Partial Identification of Economic Mobility: With an Application to the United States

TL;DR: This article develops a nonparametric partial identification approach to bound transition probabilities under various assumptions on the measurement error and mobility processes that is applied to panel data from the United States to explore short-run mobility before and after the Great Recession.

Abstract: The economic mobility of individuals and households is of fundamental interest. While many measures of economic mobility exist, reliance on transition matrices remains pervasive due to simplicity a...

Summary

Global and Epidemiological

• Perspectives on Diet and Mood F.N. Jacka Deakin University, Geelong, VIC, Australia 8 Noncommunicable diseases (NCDs) globally account for the largest burden of early mortality and are predicted to cost more than US$30 trillion over the next 20 years (Bloom et al., 2011).
• When the global burden of disease is viewed in terms of disability rather than mortality, mental and substance use disorders account for the leading cause of health-related disability worldwide, with unipolar depression alone accounting for the second highest number of years lost to disability (Murray et al., 2013).
• There are data to support such increases from the United States (Twenge et al., 2010), Britain (Collishaw et al., 2004), Taiwan (Fu et al., 2013), and Australia (O’Donnell et al., 2013), although upward trends in young people may be plateauing (Maughan et al., 2008).
• Type 2 diabetes mellitus, and cancer—are well known to be directly influenced by unhealthy diet (WHO, 2011; Swinburn et al., 2011), there is now highly consistent evidence across age groups, cultures, and countries to suggest that unhealthy diet is also a key risk factor for common mental disorders, particularly depression.
• The following presents a discussion of the change to global eating patterns and the recent literature highlighting the relationships between diet and mental health.

CHANGES TO THE FOOD SUPPLY AND GLOBAL IMPACT ON HEALTH

• Substantial changes in efficiencies of production, marketing, transport, and sale of food have had a highly detrimental impact on dietary patterns across the globe, with a widespread shift toward increased intake of fast foods and sugar-sweetened beverages (Adair and Popkin, 2005).
• In the West, dietary patterns are commonly high in saturated fats and refined sugar, with nutrient-poor and energy-dense foods contributing approximately 30% of the daily intakes of US adults (Kant, 2000).
• Alongside these changes were declines in the intake of fruit and vegetables.
• It is estimated that dietary intakes of micronutrients for early humans may have been up to 10 times that of modern humans because of the composition of wild plant foods known to be consumed by hunter-gatherers (Brand-Miller and Holt, 1998); carbohydrate consumption was almost exclusively derived from fruits and vegetables (Eaton and Eaton, 2000).
• In support of this, intervention studies performed in indigenous Australian populations have reported a pronounced reduction in risk factors for CVD, as well as metabolic abnormalities associated with diabetes, after reversion to a traditional hunter-gatherer diet containing substantial quantities of red meat from wild animals (O’Dea, 1984; O’Dea and Sinclair, 1985).

NUTRIENTS AND MENTAL HEALTH

• Before 2009, scientific data were scarce and the existing literature focused primarily on individual nutrients or foods—particularly fish and the long-chain omega-3 polyunsaturated fatty acids (n-3 PUFAs).
• In one of the first studies in this field, Hibbeln (1998) suggested that substantial cross-national variation in prevalence rates of depression may be, in part, a function of a demonstrated, strong, inverse correlation between national levels of fish consumption and national depression prevalence rates across nine countries.
• One meta-analysis examining the effect of n-3 PUFA supplementation for depressed mood concluded that there was a small beneficial effect of treatment with n-3 PUFA compared with placebo, but that this benefit was restricted to those with major depressive disorder (MDD; Appleton et al., 2010).
• Clinical studies have long observed folate deficiency and low folate status in those with clinical depression, and low folate is also associated with depression in population studies (Morris et al., 2003), even in the presence of folate fortification (Ramos et al., 2004).
• In a clinical context, low concentrations of zinc are commonly observed in patients with major depression (Swardfager et al., 2013) whereas zinc supplementation has been shown to enhance the efficacy of antidepressant therapy (Nowak et al., 2003a), with a systematic review supporting its use as an adjunctive therapy (Lai et al., 2011).

DIET QUALITY

• It is important to recognize that magnesium, folate, zinc, and long-chain fatty acids are all components of a healthy diet, found primarily in foods such as leafy green vegetables, legumes, whole grains, lean red meat, and fish.
• This association existed before and after controlling for a comprehensive range of potentially confounding factors, including sociodemographic, anthropometric, and lifestyle factors; other health behaviors; and medical history.
• Thus the hypothesis that diet is related to common mental disorders, particularly depression, is supported by studies in a wide range of countries and cultures as diverse as Spain, Norway, China, the United States, Japan, Australia, and many others.
• At the other end of the age spectrum, diet quality is also associated with mental health in adolescents and children.
• As well as both healthy and unhealthy dietary patterns during the first years of life, are associated with the risk for mental health problems in young children (Jacka et al., 2013b).

INTERVENTION STUDIES

• This new body of observational data is notable for the relative consistency of the reported relationships and the observed effect sizes.
• This is an increasingly common question in clinical practice and the general community, and it remains unanswered to date, representing a serious gap in their knowledge base.
• A systematic review examined the data from dietary interventions that have examined mental health outcomes in various populations and concluded that, although data from depressed samples are currently lacking, there is some evidence suggesting a positive impact of dietary improvement on depression (Opie et al., 2014).
• In the large PREDIMED study, older individuals randomized to an MDP compared with a low-fat diet tended to be less likely to develop depression over the period of the intervention, and this relationship was particularly pronounced for those individuals with type 2 diabetes (Sanchez-Villegas et al., 2013).
• Those in the Mediterranean diet groups also demonstrated improved cognition compared with controls (Martinez-Lapiscina et al., 2013).

CLINICAL APPLICATIONS

• This new literature provides face validity for the role of nutritional factors in the genesis and management of depression.
• The data largely fulfill the Bradford Hill criteria for causality (Jacka et al., 2012c) and are consistent and compelling.
• Moreover, although there is currently a dearth of evidence regarding the efficacy of dietary modulation to treat depression, it is clear that diet has a major impact on comorbid physical disorders that are disproportionally more common in people with depression, such as cardiovascular disorders and diabetes.
• Chronic low-grade inflammation, with accompanying oxidative stress, is a common feature of virtually all mental disorders, as well as the somatic disorders with which mental disorders are so commonly comorbid.
• Indeed, emerging data from experimental and human studies now suggest that the gut is a key pathway by which environmental factors, such as poor diet, sedentary behavior, and stress, influence the immune system and host health, with downstream effects on the risk for mental, as well as physical, disorders.

Figures

Table 1. Summary Statistics.

Table 5. Tercile Transition Matrices: Misclassification Assumptions.

Table 4. Poverty Transition Matrices: Monotonicity + Level Set Restrictions.

Table 3. Poverty Transition Matrices: Level Set Restrictions.

Table 8. Tercile Transition Matrices: Summary Mobility Measures.

Table 7. Tercile Transition Matrices: Monotonicity + Level Set Restrictions.

Table 2. Poverty Transition Matrices: Misclassification Assumptions.

Table 6. Tercile Transition Matrices: Level Set Restrictions.

Full text

Partial Identi…cation of Economic Mobility:
With an Application to the United States
Daniel L. Millimet
Southern Methodist University & IZA
Hao Li

Nanjing Audit University
Punarjit Roychowdhury
Indian Institute of Management, Indore
January 3, 2019
Abstract
The economic mobility of individuals and households is of fundamental interest. While many
measures of economic mobility exist, reliance on transition matrices remains pervasive due
to simplicity and ease of interpretation. However, estimation of transition matrices is com-
plicated by the well-acknowledged problem of measurement error in self-reported and even
administrative data. Existing methods of addressing measurement error are complex, rely
on numerous strong assumptions, and often require data from more than two periods. In
this paper, we investigate what can be learned about economic mobility as measured via
transition matrices while formally accounting for measurement error in a reasonably trans-
parent manner. To do so, we develop a nonparametric partial identi…cation approach to
bound transition probabilities under various assumptions on the measurement error and mo-
bility processes. This approach is applied to panel data from the United States to explore
short-run mobility before and after the Great Recession.
JEL: C18, D31, I32
Keywords: Partial Identi…cation, Measurement Error, Mobility, Transition Matrices, Poverty

The authors are grateful for helpful comments from the editor, Rajeev Dehejia, two anonymous referees,
Hao Dong, Elira Kuka, Essie Maasoumi, Xun Tang, and conference participants at Texas Econometrics
Camp XXI I and the LACEA-LAMES 2018 Annual Meeting. Corresponding author: Hao Li, Department of
Economics, Nanjing Audit University, Nanjing, Jiangsu, China. E-mail: lihao@nau.edu.cn.

1 Introduction
There has been substantial interest of late in intra- and inter-generational mobility. Dang
et al. (2014, p. 112) state that mobility “is currently at the forefront of policy debates
around the world.” Within the popular press, it has been noted that “social mobility ...
has become a major focus of political discussion, academic research and popular outrage in
the years since the global …nancial crisis.”
1
In this paper, we study economic mobility while
accounting for measurement error in income data. Speci…cally, we o¤er a new approach to
addressing measurement error in the estimation of transition matrices.
Measurement error in income data is known to be pervasive, even in administrative data.
In survey data, measurement error arises for two main reasons: misreporting (particularly
with retrosp ective data) and imputation of missing data (Jäntti and Jenkins 2015). It is
now taken as given that self-reported income in survey data contain signi…cant measurement
error, and that the measurement error is nonclassical in the sense that it is mean-reverting
and serially correlated (Bound et al. 2001; Kapteyn and Ypma 2007; Gottschalk and Huynh
2010). Compounding matters, Meyer et al. (2015) …nd that both problems –nonresponse
and accuracy conditional on answering – are worsening over time. In administrative data,
measurement error arises for three main reasons: misreporting (tax evasion or …ling errors),
conceptual di¤erences between the desired and available income measures, and processing
errors (Bound et al. 2001; Kapteyn and Ypma 2007; Pavlopoulos et al. 2012; Meyer et al.
2015). Even if administrative data are entirely accurate, they are only available in a handful
of developed countries.
However, existing studies of mobility either ignore the issue or utilize complex solutions
that invoke strong (and often non-transparent) identi…cation assumptions and have data
requirements that are quite limiting. The most frequent response to measurement error in
the empirical literature on mobility is to mention it as a caveat (Dragoset and Fields 2006).
While the usual assumption is that measurement error will bias measures of mobility upward,
the complexity of mobility measures along with the nonclassical nature of the measurement
1
See Washington Post (October 6, 2016) at https://www.washingtonpost.com/news/wonk/wp/
2016/10/06/striking-new-research-on-inequality-whatever-you-thought-its-worse/?utm_term=
.83d37c53195b.
1

error makes the direction of any bias uncertain. Glewwe (2012, p. 239) states that “all indices
of relative mobility tend to exaggerate mobility if income is measured with error,”yet others
o¤er a di¤erent opinion. Dragoset and Fields (2006, p. 1) contend that “very little is known
about the degree to which earnings mobility estimates are a¤ected by measurement error.”
Gottschalk and Huynh (2010, p. 302) note that “the impact of nonclassical measurement
error on mobility is less clear since mobility measures are based on the joint distribution of
reported earnings in two periods.”
Our approach to the analysis of mobility given measurement error in income data concen-
trates on the partial identi…cation of transition matrices. We provide informative bounds on
the transition probabilities under minimal assumptions concerning the measurement error
process and a variety of nonparametric assumptions on income dynamics. To our knowledge,
this is the …rst study to extend the literature on partial identi…cation to the study of transi-
tion matrices (see, e.g., Horowitz and Manski 1995; Manski and Pepper 2000).
2
Within this
environment, we …rst derive sharp b ounds on transition probabilities under minimal assump-
tions on the measurement error process. We then show how the bounds may be narrowed
by imposing more structure via shape restrictions, level set restrictions that relate transition
probabilities across observations with di¤erent attributes (Manski 1990; Lechner 1999), and
monotonicity restrictions that assume monotonic relationships between the true income and
certain observed covariates (Manski and Pepper 2000).
In contrast to existing approaches to address measurement error in studies of mobility
(discussed in Section 2), our approach has several distinct advantages. First, the assump-
tions invoked to obtain a given set of the bounds are transparent, easily understood by a
wide audience, and easy to imp ose or not impose depending on the particular context. More-
over, bounds on the elements of transition matrices extend naturally to bounds on mobility
measures derived from transition matrices. Second, our approach only requires data at two
points in time. Third, our approach is easy to implement (through our creation of a generic
2
In closely related work, Vikström et al. (2018) study the partial identi…cation of treatment e¤ects where
the outcomes are conditional transition probabilities. In their setup, measurement error is not considered.
Rather, point identi…cation fails even under randomized treatment assignment as treatment assignment is not
guaranteed to be independent of potential outcomes in future periods conditional on intermediate outcomes.
Our approach is also similar to Molinari (2008); she studies the partial identi…cation of the distribution of a
discrete variable that is observed with error.
2

Stata command).
3
Fourth, our approach extends easily to applications other than income,
such as dynamics related to consumption, wealth, occupational status, labor force status,
health, student achievement, etc.
The primary drawback to our approach is the lack of point identi…cation. Two responses
are in order. First, our approach should be viewed as a complement to, not a replacement
for, existing approaches. Indeed, one usefulness of our approach is to provide bounds with
which point estimates derived via alternative estimation techniques may be compared. Sec-
ond, many existing approaches to deal with measurement error in mobility studies end up
producing bounds even though the solutions are not couched as a partial identi…cation ap-
proach (e.g., Dang et al. 2014; Lee et al. 2017). This arises due to an inability to identify
all parameters in some structural model of observed and actual incomes.
Perhaps a secondary drawback of our approach is the focus on transition matrices to
capture mobility. Such matrices have the disadvantage of not providing a scalar measure of
mobility, simplifying spatial and temporal comparisons of mobility. While there is merit to
this critique, there are several responses. First, transition matrices are an obvious starting
point in the measurement of mobility. Jäntti and Jenkins (2015, p. 822) argue that, when
measuring mobility across two points in time, “the bivariate joint distribution of income con-
tains all the information there is about mobility, so a natural way to begin is by summarizing
the joint distribution in tabular or graphical form.” Second, transition matrices are easily
understood by policymakers and the general public and thus are frequently referenced within
these domains. Third, transition matrices allow one to examine mobility at di¤erent parts of
the income distribution (Lee et al. 2017). Finally, bounds on (scalar) measures of mobility
derived from the elements of transition matrices are easily obtained from our approach.
We illustrate our approach with an examination of intragenerational mobility in the
United States using data from the Survey of Income and Program Participation (SIPP).
Speci…cally, we examine mobility over two four-year periods, 2004 to 2008 and 2008 to 2012.
Understanding mobility patterns in the US is important as there is convincing evidence
that income inequality has been increasing in the US.
4
However, the welfare impact of this
3
Available at http://faculty.smu.edu/millimet/code.html.
4
The level of income inequality in the US has followed a U-shaped pattern over the past century (Picketty
and Saez 2003; Kopczuk et al. 2010; Atkinson and Bourguignon 2015).
3

rise depends crucially on the level of economic mobility. Shorrocks (1978, p. 1013) argues
that “evidence on inequality of incomes or wealth cannot be satisfactorily evaluated without
knowing, for example, how many of the less a- uent will move up the distribution later in
life.”More recently, Kopczuk et al. (2010, p. 91-2) conclude that “a comprehensive analysis
of disparity requires studying both inequality and mobility”as “annual earnings inequality
might substantially exaggerate the extent of true economic disparity among individuals.”
Our analysis of US mobility yields some striking results. First, we show that relatively
small amounts of measurement error leads to bounds that can be quite wide in the absence
of other information or restrictions. Second, the restrictions considered contain signi…cant
identifying power as the bounds can be severely narrowed. Third, allowing for misclassi…-
cation errors in up to 10% of the sample, we …nd that the probability of being in (out of)
poverty in 2008 conditional on being in poverty in 2004 is at least 35% (27%) under our
most restrictive set of assumptions. The probability of being in (out of) poverty in 2012
conditional on being in poverty in 2008 is at least 36% (25%) under our most restrictive set
of assumptions. Finally, the probability of being in poverty in 2008 conditional on not being
in poverty in 2004 is at least 2% and no more than 11% under our most restrictive set of
assumptions. The probability of being in poverty in 2012 conditional on not being in poverty
in 2008 is at least 4% and no more than 13% under our most restrictive set of assumptions.
The rest of the paper is organized as follows. Section 2 provides a brief review of existing
approaches to address measurement error in studies of mobility. Section 3 presents our partial
identi…cation approach. Section 4 contains the empirical application. Section 5 concludes.
2 Literature Review
Burkhauser and Couch (2009) and Jäntti and Jenkins (2015) provide excellent reviews of the
numerous mobility measures. Bound et al. (2001) and Meyer et al. (2015) o¤er excellent
surveys regarding measurement error in microeconomic data. Tamer (2010), Bontemps and
Magnac (2018), and Ho and Rosen (2017) provide in depth reviews of the recent literature
on partial identi…cation.
5
Here, we focus on approaches that have been taken to address (or
5
Within the partial identi…cation literature, our analysis is most closely related to Molinari (2008), who
posits a direct misclassi…cation approach in order to bound the distribution of a discrete variable in the
4

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Partial identification of economic mobility: with an application to the united states" ?

However, estimation of transition matrices is complicated by the well-acknowledged problem of measurement error in self-reported and even administrative data. In this paper, the authors investigate what can be learned about economic mobility as measured via transition matrices while formally accounting for measurement error in a reasonably transparent manner. 

Q2. What are the future works mentioned in the paper "Partial identification of economic mobility: with an application to the united states" ?

The authors are hopeful that future work will consider additional restrictions that may be used to further tighten the bounds on transition probabilities, as well as bounds on additional summary measures of mobility derived from the transition matrix. 

Q3. What is the scalar measure of mobility?

Several scalar measures of mobility considered in the literature are derived directly from the elements of the transition matrices. 

Q4. What are the main assumptions for identification of error-free income?

as in McGarry (1995), identification of error-free income relies on strong assumptions for identification, such as serially uncorrelated measurement error, particular functional forms, and valid instrumental variables. 

Q5. What is the disadvantage of transition matrices?

Such matrices have the disadvantage of not providing a scalar measure of mobility, simplifying spatial and temporal comparisons of mobility. 

Q6. What are the main disadvantages of the simulation-based methods?

While simulation-based methods allow for estimation of transition matrices, these methods are complex, lack transparency, rely on strong functional form and distributional assumptions, and often require more than two years of data. 

Q7. What are the effects of shape restrictions on the population transition probabilities?

Level set restrictions place equality constraints on population transition probabilities across observations with different observed attributes (Manski 1990; Lechner 1999). 

Q8. What are the disadvantages of the model?

While the authors’model has some advantages compared to earlier attempts to simulate error-free outcomes, these advantages come at a cost of increased complexity, decreased transparency of the identifying assumptions, and a need for four periods of data. 

Q9. What is the measure of mobility given by Prais (1955)?

The Prais (1955) measure of mobility captures the expected exit time from partition k and is given by11− p∗kk , k = 1, ..., K. (11)Bradbury (2016) defines measures of upward and downward mobility that account for the size of the partitions. 

Q10. How many assumptions are used to determine the probability of being in poverty in 2008?

allowing for misclassification errors in up to 10% of the sample, the authors find that the probability of being in (out of) poverty in 2008 conditional on being in poverty in 2004 is at least 35% (27%) under their most restrictive set of assumptions. 

Q11. What is the probability of income being misclassified in a period?

θ(0,0)(k,l) represents the probability of no misclassification in either period for an observation with true income in partitions k and l.99θ (0,0) (k,l) may be strictly positive even though income is misreported in either or both periods (i.e., yit 6= y∗it for at least some i and t) as long as the misreporting is not so severe as to invalidate the observed partitions (i.e., k′ = k and l′ = l regardless). 

Q12. What is the simplest way to explain the inequality constraints?

Monotonicity restrictions place inequality constraints on population transition probabilities across observations with different observed attributes (Manski and Pepper 2000; Chetverikov et al. 2018).