WPS8353
Policy Research Working Paper 8353
Identifying Catch-Up Trajectories
in Child Growth
New Methods with Evidence from Young Lives
Sam Jones
Jere R. Behrman
Hai-Anh H. Dang
Paul Anand
Development Economics
Development Data Group
February 2018
Policy Research Working Paper 8353
Abstract
Definitions of catch-up growth in anthropometric out- incorporating individual-specific intercepts and slopes, this
comes among young children vary across studies. This enables between- and within-group forms of catch-up to be
paper distinguishes between catch-up in the mean of a tested in a unified setting. The application of the proposed
group toward that of a healthy reference population versus approach reveals significant differences in the nature, extent,
catch-up within the group, associated with a narrowing and drivers of catch-up growth across the four Young Lives
of the outcome distribution. In contrast to conventional countries (Ethiopia, India, Peru, and Vietnam). In addition,
empirical approaches based on dynamic panel models, the paper shows how conclusions about catch-up are sensitive
the paper shows how catch-up can be tested via a latent to the way in which anthropometric outcomes are expressed.
growth framework. Combined with a flexible estimator
This paper is a product of the Development Data Group, Development Economics. It is part of a larger effort by the World
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Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted
at sam.jones@econ.ku.dk or hdang@worldbank.org.
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Produced by the Research Support Team
Identifying Catch-Up Trajectories in Child Growth:
New Methods with Evidence from Young Lives
Sam Jones* Jere R. Behrman Hai-Anh H. Dang Paul Anand
* Jones (sam.jones@econ.ku.dk, corresponding author) is an Associate Professor in the Department of Eco-
nomics, University of Copenhagen; Behrman is the William R. Kenan, Jr. Professor of Economics at the University of
Pennsylvania; Dang (hdang@worldbank.org; corresponding author) is an economist in the Survey Unit, Development
Data Group, World Bank, and a non-resident senior research fellow with Vietnam’s Academy of Social Sciences; Paul
Anand is a Professor at the Open University and Research Associate at HERC in Oxford University. We acknowledge
helpful comments from Thomas Sohnesen, Hanna Szymborska, Jed Friedman and Ha Nguyen. Jones was funded by
the Open University. Dang was funded by the Strategic Research Program (SRP) supported by UK Department of
International Development.
1 Introduction
It is widely recognized that early childhood represents a critical period for the development of
human capabilities Heckman (2006; 2007). Disadvantages experienced during this period can
cast a long shadow, often affecting outcomes over the life course (Black et al. 2017). Among a
range of risk factors, the effects of poor nutrition have received extensive attention, especially in
developing country contexts (de Onis et al. 2012). While both the nature and average impacts of
these risk factors are generally well known, differences in how children respond and cope with
early disadvantage are not fully understood (Engle et al. 1996). A key area of debate concerns
catch-up growth, which refers to recovery in anthropometric outcomes (height, weight) from initial
disadvantage. Identifying family and community characteristics, as well as governmental policies,
that systematically enhance resilience to shocks and promote developmental recovery remains an
important research agenda (Almond et al. 2017, Anand et al. 2016).
Our point of departure in this study is that extant deﬁnitions and associated statistical tests for
catch-up growth among children are not entirely convincing. This raises a fundamental challenge –
advances in identifying relevant factors that promote recovery are likely to be limited if the presence
(absence) of catch-up growth is not measured correctly. Indeed, until recently a prevailing view was
that catch-up growth among stunted children was only possible during the child’s ﬁrst 24 months.
In a landmark study, Adair (1999) tracked the heights of around 2,000 Filipino children from 2
to 12 years of age and found a signiﬁcant reduction in the prevalence of stunting over time – i.e.,
recovery occurred beyond 2 years of age. However, on reviewing the same evidence, Cameron et al.
(2005) conclude these results were driven by regression to the mean effects and, using alternative
methods, ﬁnd no evidence of catch-up growth in the same sample.
Methodological debates around catch-up growth remain alive (e.g., Hirvonen 2014). The aim of
this paper is to revisit the ways in which catch-up growth is deﬁned, as well as how it is measured
in practice. We contribute to the literature in four ways. First, we propose a uniﬁed set of tests
for catch-up growth. We carefully distinguish between two distinct forms of catch-up, namely: (i)
convergence of initially disadvantaged individuals within a sampled population toward the group
mean (within-group catch-up); and (ii) convergence of the group mean toward that of the median
child from a healthy external reference population (between-group catch-up).1 Existing studies in
the literature have typically focussed on either one of these forms. However, since within-group
catch-up has no necessary relation to between-group catch-up, an exclusive focus on any one form
fails to provide comprehensive insights regarding recovery dynamics.
Our second contribution it to embed the two tests of catch-up in a latent growth framework. This
1 The distinction between means and medians is important. Conventional methods to standardize anthropometric
outcomes against a reference population employ the median as the relevant measure of central tendency. At the same
time, least squares estimates of catch-up growth focus on means. So, if the mean for a standardized outcome (e.g.,
height-for-age z-score) in a sampled group approaches zero, then the average child in the sample corresponds to the
median of the reference distribution. We retain the relevant distinction between means and medians throughout.
1
framework has been used to study child development before, but it has not been widely used as a
basis to test for catch-up growth. We show how the framework allows both forms of catch-up to be
tested in a uniﬁed setting. In addition, we apply a novel estimator that relies on minimal statistical
assumptions regarding the data generating process. The merit of this procedure is demonstrated
with longitudinal data from the four Young Lives countries – Ethiopia, India, Peru and Vietnam. On
a technical level, we ﬁnd the proposed ﬁxed-effects estimator with individual slopes outperforms
conventional correlated random effects (mixed effects) models. On the substantive level, the
procedure reveals limited evidence for catch-up growth in either stature or body mass across the
four countries.
Our third contribution is to verify the extent to which estimates of catch-up are sensitive to the
ways in which anthropometric outcomes are transformed, often via standardization procedures.
This point has been acknowledged before (e.g., Leroy et al. 2015), but few studies include a
systematic comparison of the degree to which statistical tests of different forms of catch-up depend
on the speciﬁc metric or transformation chosen. We ﬁnd that externally-standardized z-scores
and associated binary transformations (e.g., stunted / not stunted) tend to produce more positive
assessments of the presence of catch-up growth compared to tests based on absolute differences.
As this may be driven by changes in the variance of the reference distribution, we propose that the
ratio between the outcome for a given child and the expected outcome for a child of the same age
and gender in a reference population is used as a complementary measure.
Our ﬁnal contribution is to extend the latent growth framework to show how the determinants
of variation in mean child size and growth velocity can be investigated via a second stage of
analysis. This provides a consistent and straightforward approach to identifying factors that account
for differences in growth trajectories across children. As applied to the Young Lives data we
ﬁnd signiﬁcant heterogeneity in the magnitude (and direction) of different drivers across the four
countries. We also ﬁnd that community characteristics explain a non-trivial share of the variance in
child size and growth velocity.
The remainder of the paper is structured as follows: Section 2 reviews conventional deﬁnitions
of catch-up and brieﬂy summarizes recent literature. Addressing the shortcomings of extant
empirical approaches, Section 3 sets out our proposed alternative procedure based on a latent
growth framework; and we discuss alternative empirical estimation strategies. Section 4 applies our
recommended approach to the Young Lives data and discusses the results. Section 5 concludes.
2
2 Identifying catch-up growth
2.1 Deﬁnitions
Existing literature on catch-up growth spans contributions across the medical, biological and social
sciences.2 Despite the huge volume of scholarship, studies do not employ a unique or consistent
deﬁnition of what actually constitutes catch-up growth. Two main uses of the concept can be
distinguished, each corresponding to different substantive research questions. The ﬁrst focuses
on whether initially disadvantaged children grow at a more rapid rate than other children in the
sample population – i.e., they converge towards the mean of the group (Wit and Boersma 2002).3
The second notion of catch-up concerns whether the average child in the sampled population is
converging toward the median outcome observed in a healthy external reference population. So, in
the ﬁrst notion, interest is on how the shape of the outcome distribution evolves for a given sample
over time. In the second notion the focus is on how the mean of this distribution evolves relative to
the median child in a healthy population. Put differently, the ﬁrst notion of catch-up focuses on
within-group processes of convergence, while the second attends to between-group processes.
To appreciate the difference between the two notions of catch-up growth it is helpful to review what
conventional empirical tests capture. Various studies of catch-up growth use some version of the
following model:
yit = α + β yi,t − j + xis γ + εit (1)
/ , t , ..., t − j, ..., 0}
where s ∈ {0
where y is an anthropometric outcome of interest; i ∈ I = {1, ..., I } indexes individual children;
t represents the child’s age at the time of measurement; x is a vector of observed explanatory
variables (e.g., household income); α is an intercept; ε is residual error; and subscript s is deﬁned
so as to encompass control variables observed at different points in time, as well as time-invariant
characteristics.
Existing studies based on this model often focus on estimates for β . When outcomes are standard-
ized by an external reference distribution, it is often argued that complete catch-up obtains when
β = 0, and no catch-up obtains when β = 1. Setting all the elements of γ to zero and assuming
only two periods (t , t − j) for simplicity, it is easy to see that least squares estimates of equation (1)
2 In recognition of this, throughout this paper we draw on studies from a wide range of ﬁelds. For now, we use the
term ‘growth’ in reference to various child development outcomes of interest (e.g., height, weight, BMI etc.). We are
speciﬁc later.
3 Related studies in this vein also seek to identify speciﬁc factors that inﬂuence variation in individual growth rates
within the given sample. See further below.
3
will yield the following slope coefﬁcient estimate:
ˆ = ∑i∈I (yi,t − y
β
¯t )(yi,t − j − y ¯t − j )
∑i∈I (yi,t − j − y ¯t − j )2
∑ (y ˜i,t − j ) 1/I ∑i∈I (y
˜i,t y ˜i,t − j )
˜i,t y
= i∈I =
∑i∈I (y ˜i,t − j )2 st2− j
st
= ρt ,t − j (2)
st − j
in which y˜i,t = (yi,t − y
¯t ) is the demeaned outcome at time t ; st is the sample standard deviation of
yt (calculated over all I sampled children observed at time t ); and ρt ,t − j is the jth order coefﬁcient
of autocorrelation.
In the case that the outcome (in each period) is standardized in relation to some reference distribution
with median µt and standard deviation σt , we note that:
y∗
i,t = (yi,t − µt )/σt
¯t − µt ) y
(yi,t − µt ) − (y ˜i,t
˜∗
=⇒ yi,t = = (3)
σt σt
In turn, the corresponding slope coefﬁcient becomes:
ˆ ∗ = ∑i∈I (y
β
˜i,t − j )/(σt σt − j )
˜i,t y
∑i∈I (y ˜i,t − j )2 /σt2
−j
σt − j ˆ st σt − j
= β= ρt ,t − j (4)
σt σt st − j
These expressions prompt three insights. First, both slope coefﬁcient estimates are directly propor-
tional to the autocorrelation coefﬁcient, which captures the speed at which yt converges towards
its mean. Thus, the consistent component of this notion of catch-up is the (average) degree of
persistence in the divergence of outcomes from the sample mean. Second, as equation (3) indicates,
the slope coefﬁcient estimates are unaffected by adding or subtracting a ﬁxed constant in any period,
such as µt – i.e., the β and β ∗ estimates are only related by the ratio of the scaling factors (standard
deviations) applied in the standardization procedure. Consequently, the slope estimates in this
formulation contain no direct information about the direction in which outcomes for the average
child in the sample are moving. It follows that neither estimate for β can be informative about
between-group catch-up, and this holds regardless of how the outcome is expressed, including
whether it is standardized by an external reference distribution such as that provided by the WHO.
Third, despite the previous points, estimates of the slope coefﬁcient will vary according to the
standardization imposed. In the case where outcomes are standardized internally (i.e., σyt ≡ st ),
then the slope coefﬁcient is the autocorrelation coefﬁcient. If raw outcomes are used, such as
height, then the slope coefﬁcient captures the autocorrelation coefﬁcient multiplied by the ratio
4
of the standard deviation of outcomes at time t to that in the previous period (see equation 2).
For externally standardized outcomes, the scaling factor will be greater than one if the relative
dispersion of the sample distribution to that of the reference population is growing over time.
Consequently, in anything other than the internally standardized case, slope coefﬁcients derived
from equation (1) will not exclusively capture the speed of convergence toward the sample mean.
In itself, this ambiguity motivates a search for alternative tests of this form of catch-up.
A second approach to identifying catch-up growth is often found in studies of disadvantaged
populations (samples). In contrast to focussing on within-group convergence, Hirvonen (2014)
recommends a shift of attention toward α in equation (1). Least squares estimates for this coefﬁcient
imply:
αˆ =y ¯t − βˆy¯t − j − x ˆ
¯is γ (5a)
ˆ ∗ = (y
α ¯t − µt ) − βˆ (y
¯t − j − µt − j ) − x ˆ
¯is γ (5b)
and where the latter estimate refers to externally-standardized outcomes. In this case, the same
author suggests the simple difference captures between-group catch-up, deﬁned as the extent of
convergence of the sample mean toward that of the reference population. In the unconditional
case, this is obtained directly from equation (5b) when γ = 0 ∧ β = 1, which is equivalent to
examining mean differences in the standardized outcome, ∆y ¯∗ . Similarly, Cameron et al. (2005)
argue that αˆ provides a meaningful measure of catch-up in stature from initial stunting. However,
they recommend β remains unconstrained in order to address potential regression to the mean
effects (see further below; also Adair 1999).
2.2 Existing literature
The above discussion distinguished between two forms of catch-up growth. Table 1 summarizes a
selection of ten recent papers in both these traditions (published in the past 5 years), all of which
refer to developing country contexts.4 From this, four general comments merit note. First, existing
studies have predominantly considered catch-up growth in relation to child height. One rationale is
that low height-for-age is associated with stunting, which is known to adversely affect outcomes
in adulthood (Almond et al. 2017, Hoddinott et al. 2013). However, low BMI-for-age values are
associated with child wasting, which is a symptom of acute under-nutrition. Shrimpton et al. (2001)
argue that adverse in utero conditions tend to affect child birth weight more than length, but in
contrast to height disadvantages, weight disadvantages can be addressed relatively more quickly in
4 The studies were identiﬁed from Google Scholar based on the combined search text: ‘ "catch-up growth" & children
& "developing countries" ’. The search period was limited to 2012-2017; only primary empirical studies undertaken
in developing countries were retained; and those concerned with catch-up growth either after speciﬁc interventions or
associated with speciﬁc medical conditions were excluded. Papers were also excluded that have a primary focus on
the determinants of variation in growth rates (e.g., Georgiadis et al. 2016; 2017). The ﬁnal list (in the table) includes
the most relevant peer-reviewed studies; but we recognize this is somewhat subjective and is likely to be selective.
5
Table 1: Summary of recent studies of catch-up growth
Study Context Outcomes Metric(s) Findings
M2012 Indonesia Height β Incomplete within catch-up
O2013 Ethiopia HAZ β Incomplete within catch-up
S2013 Ethiopia, India, Peru, HAD β Some within-catch-up,
Vietnam especially before age 5
F2014 Ethiopia, India, Peru, HAZ ¯, β
∆y Some within-catch-up, no
Vietnam between catch-up
L2014 Brazil, Guatemala, HAZ, HAD ¯
∆y Some between catch-up (HAZ),
India, Philippines, RSA persistent deﬁcits (HAD)
L2015 Ethiopia, India, Peru, HAZ, HAD ¯
∆y No between catch-up (HAZ &
Vietnam HAD)
T2015 Malawi HAZ, HAD ¯
∆y Some between catch-up (HAZ),
persistent deﬁcits (HAD)
H2016 China, South Africa, HAZ β Incomplete within catch-up
Nicaragua
Z2016 Bolivia (Amazon) Height, HAZ β Incomplete within catch-up,
persistent deﬁcits
S2017 Timor-Leste HAZ, BAZ ¯
∆y Some between catch-up in
height (HAZ), not BMI (BAZ)
Notes: Studies are classiﬁed according to the main parameter on which analysis focuses, which we
(re)interpret according to the distinction between within- and between-group convergence; ‘Metrics’
¯ is the unconditional change in the mean sample
are the parameters and refer to equation (1), where ∆y
outcome; HAZ and HAD are height-for-age z-scores and absolute deﬁcits respectively, measured relative
to a well-nourished external population; BAZ is the BMI-for-age z-score, also measured relative to a
well-nourished external population; RSA denotes South Africa.
Study abbreviations: M2012 = Mani (2012), O2013 = Outes and Porter (2013), S2013 = Schott et al.
(2013), F2014 = Fink and Rockers (2014), L2014 = Lundeen et al. (2014b), L2015 = Leroy et al. (2015),
T2015 = Teivaanmäki et al. (2015), H2016 = Handa and Peterman (2016), Z2016 = Zhang et al. (2016),
S2017 = Spencer et al. (2017).
the ﬁrst few years of life, where conditions are appropriate. As such, consideration of trends in both
height and BMI may provide valuable complementary insights. This is conﬁrmed by Spencer et al.
(2017), who ﬁnd a divergence in trends for HAZ and BAZ scores (BMI-for-age z-scores) in two
rural communities in Timor-Leste, calculated with reference to WHO (World Health Organization)
charts.5
Second, the vast majority of studies focus on only one or the other form of catch-up growth. While
this is reasonable in speciﬁc cases, it means evidence on catch-up growth (broadly deﬁned) is
often partial, and little is known about the relationship between the two forms of catch-up. This
is important because the forms of catch-up are distinct and bear no necessary (logical) relation to
5 The main research goal of Spencer et al. (2017) is to calculate community-speciﬁc growth curves. However, their
comparison of these curves against WHO standards provides an indication of between-group catch-up.
6
one another. Admittedly, a number of the studies include an analysis of recovery from stunting,
which is identiﬁed via a binary transformation of HAZ scores.6 For instance, Teivaanmäki et al.
(2015) compare the evolution of HAZ scores for children classed as not stunted, moderately stunted
and severely stunted at baseline. They ﬁnd that HAZ scores improved for children in each class,
which is indicative of between-group convergence. But they also ﬁnd that HAZ scores improved at
a somewhat faster pace for the severely stunted, which would be consistent with the presence of
within-group catch-up (see also Fink and Rockers 2014, Zhang et al. 2016). While these kinds of
stratiﬁed analyses do touch on the different forms of catch-up, their ﬁndings are rarely systematized
so as to shed light on the correspondence between alternative forms of convergence.
Third, our review of existing studies reveals there is no uniﬁed framework for testing both forms
of catch-up – i.e., the two forms are analyzed separately, often via distinct models. This raises
doubts as to whether the ﬁndings are mutually consistent. For instance, estimates of between-group
catch-up tend to focus on average changes in outcomes. In the case of conditional growth models,
these use estimates for α in equation (1) derived under the constraint that β = 1, which rules
out estimation of the autocorrelation coefﬁcient.7 More generally, the use of multiple empirical
operationalizations of catch-up growth makes it difﬁcult to consistently interpret and compare
results across different studies.
Fourth, the papers referenced in Table 1 predominantly rely on estimates for catch-up based on
some form of equation (1) (as do many earlier studies; e.g., Adair 1999, Hoddinott and Kinsey
2001). However, these estimates are prone to bias. A primary concern stems from measurement
error in the outcome variable. Assuming the error in each measurement period is white noise,
such that the observed outcome includes the true component plus error: yit = y∗ it + νit , it can be
shown that least squares estimates of β will be biased towards zero to an extent proportional to the
magnitude of measurement error in period t − j. This is a main form in which regression to the
mean effects arise (Lohman and Korb 2006, Tu and Gilthorpe 2007; 2011) and is pertinent in the
current context since anthropometric measurement error may be commonplace and relatively more
acute among younger children (e.g., where exact age is not known; see Ulijaszek and Kerr 1999,
Zhang et al. 2016).8 Such effects are also a major concern where units are stratiﬁed by baseline
outcomes (Cameron et al. 2005, Jerrim and Vignoles 2013) – e.g., the true rate of recovery from
initial stunting may be inﬂated by regression to the mean effects. Furthermore, by the mechanics of
least squares, any bias in β will directly effect estimates for α .
An additional source of bias is due to the omission of variables that are correlated with the lagged
outcome or the included set of controls (xit ). The latter play two important roles. First they can
be used to control for idiosyncratic deviations in the composition of the sample, either over time
6 Typically, a child with a height-for-age z-score below -2 is classiﬁed as stunted.
7 This is distinct from studies that estimate equation (1) where the lagged outcome is subtracted from both sides, the
reason being that in these cases it is the estimates for β that remain in focus.
8 A related challenge is noise induced by differences in the timing of growth spurts. These may be captured by richer
models – see further below.
7
or relative to the complete sample frame, in turn reducing any bias in the parameter estimates
deriving from compositional sources (e.g., an excess number of wealthier children in the actual
sample). Second, they can be used to study heterogeneity in growth patterns – e.g., via inclusion of
interaction terms or through auxiliary residual analysis (see Section 4.4). If relevant elements of xit
have been omitted then conditional estimates of catch-up may be biased, or our understanding of
growth heterogeneity will be limited.
In the context of longitudinal data, a promising response is to include individual ﬁxed effects
(Handa and Peterman 2016). Typically these account for a large amount of the variation in the
data and are able to control for all time-invariant factors (at the individual level) that inﬂuence
growth outcomes. However, the dynamic nature of the speciﬁcation makes inclusion of such terms
highly problematic. Panel estimates of equation (1) correspond to a dynamic or autoregressive
panel situation in which the hypothesis of no catch-up (convergence) implies the outcome follows
a random walk. A large econometrics literature deals with the technical problems of rigorously
identifying the parameters of dynamic panel models in the presence of (unobserved) heterogeneity.
Bond et al. (2005) review the properties of equation (1) under the simplifying assumption that the
error term follows: εit = (1 − β )αi + νit , where αi is an individual-speciﬁc constant and νit is white
noise. They show that standard OLS estimates of (1) are consistent under the null hypothesis of a
unit root (β = 1); but estimates for β will be upward biased under the alternative (see also Baltagi
2008, Bond 2002). They also show that popular solutions to this problem, such as those based on
instrumental variables or GMM techniques, are not a panacea. While alternative estimators can be
consistent, they often have low power in ﬁnite samples. Furthermore, and as is well known, ﬁnding
instruments that are valid and perform well across different contexts is a formidable task.
The challenges associated with addressing unobserved individual heterogeneity in the dynamic
panel context have not been widely appreciated in empirical work on catch-up growth. Rather,
individual (unobserved) heterogeneity is often either ignored, or is swept out by ﬁrst differencing
(Handa and Peterman 2016, Mani 2012). This contrasts with research in other ﬁelds where these
issues are often placed at center stage – e.g., as in empirical tests of the relationship between initial
ﬁrm size and subsequent growth (e.g., Ribeiro 2007), as well as in the literature on aggregate
growth (e.g., Durlauf et al. 2005). Furthermore, individual heterogeneity in child growth patterns
is not only highly plausible, such as due to signiﬁcant differences in initial conditions (e.g., birth
weight and length), but also the ways in which time-invariant child characteristics affect subsequent
growth are often of stand-alone interest. Thus, tests for catch-up that are able to take into account
individual heterogeneity – not only as an additive control but also as a determinant of variation in
growth patterns – are called for.
8
3 Latent growth methods
The previous section argued that existing studies of catch-up growth are limited in scope and often
unconvincing. Critically, a number of the main sources of potential bias affecting estimates of
equation (1) are difﬁcult to address within a dynamic panel framework. Latent growth models
provide an alternative means to analyze processes of child development. Models of this sort are
widely applied to understand child development, often to quantify and explain (socio-economic)
gradients in growth outcomes.9 In this section we show how latent growth models also provide
a unifying framework for identifying the nature and extent of catch-up growth, addressing the
shortcomings associated with the conventional framework (equation 1). In addition, we recommend
a ﬂexible empirical estimation strategy that relies on weaker assumptions than typically imposed
by conventional random or mixed effects estimators.
A general latent growth framework, which applies naturally to child growth, is as follows:
yi (ti ) = (α0 + αi ) + (β0 + βi ) f (ti ) + xit γ + εit (6a)
2 2
Var(αi ) = σα , Var(βi ) = σβ , E(αi βi | xit ) = ραβ (σα σβ ) (6b)
E(xit αi | βi ) = ρxα (σα σx ), E(xit βi | αi ) = ρxβ (σx σβ ) (6c)
Notation here is not exactly the same as before. Consistent with our interest in both between-
and within-group catch-up, y must be referenced to an external population. Also, the nature and
interpretation of α and β are altered. Both are comprised of two parts: ﬁrst, a sample average
component, taking a zero subscript; and, second, an unobserved individual-speciﬁc component,
taking the i subscript, which are mean zero by construction.10 The α terms capture the level of
the outcome variable when f (ti ) = 0, where t represents the child’s age and f (·) is an unspeciﬁed
functional form. The β terms represent the velocity of growth – i.e., the expected change in the
outcome for each unit increment in age. Thus, in the terminology of Tanner (1962), the two
unobserved components correspond to estimates of deviations from mean child size and growth
velocity respectively. Lastly, to facilitate interpretation of the α terms, the set of time-varying
controls is demeaned.
A latent growth framework is consistent with theoretical models of early child growth (see De Cao
2015). It also holds a number of advantages in comparison to dynamic panel speciﬁcations. First,
the framework permits simultaneous examination of a set of distinct null hypotheses about different
forms of catch-up growth. Estimates for the sample average slope β0 capture the mean growth
velocity in the sample. Thus, where the outcome is expressed relative to a healthy reference
population (e.g., as per externally-referenced HAZ scores), positive estimates indicate a faster pace
9 A recent example in this vein is McCrory et al. (2017). Also see Georgiadis et al. (2016) for similar analysis, using a
path model.
10 Estimation of individual heterogeneity in the growth trajectories is the ‘latent’ term in the description of the approach.
9
Table 2: Alternative combinations of hypotheses regarding catch-up growth
Between convergence?
Yes No
Yes (α0 < 0 ∧ β0 > 0) ∧ ραβ < 0 (α0 ≥ 0 ∨ β0 ≤ 0) ∧ ραβ < 0
Within
convergence? No (α0 < 0 ∧ β0 > 0) ∧ ραβ ≥ 0 (α0 ≥ 0 ∨ β0 ≤ 0) ∧ ραβ ≥ 0
Note: ‘∧’ is the logical AND operator; ‘∨’ is the logical OR operator; these conditions assume
outcomes are expressed relative to a healthy external reference population and a positive change
in the outcome always corresponds to an improved situation.
of growth relative to the reference group. Consequently, if average (or initial) size is below that
of the reference group, then a positive mean slope would indicate population-wide catch-up has
occurred. Within-group catch-up (convergence) is captured by the relationship between child size
and velocity (ραβ ). If the correlation is negative, then below-average children are growing faster
than their counterparts and the predicted distribution of outcomes is converging toward a common
mean over time. If the correlation is positive, above-average children are extending their advantage
and the predicted outcome distribution is widening over time. Thus, within-population or relative
catch-up requires: ραβ < 0, and where the latter correlation is conditional on the included controls
(xit ).
Second, unlike the conventional framework, there is no lagged outcome in the set of explanatory
variables. As a result, measurement error in the outcome variable will only reduce the overall
explanatory power of the model, but it should not bias estimates of the catch-up parameters. Third,
the absence of the lagged outcome variable means that the unobserved individual intercepts and
slopes can be retrieved directly (see below), side-stepping the various problems encountered in
dynamic panels. Fourth, the framework easily extends to multiple observation periods and, where
data permit, allows for non-linearities in growth patterns to be estimated and tested via choices for
f (ti ) (e.g., Chirwa et al. 2014).11
Based on this framework, Table 2 sets out speciﬁc hypotheses regarding the combinations of
convergence that may be identiﬁed. Formally, absolute convergence of the sample mean toward
the median of a healthy external reference population requires: α0 < 0 ∧ β0 > 0. The appropriate
null hypotheses to be taken to the data, corresponding to an absence of either form of catch-up,
are located in the bottom right cell of the table ‘(No, No)’. In the present context, the analytically
interesting alternative to the null hypotheses is that catch-up growth is occurring. As a result,
we state the null hypotheses using inequality signs, implying that one-sided statistical tests are
appropriate. Put differently, conventional tests of a general null hypothesis that, say, α0 = 0 are not
particularly informative about catch-up growth.
The outstanding issue is how these hypotheses might be tested. This requires selection of an
11 In our application (Section 4) we have four observations per individual so non-linear functional forms cannot be
¯.
considered. Rather, we use the simple linear function: f (ti ) = ti − t
10
econometric approach, yielding parameter estimates and standard errors, as well as a method to
combine tests of individual parameters into composite hypotheses about catch-up. With respect to
the ﬁrst question, existing approaches to estimation of growth trajectories correspond to special
cases of equation (6a), distinguished by speciﬁc restrictions imposed on the moments given by
equations (6b) to (6c). The most simple cross-section approach restricts all moments to zero, which
amounts to setting ∀i : αi = 0 ∧ βi = 0, and treats all forms of unobserved individual heterogeneity
as a nuisance. While this can be estimated easily via pooled OLS (POLS), many scholars consider
these restrictions excessively strong. Moreover, for present purposes, a POLS estimator does not
permit direct tests of the full set of catch-up hypotheses (Table 2).
An established alternative to pooled OLS is a mixed effects estimator, in which the unobserved
heterogeneity terms (αi , βi ) are treated as random effects, presumed to follow a multivariate normal
distribution. Speciﬁc instances of these models also permit the unknown correlation ραβ to be
estimated. Such Correlated Random Effects (CRE) estimators are a popular choice and allow the
full range of catch-up hypotheses to be tested (see Johnson 2015). Despite these attractions, in
most applications of mixed effects models the ﬁnal two moments of the general system (equation
6c) remain restricted to zero – i.e., all elements of x are treated as orthogonal to the unobserved
heterogeneity terms. In part, this assumption derives from technical challenges of convergence –
the more parameters there are to be estimated, model convergence takes signiﬁcantly longer and is
more prone to failure (Chirwa et al. 2014, Gurka et al. 2011). The trade-off is that misspeciﬁcation
of the structure of correlation between so-called ﬁxed and random effects is not innocuous and will
deliver inconsistent estimates where independence restrictions are violated (see Hausman 1978,
Jacqmin-Gadda et al. 2007).
In light of these considerations, a further option is to treat the unobserved heterogeneity terms
as ‘ﬁxed’ latent variables. While ﬁxed-effects estimators for additive terms are commonplace
and generally preferred within the economics literature, estimators allowing for ﬁxed individual-
speciﬁc slopes are less known. The latter models, often referred to as FEIS estimators (ﬁxed
effects with individual slopes), were originally introduced by Polachek and Kim (1994) but were
not widely disseminated mainly due to implementation challenges. In our case, the advantage is
that FEIS estimators impose no restrictions on any of the moments, nor do they impose speciﬁc
distributional assumptions on the latent variables. Furthermore, recent advances in computing
power and estimation algorithms mean these models can now be implemented with relative ease,
even in the context of large panels (Guimarães and Portugal 2010). We therefore review the
performance of this class of estimators, which to our knowledge have not been previously applied
to analyze latent growth.
Appendix Table A1 summarizes some of the primary advantages and disadvantages associated with
alternative econometric approaches to estimating the latent growth model. It notes that although
the FEIS model is a ﬂexible alternative to CRE models, it does introduce additional costs. First,
we expect a reduction in efﬁciency (loss of degrees of freedom) since a large number of additional
11
parameters must be estimated. As such, the gain in ﬁt of the FEIS model, relative to alternatives,
merits examination. Second, under the FEIS estimator, all time-invariant elements of x are absorbed
by the latent variables. This naturally focuses attention on the predicted effects of age-speciﬁc
changes in the elements of x (within-unit variation), holding each individual’s mean size (intercept)
and growth rate (slope) ﬁxed. As Bell and Jones (2015) explain, there is no necessary reason
why the effect of within-unit variation in x on the outcome will always be the same as the effect
of between-unit variation, one reason being that the latter reﬂects conditions at a higher level of
aggregation. In longitudinal individual data, the distinction of within- versus between-unit variation
effectively captures the difference between transitory (short-run) and permanent (long-run) effects.
So, applying an FEIS estimator to equation (6a) would effectively address bias from unobserved
static heterogeneity. But the downside is these estimates do not yield direct insights into the effects
of long-run elements of x on the outcome, including time-invariant characteristics. This drawback
is not crucial for testing the basic hypotheses about catch-up. Even so, in Section 4.4 we show how
further insight can be gained by examining the correlates of the estimated ﬁxed effects in a second
stage analysis.
The remaining issue is that hypotheses about catch-up (Table 2) are composite in nature. In order to
evaluate claims about catch-up, we must combine results from various individual hypothesis tests,
while also minimizing type I and II errors. Following Wilkinson (1951), the maximum probability
(maxP) associated with each individual null hypothesis represents a straightforward omnibus test
of whether they can be rejected in all cases simultaneously (i.e., it is a test of their disjunction).
We use this approach here.12 To address type II errors from multiple testing, we further adjust
the conﬁdence level using the conservative Bonferonni procedure, which is robust to arbitrary
dependence between the individual tests (Clarke and Hall 2009). So, to test for between-group
convergence, we take the highest probability of the separate one-sided nulls (α0 ≥ 0; β0 ≤ 0) and
adjust the chosen conﬁdence level by a factor of two. Similarly, to test for both within- and
between-convergence, we take the highest probability of the three one-sided nulls and adjust the
chosen conﬁdence level by a factor of three.
4 Application to Young Lives
The rest of this paper applies the proposed approach to the rich longitudinal data on child develop-
ment collected under the Young Lives (YL) initiative, which covers four low-income countries.13
Section 4.1 brieﬂy introduces the data; Section 4.2 reports the baseline results for tests of the
12 Various alternative omnibus tests have been proposed. However, many of these are designed for cases where the
number of hypotheses to be tested (jointly) is very large (e.g., Roback and Askins 2005). Also, various other omnibus
tests, such as Fisher’s combined probability test, essentially provide a kind of average probability and therefore can
lead to a rejection of the null even if some individual probabilities (hypotheses) are insigniﬁcant.
13 The YL data have been used extensively before, therefore detailed introduction is not necessary. For further details
and examples see (inter alia) Barnett et al. (2012), Crookston et al. (2013), Lundeen et al. (2014a).
12
catch-up hypotheses (Table 2), focussing on conventional HAZ scores; Section 4.3 considers the
sensitivity of these results across alternative metrics of height as well as BMI; and Section 4.4
explores speciﬁc time-invariant child characteristics associated with child size and growth velocity.
4.1 Data
We use the four latest public-use rounds of the YL data and focus exclusively on the youngest
cohort, who were approximately one year old in the ﬁrst round and around 12 years old in the
fourth round.14 Table 3 summarizes the analytical dataset we use hereafter. We have removed
children who have missing data for their age (1,116 observations removed), or who were under
6 months old in the ﬁrst round (165 observations removed). Also, due to the longitudinal nature
of the intended analysis, we excluded children observed in fewer than three of the four rounds
(452 observations removed). This yields a ﬁnal dataset containing 30,515 observations on 7,681
children.15 As Part (a) of the table shows, each round contains data on almost 2,000 children in all
four countries. The table also indicates that the sample is well balanced by gender, and there are
only small differences in the ages of sampled children (given as the age in months divided by 12)
across countries in each collection round.
The descriptive statistics in Table 3(a) report averages across rounds for a range of growth outcomes.
Child height (in cm) and weight are measured directly, from which the BMI (body mass index) is
derived.16 Notably, average heights are broadly comparable across the countries, but mean BMIs
differ – e.g., contrast Peru (17.6) and India (14.7). All remaining outcomes are expressed in relation
to expectations for a child of the same age and gender in a healthy external reference population.
For this, we use the latest WHO child growth standards and follow Vidmar et al. (2004; 2013) to
estimate relevant healthy population medians and standard deviations.
Denoting the observed outcome (i.e., height, BMI) for child i at age t as yit , and the corresponding
median expected outcome estimated from the reference distribution as θ ¯t , we construct the following
¯t )/ Var(θt ), which yield conventional z-scores
indicators: (i) standardized differences, (yit − θ
14 To enhance replication, we use the merged or constructed YL datasets, available at: https://www.younglives.
org.uk/content/young-lives-rounds-1-4-constructed-files. The older cohorts are excluded since aca-
demic analysis of growth recovery (faltering) has typically focussed on the growth trajectories of younger children
(e.g., Leroy et al. 2014, Victora et al. 2010). Furthermore, children from the older cohorts attain puberty during the
data collection period, which is associated with a rapid spurt in growth. As such, the simplifying assumption of a
linear (individual) growth trajectory, which is necessary given we have a maximum of four observations per child,
is problematic for older cohorts. For the younger cohort, a small share (less than one-third) of girls have reached
menarche by the fourth round (for further discussion see Schott et al. 2017).
15 Given the structure of the model, children observed in just three rounds can be retained in straightforward fashion.
16 We follow Crookston et al. (2013), among others, and adjust the raw heights and BMIs collected in the ﬁrst
round to account for differences in ages at the time of observation. As Victora et al. (2010) note, HAZ patterns in
disadvantaged communities tend to decline sharply from birth until around 20 months, after which a more gradual
pattern of change is found. This correction therefore addresses any non-linearities in the relation between outcomes
and child age in the ﬁrst round, and strengthens the general assumption of a linear trend in outcomes as applied in
our regression estimates. However, this does not meaningfully alter our results.
13
Table 3: Descriptive statistics, by country
Ethiopia India Peru Vietnam
Mean (st.err.) Mean (st.err.) Mean (st.err.) Mean (st.err.)
(a) Averages (Rounds 1 to 4):
No. children 1,882 (0.08) 1,920 (0.11) 1,913 (0.30) 1,914 (0.23)
Child’s age (years) 6.6 (0.05) 6.6 (0.05) 6.5 (0.05) 6.6 (0.05)
Female (%) 47.2 (0.58) 46.4 (0.57) 49.7 (0.57) 48.7 (0.57)
Household size 6.0 (0.02) 5.3 (0.02) 5.5 (0.02) 4.7 (0.02)
Height 109.6 (0.30) 108.8 (0.29) 109.4 (0.30) 110.5 (0.31)
Height z-score (HAZ) -1.4 (0.01) -1.4 (0.01) -1.3 (0.01) -1.1 (0.01)
Height ratio (HAR, %) -5.8 (0.06) -6.1 (0.05) -5.3 (0.05) -4.7 (0.05)
Height difference (HAD) -6.9 (0.07) -7.3 (0.07) -6.1 (0.07) -5.6 (0.07)
Not stunted (%) 71.0 (0.52) 69.9 (0.52) 74.7 (0.50) 79.2 (0.46)
Not severely stunted (%) 90.8 (0.33) 93.3 (0.29) 93.6 (0.28) 96.0 (0.22)
BMI 14.9 (0.02) 14.7 (0.03) 17.6 (0.03) 15.8 (0.03)
BMI z-score (BAZ) -1.1 (0.01) -1.2 (0.01) 0.6 (0.01) -0.5 (0.01)
BMI ratio (BAR, %) -8.9 (0.12) -9.7 (0.17) 8.1 (0.16) -3.4 (0.14)
BMI difference (BAD) -1.5 (0.02) -1.6 (0.03) 1.3 (0.03) -0.6 (0.02)
BMI not low (%) 69.3 (0.53) 64.9 (0.54) 98.3 (0.15) 85.6 (0.40)
Healthy BMI (%) 62.1 (0.56) 60.9 (0.56) 66.2 (0.54) 75.8 (0.49)
(b) Long differences (Round 4 - Round 1):
No. children -19 (0.00) -23 (0.00) -70 (0.00) -46 (0.00)
Child’s age (years) 11.1 (0.01) 11.0 (0.01) 10.9 (0.01) 11.2 (0.01)
Female (%) 0.0 (0.00) 0.0 (0.00) 0.0 (0.00) 0.0 (0.00)
Household size 0.1 (0.06) -0.6 (0.06) -0.5 (0.06) -0.4 (0.04)
Height 69.5 (0.16) 68.3 (0.16) 71.2 (0.15) 71.9 (0.17)
Height z-score (HAZ) 0.0 (0.04) -0.2 (0.03) 0.4 (0.02) 0.0 (0.02)
Height ratio (HAR, %) -1.9 (0.13) -2.4 (0.11) -0.1 (0.10) -1.3 (0.10)
Height difference (HAD) -6.4 (0.15) -6.7 (0.15) -3.6 (0.15) -4.5 (0.16)
Not stunted (%) 7.8 (1.29) -0.5 (1.23) 10.9 (1.06) 0.0 (1.00)
Not severely stunted (%) 11.6 (0.97) 3.7 (0.75) 6.0 (0.70) -0.5 (0.58)
BMI -1.0 (0.05) 0.6 (0.11) 1.9 (0.07) 1.1 (0.06)
BMI z-score (BAZ) -1.2 (0.04) -0.3 (0.03) -0.2 (0.03) -0.2 (0.03)
BMI ratio (BAR, %) -12.1 (0.30) -2.7 (0.61) 3.7 (0.43) -1.1 (0.35)
BMI difference (BAD) -2.2 (0.05) -0.6 (0.11) 0.7 (0.07) -0.2 (0.06)
BMI not low (%) -36.9 (1.40) -13.9 (1.40) 0.8 (0.44) -13.4 (1.04)
Healthy BMI (%) -19.2 (1.58) -15.8 (1.48) 8.5 (1.48) -18.2 (1.31)
Note: aside from raw height and BMI (body mass index), all anthropometric outcomes are calculated with reference to a healthy
external population; stunting is given by a HAZ score below -2; severe stunting is given by a HAZ score below -3; a low BMI
is below the 5th percentile of the reference distribution; a healthy BMI lies between the 5th and 85th percentiles; see text for
further description of height and BMI transformations.
Source: own estimates.
14
(e.g., the height-for-age z-score, HAZ; BMI-for-age is denoted BAZ); (ii) raw difference scores,
yit − θ¯t , yielding a metric of absolute divergence in outcomes; and (iii) the ratio of outcomes,
¯t − 1. Although not employed widely, the latter outcome ratios address a concern that absolute
yit /θ
differences can widen over time, even if growth rates are held constant. Formally, if initial values on
an outcome for two individuals strictly differ yi0 > y j0 but grow at the same rate (g > 0) thereafter,
the absolute difference, ∆t = (yi0 − y j0 )(1 + g)t , will increase with t . In contrast, the ratio yit /y jt
will be constant, reﬂecting only the initial relative difference. Consequently, the behavior of the
ratio over time is expected to be more stable and meaningful.
A reason for expressing the same underlying outcomes in different ways is to investigate the
sensitivity of measures of catch-up to alternative formulations (for elaboration see Section 4.3).
Part (b) of Table 3 reports sample averages of long differences between the ﬁnal and ﬁrst rounds
(calculated in pairwise fashion over each child). As expected, all countries display signiﬁcant gains
in mean raw height, ranging from 68 to 72 cms over around 11 years. However, the remaining
height metrics paint a diverse picture. Both the difference and ratio metrics (HAD and HAR) show
a deterioration in outcomes over time relative to the reference population in all four countries. In
contrast, the standardized height metric (HAZ) points to small improvements in all countries other
than India. The metrics of stunting, here valued positively, are based on standard HAZ thresholds.
They show either improvement or no signiﬁcant change across all countries over the four rounds.
Turning to the BMI scores, we note that in contrast to the other three countries, Peru’s mean
standardized score (BAZ) is signiﬁcantly greater than that of the reference population, implying
children in the sample are heavier than expected (given their height and age). The long differences
for the BMI scores also reveal the Peruvian sample to be an exception. While all other countries
tend to show some deterioration in their BMI scores over time relative to the healthy external
population (regardless of the indicator chosen), the same scores in Peru tend to increase. The only
exception is the BAZ score for Peru, which records a small but not statistically signiﬁcant decline.
For reference, Appendix Figures A1 to A6 illustrate trends in the various height and BMI metrics
for the four countries over the four rounds. Each ﬁgure plots the predicted linear trend (by age) for
the 5th percentile, the median and the 95th percentile of the score distributions. The ﬁgures conﬁrm
the overall pattern in aggregate outcomes described in Table 3(b). However, they additionally reveal
important differences in how the shape of the distributions appear to be evolving. For example, the
HAZ plots (Figure A1) for Ethiopia and India indicate the distribution is narrowing over time (with
age), which would be consistent with a process of within-group convergence in z-scores. However,
the corresponding HAD plots (Figure A3) reveal clear distributional divergence; and the HAR plots
(Figure A2) indicate more moderate change – slight convergence in the case of Ethiopia and slight
divergence in India. Charts for the BMI measures also indicate signiﬁcant heterogeneity across the
countries. We explore these differences further in Section 4.3.
15
4.2 Baseline results
Our baseline results concerning catch-up growth in the four YL countries are summarized in Table
4, which takes height-for-age z-scores (HAZ) as the primary outcome of interest.17 Results for
three different estimators of equation (6a) are reported. In each case, the same speciﬁcation is
employed and we include a set of control variables spanning a range of both ﬁxed and time-varying
characteristics. These are inter alia: the child’s gender; whether children come from the majority
ethno-linguist group (ELG) of the region of residence; whether they have older siblings; if the
location is urban; the age and highest level of education of their mother (at time of birth); a
household wealth index; household size; and household exposure to shocks.18 For comparison,
Appendix Table A2 reports the same results, excluding the additional control variables. These are
highly similar and thus do not merit separate discussion.
Before reviewing the results in detail, the scaling of the age variable merits comment. It is well
known that estimates of both the intercept and the slope-intercept covariance in latent growth
models are sensitive to the point at which one sets t = 0 (Biesanz et al. 2004, Stoel and Van
Den Wittenboer 2003). While different choices can help address different analytical questions, Tu
and Gilthorpe (2011) demonstrate that age/time must be demeaned (i.e., one must use ti∗ = ti − t ¯) to
avoid distortion of the null hypothesis of zero correlation between the slope and the intercept (see
also Blance et al. 2005, Tu and Gilthorpe 2007). We apply this adjustment here, thus ensuring a
clean interpretation of the estimated slope-intercept correlation coefﬁcient under the null hypothesis
that it is equal to zero. Consequently, the individual intercept terms (α0 + αi ) represent the expected
outcome for the child at around 6.5 years of age.
For each country and estimator, Table 4 reports estimates for the three parameters of interest: the
mean outcome in the sample (i.e., the average difference in stature relative to the reference median,
α0 ), the mean growth velocity (β0 ), and the conditional correlation between the estimated latent
variables (ραβ ). In the case of the POLS estimator, the latent variables are ignored and thus not
estimated. Even so, a proxy for individual-speciﬁc slopes can be derived by interacting the child’s
age with the set of time-invariant controls. In turn, the correlation between these slopes and the
estimated conditional mean size of each child yields a crude ﬁrst-pass approximation to ραβ . For
the CRE estimator, the same correlation coefﬁcient is estimated directly as a model parameter. For
the FEIS estimator, implemented in Stata via the reghdfe command, the ﬁxed effects are retrieved
after estimation (see Guimarães and Portugal 2010). In turn, the correlation coefﬁcient is calculated
directly from the estimated ﬁxed effects.19
17 HAZ scores are chosen as they are the de facto standard in the literature (e.g., see Table 1).
18 All control variables are demeaned (at the country-level) to assist comparison of estimates. The variables are chosen
following previous studies and in accordance with variables that are common to all countries and rounds of the YL
datasets. For further discussion see Section 4.4.
19 To adjust for measurement error, the ﬁxed effects are shrunk toward the sample mean of zero in accordance with
the number of observations used to estimate each effect. This corresponds to an empirical Bayes procedure (see
the discussion in Koedel et al. 2015), yielding a shrinkage factor of 80% in the majority of cases where the child is
16
Table 4: Estimates of catch-up growth in HAZ scores (conditional model)
Between Within Both
α0 β0 ραβ H0α0 H0β0 ? H0ραβ ? ?
Ethiopia POLS -1.390 -0.016 -0.246 0.00 0.99 N 0.00 Y N
(0.022) (0.004) (0.023)
CRE -1.402 -0.002 -0.529 0.00 0.69 N 0.00 Y N
(0.022) (0.004) (0.023)
FEIS -1.381 0.003 -0.303 0.00 0.20 N 0.00 Y N
(0.023) (0.003) (0.023)
India POLS -1.438 -0.034 0.367 0.00 0.99 N 0.99 N N
(0.020) (0.004) (0.023)
CRE -1.451 -0.025 -0.295 0.00 0.99 N 0.00 Y N
(0.020) (0.003) (0.023)
FEIS -1.440 -0.015 -0.193 0.00 0.99 N 0.00 Y N
(0.021) (0.003) (0.023)
Peru POLS -1.290 0.020 0.335 0.00 0.00 Y 0.99 N N
(0.019) (0.004) (0.023)
CRE -1.327 0.045 -0.067 0.00 0.00 Y 0.00 Y Y
(0.019) (0.003) (0.023)
FEIS -1.244 0.046 -0.130 0.00 0.00 Y 0.00 Y Y
(0.022) (0.003) (0.023)
Vietnam POLS -1.128 -0.005 0.347 0.00 0.91 N 0.99 N N
(0.019) (0.004) (0.023)
CRE -1.130 0.003 0.024 0.00 0.13 N 0.85 N N
(0.019) (0.003) (0.023)
FEIS -1.124 0.020 0.036 0.00 0.00 Y 0.94 N N
(0.023) (0.002) (0.023)
Note: the ﬁrst three columns report parameter estimates (and cluster-robust standard errors, in parentheses) from
alternative estimators (POLS, CRE and FEIS); all estimates include a set of time-varying controls; columns denoted ‘H0’
report the probability associated with individual tests of the one-sided null hypotheses associated with each parameter
(see Table 2); columns denoted ‘?’ report the conclusion of (composite) hypotheses regarding speciﬁc forms of catch-up
growth; probability estimates larger than 0.99 have been rounded down to the latter value.
Source: own estimates.
17
Based on the parameter estimates, we test the catch-up hypotheses. Since these are built up
from individual hypothesis tests, against each parameter we report the probability associated with
the relevant one-sided null (denoted H0). To determine the speciﬁc combination of catch-up
possibilities revealed by the procedure, we then combine the individual probabilities using the
maxP methodology (Section 3). These results are summarized in the ‘?’ columns, where a N(o)
suggests we cannot reject the null hypothesis of no catch-up at the 5% conﬁdence level; and a
Y(es) indicates the presence of statistical evidence for a given form of catch-up. Given the chosen
direction of the null hypotheses, we do not test for the presence of growth divergence.
Five main ﬁndings stand out. First, looking across the different estimators, the qualitative conclu-
sions are broadly similar. However, parameter estimates are not identical within each country. In
some cases, the estimates switch signs depending on the estimator chosen – e.g., mean growth
velocity in Ethiopia is negative and signiﬁcant under the POLS estimator (-0.016), but positive
(0.003) and insigniﬁcant under the FEIS estimator. Furthermore, in all estimates, the FEIS estimates
for β0 are numerically larger than those taken from either the POLS or CRE models. Since the FEIS
estimator imposes fewer assumptions, it is expected to be consistent under a wider range of condi-
tions. In other words, the POLS and CRE estimators appear to yield downward-biased estimates of
mean growth velocity, perhaps driven by unobserved individual heterogeneity. Additionally, the
FEIS model displays the best goodness-of-ﬁt. As reported in Appendix Table A3, the regression
root mean square error (RMSE) and AIC criterion substantially improve (decline) as we move from
the POLS to the CRE estimator; and in most cases they improve again when moving from the CRE
to the FEIS estimator. The same table shows that estimation speed is much faster under the FEIS
estimator compared to the CRE model, by a factor of around 100. Even more critically, Hausman
tests comparing the CRE and FEIS results consistently reject the hypothesis that the estimates for
the common parameters (namely, γ ) are equal under both estimators. Taken together, these results
suggest the FEIS approach is to be preferred. Thus, unless otherwise indicated, we focus on these
results in our interpretation and henceforth.
Second, as expected given the purpose of the YL initiative is to study development trajectories
among generally disadvantaged groups, the estimate for standardized mean stature (α ˆ 0 ) is below
that of the reference population in all four countries. Consequently, a necessary precondition for
between-population catch-up growth is fulﬁlled and the null hypothesis H0α0 : α0 ≥ 0 is consistently
rejected. Third, in keeping with the long difference results in Table 3(b), the pace of mean growth
relative to the reference population (β ˆ0 ) differs across the four countries. In Peru and Vietnam,
mean height z-scores in the sample are growing faster than in the reference population, implying we
can reject the null hypothesis that β0 ≤ 0. In Ethiopia and India, by contrast, we ﬁnd no evidence
of above-normal growth rates. Thus, according to the FEIS results, between-group catch-up has
taken place in Peru and Vietnam, but not elsewhere.
Fourth, evidence for within-group catch-up is also mixed. While the estimates for ραβ generally
observed across all four rounds. By construction, this procedure does not affect the correlation coefﬁcient.
18
are closer to zero under the FEIS estimator as compared to at least one of the other estimators,
the FEIS correlation coefﬁcients are negative and signiﬁcant in all locations other than Vietnam.
It follows there is somewhat more consistent evidence of within-group catch-up of HAZ scores
in the YL countries (i.e. in three of the four countries) than there is evidence of between-group
catch-up (in two of the four countries). Taking the two forms of catch-up together, as tested in the
last column of the table, only Peru displays evidence of both between- and within-group catch-up
simultaneously.
Finally, while the above tests concern the presence of catch-up growth, they do not speak to its
magnitude. Looking more closely at the size of the point estimates, we note the pace of between-
and within-group catch-up is (at best) moderate and incomplete. For instance, while Peru displays
the largest point estimate for β0 , it would take around 27 years (on the same trajectory) for the mean
HAZ score in this sample to converge to zero. Figure 1 combines the preferred FEIS results for each
country and plots simulated HAZ score trajectories by age. Alongside the trend of the YL sample
mean, we plot the expected HAZ trends at both plus and minus two standard deviations of the
estimated child-speciﬁc ﬁxed-effects distribution (i.e., α0 ± 2σα ), taking into account the estimated
correlation with the slope effect. The results visually conﬁrm the slow pace of between-group
catch-up – i.e., over the 11 years spanned by the data, the mean (predicted) HAZ for Peru and
Vietnam has increased by less than 0.5 of a standard deviation. Similarly, for those countries
displaying within-country catch-up, this also proceeds slowly. The estimate for ραβ is largest in the
case of Ethiopia. But even for this sample, the distance between the trends located at plus/minus two
standard deviations declines by approximately 30% from ages 1 to 12; and these two trends would
take about 30 years (on the same trajectories) to converge. These additional results underscore the
merit of applying a latent growth model framework; but they also highlight the limited magnitude
of catch-up in HAZ scores among the four YL countries.
4.3 Sensitivity analysis
The previous section focussed on results for HAZ scores only. While these scores are widely used
to investigate catch-up growth, they are not the only viable metric. As noted above (Sections 2
and 4.1), previous studies use different transformations of height and BMI; and these may yield
different conclusions about changes over time (also see Le and Behrman 2017, Zhang et al. 2016).
To give one example, changes in externally standardized outcomes (e.g., HAZ scores) will reﬂect
some combination of differences in heights and changes to the shape of the reference distribution
(Cameron et al. 2005). Leroy et al. (2015) argue that changes in HAZ scores may provide overly
optimistic evidence about growth catch-up because the dispersion of heights tends to increase
with age – i.e., even if the mean absolute height difference between a sample and the median of a
reference population remains constant, the corresponding z-score difference will tend to fall. As a
result, the authors propose catch-up should be evaluated using absolute differences in height-for-age
19
Figure 1: Estimated HAZ score trajectories, by country
Ethiopia India
1
0
-1
-2
-3
Peru Vietnam
1
0
-1
-2
-3
1 3 5 7 9 11 1 3 5 7 9 11
2 4 6 8 10 12 2 4 6 8 10 12
Age (years)
+2 SD YL mean -2 SD
Note: trends are derived from the FEIS estimates reported in Table 4; the blue line (YL mean) is the estimate
of the mean HAZ score for the sample; the two red lines are estimated at ± two standard deviations (SDs) of
the estimated individual ﬁxed-effects distributions; dashed horizontal line at y = 0 is the reference median.
Source: own estimates.
(HAD) between the sample and a healthy reference population median. Based on this metric,
the authors ﬁnd evidence for growth faltering (deterioration in relative heights) across many low
income countries, including those surveyed by the Young Lives project.
In light of these concerns, we investigate the extent to which the conclusions of Section 4.2 are
sustained when other anthropometric outcomes are employed. To do so, we re-run the same analysis
but now use a range of height-for-age transformations. The results are reported in Table 5, which
replicates the format of Table 4 in summary form (dropping the probability values) and focussing
uniquely on estimates from the FEIS model.20 In line with the discussion around Table 3(b), as well
as Figures A1–A3, different height transformations do yield substantively different conclusions.
A key ﬁnding is that the HAD metric yields a much less positive view of catch-up. Based on this
metric, there is no evidence of any form of catch-up in any country over time. Moreover, there
appears to be material within-group divergence – estimates for ραβ in Table 5 are at least 39%
20 The ﬁrst set of results shown in Table 5 repeats those of Table 4. A complete set of estimates, based on other
estimators, is available on request.
20
for the HAD metric. As already argued, this ﬁnding may reﬂect the arithmetic properties of the
HAD measure, by which initial differences grow in absolute magnitude over time. The HAR metric
corrects for this problem, but remains unaffected by changes in the distribution of the reference
population. These results reported in Table 5 for the height ratio are generally less pessimistic
than those based on absolute differences. Nonetheless, the HAR estimates continue to provide
no evidence of within-group catch-up, and between-group catch-up only obtains for the Peruvian
sample.
We also investigate catch-up growth based on various measures of BMI. These are shown in Table
6, which also only uses the FEIS estimator and summarizes results for various transformations.
Here, the ﬁndings appear less sensitive to the particular metric chosen. There is no evidence of
between-group catch-up for the continuous variables (BAZ, BAD, BAR); and evidence of within-
group convergence is limited to Ethiopia (BAZ and BAR metrics) and Peru (BAZ only). The greater
consistency of ﬁndings in this case may well reﬂect the properties of BMI. Since both height and
weight are growing in a healthy reference population, changes in (median) BMI are expected to
be fairly stable between ages one and 12. Also, the distribution of body mass shows rather less
variation (changes in variance) across different age groups compared to the distribution of heights.
Thus, for the BMI measure changes in dispersion are likely to play a smaller role, in turn explaining
the broad similarity of ﬁndings across the continuous transformations.
The same analysis can be applied to binary indicators, in which case the outcome is a dummy
variable and the speciﬁcation becomes a linear probability model. Corresponding results are shown
in the bottom portions of Tables 5 and 6), where the binary indicators take a value of one if a
child is within a healthy group (this construction ensures the hypothesis tests can be applied in
the same way as before). Three points stand out. First, consistent with the HAZ results, there is
clear evidence of both between- and within-group catch-up from stunting (however deﬁned) in
the Peruvian sample. Second, all other countries show some sign of a reduction in the prevalence
of stunting – i.e., there is some between-group catch-up. But this depends on the speciﬁc metric
chosen. In Ethiopia, there is a reduction in both the prevalence of stunting (HAZ < -2) and severe
stunting (HAZ < -3). In India, we only see a fall in severe stunting; and in Vietnam we observe
a reduction in stunting, but not severe stunting. Indeed, all countries other than Vietnam show
evidence of between- and within-group catch-up from severe stunting. However, this may well
reﬂect the lower prevalence of severe stunting in Vietnam to begin with (see Table 3). Third, there
is limited evidence of catch-up from wasting. Indeed, none of the countries provides evidence of
either between- or within-group catch-up from very low BMI. Children from the Peru sample show
some increase in the share of children with a healthy BMI. However, this is likely to be driven by a
fall in those with a very high initial BMI (obese).
As a ﬁnal exercise, Table 7 pulls together the results from the various hypothesis tests examined
above (for the FEIS estimator). Similar to Table 2, it counts the number of test results corresponding
to each combination of within and between catch-up. Part (a) of the table refers to the ﬁve height
21
Table 5: Estimates of catch-up growth in height-for-age outcomes (conditional model)
Between Within Both
α0 β0 ? ραβ ? ?
Externally standardized difference (HAZ):
Ethiopia -1.38 0.00 N -0.30 Y N
India -1.44 -0.01 N -0.19 Y N
Peru -1.24 0.05 Y -0.13 Y Y
Vietnam -1.12 0.02 Y 0.04 N N
Ratio to reference median (HAR×100):
Ethiopia -5.76 -0.13 N -0.01 N N
India -6.07 -0.22 N 0.14 N N
Peru -5.36 0.10 Y 0.02 N N
Vietnam -4.71 -0.01 N 0.37 N N
Difference to reference median (HAD):
Ethiopia -6.89 -0.52 N 0.53 N N
India -7.29 -0.61 N 0.64 N N
Peru -6.54 -0.18 N 0.39 N N
Vietnam -5.60 -0.28 N 0.76 N N
Not stunted (HAZ > -2):
Ethiopia -26.75 0.56 Y 0.06 N N
India -27.85 -0.12 N 0.12 N N
Peru -22.22 1.06 Y -0.30 Y Y
Vietnam -18.45 0.39 Y 0.04 N N
Not severely stunted (HAZ > -3):
Ethiopia -9.02 0.76 Y -0.28 Y Y
India -6.59 0.41 Y -0.11 Y Y
Peru -5.33 0.59 Y -0.54 Y Y
Vietnam -3.89 0.01 N 0.05 N N
Note: columns α0 , β0 , ραβ report parameter estimates based on the FEIS estimator; all
speciﬁcations include a set of time-varying controls; columns denoted ‘?’ report the
decision (N = no; Y = yes) of whether we can reject the (composite) null hypotheses
regarding the presence of speciﬁc forms of catch-up growth.
Source: own estimates.
22
Table 6: Estimates of catch-up growth in BMI-for-age outcomes (conditional model)
Between Within Both
α0 β0 ? ραβ ? ?
Externally standardized difference (BAZ):
Ethiopia -1.08 -0.13 N -0.09 Y N
India -1.24 -0.02 N 0.23 N N
Peru 0.59 -0.02 N -0.10 Y N
Vietnam -0.51 -0.03 N 0.37 N N
Ratio to reference median (BAR×100):
Ethiopia -8.98 -1.33 N -0.05 Y N
India -9.67 -0.23 N 0.68 N N
Peru 8.33 0.34 N 0.30 N N
Vietnam -3.45 -0.18 N 0.62 N N
Difference to reference median (BAD):
Ethiopia -1.50 -0.25 N 0.00 N N
India -1.58 -0.05 N 0.71 N N
Peru 1.37 0.08 N 0.38 N N
Vietnam -0.57 -0.04 N 0.66 N N
BMI-for-age is not low (> 5th percentile):
Ethiopia -25.80 -4.69 N 0.36 N N
India -30.05 -1.26 N 0.14 N N
Peru 3.42 -0.08 N -0.03 N N
Vietnam -9.36 -1.66 N 0.41 N N
BMI-for-age is healthy (5th – 85th percentile):
Ethiopia -17.91 -3.53 N 0.18 N N
India -19.04 -1.50 N 0.10 N N
Peru -12.29 0.45 Y -0.22 Y Y
Vietnam -4.13 -2.01 N 0.32 N N
Note: columns α0 , β0 , ραβ report parameter estimates based on the FEIS estimator; all
speciﬁcations include a set of time-varying controls; columns denoted ‘?’ report the
decision (N = no; Y = yes) of whether we can reject the (composite) null hypotheses
regarding the presence of speciﬁc forms of catch-up growth.
Source: own estimates.
23
measures; and part (b) of the table refers to the ﬁve BMI measures. It shows that while there is
certainly no overwhelming evidence for catch-up of either form, around half of all tests (models)
reveal at least some form of catch-up in stature. In contrast, 4 in every 5 tests for the BMI outcomes
reveal no form of catch-up whatsoever. The disconnect between the ﬁndings for these two (related)
outcomes is striking and merits future research. The table also provides insights regarding the
relationship between the two forms of catch-up. Fisher’s exact test of their relationship suggests we
cannot reject the null hypothesis of independence. This supports the contention that the different
forms of catch-up are distinct and merit separate attention.
To conclude this subsection, it is evident that estimates of both forms of catch-up are sensitive to
the choice of metric. Indeed, how outcomes are expressed often seems rather more fundamental
than the choice of regression estimator, especially if we exclude a naïve POLS approach (i.e.,
the CRE and FEIS estimators yield qualitatively similar conclusions). This concern applies
particularly to outcomes associated with height. Here, conclusions regarding catch-up based
on a ratio transformation tend to fall between the more optimistic results deriving from z-score
transformations (e.g., HAZ and binary measures of stunting) and the pessimistic results that emerge
when we use absolute gaps (HAD). Differences between outcome metrics reﬂect a range of factors,
including (inter alia): the effect of changes in the shape of the reference distribution; alternative
ways in which height and weight are combined; how initial gaps compound over time; and the
focus on particular aspects of the outcome distribution. Since the construction choices underlying
speciﬁc outcomes often correspond to different substantive research interests, there is unlikely to
be a universal ‘best’ metric or transformation. Nonetheless, in the absence of a strong a priori
justiﬁcation, the evidence presented here suggests researchers should be wary of uniquely relying
on any single outcome indicator to make general claims about catch-up growth.
4.4 Growth modiﬁers
Thus far, our analysis has concentrated on the broad direction and magnitude of catch-up growth
(if any). However, we have not considered which factors might account for variation in individual
child growth trajectories within each sample. In the existing literature, one approach to doing so
focuses on simpliﬁed versions of equation (1). Speciﬁcally, variation in the estimated residuals
is analyzed: εˆit = yit − E(yit | yi,t −1 , xis ), including their association with later outcomes such as
cognitive achievement (e.g., Crookston et al. 2013, Gandhi et al. 2011, Kuklina et al. 2006, Schott
et al. 2013). Another approach, based on direct models of growth trajectories, is to consider
potential heterogeneity in the β coefﬁcient in equation (6a), which can be examined by including
interaction terms between age and relevant explanatory variables (e.g., Rieger and Trommlerová
2016, Rubio-Codina et al. 2015).
Our extension borrows on elements of both approaches. As discussed by Cole et al. (2010),
individual growth curves can be characterized by a small number of parameters. In our case, given
24
Table 7: Frequency tabulation of hypotheses tests, by outcome type
(a) Height outcomes:
Between?
Within? Yes No Total
Yes 5 2 7
No 4 9 13
Total 9 11 20
(b) BMI outcomes:
Between?
Within? Yes No Total
Yes 1 3 4
No 0 16 16
Total 1 19 20
Note: table reports counts of between-group and
within-group null hypothesis test results, derived from
Tables 5 and 6, covering all countries.
Source: own estimates.
we assume a linear speciﬁcation, the individual growth trajectories are simply deﬁned by child
mean size: α ˜i = α ˆ0 + α ˆ0 + β
˜i = β
ˆ i ; and linear growth velocity: β ˆi . So, rather than considering
variation in β , alone, we use the FEIS estimates for both variables and treat them as dependent
variables. Concretely, this stage of analysis is based on regressions of the following form where
children are the unit of analysis (in cross-section):
˜ i = δ0 + x
α ¯i δ1 + δ2c + ηi (7a)
˜i = θ0 + x
β ¯ θ1 + θ2c + νi (7b)
i
Estimates for δ1 indicate the impact of the mean values of the vector x, deﬁned as before, on
average child size. Estimates for θ1 give the corresponding impacts on growth velocity, indicating
how the permanent elements of x modify the pace of child development. Recall the estimated latent
variables absorb the effects of all time-invariant child characteristics. So, this analysis seeks to
quantify the extent to which these are accounted for by observed child characteristics. In addition,
community-level ﬁxed effects can be added to the regression to capture how differential access to
public services and other features of the broad local environment (outside the family) systematically
affect child growth trajectories. These are captured by the parameters δ2c , θ2c , where c is an index
for the community.21
This kind of second stage analysis has been applied in other contexts. Acemoglu et al. (2009), for
21 When treated as ﬁxed effects, we cannot identify which speciﬁc features of the local environment (community) alter
child growth trajectories. Nonetheless, as we show, this approach helps quantify the overall explanatory contribution
of variables at this level.
25
example, investigate the modernization hypothesis by regressing a set of country-speciﬁc ﬁxed
effects against time-invariant country characteristics; and Brand and Davis (2011) use a similar
method to assess heterogeneity in the effects of a college education on fertility. One challenge
in using this two-stage approach is that αi and βi are estimated with error. To address this, we
follow the advice of Lewis and Linzer (2005) who suggest that, in small samples, unweighted OLS
regressions combined with Efron or bootstrap consistent standard errors are generally reliable when
the dependent variables are themselves estimates.
For the remainder of the paper we focus on the height and BMI ratio outcomes. We do so as the
ratio transformation represents the lesser of two extremes – the ratio of outcomes in the sample
to that of the median in the reference distribution is not affected by (independent) changes in the
variance of the reference distribution, nor does it mechanically compound initial differences. Tables
8 to 11 summarize our results for the HAR and BAR outcomes, covering each country separately.
Appendix Tables A4 to A7 report the same analysis for the corresponding z-scores (HAZ and BAZ).
In each table, the ﬁrst group of the columns (Ia-Ie) refer to the height outcomes; and the ﬁnal set
of columns refer to body mass outcomes. Regression results for mean child size are reported in
columns Ia, Ib, IIa and IIb where the ﬁrst in each set excludes community ﬁxed effects. Columns Ic,
Id, IIc and IId report the same regressions for growth velocity. Lastly, columns Ie and IIe add the
estimates for child size (α˜ i ) to the velocity regression, reﬂecting the potential correlation between
these two variables (as captured by ραβ ; see above). This extension is informative because it helps
differentiate between the contributions of observed and unobserved individual determinants of
growth velocity.
We highlight four main ﬁndings from these results. First, a number of the coefﬁcient estimates are
consistent with previous literature. In particular, household wealth displays a signiﬁcant positive
relationship with both size and velocity across most countries.22 Similarly, the highest school grade
attained by the mother, here transformed into standard deviation units, shows a signiﬁcant positive
association with predicted child size (especially height), but a less systematic relation with growth
velocity. Overall, the implication is that children from more advantaged households show enhanced
average development outcomes and would appear to extend their advantage over time relative to
other children in the sample. However, these advantages generally do not imply that children from
(relatively) wealthy families catch up toward the median of the external reference population. This
is indicated in the ﬁnal row of the table, which calculates the predicted size and growth velocity
among children from households that are two standard deviations above the country mean wealth
index level. Across all the regressions for height, the adjusted estimates of mean stature remain in
the negative domain; and only in Peru and Vietnam is mean growth velocity in the positive domain
for the more wealthy households. In this sense, between-group catch-up remains elusive in both
Ethiopia and India, even for children from more advantaged households in the sample.
22 The wealth index is constructed as the standardized ﬁrst principal component of three separate indexes, which are
provided in the YL datasets. These pertain to ownership of consumer durables, housing quality, and access to basic
amenities within the household (e.g., clean water).
26
Table 8: Size and velocity regressions, Ethiopia
Height ratios BMI ratios
Size Size Velocity Velocity Velocity Size Size Velocity Velocity Velocity
Ia Ib Ic Id Ie IIa IIb IIc IId IIe
Mean -5.781∗∗∗ -5.774∗∗∗ -0.129∗∗∗ -0.129∗∗∗ -0.129∗∗∗ -8.988∗∗∗ -8.998∗∗∗ -1.328∗∗∗ -1.326∗∗∗ -1.327∗∗∗
(0.074) (0.074) (0.009) (0.010) (0.008) (0.107) (0.121) (0.021) (0.024) (0.022)
Female 0.375∗∗ 0.412∗∗∗ -0.027 -0.020 -0.019 -1.429∗∗∗ -1.475∗∗∗ -0.111∗∗∗ -0.126∗∗∗ -0.150∗∗∗
(0.155) (0.148) (0.019) (0.018) (0.019) (0.212) (0.247) (0.040) (0.037) (0.040)
Urban -0.265 1.096 0.014 0.111 0.115 -0.102 1.330 0.105 0.324 0.345
(0.258) (0.832) (0.031) (0.146) (0.136) (0.368) (1.681) (0.067) (0.335) (0.294)
Not ﬁrst born -0.032 -0.122 -0.018 -0.002 -0.002 -0.535 -0.273 0.187∗∗∗ 0.177∗∗∗ 0.173∗∗∗
(0.217) (0.190) (0.028) (0.026) (0.032) (0.354) (0.390) (0.063) (0.055) (0.056)
Majority ELG 0.358∗ 0.257 -0.038 0.027 0.028 -1.236∗∗∗ -0.179 -0.260∗∗∗ -0.084 -0.086
(0.183) (0.310) (0.023) (0.035) (0.033) (0.301) (0.433) (0.054) (0.081) (0.075)
Wealth index 1.143∗∗∗ 1.147∗∗∗ -0.005 -0.024 -0.020 -0.089 -0.569∗∗ -0.116∗∗∗ -0.083∗∗ -0.092∗∗
(0.130) (0.130) (0.018) (0.019) (0.017) (0.196) (0.244) (0.036) (0.038) (0.039)
Mother’s edu. 0.228∗ 0.186 -0.038∗∗ -0.029∗ -0.028∗ 0.168 -0.196 0.000 0.025 0.022
27
(0.117) (0.121) (0.016) (0.016) (0.017) (0.204) (0.231) (0.038) (0.037) (0.037)
Mother’s age 0.004 0.006 -0.004∗∗ -0.004∗∗ -0.004∗∗ 0.020 0.017 0.004 0.002 0.002
(0.014) (0.015) (0.002) (0.002) (0.002) (0.027) (0.023) (0.004) (0.004) (0.004)
Sibs 0-5 -0.495∗∗ -0.492∗∗ -0.104∗∗∗ -0.057∗∗ -0.059∗∗ -0.647∗∗ -0.399 -0.027 -0.198∗∗∗ -0.204∗∗∗
(0.212) (0.195) (0.025) (0.026) (0.025) (0.320) (0.326) (0.056) (0.050) (0.050)
Sibs 6-12 -0.063 -0.054 -0.062∗∗ -0.051∗ -0.052∗ -0.700∗∗ -0.796∗∗ 0.028 0.025 0.012
(0.178) (0.196) (0.029) (0.027) (0.027) (0.327) (0.332) (0.059) (0.048) (0.053)
Child FE (std.) - - - - -0.013 - - - - -0.086∗∗∗
(0.011) (0.029)
Obs. 1,845 1,845 1,845 1,845 1,845 1,846 1,846 1,846 1,846 1,846
RMSE 3.10 3.03 0.39 0.38 0.38 5.23 5.01 0.91 0.82 0.81
Adj. R2 0.13 0.17 0.02 0.09 0.09 0.05 0.13 0.04 0.22 0.23
WI×2 + Mean -3.50 -3.48 -0.14 -0.18 -0.17 -9.17 -10.14 -1.56 -1.49 -1.51
Community FE? No Yes No Yes Yes No Yes No Yes Yes
signiﬁcance: ∗ 10%, ∗∗ 5%, ∗∗∗ 1%.
Note: table reports selected coefﬁcient estimates from auxiliary analyses, where the dependent variable is denoted in the column header and is derived from the FEIS
estimates (Tables 5 and 6); individual children are the unit of observation and all covariates are taken as averages across rounds (demeaned); standard errors are based on 100
bootstrap iterations; community ﬁxed effects are included as denoted in the ﬁnal row.
Source: own estimates.
Table 9: Size and velocity regressions, India
Height ratios BMI ratios
Size Size Velocity Velocity Velocity Size Size Velocity Velocity Velocity
Ia Ib Ic Id Ie IIa IIb IIc IId IIe
Mean -6.061∗∗∗ -6.061∗∗∗ -0.222∗∗∗ -0.222∗∗∗ -0.222∗∗∗ -9.682∗∗∗ -9.681∗∗∗ -0.224∗∗∗ -0.223∗∗∗ -0.222∗∗∗
(0.065) (0.066) (0.009) (0.009) (0.008) (0.178) (0.139) (0.036) (0.039) (0.033)
Female 0.272∗∗ 0.195 0.010 0.011 0.008 -0.087 -0.138 0.061 0.044 0.067
(0.132) (0.131) (0.017) (0.015) (0.017) (0.272) (0.257) (0.069) (0.073) (0.059)
Urban -0.163 -0.112 0.065∗∗∗ -0.024 -0.023 0.730 -0.843 0.123 -0.340 -0.200
(0.203) (0.674) (0.024) (0.087) (0.094) (0.500) (1.569) (0.091) (0.315) (0.298)
Not ﬁrst born -0.476∗∗∗ -0.390∗∗ -0.048∗∗ -0.047∗∗ -0.042∗ -0.249 -0.326 -0.048 -0.071 -0.017
(0.165) (0.164) (0.022) (0.022) (0.022) (0.333) (0.385) (0.069) (0.090) (0.059)
Majority ELG -0.255∗∗ -0.133 0.000 -0.015 -0.014 -1.010∗∗∗ -0.540 -0.284∗∗∗ -0.169∗∗∗ -0.079
(0.130) (0.140) (0.017) (0.017) (0.014) (0.311) (0.329) (0.072) (0.057) (0.058)
Wealth index 0.662∗∗∗ 0.642∗∗∗ 0.047∗∗∗ 0.060∗∗∗ 0.052∗∗∗ 1.458∗∗∗ 1.312∗∗∗ 0.088 0.072 -0.146
(0.101) (0.108) (0.011) (0.013) (0.013) (0.250) (0.245) (0.067) (0.065) (0.126)
Mother’s edu. 0.223∗∗ 0.170∗ -0.002 -0.008 -0.010 0.487 0.487 0.144 0.139 0.059
28
(0.107) (0.099) (0.011) (0.012) (0.013) (0.366) (0.319) (0.101) (0.099) (0.038)
Mother’s age 0.046∗∗∗ 0.050∗∗∗ -0.006∗∗∗ -0.006∗∗ -0.006∗∗∗ -0.006 -0.000 -0.002 -0.002 -0.002
(0.015) (0.018) (0.002) (0.002) (0.002) (0.036) (0.033) (0.008) (0.008) (0.007)
Sibs 0-5 -0.038 0.029 0.048∗ 0.051∗∗ 0.051∗∗ -0.068 -0.116 -0.102 -0.148 -0.128
(0.223) (0.217) (0.027) (0.025) (0.025) (0.587) (0.538) (0.108) (0.146) (0.104)
Sibs 6-12 -0.015 0.090 0.019 0.003 0.002 -0.491 -0.189 -0.132∗∗ -0.092 -0.061
(0.195) (0.196) (0.023) (0.023) (0.019) (0.425) (0.380) (0.063) (0.065) (0.070)
Child FE (std.) - - - - 0.040∗∗∗ - - - - 1.260∗∗
(0.009) (0.511)
Obs. 1,913 1,913 1,913 1,913 1,913 1,913 1,913 1,913 1,913 1,913
RMSE 2.90 2.86 0.35 0.33 0.33 7.30 7.28 1.79 1.79 1.32
Adj. R2 0.08 0.11 0.05 0.13 0.14 0.07 0.08 0.02 0.03 0.47
WI×2 + Mean -4.74 -4.78 -0.13 -0.10 -0.12 -6.77 -7.06 -0.05 -0.08 -0.51
Community FE? No Yes No Yes Yes No Yes No Yes Yes
signiﬁcance: ∗ 10%, ∗∗ 5%, ∗∗∗ 1%.
Note: table reports selected coefﬁcient estimates from auxiliary analyses, where the dependent variable is denoted in the column header and is derived from the FEIS
estimates (Tables 5 and 6); individual children are the unit of observation and all covariates are taken as averages across rounds (demeaned); standard errors are based on 100
bootstrap iterations; community ﬁxed effects are included as denoted in the ﬁnal row.
Source: own estimates.
Table 10: Size and velocity regressions, Peru
Height ratios BMI ratios
Size Size Velocity Velocity Velocity Size Size Velocity Velocity Velocity
Ia Ib Ic Id Ie IIa IIb IIc IId IIe
Mean -5.354∗∗∗ -5.351∗∗∗ 0.098∗∗∗ 0.097∗∗∗ 0.097∗∗∗ 8.332∗∗∗ 8.336∗∗∗ 0.343∗∗∗ 0.343∗∗∗ 0.343∗∗∗
(0.079) (0.071) (0.009) (0.009) (0.010) (0.181) (0.209) (0.036) (0.037) (0.034)
Female -0.094 -0.125 0.038∗∗ 0.031 0.031 -1.335∗∗∗ -1.333∗∗∗ -0.092 -0.089 -0.038
(0.125) (0.145) (0.019) (0.019) (0.021) (0.391) (0.376) (0.070) (0.086) (0.072)
Urban 0.367∗ -0.211 -0.022 0.011 0.011 -0.982∗ -1.421∗∗ 0.164 0.050 0.106
(0.212) (0.257) (0.030) (0.038) (0.032) (0.558) (0.657) (0.104) (0.130) (0.133)
Not ﬁrst born 0.045 -0.063 -0.033 -0.035 -0.035 -0.437 -0.715 -0.003 -0.021 0.007
(0.155) (0.189) (0.028) (0.025) (0.025) (0.555) (0.516) (0.100) (0.087) (0.093)
Majority ELG 0.355∗ 0.069 -0.043 -0.036 -0.036 -0.546 -0.099 0.117 0.153 0.157
(0.187) (0.167) (0.027) (0.032) (0.027) (0.572) (0.532) (0.105) (0.129) (0.121)
Wealth index 0.913∗∗∗ 0.824∗∗∗ 0.022∗ 0.048∗∗∗ 0.050∗∗∗ 2.107∗∗∗ 1.619∗∗∗ 0.362∗∗∗ 0.248∗∗∗ 0.185∗∗∗
(0.084) (0.100) (0.012) (0.018) (0.015) (0.248) (0.279) (0.045) (0.061) (0.056)
Mother’s edu. 0.530∗∗∗ 0.471∗∗∗ -0.010 -0.008 -0.007 0.435 0.454∗ -0.012 0.001 -0.017
29
(0.091) (0.084) (0.013) (0.014) (0.014) (0.302) (0.269) (0.054) (0.061) (0.059)
Mother’s age 0.019∗ 0.024∗∗ -0.004∗∗ -0.004∗∗ -0.004∗∗ 0.040 0.046 -0.013∗∗ -0.011∗ -0.013∗
(0.011) (0.011) (0.002) (0.002) (0.002) (0.037) (0.032) (0.006) (0.006) (0.006)
Sibs 0-5 0.076 0.087 -0.076∗∗∗ -0.076∗∗∗ -0.076∗∗∗ 0.003 -0.140 -0.270∗∗∗ -0.289∗∗∗ -0.283∗∗∗
(0.167) (0.172) (0.024) (0.027) (0.023) (0.508) (0.450) (0.102) (0.105) (0.093)
Sibs 6-12 -0.669∗∗∗ -0.496∗∗∗ 0.023 -0.002 -0.003 -0.523 0.063 0.107 0.154 0.151
(0.192) (0.184) (0.035) (0.027) (0.031) (0.512) (0.457) (0.105) (0.104) (0.116)
Child FE (std.) - - - - -0.007 - - - - 0.354∗∗∗
(0.015) (0.057)
Obs. 1,916 1,916 1,916 1,916 1,916 1,921 1,921 1,921 1,921 1,921
RMSE 2.85 2.78 0.42 0.41 0.41 8.67 8.28 1.63 1.62 1.59
Adj. R2 0.31 0.35 0.02 0.05 0.05 0.10 0.17 0.11 0.13 0.16
WI×2 + Mean -3.53 -3.70 0.14 0.19 0.20 12.55 11.57 1.07 0.84 0.71
Community FE? No Yes No Yes Yes No Yes No Yes Yes
signiﬁcance: ∗ 10%, ∗∗ 5%, ∗∗∗ 1%.
Note: table reports selected coefﬁcient estimates from auxiliary analyses, where the dependent variable is denoted in the column header and is derived from the FEIS
estimates (Tables 5 and 6); individual children are the unit of observation and all covariates are taken as averages across rounds (demeaned); standard errors are based on 100
bootstrap iterations; community ﬁxed effects are included as denoted in the ﬁnal row.
Source: own estimates.
Table 11: Size and velocity regressions, Vietnam
Height ratios BMI ratios
Size Size Velocity Velocity Velocity Size Size Velocity Velocity Velocity
Ia Ib Ic Id Ie IIa IIb IIc IId IIe
Mean -4.709∗∗∗ -4.709∗∗∗ -0.012∗ -0.012∗ -0.012∗ -3.433∗∗∗ -3.429∗∗∗ -0.181∗∗∗ -0.181∗∗∗ -0.182∗∗∗
(0.060) (0.071) (0.007) (0.006) (0.007) (0.157) (0.183) (0.026) (0.026) (0.023)
Female 0.187 0.139 0.005 0.006 0.001 -1.764∗∗∗ -1.720∗∗∗ -0.159∗∗∗ -0.162∗∗∗ -0.016
(0.123) (0.135) (0.014) (0.013) (0.013) (0.348) (0.343) (0.049) (0.048) (0.039)
Urban 1.313∗∗∗ 0.843 -0.001 0.013 -0.016 4.646∗∗∗ 6.147∗∗∗ 0.177∗∗ 0.195 -0.327
(0.213) (0.653) (0.023) (0.087) (0.090) (0.602) (1.826) (0.076) (0.279) (0.235)
Not ﬁrst born -0.451∗∗ -0.383∗∗ -0.019 -0.017 -0.004 -1.131∗∗ -1.125∗∗ -0.154∗∗ -0.138∗∗ -0.043
(0.182) (0.163) (0.019) (0.020) (0.018) (0.478) (0.477) (0.070) (0.063) (0.061)
Majority ELG 1.272∗∗∗ 0.294 0.005 -0.050 -0.060 -1.473∗∗∗ 2.536∗∗ 0.013 -0.285∗ -0.500∗∗∗
(0.183) (0.318) (0.019) (0.053) (0.052) (0.452) (1.053) (0.062) (0.171) (0.176)
Wealth index 0.648∗∗∗ 0.893∗∗∗ 0.054∗∗∗ 0.057∗∗∗ 0.027∗∗ 0.948∗∗∗ 1.550∗∗∗ 0.226∗∗∗ 0.291∗∗∗ 0.159∗∗∗
(0.107) (0.108) (0.012) (0.014) (0.012) (0.266) (0.311) (0.040) (0.045) (0.036)
Mother’s edu. 0.567∗∗∗ 0.580∗∗∗ 0.023∗∗ 0.014 -0.006 0.815∗∗∗ 0.808∗∗ -0.010 -0.002 -0.071∗∗
30
(0.108) (0.104) (0.010) (0.012) (0.013) (0.313) (0.316) (0.044) (0.043) (0.036)
Mother’s age -0.017 -0.024 -0.006∗∗∗ -0.005∗∗∗ -0.004∗∗ 0.040 -0.001 0.012∗∗ 0.009∗ 0.009∗∗
(0.014) (0.015) (0.001) (0.001) (0.002) (0.037) (0.034) (0.005) (0.005) (0.004)
Sibs 0-5 -0.895∗∗∗ -0.459 -0.026 -0.013 0.003 0.528 0.324 -0.136 -0.091 -0.119
(0.284) (0.293) (0.031) (0.034) (0.030) (0.687) (0.729) (0.100) (0.088) (0.097)
Sibs 6-12 -0.434 0.022 -0.055∗∗ -0.046∗ -0.046∗∗ -1.097 0.212 -0.230∗∗∗ -0.104 -0.122
(0.267) (0.289) (0.027) (0.026) (0.023) (0.684) (0.638) (0.087) (0.090) (0.087)
Child FE (std.) - - - - 0.115∗∗∗ - - - - 0.684∗∗∗
(0.010) (0.028)
Obs. 1,908 1,908 1,908 1,908 1,908 1,909 1,909 1,909 1,909 1,909
RMSE 2.85 2.74 0.30 0.30 0.28 7.47 7.19 1.07 1.04 0.85
Adj. R2 0.29 0.35 0.07 0.10 0.19 0.14 0.20 0.10 0.14 0.44
WI×2 + Mean -3.41 -2.92 0.10 0.10 0.04 -1.54 -0.33 0.27 0.40 0.14
Community FE? No Yes No Yes Yes No Yes No Yes Yes
signiﬁcance: ∗ 10%, ∗∗ 5%, ∗∗∗ 1%.
Note: table reports selected coefﬁcient estimates from auxiliary analyses, where the dependent variable is denoted in the column header and is derived from the FEIS
estimates (Tables 5 and 6); individual children are the unit of observation and all covariates are taken as averages across rounds (demeaned); standard errors are based on 100
bootstrap iterations; community ﬁxed effects are included as denoted in the ﬁnal row.
Source: own estimates.
The second insight, which follows directly, is the presence of notable differences between countries
in the magnitude (and signiﬁcance) of different explanatory factors. For instance, there is consider-
able heterogeneity in the relative trajectories of boys and girls within each sample. In both Ethiopia
and Vietnam, girls show a larger negative average body mass difference to the gender-speciﬁc
reference population, and also show more rapid divergence compared to boys over time. In India, by
contrast, no systematic height or body mass differences are found between boys and girls, at least
after controlling for community characteristics (Table 9). Nonetheless, a distinctive insight from
India is the strong relationship between birth order and stature. In keeping with recent evidence
due to Jayachandran and Pande (2017), ﬁrst borns are both taller and grow faster than their younger
siblings. Results for the Peruvian sample indicate no birth order effects; younger sibs in Ethiopia
tend to gain body mass faster than ﬁrst borns; but younger sibs in Vietnam show a lower BMI than
their older sibs, and this gap grows over time.
Third, community characteristics play a material role in explaining differences in growth trajectories.
This is seen in two ways. First, inclusion of the community ﬁxed effects leads to material changes
in the regression coefﬁcient estimates in may cases. For instance, in the estimates that exclude
community ﬁxed effects, we ﬁnd that children from the region’s majority ethno-linguistic group
display different growth trajectories to other children, especially in India and Vietnam. However,
these differences become substantially moderated once community ﬁxed effects are included.
Second, the (adjusted) R2 estimates, reported in the footer of the regression tables, all increase
on inclusion of the community ﬁxed effects. Indeed, the increase relative to the same regression
without the ﬁxed effects represents a lower bound on the variance share attributable to variables
at this level. This varies across models, but is around 5 percentage points in most cases, which is
material in relation to the total (adjusted) R2 .
Lastly, notwithstanding the above, we recognize that the included covariates explain a comparatively
small share of the overall variation in the estimated latent variables. Again, this is shown by the
(adjusted) R2 estimates. For the mean size (height) ratio estimates, less than one-third of the
variance can be accounted for across all the estimates. While some of this may be attributable to
measurement error, it is reasonable to conclude that the set of available controls does not provide a
comprehensive picture of differences in stature and body mass between children. The same general
point extends to the velocity estimates. However, in a number of cases, the explanatory power of
the regression increases signiﬁcantly when the latent child ﬁxed effect (size) is added. For instance,
the unexplained component of growth in BMI differences in India falls from 98% to 53% once α ˜i
is added. This indicates that unobserved determinants of child development, perhaps including
genetic variation and other aspects of the home environment, are fundamental. Moreover, inclusion
of α˜ i often alters the magnitude (and even sign) of some observed coefﬁcients. This points out
that results for the latter may be biased by omitted variables, even after controlling for unobserved
community characteristics – e.g., wealth may be correlated with unobserved household conditions.
Thus, we should be wary about drawing strong conclusions about the potential impact of speciﬁc
31
policy interventions based on descriptive features of growth trajectories. Put differently, while these
regressions are of descriptive interest, they are unlikely to yield precise causal estimates.
5 Conclusion
The aim of this study was to revisit how catch-up growth in children is deﬁned and measured. Our
motivation was that existing deﬁnitions tend to focus on speciﬁc dimensions of catch-up – namely
either between-group or within-group convergence, but not both. We argued that existing studies
also tend to focus on speciﬁc outcomes (e.g., HAZ) and that conventional dynamic panel estimates
of catch-up are prone to bias. In contrast, we suggested that a latent growth framework is an
attractive alternative. It avoids many of the technical problems with dynamic panel approaches and
provides a uniﬁed setting in which to test for both between- and within-group catch-up. We outlined
how the latent growth framework can be estimated in practice, proposing that a ﬁxed-effects and
individual slopes (FEIS) estimator imposes minimal assumptions relative to alternatives such as a
correlated random effects (mixed effects) model. In addition, we showed how composite hypotheses
regarding catch-up can be tested by combining probability values from the regression estimates.
In our application of this approach, we compared the performance of various latent growth esti-
mators using four rounds of the Young Lives data collected in (sentinel sites) in Ethiopia, India,
Peru and Vietnam. All of the samples come from relatively disadvantaged populations and, as
expected, on average the sampled children display lower height and body mass than those from a
healthy international reference population. On the basis of externally-standardized HAZ scores,
our latent growth estimates revealed there has been statistically signiﬁcant catch-up growth in
Peru and Vietnam, but that the speed of catch-up has been slow. All countries other than Vietnam
were also found to display some degree of within-group catch-up, but again this appears to have
proceeded at a relatively moderate pace. Together, and as depicted in Figure 1, these results conﬁrm
the importance of conceptually and practically distinguishing between within- and between-group
catch-up. Indeed, while the (predicted) distribution of HAZ scores for the sample in India is
narrowing, the overall trend for the sample is declining. Only in Peru do we ﬁnd evidence for both
forms of catch-up simultaneously.
In terms of technical insights, we found that the FEIS estimator outperforms existing alternatives.
Estimates based on a POLS or CRE estimator were found to be inconsistent (biased) and showed
poorer goodness-of-ﬁt. At the same time, conclusions about catch-up were found to be highly
sensitive to the outcome metric chosen, especially in the case of height outcomes. As found
elsewhere, evidence on catch-up based on HAZ scores tends to be more optimistic than evidence
based on absolute or relative height differences. We argued that while there is unlikely to be a
single ‘best’ metric, the choice of outcome metric is far from trivial and needs to be justiﬁed in
relation to substantive research questions. Moreover, checking whether ﬁndings about catch-up
32
hold across alternative metrics can be very helpful. Indeed, our analysis showed a clear disconnect
in the evidence for catch-up based on metrics of stature vs BMI (see Table 7).
Lastly, we extended the analysis to show how the framework can be used to identify systematic
drivers of heterogeneity in both size and growth velocity. This corresponded to a second analytical
stage, in which the estimated latent variables (i.e., the FEIS intercept and slope ﬁxed effects) were
regressed on (time-invariant) individual characteristics such as gender, household wealth and the
mother’s level of education as well as community ﬁxed effects. These results revealed signiﬁcant
diversity across the four countries. However, higher household wealth is often associated with
more rapid growth, enabling children to extend initial advantages. Also, community ﬁxed effects
appear to play a material role in inﬂuencing growth trajectories. In sum, a latent growth framework
combined with a ﬂexible FEIS estimation procedure provides a rich and practical empirical basis,
both to distinguish between alternative forms of catch-up and also to investigate factors affecting
the heterogeneity in child growth patterns
33
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39
A Appendix: Supplementary material
A.1 Tables
Table A1: Summary of properties of alternative latent growth regression estimators
Estimator Key assumptions Strengths Weaknesses
POLS σα = σβ = ραβ = 0 Simple Restrictive assumptions do
not permit direct tests of
within-group catch-up
CRE ρxβ = ρxα = 0 Permits direct tests of Random effects assumed to
αi ∼ N 2)
(0, σα all catch-up hypotheses follow normal distribution;
practical restrictions on
βi ∼ N 2)
(0, σβ
covariance structure;
convergence can fail / be slow
FEIS – Minimal assumptions; Measurement error in ﬁxed
controls for all time effects; no direct tests of
invariant factors on size individual time invariant
and growth velocity; factors
rapid convergence
Note: ‘Assumptions’ refer to the practical restrictions applied to equations (6a)–(6c) in standard applications
of the different estimators.
40
Table A2: Estimates of catch-up growth in HAZ scores (unconditional model)
Between Within Both
α0 β0 ραβ H0α0 H0β0 ? H0ραβ ? ?
Ethiopia POLS -1.383 0.005 . 0.00 0.96 N . . .
(0.023) (0.003)
CRE -1.389 0.007 -0.473 0.00 0.01 Y 0.00 Y Y
(0.023) (0.003) (0.023)
FEIS -1.381 0.003 -0.297 0.00 0.20 N 0.00 Y N
(0.023) (0.003) (0.023)
India POLS -1.438 -0.011 . 0.00 0.99 N . . .
(0.021) (0.003)
CRE -1.447 -0.009 -0.275 0.00 0.99 N 0.00 Y N
(0.022) (0.003) (0.023)
FEIS -1.439 -0.007 -0.191 0.00 0.99 N 0.00 Y N
(0.021) (0.003) (0.023)
Peru POLS -1.275 0.041 . 0.00 0.99 N . . .
(0.022) (0.002)
CRE -1.284 0.041 -0.073 0.00 0.00 Y 0.00 Y Y
(0.023) (0.002) (0.023)
FEIS -1.273 0.039 -0.029 0.00 0.00 Y 0.10 N N
(0.022) (0.002) (0.023)
Vietnam POLS -1.126 0.005 . 0.00 0.99 N . . .
(0.022) (0.002)
CRE -1.129 0.006 0.056 0.00 0.00 Y 0.99 N N
(0.022) (0.002) (0.023)
FEIS -1.126 0.016 0.036 0.00 0.00 Y 0.94 N N
(0.022) (0.002) (0.023)
Note: see Table 4; no additional covariates included in the models; probability estimates larger than 0.99 have been
rounded down to the latter value.
Source: own estimates.
41
Table A3: Goodness-of-ﬁt metrics, HAZ scores
Unconditional Conditional
Eth. Ind. Peru Viet. Eth. Ind. Peru Viet.
POLS estimates:
Time 0.04 0.01 0.01 0.01 0.33 0.19 0.16 0.19
RMSE 1.25 1.13 1.12 1.11 1.21 1.06 0.94 0.98
Adj. R2 0.00 0.00 0.02 0.00 0.07 0.11 0.28 0.21
AIC 2.4e+04 2.3e+04 2.3e+04 2.3e+04 2.3e+04 2.2e+04 1.7e+04 2.1e+04
CRE estimates:
Time 6.88 6.64 6.86 7.22 58.41 62.07 53.87 116.30
RMSE 0.64 0.49 0.44 0.40 0.52 0.39 0.33 0.40
AIC 2.2e+04 2.0e+04 1.8e+04 1.8e+04 2.1e+04 1.9e+04 1.4e+04 1.7e+04
FEIS estimates :
Time 0.41 0.46 0.58 0.52 0.46 0.55 0.68 0.58
RMSE 0.54 0.41 0.37 0.33 0.54 0.40 0.30 0.33
Adj. R2 0.62 0.74 0.78 0.82 0.62 0.74 0.81 0.82
AIC 1.2e+04 7.9e+03 6.4e+03 4.8e+03 1.2e+04 7.8e+03 2.9e+03 4.7e+03
Hausman . . . . 0.00 0.00 0.00 0.00
Note: Time is the execution speed of the regression command, in seconds; Hausman gives the probability
associated with a Hausman test where the null hypothesis is that the CRE estimator is consistent (and efﬁcient)
versus the alternative hypothesis which is that the FEIS estimator is consistent.
Source: own estimates.
42
Table A4: Size and velocity regressions, Ethiopia
Height (HAZ) BMI (BAZ)
Size Size Velocity Velocity Velocity Size Size Velocity Velocity Velocity
Mean -5.781∗∗∗ -5.774∗∗∗ -0.129∗∗∗ -0.129∗∗∗ -0.129∗∗∗ -8.988∗∗∗ -8.998∗∗∗ -1.328∗∗∗ -1.326∗∗∗ -1.327∗∗∗
(0.074) (0.074) (0.009) (0.010) (0.008) (0.107) (0.121) (0.021) (0.024) (0.022)
Female 0.375∗∗ 0.412∗∗∗ -0.027 -0.020 -0.019 -1.429∗∗∗ -1.475∗∗∗ -0.111∗∗∗ -0.126∗∗∗ -0.150∗∗∗
(0.155) (0.148) (0.019) (0.018) (0.019) (0.212) (0.247) (0.040) (0.037) (0.040)
Urban -0.265 1.096 0.014 0.111 0.115 -0.102 1.330 0.105 0.324 0.345
(0.258) (0.832) (0.031) (0.146) (0.136) (0.368) (1.681) (0.067) (0.335) (0.294)
Not ﬁrst born -0.032 -0.122 -0.018 -0.002 -0.002 -0.535 -0.273 0.187∗∗∗ 0.177∗∗∗ 0.173∗∗∗
(0.217) (0.190) (0.028) (0.026) (0.032) (0.354) (0.390) (0.063) (0.055) (0.056)
Majority ELG 0.358∗ 0.257 -0.038 0.027 0.028 -1.236∗∗∗ -0.179 -0.260∗∗∗ -0.084 -0.086
(0.183) (0.310) (0.023) (0.035) (0.033) (0.301) (0.433) (0.054) (0.081) (0.075)
Wealth index 1.143∗∗∗ 1.147∗∗∗ -0.005 -0.024 -0.020 -0.089 -0.569∗∗ -0.116∗∗∗ -0.083∗∗ -0.092∗∗
(0.130) (0.130) (0.018) (0.019) (0.017) (0.196) (0.244) (0.036) (0.038) (0.039)
Mother’s edu. 0.228∗ 0.186 -0.038∗∗ -0.029∗ -0.028∗ 0.168 -0.196 0.000 0.025 0.022
(0.117) (0.121) (0.016) (0.016) (0.017) (0.204) (0.231) (0.038) (0.037) (0.037)
43
Mother’s age 0.004 0.006 -0.004∗∗ -0.004∗∗ -0.004∗∗ 0.020 0.017 0.004 0.002 0.002
(0.014) (0.015) (0.002) (0.002) (0.002) (0.027) (0.023) (0.004) (0.004) (0.004)
Sibs 0-5 -0.495∗∗ -0.492∗∗ -0.104∗∗∗ -0.057∗∗ -0.059∗∗ -0.647∗∗ -0.399 -0.027 -0.198∗∗∗ -0.204∗∗∗
(0.212) (0.195) (0.025) (0.026) (0.025) (0.320) (0.326) (0.056) (0.050) (0.050)
Sibs 6-12 -0.063 -0.054 -0.062∗∗ -0.051∗ -0.052∗ -0.700∗∗ -0.796∗∗ 0.028 0.025 0.012
(0.178) (0.196) (0.029) (0.027) (0.027) (0.327) (0.332) (0.059) (0.048) (0.053)
Child FE (std.) - - - - -0.013 - - - - -0.086∗∗∗
(0.011) (0.029)
Obs. 1,845 1,845 1,845 1,845 1,845 1,846 1,846 1,846 1,846 1,846
RMSE 3.10 3.03 0.39 0.38 0.38 5.23 5.01 0.91 0.82 0.81
Adj. R2 0.13 0.17 0.02 0.09 0.09 0.05 0.13 0.04 0.22 0.23
WI×2 + Mean -3.50 -3.48 -0.14 -0.18 -0.17 -9.17 -10.14 -1.56 -1.49 -1.51
Community FE? No Yes No Yes Yes No Yes No Yes Yes
signiﬁcance: ∗ 10%, ∗∗ 5%, ∗∗∗ 1%.
Note: table reports selected coefﬁcient estimates from auxiliary analyses, where the dependent variable is denoted in the column header and is derived from the FEIS
estimates (Tables 5 and 6); individual children are the unit of observation and all covariates are taken as averages across rounds (demeaned); standard errors are based on 100
bootstrap iterations; community ﬁxed effects are included as denoted in the ﬁnal row.
Source: own estimates.
Table A5: Size and velocity regressions, India
Height (HAZ) BMI (BAZ)
Size Size Velocity Velocity Velocity Size Size Velocity Velocity Velocity
Mean -6.061∗∗∗ -6.061∗∗∗ -0.222∗∗∗ -0.222∗∗∗ -0.222∗∗∗ -9.682∗∗∗ -9.681∗∗∗ -0.224∗∗∗ -0.223∗∗∗ -0.222∗∗∗
(0.065) (0.066) (0.009) (0.009) (0.008) (0.178) (0.139) (0.036) (0.039) (0.033)
Female 0.272∗∗ 0.195 0.010 0.011 0.008 -0.087 -0.138 0.061 0.044 0.067
(0.132) (0.131) (0.017) (0.015) (0.017) (0.272) (0.257) (0.069) (0.073) (0.059)
Urban -0.163 -0.112 0.065∗∗∗ -0.024 -0.023 0.730 -0.843 0.123 -0.340 -0.200
(0.203) (0.674) (0.024) (0.087) (0.094) (0.500) (1.569) (0.091) (0.315) (0.298)
Not ﬁrst born -0.476∗∗∗ -0.390∗∗ -0.048∗∗ -0.047∗∗ -0.042∗ -0.249 -0.326 -0.048 -0.071 -0.017
(0.165) (0.164) (0.022) (0.022) (0.022) (0.333) (0.385) (0.069) (0.090) (0.059)
Majority ELG -0.255∗∗ -0.133 0.000 -0.015 -0.014 -1.010∗∗∗ -0.540 -0.284∗∗∗ -0.169∗∗∗ -0.079
(0.130) (0.140) (0.017) (0.017) (0.014) (0.311) (0.329) (0.072) (0.057) (0.058)
Wealth index 0.662∗∗∗ 0.642∗∗∗ 0.047∗∗∗ 0.060∗∗∗ 0.052∗∗∗ 1.458∗∗∗ 1.312∗∗∗ 0.088 0.072 -0.146
(0.101) (0.108) (0.011) (0.013) (0.013) (0.250) (0.245) (0.067) (0.065) (0.126)
Mother’s edu. 0.223∗∗ 0.170∗ -0.002 -0.008 -0.010 0.487 0.487 0.144 0.139 0.059
(0.107) (0.099) (0.011) (0.012) (0.013) (0.366) (0.319) (0.101) (0.099) (0.038)
44
Mother’s age 0.046∗∗∗ 0.050∗∗∗ -0.006∗∗∗ -0.006∗∗ -0.006∗∗∗ -0.006 -0.000 -0.002 -0.002 -0.002
(0.015) (0.018) (0.002) (0.002) (0.002) (0.036) (0.033) (0.008) (0.008) (0.007)
Sibs 0-5 -0.038 0.029 0.048∗ 0.051∗∗ 0.051∗∗ -0.068 -0.116 -0.102 -0.148 -0.128
(0.223) (0.217) (0.027) (0.025) (0.025) (0.587) (0.538) (0.108) (0.146) (0.104)
Sibs 6-12 -0.015 0.090 0.019 0.003 0.002 -0.491 -0.189 -0.132∗∗ -0.092 -0.061
(0.195) (0.196) (0.023) (0.023) (0.019) (0.425) (0.380) (0.063) (0.065) (0.070)
Child FE (std.) - - - - 0.040∗∗∗ - - - - 1.260∗∗
(0.009) (0.511)
Obs. 1,913 1,913 1,913 1,913 1,913 1,913 1,913 1,913 1,913 1,913
RMSE 2.90 2.86 0.35 0.33 0.33 7.30 7.28 1.79 1.79 1.32
Adj. R2 0.08 0.11 0.05 0.13 0.14 0.07 0.08 0.02 0.03 0.47
WI×2 + Mean -4.74 -4.78 -0.13 -0.10 -0.12 -6.77 -7.06 -0.05 -0.08 -0.51
Community FE? No Yes No Yes Yes No Yes No Yes Yes
signiﬁcance: ∗ 10%, ∗∗ 5%, ∗∗∗ 1%.
Note: table reports selected coefﬁcient estimates from auxiliary analyses, where the dependent variable is denoted in the column header and is derived from the FEIS
estimates (Tables 5 and 6); individual children are the unit of observation and all covariates are taken as averages across rounds (demeaned); standard errors are based on 100
bootstrap iterations; community ﬁxed effects are included as denoted in the ﬁnal row.
Source: own estimates.
Table A6: Size and velocity regressions, Peru
Height (HAZ) BMI (BAZ)
Size Size Velocity Velocity Velocity Size Size Velocity Velocity Velocity
Mean -5.354∗∗∗ -5.351∗∗∗ 0.098∗∗∗ 0.097∗∗∗ 0.097∗∗∗ 8.332∗∗∗ 8.336∗∗∗ 0.343∗∗∗ 0.343∗∗∗ 0.343∗∗∗
(0.079) (0.071) (0.009) (0.009) (0.010) (0.181) (0.209) (0.036) (0.037) (0.034)
Female -0.094 -0.125 0.038∗∗ 0.031 0.031 -1.335∗∗∗ -1.333∗∗∗ -0.092 -0.089 -0.038
(0.125) (0.145) (0.019) (0.019) (0.021) (0.391) (0.376) (0.070) (0.086) (0.072)
Urban 0.367∗ -0.211 -0.022 0.011 0.011 -0.982∗ -1.421∗∗ 0.164 0.050 0.106
(0.212) (0.257) (0.030) (0.038) (0.032) (0.558) (0.657) (0.104) (0.130) (0.133)
Not ﬁrst born 0.045 -0.063 -0.033 -0.035 -0.035 -0.437 -0.715 -0.003 -0.021 0.007
(0.155) (0.189) (0.028) (0.025) (0.025) (0.555) (0.516) (0.100) (0.087) (0.093)
Majority ELG 0.355∗ 0.069 -0.043 -0.036 -0.036 -0.546 -0.099 0.117 0.153 0.157
(0.187) (0.167) (0.027) (0.032) (0.027) (0.572) (0.532) (0.105) (0.129) (0.121)
Wealth index 0.913∗∗∗ 0.824∗∗∗ 0.022∗ 0.048∗∗∗ 0.050∗∗∗ 2.107∗∗∗ 1.619∗∗∗ 0.362∗∗∗ 0.248∗∗∗ 0.185∗∗∗
(0.084) (0.100) (0.012) (0.018) (0.015) (0.248) (0.279) (0.045) (0.061) (0.056)
Mother’s edu. 0.530∗∗∗ 0.471∗∗∗ -0.010 -0.008 -0.007 0.435 0.454∗ -0.012 0.001 -0.017
(0.091) (0.084) (0.013) (0.014) (0.014) (0.302) (0.269) (0.054) (0.061) (0.059)
45
Mother’s age 0.019∗ 0.024∗∗ -0.004∗∗ -0.004∗∗ -0.004∗∗ 0.040 0.046 -0.013∗∗ -0.011∗ -0.013∗
(0.011) (0.011) (0.002) (0.002) (0.002) (0.037) (0.032) (0.006) (0.006) (0.006)
Sibs 0-5 0.076 0.087 -0.076∗∗∗ -0.076∗∗∗ -0.076∗∗∗ 0.003 -0.140 -0.270∗∗∗ -0.289∗∗∗ -0.283∗∗∗
(0.167) (0.172) (0.024) (0.027) (0.023) (0.508) (0.450) (0.102) (0.105) (0.093)
Sibs 6-12 -0.669∗∗∗ -0.496∗∗∗ 0.023 -0.002 -0.003 -0.523 0.063 0.107 0.154 0.151
(0.192) (0.184) (0.035) (0.027) (0.031) (0.512) (0.457) (0.105) (0.104) (0.116)
Child FE (std.) - - - - -0.007 - - - - 0.354∗∗∗
(0.015) (0.057)
Obs. 1,916 1,916 1,916 1,916 1,916 1,921 1,921 1,921 1,921 1,921
RMSE 2.85 2.78 0.42 0.41 0.41 8.67 8.28 1.63 1.62 1.59
Adj. R2 0.31 0.35 0.02 0.05 0.05 0.10 0.17 0.11 0.13 0.16
WI×2 + Mean -3.53 -3.70 0.14 0.19 0.20 12.55 11.57 1.07 0.84 0.71
Community FE? No Yes No Yes Yes No Yes No Yes Yes
signiﬁcance: ∗ 10%, ∗∗ 5%, ∗∗∗ 1%.
Note: table reports selected coefﬁcient estimates from auxiliary analyses, where the dependent variable is denoted in the column header and is derived from the FEIS
estimates (Tables 5 and 6); individual children are the unit of observation and all covariates are taken as averages across rounds (demeaned); standard errors are based on 100
bootstrap iterations; community ﬁxed effects are included as denoted in the ﬁnal row.
Source: own estimates.
Table A7: Size and velocity regressions, Vietnam
Height (HAZ) BMI (BAZ)
Size Size Velocity Velocity Velocity Size Size Velocity Velocity Velocity
Mean -4.709∗∗∗ -4.709∗∗∗ -0.012∗ -0.012∗ -0.012∗ -3.433∗∗∗ -3.429∗∗∗ -0.181∗∗∗ -0.181∗∗∗ -0.182∗∗∗
(0.060) (0.071) (0.007) (0.006) (0.007) (0.157) (0.183) (0.026) (0.026) (0.023)
Female 0.187 0.139 0.005 0.006 0.001 -1.764∗∗∗ -1.720∗∗∗ -0.159∗∗∗ -0.162∗∗∗ -0.016
(0.123) (0.135) (0.014) (0.013) (0.013) (0.348) (0.343) (0.049) (0.048) (0.039)
Urban 1.313∗∗∗ 0.843 -0.001 0.013 -0.016 4.646∗∗∗ 6.147∗∗∗ 0.177∗∗ 0.195 -0.327
(0.213) (0.653) (0.023) (0.087) (0.090) (0.602) (1.826) (0.076) (0.279) (0.235)
Not ﬁrst born -0.451∗∗ -0.383∗∗ -0.019 -0.017 -0.004 -1.131∗∗ -1.125∗∗ -0.154∗∗ -0.138∗∗ -0.043
(0.182) (0.163) (0.019) (0.020) (0.018) (0.478) (0.477) (0.070) (0.063) (0.061)
Majority ELG 1.272∗∗∗ 0.294 0.005 -0.050 -0.060 -1.473∗∗∗ 2.536∗∗ 0.013 -0.285∗ -0.500∗∗∗
(0.183) (0.318) (0.019) (0.053) (0.052) (0.452) (1.053) (0.062) (0.171) (0.176)
Wealth index 0.648∗∗∗ 0.893∗∗∗ 0.054∗∗∗ 0.057∗∗∗ 0.027∗∗ 0.948∗∗∗ 1.550∗∗∗ 0.226∗∗∗ 0.291∗∗∗ 0.159∗∗∗
(0.107) (0.108) (0.012) (0.014) (0.012) (0.266) (0.311) (0.040) (0.045) (0.036)
Mother’s edu. 0.567∗∗∗ 0.580∗∗∗ 0.023∗∗ 0.014 -0.006 0.815∗∗∗ 0.808∗∗ -0.010 -0.002 -0.071∗∗
(0.108) (0.104) (0.010) (0.012) (0.013) (0.313) (0.316) (0.044) (0.043) (0.036)
46
Mother’s age -0.017 -0.024 -0.006∗∗∗ -0.005∗∗∗ -0.004∗∗ 0.040 -0.001 0.012∗∗ 0.009∗ 0.009∗∗
(0.014) (0.015) (0.001) (0.001) (0.002) (0.037) (0.034) (0.005) (0.005) (0.004)
Sibs 0-5 -0.895∗∗∗ -0.459 -0.026 -0.013 0.003 0.528 0.324 -0.136 -0.091 -0.119
(0.284) (0.293) (0.031) (0.034) (0.030) (0.687) (0.729) (0.100) (0.088) (0.097)
Sibs 6-12 -0.434 0.022 -0.055∗∗ -0.046∗ -0.046∗∗ -1.097 0.212 -0.230∗∗∗ -0.104 -0.122
(0.267) (0.289) (0.027) (0.026) (0.023) (0.684) (0.638) (0.087) (0.090) (0.087)
Child FE (std.) - - - - 0.115∗∗∗ - - - - 0.684∗∗∗
(0.010) (0.028)
Obs. 1,908 1,908 1,908 1,908 1,908 1,909 1,909 1,909 1,909 1,909
RMSE 2.85 2.74 0.30 0.30 0.28 7.47 7.19 1.07 1.04 0.85
Adj. R2 0.29 0.35 0.07 0.10 0.19 0.14 0.20 0.10 0.14 0.44
WI×2 + Mean -3.41 -2.92 0.10 0.10 0.04 -1.54 -0.33 0.27 0.40 0.14
Community FE? No Yes No Yes Yes No Yes No Yes Yes
signiﬁcance: ∗ 10%, ∗∗ 5%, ∗∗∗ 1%.
Note: table reports selected coefﬁcient estimates from auxiliary analyses, where the dependent variable is denoted in the column header and is derived from the FEIS
estimates (Tables 5 and 6); individual children are the unit of observation and all covariates are taken as averages across rounds (demeaned); standard errors are based on 100
bootstrap iterations; community ﬁxed effects are included as denoted in the ﬁnal row.
Source: own estimates.
A.2 Figures
Figure A1: Trends in height-for-age z-scores (HAZ), by country
Ethiopia India
2
0
-2
-4
Peru Vietnam
2
0
-2
-4
1 3 5 7 9 11 1 3 5 7 9 11
2 4 6 8 10 12 2 4 6 8 10 12
Age (years)
5th Median 95th
Notes: trends estimated via quantile regression.
Source: own estimates
47
Figure A2: Trends in height-for-age ratios (HAR), by country
Ethiopia India
5
0
-5
-15 -10
Peru Vietnam
5
0
-5
-15 -10
1 3 5 7 9 11 1 3 5 7 9 11
2 4 6 8 10 12 2 4 6 8 10 12
Age (years)
5th Median 95th
Notes: trends estimated via quantile regression.
Source: own estimates
48
Figure A3: Trends in height-for-age differences (HAD), by country
Ethiopia India
10
0
-10
-20
Peru Vietnam
10
0
-10
-20
1 3 5 7 9 11 1 3 5 7 9 11
2 4 6 8 10 12 2 4 6 8 10 12
Age (years)
5th Median 95th
Notes: trends estimated via quantile regression.
Source: own estimates
49
Figure A4: Trends in BMI-for-age z-scores (BAZ), by country
Ethiopia India
2
0
-2
-4
Peru Vietnam
2
0
-2
-4
1 3 5 7 9 11 1 3 5 7 9 11
2 4 6 8 10 12 2 4 6 8 10 12
Age (years)
5th Median 95th
Notes: trends estimated via quantile regression.
Source: own estimates
50
Figure A5: Trends in BMI-for-age ratios (BAR), by country
Ethiopia India
40
20
0
-20
Peru Vietnam
40
20
0
-20
1 3 5 7 9 11 1 3 5 7 9 11
2 4 6 8 10 12 2 4 6 8 10 12
Age (years)
5th Median 95th
Notes: trends estimated via quantile regression.
Source: own estimates
51
Figure A6: Trends in BMI-for-age differences (BAD), by country
Ethiopia India
10
5
0
-5
Peru Vietnam
10
5
0
-5
1 3 5 7 9 11 1 3 5 7 9 11
2 4 6 8 10 12 2 4 6 8 10 12
Age (years)
5th Median 95th
Notes: trends estimated via quantile regression.
Source: own estimates
52