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New approaches to estimating the childhealth-parental income relationship.∗
Brenda Gannona David Harrisb Mark. N. Harrisc
Leandro M. Magnussond Bruce Hollingsworthe Brett Inderb
Pushkar Maitrab Luke Munforda
October 8, 2015
Abstract
This paper exploits two new alternative approaches to estimate the childhealth-parental income gradient, using both a threshold model and a moreparsimonious random parameters model, applied to the Health Survey forEngland data 2008-2012. We build on previous research and test the ap-propriateness of the usual standard age categories (0-3, 4-8, 9-12 and 13-17)exploited in the literature and for policy intervention. Our threshold methodestimates different age categories and higher income gradient for children agedbetween 6 and 8 years old. We further extend our analysis to allow for cohorteffects. We find that a higher income is required to improve young children’shealth aged 0-2 post 2010. We discuss the relevant reasons and policy impli-cations – most notably that there are socioeconomic child health inequalitiesexasperated by the recent recession and inequity in the distribution of healthinterventions towards those most in financial need.
JEL Classification: I14, I18, C24, C25.
aUniversity of Manchester, UK;
bMonash University, Australia;
cCurtin University, Australia;
dUniversity of Western
Australia, Australia;eUniversity of Lancaster, UK.
∗We would like to thank seminar participants at Curtin University for useful comments andsuggestions. We would also like to thank Yiu-Shing Lau for research assistance. Finally, we thankthe Australian Research Council for their generous support. The usual caveats apply.
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1 Introduction and Background
The study of child health and parental income has received much attention in recent
years. A good start to a person’s health may well be influenced by their parents’
income. The mechanism through which parental income may affect child health is
theoretically represented by the Grossman (2000) model in which health is viewed as
an investment that is produced by families using health inputs. Utility is maximised
subject to a production function, prices and budget constraints. The mechanism
from income may occur in three ways. First, an endowment of health when born
that depreciates over time (Case, Lubotsky, and Paxson (2002)). Second, access to
health care; if children from poorer families in deprived areas are less able to access
medical care, then a policy for improved access to health care services is required,
see Apouey and Geoffard (2013). Third, education; it has been shown that poor
health in childhood is associated with lower lifetime human capital accumulation
(low educational attainment and poor lifetime health outcomes), and, consequently
poor labor market outcomes as adults (Currie and Hyson (1999), Case, Lubotsky,
and Paxson (2002), Currie (2004)).
Much of the recent literature using data from a number of different countries has
been concerned with whether the child health-income gradient varies by age of the
child or not (Case, Lubotsky, and Paxson (2002), Cameron and Williams (2009),
Park (2010), Currie, Shields, and Price (2007), Khanam and Nghiem (2009), Case,
Lee, and Paxson (2008)). Focussing on the evidence from UK, there appears to be
no consensus among researchers as to whether household income has any effect on
child health, and if it does, whether the effect varies with the child’s age. Currie,
Shields, and Price (2007) using the pooled data from the 1997 to 2002 Health Surveys
for England (HSE) also find a statistically significant effect on child health but the
gradient is smaller compared to that found in the US and furthermore they find no
evidence that this gradient increases for older children - in fact they find it decreases
after age 8. They conclude that, at least in the context of England, family income
is not a major determinant of child health. Similar results are obtained by West
(1997) using data from the 1991 British census and Burgess, Propper, and Rigg
(2004), using a regional cohort of children drawn from the Avon Longitudinal Study
of Parents and Children. More recently, however, Case, Lee, and Paxson (2008)
have re-examined the US and English data, comparing similar data (in terms of
time periods). They use the same methods as Currie, Shields, and Price (2007) and
find that the income gradient in children’s health increases with age by the same
amount in the two countries. Their main finding is that the gradient increases up
to age 12.
One important point, common across all of the existing literature, is that the age
bands are exogenously fixed. For very recent examples of this, see Kruk (2013) and
Fletcher and Wolfe (2014). Typically the extant literature has used the following age
groupings 0–3, 4–8, 9–12 and 13 and higher. There is, however, no particular medical
or theoretical reason for choosing these particular age bands. Indeed Apouey and
Geoffard (2013) argue that using those set age groups does not describe the evolution
of the health income gradient with age. The authors state that this knowledge is
crucial for implementing an appropriate health policy intervention that aims to
mitigate social inequalities. They therefore examine the evolution of the gradient
between ages instead of just across age groups. They achieve this by estimating
linear probability models, for good health versus fairly good or poor health, with 17
different interaction terms of income and age.
We build on the existing literature to estimate more flexible models to test whether
the income-health relationship varies with age by allowing the data to endogenously
determine the age groups. We hypothesize that the use of predetermined age groups
is giving a incomplete picture of the relationship between income and child health.
We also suggest a more parsimonious and generalised approach by treating the in-
come effect as a random parameter, where the randomness is related to the actual
age of the child. Indeed, our results show that the gradient is highly heterogenous,
but here is represented in a very parsimonious way: instead of adding dummy vari-
ables for each age (or age group), this method involves estimating only one additional
parameter. This is a novel use of random parameter models, mimicking threshold
effects.
3
We apply our models to the Health Survey for England (HSE) 2008–2012 data, a
repeated cross section of data. The initial results using pooled HSE dataset and
individual years are quite divergent. We find only one or at most two thresholds
when using year specific data, but three thresholds when we use pooled data, i.e.
the income effects are different when using pooled data instead of year data. This is
an important finding that is critical to address further, since it suggests that there
may be some other threshold effects involved, e.g. if children born from different
cohorts require more income than others. So we adjust our threshold model to also
search for cohort effects. Our findings clearly suggest the importance to condition
on both unknown age and cohort effects.
Overall, our results show that allowing the age groupings to be endogenously deter-
mined makes a difference. With the standard exogenously imposed groupings, there
is evidence of a gradient up to age 4 and at the same level up to age 8, that decreases
slightly after age 8 and even more so after age 12. This could lead policy makers
to conclude that the only major difference is for those before age 8, where the rela-
tionship between health and income is then stronger. However, with endogenously
determined age groups (firstly without the additional cohort effect), the gradient
change occurs at different points, at age 6, then 8 and 12. More interestingly, we
find a higher gradient for those born in 2010 or later suggesting that the income
effect for those aged 2 and under varies according to the child’s cohort. The results
indicate that the effect of income is higher among the 6–8 age group, compared to
the 0–6 age group. This is a significant finding because most children will commence
school around age 6. It suggests a significant role for policy to attenuate the effect
of income on health. While we find similar results after age 8, compared to the
previous literature, the heterogenous effect below age 8 would have been completely
masked if we used the standardised age bands. The results indicate no major impact
of income on health, and in a health system where there is free GP care, this is an
encouraging result. Furthermore, our results confirm those of previous authors (e.g.
Currie, Shields, and Price (2007) and Case, Lee, and Paxson (2008)), along with
our additional finding for those aged under 6.1
1Currie, Shields, and Price (2007) find coefficients of -0.146, -0.212, -0.196 and -0.174 at ages
4
One of our most interesting findings is that there is a differential effect at age 0–2 in
2010 and after. We argue that this is most likely due to macroeconomic conditions
arising from the financial crisis and rising inequality. The results post 2010 indicate
there is an important difference in income required to improve young children’s
health, from birth to age 2 since 2010. This finding is a significant new contribution
to the literature and a very promising indicator for policy implications since 2010
and going forward.
2 Methodology
The ordered probit forms the basis of most research in this area, for example, see
Currie, Shields, and Price (2007). As a starting point, consider a standard latent
regression for parent assessed child health (H∗) of the form
H∗i = x′iβ + εi (1)
where x are a standard set of controls (with no constant term, and for the time-
being omitting income), β is a vector of unknown coefficients and ε a standard
normally distributed random disturbance term. In the usual ordered probit (OP)
set-up, latent health (H∗) is translated into observed reported health H, with j =
0, ..., J − 1 reported outcomes (where J is the total number of outcomes), via the
standard ordered probit mapping H∗ to the inherent boundary parameters µ =
µ1, ..., µJ−1, µ1 < µ2< . . . < µJ−1 in the ordered probit model, see Greene and
Hensher (2010).
The usual approach in the literature is to include (log) parental income into equation
(1), such that
H∗i = x′iβ + γ ln yi + εi (2)
less than 3, 4-8, 9-12 and 13+ and conclude there is no major impact of income on child health.Case, Lee, and Paxson (2008) then find a gradient up to age 12, with coefficients -0.141, -0.207,-0.229 and -0.180 in the same age groups. Again, this shows there is not a major gradient in theUK. Later on, we will show how we find coefficients of -0.169, -0.186, -0.164 and -0.136 at ages lessthan 6, 6-8, 9-12 and age 13+.
5
but importantly to split the ln y variable into four groups such that
H∗i = x′iβ +4∑
m=1
γm (ln yi ×Dm) + εi (3)
where the respective indices m = 1, . . . , 4, represent the age bands of the child,
which are 0-3, 4-8, 9-12, 13-17; and D1 is an indicator function for whether the child
is aged between 0 and 3 (and so on), such that∑
m (ln yi ×Dm) ≡ ln yi. This latter
restriction can be simply imposed by specifying D4 = 1 − D1 − . . . − D3. Finally,
any differences across γm are then taken as differential effects of parental income
according to the age of the child as in Case, Lubotsky, and Paxson (2002), and
Currie, Shields, and Price (2007). 2
2.1 Approach 1: Threshold effects
The child health-income gradient relationships is essentially a threshold model,
where these thresholds are determined by the age of the child. The thresholds cap-
ture the possbile several discontinuities in the linear relation between child health H∗
and parental income ln y. The literature invariably imposes the number of thresh-
olds values (3) and their location a priori at ages 0-3, 4-8, 9-12 and 13-17. What
we suggest in this paper is to estimate the number of thresholds (if any), together
with their position. Clearly, any erroneous imposition of thresholds values and/or
incorrect positions of the thresholds may likely lead to biased results and inaccurate
policy inference.
Gannon, Harris, and Harris (2014) recently consider the issue of determining both
the number and position of any thresholds in a nonlinear model (such as the ordered
probit model as appropriate for our ordered, categorical, dependent variable). Their
paper shows how the combination of grid-search techniques along with the use of
information criteria addresses this issue. Full details of the suggested procedure are
2It could be argued that there are also age effects for other socioeconomic indicators such aseducation, and therefore these should also be included as differential effects on health. The pointof this paper, however, is to look at child health-parental income inequality only and to obtain aconsistent comparison across studies. Therefore we do not explore this route.
6
provided in Gannon, Harris, and Harris (2014); however, here we summarise the
essential ideas. Their procedure involves estimating parameters and thresholds in
models of the form
H∗i = x′iβ +M∑m=1
γm (ln yi ×Dm) + εi (4)
and to optimally choose the M − 1 thresholds (τ1 < . . . < τM−1) that define the
dummy variables Dm. Let M∗ be the chosen hypothesized number of thresholds.
We estimate all possible m∗ = 0, 1, . . . ,M∗ threshold models and choose the one
that minimises the BIC (the Bayesian Information Criteria). In practice, Gannon,
Harris, and Harris (2014) suggest a sequential procedure, starting with a small M∗
and increasing its value if necessary. We reinforce here, the procedure simultaneously
searches for all thresholds and selects optimal values for both the number and position
of these. Thus this approach is ideally suited to estimating such thresholds in the
child health-parental income relationship.
2.2 Approach 2: A random parameters approach
As an alternative approach and for robustness, we now consider augmenting equation
(??) with income as per equation (??) but replacing γ with γt such that
H∗i = x′iβ + γt ln yi + εi. (5)
Importantly, we allow the coefficient on ln y to vary by child age group t (we have
t = 0, . . . , 17) such that
γt = γ + αt (6)
where αt are assumed to be random draws from N (0, σ2α) and γ is the average effect
of income.
This model can be estimated by simulation, where we draw r = 1, . . . , R normal
variates of αt from N (0, σ2α).3 This method can be interpreted as the generalization
3In estimation we use a Halton sequence (Train (2003)) of length R = 1, 000.
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of threshold approach outlined above, with the advantage of being more “parsimo-
nious”, as estimation requires only one additional parameter, namely σ2α.
Note that such an approach is akin to a panel data set-up, where t indexes the panel
and not i, and where each t group will have potentially different sample sizes (Nt).
Moreover due to the dependence within each t category arising from the common
αt, the likelihood for a group of t observations is now the product of the sequence
of the OP probabilities corresponding to the observed health outcome, where these
depend on the rth draw of αt, αrt :
pi(β, σ2
α, αrt
)=J−1∏j=0
[Pr (Hi = j |ln yi,xi, αrt )]dij . (7)
The simulated log likelihood function is therefore
`∗(β, γ, σ2
α
)=
T∑t=0
log
{1
R
R∑t=1
[Nt∏i=1
pi(β, σ2
α, αrt
)]}. (8)
Ex post, conditional on the data, age group-specific estimates of γt are available as
(Train (2003); Greene (2007))
γt = γ +1
R
R∑r=1
αrtωrt , where ωrt =
Nt∏i=1
pi
(β, σ2
α, αrt
)1R
∑Rr=1
[Nt∏i=1
pi
(β, σ2
α, αrt
)] , (9)
which are draws from the conditional distribution of γt.
We also consider an augmented RP approach. Due to the temporal proximity of the
γt parameters, it is likely that neighbouring ones will be related. That is, a priori
one would expect the differing effects to evolve slowly over time, and not necessarily
be subject to abrupt jumps at integer values of the child’s age. To this end, we
extend equation (6) such that
γt = γ + αt (10)
αt = ραt−1 + ut.
8
This extension to RP modelling is, to the best of the authors’ knowledge, original,
and the first time applied in the literature.
3 Data
The data are from the Health Surveys for England (HSE) pooled over the years
2008 − 2012. This is a series of repeated cross-section annual surveys designed to
measure health and health related behaviours in adults and children. The data are
collected at the household level and provide much information about relationships
between individuals in the household. For each household, we matched data from
the children with that obtained from both parents. The survey interviews adults and
a maximum of two randomly selected children from each household. For children
below the age of 13 the parents are questioned with the child present. The data
include a core and a boosted sample - the latter includes a further sample of children
only, so we only use the core sample in our analysis as there is no information on
their parents’ income and other relevant characteristics in the boosted sample.
To measure child health, as with most of the literature, we utilise the standard
parent-assessed general health variable, (1) very good; (2) good; (3) fair; (4) bad; and
(5) very bad. Due to the low number of responses to (4) and (5), the proportions in
the lower categories are combined to give us 4 categories overall, which are decreasing
in health, although we note that our techniques are similarly applicable to other
outcome measures.
For the income variable, following previous literature, for example, Currie, Shields,
and Price (2007), we use current total pre-tax annual family income, measured in
31 bands, from < £520 to > £150,000 as illustrated in Table 1. Using the midpoint
of these bands, we then obtain the log of income which is deflated to 2005 prices,
resulting in a pseudo-continuous measure of family income. We also follow recent
literature, as Khanam and Nghiem (2009) and references therein, and include a
relatively standard set of controls: sex of child, ethnicity, log of household size,
age of mother and father, indicator of absence of father from household, mother’s
9
and father’s education, mother’s and father’s employment status. The full set of
summary statistics are presented in Table 2.
Our sample consists of 9,613 observations in total, in which less than 1% of the
children are in bad or very bad health, 4% are in fair health, whilst 33% and 62%
are, respectively, in good and very good health; average child age is just under 8;
and 50% percent are male; average annual pre-tax income is some £25,015; and
average number of people in the household is just under 4.
4 Results
4.1 Threshold Models
To investigate the threshold effects, we pool the data for all years from 2008 to 2012
and we include year dummies to capture any macroeconomic effects. We include
all age groups into one model, each interacted with income. Other authors have
run separate regressions for age group, included an income variable with no age
interaction. We do not include the age variable by itself in the regressions due to
collinearity. In all models, we estimate income at constant prices. For the imposed
regimes we allow for one, four, and seventeen regimes. These regimes are consistent
with the majority of the existing literature, i.e. the four regime imposed model has
thresholds at 3, 8, and 12 years of age.4 Our results are presented in Table 3 and
Figure 1.
Insert Figure 1 about here
In Table 3, we present the log-likelihood scores along with the information criteria,
BIC.5 We also present the location of the thresholds (τ) and the estimated income
4M* is reached once we have obtained our optimal model, based on BIC criteria. We recom-mended searching for one extra regime after the optimal is achieved.
5Note that we do not report t-stats or standard errors in the Tables, for ease of exposition.These are available on request. Instead, in our Tables we simply report the significance as *** etc.
10
coefficients for the various regimes.6 Note that in each column of the estimated
regime block (m = 1, .., 5), the results are those for the optimal M∗; for example,
the column M = 4 considers all possible models in which there are 3 thresholds
which generate four income gradient effects.
In light of the previous literature, a researcher may then wish to consider the existing
exogenously determined age groupings, with thresholds at 3, 8, and 12. In this case,
the model suggest that there is a positive, and statistically significant, relationship
between family income and child health. Recall child health is an ordinal response,
where lower values relate to better self-assessed health. We observe roughly equal
coefficients up to age 12 (all approximately -0.17), and then a reduction thereafter.
Under our threshold approach, if we assume only one threshold, our findings indicate
that the single threshold should be located at age 12. Allowing three regimes adds
an additional threshold at age 14, four regimes indicates a further threshold at age
6, and finally allowing five regimes suggests the first threshold should occur at age
1.7 Comparing among only the endogenously estimated thresholds, our findings
show that M = 4 is the optimal number of regimes according to BIC, and that
the thresholds should be located at age 6, 8, and 12. These results are essentially
consistent with the imposed regimes model, the only difference being the estimated
regime model indicates the lowest threshold should be at age 6, and not at age 3 (as
assumed in the imposed regime). The remaining two thresholds located at 8 and 12
are the same as imposed thresholds. We also observe that the BIC for the estimated
regimes is lower than the corresponding BIC for the imposed regimes, indicating
that the estimated (regardless of how may regimes we specify) outperform the fixed
threshold regressions.
These are promising results from our threshold models but we will later show how
introducing cohort effects will increase this disparity between our endogenous model
6For reasons of space we only present the coefficients for the income variables. Full coefficientresults are available on request, but are in general accordance with those expected and found inthe previous literature.
7Whilst having a threshold at age 1 does not make much intuitive sense, this is where the datasuggests the first threshold should go to minimise BIC.
11
and the standard exogenous models even further. So far our results should be
reassuring to researchers in this field of child income and health, that the standard
exogenous categories appear to be relevant for the policy intervention perspective.
Next, we present results from the parsimonious random parameter approach.
4.2 Random Parameters Models
The full set of results from our standard random parameter models without cohort
effects are presented in Table 4. Our key results relate to the estimates of γ and σα.
We find that the RP and RP-AR(1) show the similar estimates for these parameters:
ˆγ=-0.158 and σα=0.018 (t-stats: -8.307 and 5.513) for the RP and ˆγ=-0.156 and
σα=0.016 for the RP-AR(1) model (t-stats: -8.229 and 6.044). Thus, once again,
income has a positive effect on health.
However, importantly for us, there is clear evidence of an income effect varying
with child age. Figure 2 plots the age specific effect from the estimated thresholds
and random parameters (RP) models. Results vary by age but the general trend
is consistent with our threshold model results: the ages 0-3 and 4-8 show a higher
income effect overall. There is some disparity at age 6-8, showing a higher income
effect, and this is consistent with results from our threshold model. Then after age
12, the effect is lower again. It is interesting to note that both our parsimonious
and threshold models provide similar results.
Insert Figure 2 about here
Note that both the RP and RP-AR(1) models provide similar results, as shown
in Table 4 and Figure 2.8 This indicates that temporal proximity between the
parameters is not a huge issue in this case, but nonetheless this innovative approach
should be taken into account in the model. At this point, we show consistent results
between the threshold model and RP models, but bearing in mind that we may need
8Full results on the income coefficients are available upon request.
12
to adjust for cohort differences to provide a full and precise testing of the exogenous
categories in our analysis.
5 Stability of the gradient across time
In order to study if the gradient is stable across time, we run separate regressions
for each year. In Tables 5 to 9 we present the results from the our threshold models
(Estimated Regimes) for each separate year 2008 to 2012 together with the results
from the traditional benchmark models (Imposed Regimes).
The 2008 data suggest that the optimal model has two regimes, with the threshold
occuring at age 12. Consistent with the literature, we observe stronger effects of
income on health for younger children (coefficients of 0.171> 0.138). For the years
2009, 2010 and 2012 (Tables 6 to 9) we again find support for the model with two
estimated age-groups as the optimal model, while for 2011 the optimal model has
three age-groups.
Whilst the finding of one threshold is consistent for almost all the years, the mag-
nitudes of the gradient as well as the age at which the threshold occurs, however,
are not. For example, in 2008 we find evidence to suggest that age 12 is where the
change occurs, whereas this is age 8 in 2009, age 12 in 2010, ages 1 and 9 in 2011, and
age 13 in 2012. The period that we consider, 2008-2012, include the recent financial
crisis, and hence income is likely to fluctuate over the years, although we deflated
income to 2005 and controlled by year dummies to help alleviate this limitation.
6 Cohort Effects
6.1 Extensions for cohort effects
Estimating our models both on the pooled data set, and then on the individual years,
gave quite distinct results, even after for controlling for year effects, indicating the
13
possible presence of cohort effects. Theoretically, the question is “does the child
health-parental income relationship remain stable or shift over time”? The presence
of cohort effects would imply an unstable relationship in the gradient due to, for
example, a macroeconomic effect such as the financial crisis of 2008.
As we have ages of the child ranging from 0, 1, . . . , 17, a more flexible approach
allows us to condition on up to 17 + 4 = 21 potential cohort effects (where 4 is
the number of years of pooled data minus 1). A priori, however, the number and
location of thresholds in the child health-income relationship occur across differing
birth cohorts is unknown. To incorporate threshold cohort effects in the analysis,
we extend equation (4) to
H∗i = x′iβ +M∑m=1
C∑c=1
γmc (ln yi ×Dm ×Dc) + εi (11)
where c = 1, 2, . . . , C indexes the cohorts. The search procedure to find the optimal
number of group-age and cohort effects{M, C
}are carried out in the same way as
before with the inclusion of an additional dimension in the grid search. Again, to
ensure the appropriate adding-up restraints (of these dummy variables), we require
that DM = 1−D1 − . . .−DM−1 and DC = 1−D1 − . . .−DC−1.
We extend the previous RP model by allowing for cohort and child age effects
simultaneously. Now, instead of equations (6) and (10), we model heterogeneity at
the combined child age and cohort level such that
γtc = γ + αtc. (12)
To create our cohort variables, we create 21 dummy variables based on the cohort
a child is born in - for example, a 17 year old in our data 2008-2012, would have
been born in 1991-1995, so for each age group we have 5 possible cohorts. Those in
cohort 0, could only by definition be aged 17 - i.e. those born in the earliest possible
year of 1991. However, those in cohort 2 could have been born in 1991 or 1992, and
hence aged 16 or 17. We provide a graphical representation of our variables for ease
of interpretation, in Table 10.
14
Table 10 about here
6.2 Cohort effects results
The results for the cohort effects are presented in Tables 11 and 12. These tables
only report the results from a range of cohort effect models, with M = 4, but
with differing values of C∗;9 the former contains the summary statistics (likelihood
functions, BIC metric and position of the cohort thresholds) and the latter the
coefficient results.
Table 11 shows that the favoured model with cohort effects is, as noted, M = 4 age
regime model, combined with C = 2 cohort effects. We note there that, regardless of
the number of the possible cohorts, the optimal number of age thresholds and their
positions are very similar to previously found, that is, 4 age-groups at age ranges
< 6, 7−8, 9−12 and > 12. The only cohort threshold is located at the 19th cohort,
corresponding to children born in 2010. By definition, children born in cohorts 19,
20 or 21 must be aged 0, 1, or 2, and, therefore, these individuals can never be in
the last three of the four estimated age groups. For example, a 1 year old will never
enter into the 6-8 age category. 10
What is interesting though, is that this difference between 0-2 or 0-3 and those aged
4-8 became more prominent in 2010 and after, and this difference is statistically
significant. The coefficients of -0.165 before 2010 and -0.188 post 2010 indicates
higher income is required for improving the health of children born after 2010.
This finding is a significant new contribution to the literature and a very promising
indicator for policy implications.
Figure 3 about here
9The maximum number of cohorts and age-group effects considered jointly for implementingthe grid-search are C∗ = 3 and M∗ = 5.
10We refer the reader to Figure 10 that shows the number of age years included into each cohort.It is possible that after cohort 18, the data are more sparse and hence this could lead to theconclusion that there are data issues and overfitting. However, our samples are quite large andhence this is not an issue in our model.
15
Including cohort effects into our random parameters approach yields the results
presented in Table 13 and cohort-age specific parameters illustrated in Figure 4.
Firstly, we note that the null hypothesis of an age-constant income effect is clearly
rejected, with σα = 0.015 (t = 6.3), and again marginal evidence that neighbouring
parameters are related (ρ = −0, 39). Figure 4 plots the estimated cohort-age effects.
Note that for each age there are five different cohort entries, and each cohort will
have a minimum length of 1 and maximum of five, as explained previously (see
Figure 1). Firstly we note that once more, we robustly find a non-constant effect
of income with respect to age, and moreover one that increases (i.e. becomes less
negative) with age. For any particular age category, the range of coefficients across
this gives an estimate of presence of any cohort effects. Thus we see that there is
relatively little heterogeneity from ages 8 and upwards, but there is much evidence
of this prior to age 8, confirming the results from the previous analysis. In fact the
RP approach would tend to suggest stronger cohort effects.
Figure 4 about here
In general, our models indicate that the current age groups are appropriate for
estimation of health outcome models for children, and show that since 2010, there
is an even higher need for policy intervention for the 0-2 age group. We have
provided innovative models for analysing the child health-parental income gradient.
As we noted in our introduction, the only other paper to consider estimation of this
relationship by each age was by Apouey and Geoffard (2013) where they introduced
the idea of including the age effect on the gradient analysis. We are now further
exploring these age effects using new and suitable methods to incorporate cohort
effects. Our results indicates that only when cohort effects are taken into account,
then can we consider which health policies and interventions have or have not worked
to alleviate child health inequality.
We note our models are based on associations only and we have not accounted for
any possible endogeneity of income. According to Apouey and Geoffard (2013) this
could occur if there is reverse causation from child health to income, if parents
16
cannot work due to ill health of a child. Or it could occur if we have omitted any
important variables from the analysis, e.g. parents health. Apouey and Geoffard
(2013) investigate these issues by re-estimating with a sample with no ill children
in the household and find no evidence of reverse causation. They also expanded
their set of controls to include parents health, and while these do impact on child’s
health, they do not affect the income gradient. Therefore our results should not be
adversely affected by possible endogeneity.
Likelihood ratio or Wald tests clearly support the statistically significance of in-
come effects, although the effects become less pronounced with age. Is this decrease
economically meaningful though? The marginal effects from our models (available
upon request) clearly mimic the pattern exhibited by the coefficients, varying sig-
nificantly by age of the child. Any economically meaningful differences in these
across ages, however, are remains subjective. The primary results in Case, Lubot-
sky, and Paxson (2002) (Table 2, Controls 1) increasing in age coefficients from a
low (in absolute terms) of −0.18 to a high of −0.32 is deemed evidence that the
gradient effect “becomes more pronounced as children age” (Abstract, page 1308).
On the other hand, for the U.K. Currie, Shields, and Price (2007), in the range of
(−0.146,−0.212) , Table 1, Controls 1, page 220, find “...no evidence that the slope
of the gradient increases with child age” (Abstract, page 213). Dependent on the
particular technique employed, we find an equivalent, as with the Case, Lubotsky,
and Paxson (2002), range of around −0.18 to −0.13.
7 Policy discussion
Our findings are of direct relevance to the evolving health income inequalities for
children in England. The 0-2 age group is viewed as an important window of oppor-
tunity to make long term impacts on child nutritional status and health (McKenna,
Chalabi, Epstein, and Claxton (2010)). In England, the public health white paper:
Healthy Lives, Healthy People 2010 emphasised the importance of giving all children
a healthy start to life. The Marmot Review (2010), a strategic review of health in-
17
equalities in England post 2010, stated that income related health inequalities were
increasingly evident over the last decade. To reduce this, there was a renewed need
for initiatives to aim to decrease the effects of income on health. Interventions were
needed to improve health behaviour especially of the lower income level households.
Further the context behind the Child Poverty Act 2010 noted that the impact of
inequalities on the quality and quantity of provision of health and social care can
account for 20% of total costs of health care, 15% of total costs of social security
benefits and 9.4% of GDP, another reason for policymakers to decrease these socioe-
conomic health inequalities. In 2012, Health Maps Atlas of Variation for Children’s
Services showed large increases in use of services by children, e.g. since 2009/2010
there had been a five-fold variation in use of Asthma services compared to a four-
fold variation before that. The maps were published to highlight the unjustified
variation in most essential services for childcare across the country. The direction of
effect is not possible to derive from these maps, i.e. did more children need services
due to lower income, or were more services put in place and therefore availed of by
families and were these services equally distributed in terms of socioeconomic need
and local deprivation? Nonetheless, the maps serve as a further useful pointer that
health inequalities do exist.
During the late 1990s and from 2000 onwards, a range of policies were implemented.
Healthy Child Program 2009 advocated a range of new policies. In 2010, a new
policy, Getting It Right for Children and Young People stated that services provided
by the National Health Services were indicated as patchy and greater integration
needed. Later developments included in 2014, the offer of free school meals to all
pupils in reception year, year 1 and year 2 in state funded schools in England;
between 2010 and 2015, 4,200 Health Visitors were to be recruited and trained
to increase support and information available to families; between 2010-2015, aim
was to double number of places on Family Nurse Partnership to support vulnerable
mothers – give young first time family mothers a family nurse, who can help them
prepare for parenthood and support them until their child is 2.
Given the high emphasis on the relationship between parental income and child
18
health, a number of strategic initiatives have been implemented over the last decade
and are continuously improving. The more recent changes to policy indicate the
increased need for such careful intervention. For example, the Child and Young
People health outcomes framework now sets new quality standards and clinical indi-
cators. Furthermore in October, 2015, local authorities will take over responsibility
for planning and paying for public health services for babies and children up to age
5 years, as they know needs best and are able to bring a range of different services to
children and families and have more opportunities to reduce the health inequalities
in their area (Department of Health website).
All of these evolving and changing policy interventions indicate to us that there
is still a need for policy to reduce child health income inequality, as indicated by
our results in this paper. This still does not answer the question, why do we find
different results for children under age 2, before and after 2010? One possible answer
is that socio economic inequalities have been increasingly more common for women,
according to HSE data exploited as part of the economic framework for analysing
health inequalities in England, 2009, for the Marmot Review. This hypothesis was
set out by McKenna, Chalabi, Epstein, and Claxton (2010) for the economic task
group of the health inequalities commission for this review, and the Marmot review
had emphasised that further research was needed to understand the mechanisms
and dynamics between income and health behaviour, but the overall conclusion was
that there was a need to integrate equity into health priority setting.
8 Conclusion
This paper considered two new alternative approaches for estimating the relation-
ship between parental income and child health. We demonstrated how to estimate
the thresholds in the relationship across age groups and with a parsimonious ran-
dom parameters model. In our case, the random parameters are not defined over
individuals, but over the variable defining the nonlinear effects. This is parsimo-
nious in the sense that it requires, in our case, estimation of just one additional
19
parameter (the variance of the random parameters), compared to a baseline model
with no such effects. Ex post however, it is possible to calculate the means and stan-
dard deviations of the conditional distributions of these group-varying parameters.
In addition, we estimate innovative cohort effects in both models, to allow for any
potential shifts in these relationships over time. Our results broadly confirm those
of previous literature, in that the exogenous age categories that have been estimated
are generally 0-3, 4-8, 9-12 and 13-17. The standard age groups imposed previously
are approximately correct but do disguise some heterogeneity before age 8, but not
post age 8. All approaches tend to support this conclusion. We find threshold ef-
fects at age 6, 8 and 12, and an additional cohort effect at age 0-2 for those born
after 2010. This is a new finding to the mechanisms of child health and income in
the literature. However, when we estimate the models using these new age cate-
gories, the coefficients on income are generally the same as those when estimated
with the original exogenous age categories. This is an interesting conclusion and will
reassure researchers who have used these exogenous age groups in their estimation,
that these have now been validated using our two alternative estimation techniques
that allowed the data to determine the endogenous categories. Furthermore, our ap-
proaches are applicable to other health or indeed any outcomes in a discrete choice
or limited dependent variable model and flexible enough to accommodate nonlinear
effects on one or several variables in the model and therefore are widely applicable
in any applied economics context.
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A Tables
Table 1: Income bands
HSE 2008-2012Code Income Band Code Income Band
1 < £520 17 £33,800 - £36,4002 £520 - £1,600 18 £36,400 - £41,6003 £1,600 - £2,600 19 £41,600 - £46,8004 £2,600 - £3,600 20 £46,800 - £52,0005 £3,600 - £5,200 21 £52,000 - £60,0006 £5,200 - £7,800 22 £60,000 - £70,0007 £7,800 - £10,400 23 £70,000 - £80,0008 £10,400 - £13,000 24 £80,000 - £90,0009 £13,000 - £15,600 25 £90,000 - £100,000
10 £15,600 - £18,200 26 £100,000 - £110,00011 £18,200 - £20,800 27 £110,000 - £120,00012 £20,800 - £23,400 28 £120,000 - £130,00013 £23,400 - £26,000 29 £130,000 - £140,00014 £26,000 - £28,600 30 £140,000 - £150,00015 £28,600 - £31,200 31 > £150,00016 £31,200 - £33,800
.
23
Table 2: Selected Descriptive Statistics, 2008 - 2012
Variable Mean Std. Dev. Minimum Maximum
Self Reported Health 1.439 0.612 1 4Very Good 62%Good 33%Fair 4%Bad/Very Bad 1%
Child’s Age 7.982 5.141 0 17Log of Family Income 10.127 0.847 5.382 11.837Male 0.505 0.500Black 0.036 0.187Asian 0.065 0.247Other minority 0.066 0.248Mother is employed 0.652 0.476
NVQ Levels 1, 2 0.312 0.463NVQ Levels 3, 4, 5 0.550 0.498
Father is employed 0.569 0.497NVQ Levels 1, 2 0.376 0.485NVQ Levels 3, 4, 5 0.169 0.348is absent 0.371 0.483
Log of Household size 1.330 0.266 0.693 2.303Number of Observations 9,613
NVQ stands for National Vocation Qualification.
24
Table 3: 2008 - 2012 Optimal Values
Estimated Regimes (Endogenous) Imposed Regimes (Exogenous)2 3 4 5 1 4 17
Log-like. -7801.093 -7796.541 -7789.432 -7784.882 -7872.724 -7797.587 -7775.233BIC(M) 15794.77 15794.84 15789.79 15789.86 15928.86 15806.10 15889.79
Threshold Parametersτ1 12 12 6 6 N/A 3 -τ2 14 8 8 8 -τ3 12 12 12 -τ4 14 -
Income Coefficientsγ1 -0.169∗∗∗ -0.169∗∗∗ -0.169∗∗∗ -0.169∗∗∗ -0.147∗∗∗ -0.171∗∗∗ -γ2 -0.135∗∗∗ -0.144∗∗∗ -0.186∗∗∗ -0.186∗∗∗ -0.173∗∗∗ -γ3 -0.130∗∗∗ -0.164∗∗∗ -0.164∗∗∗ -0.164∗∗∗ -γ4 -0.136∗∗∗ -0.145∗∗∗ -0.136∗∗∗ -γ5 -0.130∗∗∗ -
Number of Observations: 9,613. ∗ p < 0.10; ∗∗ p < 0.05; ∗∗∗ p < 0.01, τi and γi are not presented
for the M = 17 model for reasons of space. Optimal model is highlighted in bold.
25
Table 4: 2008 - 2012 Random Parameter Models
RP RP–AR(1)
Log of family Income -0.158∗∗∗ -0.156∗∗∗
Male 0.037 0.037Black 0.131∗∗ 0.130∗∗
Asian 0.238∗∗∗ 0.239∗∗∗
Minority 0.102∗∗ 0.103∗∗
Mother NVQ Levels 1, 2 -0.004 -0.004NVQ Levels 3, 4, 5 -0.129∗∗∗ -0.128∗∗∗
Father NVQ Levels 1, 2 -0.014 -0.013NVQ Levels 3, 4, 5 -0.130∗∗∗ -0.129∗∗
Father Employment 0.010 0.011Mother Employment -0.053∗ -0.054∗
Father Absence -0.024 -0.024log of family size 0.133∗∗ 0.132∗∗
µ0 -1.222∗∗∗ -1.230∗∗∗
µ1 0.169 0.162µ1 1.038∗∗∗ 1.031∗∗∗
σα 0.018∗∗∗ 0.016∗∗∗
ρ -0.246∗
Number of Observations: 9,613. ∗ p < 0.10; ∗∗ p < 0.05; ∗∗∗ p < 0.01.
26
Table 5: 2008 Optimal Values
Estimated Regimes (Endogenous) Imposed Regimes (Exogenous)2 3 4 5 1 4 17
Log-like. -2642.160 -2639.456 -2638.698 -2637.420 -2663.002 -2642.802 -2635.209BIC(M) 5421.531 5424.194 5430.749 5436.265 5455.144 5438.957 5536.769
Threshold Parametersτ1 12 12 6 1 N/A 3 -τ2 14 12 6 8 -τ3 14 12 12 -τ4 14 -
Income Coefficientsγ1 -0.171∗∗∗ -0.171∗∗∗ -0.167∗∗∗ -0.176∗∗∗ -0.153∗∗∗ -0.168∗∗∗ -γ2 -0.138∗∗∗ -0.154∗∗∗ -0.174∗∗∗ -0.163∗∗∗ -0170∗∗∗ -γ3 -0.132∗∗∗ -0.153∗∗∗ -0.173∗∗∗ -0.171∗∗∗ -γ4 -0.132∗∗∗ -0.153∗∗∗ -0.140∗∗∗ -γ5 -0.131∗∗∗ -
Number of Observations: 3201. ∗ p < 0.10; ∗∗ p < 0.05; ∗∗∗ p < 0.01, τi and γi are not presented
for the M = 17 model for reasons of space. Optimal model is highlighted in bold.
Table 6: 2009 Optimal Values
Estimated Regimes (Endogenous) Imposed Regimes (Exogenous)2 3 4 5 1 4 17
Log-like. -835.718 -832.378 -829.788 -828.675 -847.047 -833.424 -822.030BIC(M) 1789.923 1790.212 1792.003 1796.746 1805.611 1799.274 1874.063
Threshold Parametersτ1 8 7 7 1 N/A 3 -τ2 8 8 7 8 -τ3 15 8 12 -τ4 15 -
Income Coefficientsγ1 -0.298∗∗∗ -0.293∗∗∗ -0.299∗∗∗ -0.312∗∗∗ -0.250∗∗∗ -0.297∗∗∗ -γ2 -0.261∗∗∗ -0.345∗∗∗ -0.351∗∗∗ -0.292∗∗∗ -0.308∗∗∗ -γ3 -0.260∗∗∗ -0.271∗∗∗ -0.349∗∗∗ -0.276∗∗∗ -γ4 -0.242∗∗∗ -0.270∗∗∗ -0.256∗∗∗ -γ5 -0.240∗∗∗ -
Number of observations: 1064. ∗ p < 0.10; ∗∗ p < 0.05; ∗∗∗ p < 0.01, τi and γi are not presented
for the M = 17 model for reasons of space. Optimal model is highlighted in bold.
27
Table 7: 2010 Optimal Values
Estimated Regimes (Endogenous) Imposed Regimes (Exogenous)2 3 4 5 1 4 17
Log-like. -1540.103 -1537.266 -1536.312 -1534.966 -1552.706 -1539.530 -1527.711BIC(M) 3208.379 3210.243 3215.875 3220.723 3226.046 3222.311 3304.228
Threshold Parametersτ1 12 6 6 1 N/A 3 -τ2 12 11 3 8 -τ3 14 5 12 -τ4 12 -
Income Coefficientsγ1 -0.163∗∗∗ -0.157∗∗∗ -0.156∗∗∗ -0.152∗∗∗ -0.136∗∗∗ -0.160∗∗∗ -γ2 -0.132∗∗∗ -0.174∗∗∗ -0.176∗∗∗ -0.172∗∗∗ -0.163∗∗∗ -γ3 -0.133∗∗∗ -0.144∗∗∗ -0.150∗∗∗ -0.169∗∗∗ -γ4 -0.126∗∗∗ -0.174∗∗∗ -0.132∗∗∗ -γ5 -0.134∗∗∗ -
Number of Observations: 1881. ∗ p < 0.10; ∗∗ p < 0.05; ∗∗∗ p < 0.01 . τi and γi are not presented
for the M = 17 model for reasons of space. Optimal model is highlighted in bold.
Table 8: 2011 Optimal Values
Estimated Regimes (Endogenous) Imposed Regimes (Exogenous)2 3 4 5 1 4 17
Log-like. -1347.727 -1343.943 -1340.812 -1338.372 -1366.053 -1345.084 -1332.106BIC(M) 2822.146 2822.029 2823.219 2825.792 2851.345 2831.764 2910.141
Threshold Parametersτ1 9 1 1 1 N/A 3 -τ2 9 2 2 8 -τ3 9 8 12 -τ4 12 -
Income Coefficientsγ1 -0.144∗∗∗ -0.166∗∗∗ -0.164∗∗∗ -0.164∗∗∗ -0.105∗∗∗ -0.149∗∗∗ -γ2 -0.106∗∗∗ -0.139∗∗∗ -0.112∗∗∗ -0.112∗∗∗ -0.143∗∗∗ -γ3 -0.106∗∗∗ -0.142∗∗∗ -0.143∗∗∗ -0.122∗∗∗ -γ4 -0.104∗∗∗ -0.119∗∗∗ -0.099∗∗∗ -γ5 -0.096∗∗∗ -
Number of observations: 1724. ∗ p < 0.10; ∗∗ p < 0.05; ∗∗∗ p < 0.01 τi and γi are not presented
for the M = 17 model for reasons of space. Optimal model is highlighted in bold.
28
Table 9: 2012 Optimal Values
Estimated Regimes (Endogenous) Imposed Regimes (Exogenous)2 3 4 5 1 4 17
Log-like. -1387.907 -1384.747 -1381.136 -1376.782 -1402.967 -1386.160 -1376.540BIC(M) 2902.690 2903.834 2905.275 2908.832 2925.348 2914.124 2999.371
Threshold Parametersτ1 13 8 5 4 N/A 3 -τ2 13 8 5 8 -τ3 13 8 12 -τ4 13 -
Income Coefficientsγ1 -0.178∗∗∗ -0.186∗∗∗ -0.180∗∗∗ -0.182∗∗∗ -0.161∗∗∗ -0.185∗∗∗ -γ2 -0.135∗∗∗ -0.169∗∗∗ -0.202∗∗∗ -0.157∗∗∗ -0.188∗∗∗ -γ3 -0.137∗∗∗ -0.169∗∗∗ -0.200∗∗∗ -0.170∗∗∗ -γ4 -0.137∗∗∗ -0.166∗∗∗ -0.143∗∗∗ -γ5 -0.135∗∗∗ -
Number of observations: 1743. ∗ p < 0.10; ∗∗ p < 0.05; ∗∗∗ p < 0.01 τi and γi are not presented
for the M = 17 model for reasons of space. Optimal model is highlighted in bold.
29
Tab
le10
:C
ohor
ts19
91–2
012
Bor
n19
9119
9219
9319
9419
9519
9619
9719
9819
9920
0020
0120
0220
0320
0420
0520
0620
0720
0820
0920
1020
1120
12C
0C
1C
2C
3C
4C
5C
6C
7C
8C
9C
10C
11C
12C
13C
14C
15C
16C
17C
18C
19C
20C
21A
ge 020
0820
0920
1020
1120
121
2008
2009
2010
2011
2012
220
0820
0920
1020
1120
123
2008
2009
2010
2011
2012
420
0820
0920
1020
1120
125
2008
2009
2010
2011
2012
620
0820
0920
1020
1120
127
2008
2009
2010
2011
2012
820
0820
0920
1020
1120
129
2008
2009
2010
2011
2012
1020
0820
0920
1020
1120
1211
2008
2009
2010
2011
2012
1220
0820
0920
1020
1120
1213
2008
2009
2010
2011
2012
1420
0820
0920
1020
1120
1215
2008
2009
2010
2011
2012
1620
0820
0920
1020
1120
1217
2008
2009
2010
2011
2012
30
Table 11: 2008 - 2012 Pooled data: Estimated Cohorts
Cohort Regimes (Endogenous)C = 1 C = 2 C = 3
Log-likelihood -7789.432 -7784.723 -7782.997BIC(M,C) 15789.79 15789.55 15795.27
Cohort Thresholdsκ1 − 19 2κ2 19κ3 -
Notes: number of age groups is 4.
Table 12: 2008 - 2012 Pooled data: Optimal Values for M = 4 Estimated Regimeswith Cohort Effects
C = 1; γ (age) C = 2; γ (age, cohort) C = 3; γ (age, cohort)
γ(age ≤ 6) -0.169 γ(age ≤ 6, cohort ≤ 19)-0.165 γ(age ≤ 6, 2 < cohort ≤ 19) -0.166− − γ(age ≤ 6, cohort > 19)-0.188 γ(age ≤ 6, cohort > 19) -0.189γ(7 ≤ age ≤ 8) -0.186 γ(7 ≤ age ≤ 8) -0.185 γ(7 ≤ age ≤ 8, 2 < cohort ≤ 19) -0.185γ(9 ≤ age ≤ 12) -0.164 γ(9 ≤ age ≤ 12) -0.163 γ(9 ≤ age ≤ 12, 2 < cohort ≤ 19)-0.163γ(age ≥ 13) -0.136 γ(age ≥ 13) -0.135 γ(age ≥ 13, cohort ≤ 2) -0.127
− γ(age ≥ 13, 2 < cohort ≤ 19) -0.138
Notes: all coefficients significant at 5% level. Number of observations: 9,613.
31
Table 13: 2008 - 2012 Random Parameter Models: With Cohort Effects
RP RP–AR(1)
Log of family Income -0.158∗∗∗ -0.159∗∗∗
Male 0.035 0.035Black 0.127∗ 0.126∗∗
Asian 0.235∗∗∗ 0.236∗∗∗
Minority 0.093∗ 0.096∗
Mother NVQ Levels 1, 2 -0.008 -0.008NVQ 3, 4, 5 -0.131∗∗∗ -0.130∗∗∗
Father NVQ 1, 2 -0.018 -0.019NVQ 3, 4, 5 -0.135∗∗ -0.135∗∗∗
Father Employment 0.010 0.012Mother Employment -0.047 -0.047Father Absence -0.026 -0.024log of family size 0.137∗∗∗ 0.139∗∗∗
µ0 -1.213∗∗∗ -1.222∗∗∗
µ1 0.179 0.170µ1 1.047∗∗∗ 1.038∗∗∗
σα 0.018∗∗∗ 0.015∗∗∗
ρ -0.387
32
B Figures
Figure 1: Gradient: Fixed and Estimated Thresholds Models
-0.20
-0.19
-0.18
-0.17
-0.16
-0.15
-0.14
-0.13
-0.12
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Imposed
Estimated
33
Figure 2: Gradient: Random Parameters and Estimated Thresholds Models
-0.20
-0.19
-0.18
-0.17
-0.16
-0.15
-0.14
-0.13
-0.12
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
RP
RP-AR(1)
Estimated
34
Figure 3: Gradient: Estimated Age-Cohort Thresholds Models
-0.20
-0.19
-0.18
-0.17
-0.16
-0.15
-0.14
-0.13
-0.12
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Imposed
Estimated
Cohort ≤ 19
Cohort >19
35
Fig
ure
4:G
radie
nt:
Ran
dom
Par
amet
ers
wit
hA
gean
dC
ohor
tT
hre
shol
ds
-0.20
-0.19
-0.18
-0.17
-0.16
-0.15
-0.14
-0.13
-0.12
01
23
45
67
89
10
11
12
13
14
15
16
17
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20
36