New approaches to estimating the child health-parental ... · New approaches to estimating the child health-parental income relationship. ... We further extend our analysis to allow

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.

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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+.

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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.

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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.

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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

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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.

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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.

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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.

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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

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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.

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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.

References

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the UK,” Journal of Health Economics, 32(4), 715–727.

Burgess, S., C. Propper, and J. Rigg (2004): “The Impact of Low-Income on

Child Health: Evidence from a Birth Cohort Study,” Discussion Paper 04/098,

CMPO Working Paper Series.

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Cameron, L., and J. Williams (2009): “Is the Relationship between Socioeco-

nomic Status and Health Stronger for Older Children in Developing Countries?,”

Demography, 46(2), 303–324.

Case, A., D. Lee, and C. Paxson (2008): “The income gradient in children’s

health: A comment on Currie, Shields andWheatley-Price,” Journal of Health

Economics, 27, 801 – 807.

Case, A., D. Lubotsky, and C. Paxson (2002): “Economic Status and Health

in Childhood: The Origins of the Gradient,” American Economic Review, 92(5),

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Currie, A., M. A. Shields, and S. W. Price (2007): “The Child Health/Family

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– 32.

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Fletcher, J., and B. Wolfe (2014): “Increasing our understanding of the

health-income gradient in children,” Health Economics, 23(4), 473–486.

Gannon, B., D. Harris, and M. Harris (2014): “Threshold Effects in Nolinear

Models with an Application to the Social Capital-Retirement-Health Relation-

ship,” Health Economics, 23(9), 1072–1083.

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Greene, W., and D. Hensher (2010): Modeling Ordered Choices. Cambridge

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Khanam, R., and H. L. C. Nghiem (2009): “Child Health and the Income

Gradient: Evidence from Australia,” Journal of Health Economics, 28, 805–817.

Kruk, K. (2013): “Parental income and the dynamics of health inequality in early

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McKenna, C., Z. Chalabi, D. Epstein, and K. Claxton (2010): “Budgetary

policies and available actions: A generalisation of decision rules for allocation and

research decisions,” Journal of Health Economics, 29(1), 170 – 181.

Park, C. (2010): “Children’s Health Gradient in DevelopingCountries: Evidence

from Indonesia,” Journal of Economic Develpoment, 35(4), 25–44.

Train, K. (2003): Discrete Choice Methods with Simulation. Cambridge University

Press.

West, P. (1997): “Health Inequalities in the Early Years: Is there Equalization in

Youth?,” Social Science and Medicine, 44, 833 – 858.

22

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


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


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




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



Number of observations: 1064. ∗ p < 0.10; ∗∗ p < 0.05; ∗∗∗ p < 0.01, τi and γi are not presented


27



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



Number of Observations: 1881. ∗ p < 0.10; ∗∗ p < 0.05; ∗∗∗ p < 0.01 . τi and γi are not presented




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



Number of observations: 1724. ∗ p < 0.10; ∗∗ p < 0.05; ∗∗∗ p < 0.01 τi and γi are not presented


28



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



Number of observations: 1743. ∗ p < 0.10; ∗∗ p < 0.05; ∗∗∗ p < 0.01 τi and γi are not presented


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

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Ran

dom

Par

amet

ers

wit

hA

gean

dC

ohor

tT

hre

shol

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-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

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New approaches to estimating the child health-parental ... · New approaches to estimating the child health-parental income relationship. ... We further extend our analysis to allow