|**IGC**|**IGRC**|
|**All Children Sample**|||||||
|**Father's Schooling**|**-8.27%**|**-10.30%**|**-29.22%**|**-9.76%**|**-11.10%**|**-15.73%**|
|**Mother's Schooling**|**-11.16%**|**-10.20%**|**-36.27%**|**-18.96%**|**-15.70%**|**-25.60%**|
|**Average Schooling**|**-5.14%**|**-5.09%**|**-21.86%**|**-11.65%**|**-13.37%**|**-20.58%**|
|**Father-Sons**|**-5.05%**|**-8.46%**|**-26.72%**|**-1.63%**|**-3.04%**|**-7.54%**|
|**Mother-Daughters**|**-14.16%**|**-8.05%**|**-41.44%**|**-14.28%**|**-8.68%**|**-20.37%**|

|**Table 5B: Bi**|**as in Relative**<br>**BAN**|**Mobility: E**<br>**GLADESH**|**stimates fro**<br>|**m 16-60 Year**|**s Age Sam**<br>**INDIA**|**ple**|
|---|---|---|---|---|---|---|
||**Rank Corr.**|**IGC**|**IGRC**|**Rank Corr.**|**IGC**|**IGRC**|
|**All Children Sample**|||||||
|**Father's Schooling**|**-6.14%**|**-7.7%**|**-21.23%**|**-6.72%**|**-9.8%**|**-13.47%**|
|**Mother's Schooling**|**-7.63%**|**-7.5%**|**-27.67%**|**-17.33%**|**-15.2%**|**-24.55%**|
|**Average Schooling**|**-1.4%**|**-1.26%**|**-11.13%**|**-9.09%**|**-12.54%**|**-19.09%**|
|**Father-Sons**|**-5.1%**|**-7.26%**|**-21.47%**|**-0.58%**|**-2.41%**|**-6.88%**|
|**Mother-Daughters**|**-8.14%**|**-4.54%**|**-30.84%**|**-7.3%**|**-5.34%**|**-13.17%**|

NOTES: (3) The numbers in the table are the percentage bias as defined in Table 3. (2) IGC provides estimate of Pearson correlation, IGRC is intergenerational Regression Coefficient. (3) Average schooling is the average of mother's and father's schooling.
**Table 6:  Truncation Bias in the Intercepts (13-60 Years)**

||**(δ0)**|**(ρ0)**<br>|**(β0)**||
|---|---|---|---|---|
||**Full**<br>**Coresident**|**Full**<br>**Coresident**|**Full**<br>**C**|**oresident**|
|**All Children Sample**|**BA**|**NGLADESH**|||
|**Father's Schooling**|0.247<br>0.26|0.741<br>1.016|3.225|3.955|
|**Bias**|**4.79%**|**27.05%**|**18%**||
|**Mother's Schooling**|0.232<br>0.256|0.869<br>1.134|3.813|4.435|
|**Bias**|**9.52%**|**23%**|**14.04%**||
|**Average Schooling**|0.234<br>0.247|0.751<br>0.999|3.303|3.896|
|**Bias**|**5.32%**|**25%**|**15.22%**||
|**Father-Sons**|0.297<br>0.262|0.847<br>0.980|3.994|3.999|
|**Bias**|**-13.38%**|**14%**|**1.37%**||
|**Mother-Daughters**|0.170<br>0.237|0.770<br>1.200|3.002|4.218|
|**Bias**|**28.39%**|**35.82%**|**28.83%**||
|**All Children Sample**||**INDIA**|||
|**Father's Schooling**|0.276<br>0.295|0.838<br>1.037|4.113|4.987|
|**Bias**|**6.53%**|**19.18%**|**17.53%**||
|**Mother's Schooling**|0.273<br>0.309|1.056<br>1.25|5.185|6.012|
|**Bias**|**11.58%**|**15.49%**|**13.76%**||
|**Average Schooling**|0.266<br>0.290|0.855<br>1.062|4.199|5.110|
|**Bias**|**8.23%**|**19.48%**|**17.83%**||
|**Father-Sons**|0.350<br>0.326|1.097<br>1.179|5.321|5.581|
|**Bias**|**-7.32%**|**6.95%**|**4.67%**||
|**Mother-Daughters**|0.146<br>0.164|0.776<br>0.878|3.603|4.137|
|**Bias**|**11.06%**|**11.68%**|**12.90%**||

NOTES: (1) **( δ0 )** is the intercept of rank-rank, **( ρ0 )** is the intercept of IGC regression, and **( β0 )** is the intercept of IGRC regression.
**Table 7A: Truncation Bias in the Intercept: Estimates from 13-50 Years Age Sample**

||**BA**|**NGLADESH**|||**INDIA**||
|---|---|---|---|---|---|---|
||**(δ0)**|**(ρ0)**|**(β0)**|**(δ0)**|**(ρ0)**|**(β0)**|
|**All Children Sample**|||||||
|**Father's Schooling**|**4.93%**|**26.97%**|**18.44%**|**6.76%**|**19.44%**|**17.78%**|
|**Mother's Schooling**|**9.27%**|**23.14%**|**13.91%**|**11.71%**|**15.63%**|**13.90%**|
|**Average Schooling**|**5.19%**|**24.67%**|**15.11%**|**8.43%**|**19.69%**|**18.03%**|
|**Father-Sons**|**-13.28%**|**13.69%**|**1.40%**|**-7.01%**|**7.24%**|**4.97%**|
|**Mother-Daughters**|**28.62%**|**35.64%**|**28.62%**|**10.96%**|**11.76%**|**12.92%**|
|**Table 7B: Truncati**|**on Bias in the**<br>**BA**|**Intercept:**<br>**NGLADESH**|**Estimates**<br>|**from 16-60**|**Years Ag **<br>**INDIA**|**e Sample**|
||**(δ0)**|**(ρ0)**|**(β0)**|**(δ0)**|**(ρ0)**|**(β0)**|
|**All Children Sample**|||||||
|**Father's Schooling**|**2.66%**|**28.31%**|**23.40%**|**5.11%**|**20.98%**|**20.40%**|
|**Mother's Schooling**|**6.81%**|**24.45%**|**19.29%**|**11.61%**|**17.07%**|**16.47%**|
|**Average Schooling**|**1.59%**|**24.63%**|**19.53%**|**7.15%**|**21.40%**|**20.83%**|
|**Father-Sons**|**-18.5%**|**13.37%**|**4.02%**|**-11.01%**|**7.38%**|**5.61%**|
|**Mother-Daughters**|**27.42%**|**37.30%**|**35.71%**|**0.41%**|**6.80%**|**12.94%**|

NOTES: (1) **( δ0 )** is the intercept of rank-rank, **( ρ0 )** is the intercept of IGC regression, and **( β0 )** is the intercept of IGRC regression. (2) The reported numbers are the percentage bias as defined in Table 3.
**Table 8A: Bias in Expected Years of Schooling Conditional on Parental Schooling Rank**

||**BANGL**|**ADESH**|**IN**|**DIA**|
|---|---|---|---|---|
||**P25**|**P75**|**P25**|**P75**|
|**All Children Sample**|||||
|**Father's Schooling**|**18.46%**|**8.19%**|**17.53%**|**4.31%**|
|**Mother's Schooling**|**14.04%**|**8.13%**|**13.76%**|**20.19%**|
|**Average Schooling**|**15.22%**|**5.67%**|**17.83%**|**9.30%**|
|**Father-Sons**|**1.37%**|**-1.52%**|**4.67%**|**0.16%**|
|**Mother-Daughters**|**28.83%**|**16.17%**|**12.90%**|**23.88%**|
|**Table 8B: Bias in Expec**|**ted Normalized Scho**<br>**BANGL**|**oling Condition**<br>**ADESH**|**al on Parental S**<br>**IN**|**chooling Rank**<br>**DIA**|
||**P25**|**P75**|**P25**|**P75**|
|**All Children Sample**|||||
|**Father's Schooling**|**27.1%**|**17.9%**|**19.2%**|**6.2%**|
|**Mother's Schooling**|**23.3%**|**18.1%**|**15.5%**|**21.8%**|
|**Average Schooling**|**24.8%**|**16.4%**|**19.5%**|**11.1%**|
|**Father-Sons**|**13.7%**|**11.2%**|**7%**|**2.6%**|
|**Mother-Daughters**|**35.8%**|**24.4%**|**11.7%**|**22.8%**|

**Table 8B: Bias in Expected Normalized Schooling Conditional on Parental Schooling Rank**

NOTES: (1) P25 is the 25th percentile of parental education rank defined in terms of a given indicator such as father's schooling. P75 is similarly defined. (2) Normalized schooling is years of schooling divided by its standard deviation. (3) The numbers reported are the percentage bias as defined in Table 3.
**Table 9:  Truncation Bias in Absolute Mobility**

||**P**|**25**|**P**|**75**|
|---|---|---|---|---|
||**Rank-based**|**IGRC-based**|**Rank-based**|**IGRC-based**|
||||**BANGLADESH**||
|**All Children Sample**|||||
|**Father's Schooling**|**-0.7%**|**-22%**|**0.2%**|**1.90%**|
|**Mother's Schooling**|**-2.2%**|**4.7%**|**-2.4%**|**-8.1%**|
|**Average Schooling**|**-6%**|**-25.9%**|**-1.4%**|**-8.7%**|
|**Father-Sons**|**-12.9%**|**-3.9%**|**-5.7%**|**4.2%**|
|**Mother-Daughters**|**7.2%**|**-10.1%**|**2.5%**|**-10.4%**|
||||**INDIA**||
|**All Children Sample**|||||
|**Father's Schooling**|**1.60%**|**-25.5%**|**-3.8%**|**5.70%**|
|**Mother's Schooling**|**0.9%**|**-10.3%**|**1.2%**|**16.1%**|
|**Average Schooling**|**2.40%**|**-0.1%**|**-1%**|**5.70%**|
|**Father-Sons**|**-7.8%**|**-8.8%**|**-6.4%**|**-3.7%**|
|**Mother-Daughters**|**-4.5%**|**-7%**|**-0.3%**|**-9.3%**|

NOTES: (1) Rank-based absolute mobility is defined following Chetty et al. (2014). The IGRC-based absolute mobility is the expected rank of the predicted years of schooling using equation (5) in the actual schooling distribution of children. (2) The reported numbers are the percentage bias estimates as defined in Table 3.

**Figure 1: Child’s Education and his/her probability of non-residency in Bangladesh and India**

<!-- Start of picture text --> Figure 1a: Probability of non-residency in Bangladesh Figure 1b: Probability of non-residency in India<br>0 5 10 15 20 0 5 10 15 20<br>Child's Education Child's Education<br>Female Male Female Male<br>.7<br>.6<br>.5<br>.4<br>.3<br>.2<br>.9<br>.8<br>.7<br>.6<br>.5<br><!-- End of picture text -->