LifeSim: A Lifecourse Dynamic Microsimulation Model of the Millennium Birth Cohort in England

Our microsimulation model is a discrete event simulation in discrete time (annual periods), which progresses 100,000 individuals through each year of their lives from birth in the year 2000 to death. From ages 0 to 18 it closely follows observed Millennium Cohort Study (MCS) data, and thereafter predicts the annual evolution of each life outcome based on the current values of relevant characteristics and outcomes, which in term depend on lagged values.³ This kind of model can be seen as a pragmatic compromise between a simpler Markov model structure, which has no “memory” or dependence upon lagged values, and a more complicated agent-based model structure, which explicitly models interactions between individuals and how individual behaviour may depend upon the macro-level policy environment as well as the behaviour of others. Allowing dependence upon lagged values allows a rich analysis of the dynamic clustering and compounding of multiple outcomes over time, while setting aside agent-based interactions keeps the model tractable, even when modelling a relatively large number of outcomes.

The model links together a diverse set of individual-level life outcomes of interest to policymakers (Figure 1). By using rich observational data from the MCS, our model provides information on various aspects of human capital development in childhood - including social skills, cognitive skills, and health behaviour (teenage smoking) - and then extrapolates later life outcomes across economic, social, and health domains for the rest of the lifecourse. For simplicity and concreteness we focus on one important and readily measurable dimension of social skills - conduct problems - as proxied by two separate parent reported measures. Child conduct is related to self-control and regulation, which have been shown to matter in many aspects of life, including wellbeing, income, employment, crime and health outcomes (Goodman et al., 2015). We also model mental illness and health-related quality of life during childhood, using external datasets (Mental Health of Children and Young People Great Britain, and a dataset by Love-Koh et al. (2015)).

Figure 1

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Let $i=1,\mathrm{..100},000$ index the individual children in the cohort. Let yearly time periods also corresponding to the age of children be indexed as $age=0,1,\mathrm{..18}..T$ where $age=18$ marks the end of childhood, and $T$ is the last time period in which there are any cohort members still alive (which we assume to be 100, since small number problems make predictions decreasingly reliable at older ages). Let $X_i$ be the vector of initial conditions assumed to be constant for child $i$ (e.g. individual and family characteristics at birth or other early time period – if data at birth on the condition is not available); let ${\theta }_{i,age}$ be an age-specific vector of child Strengths and Difficulties Questionnaire (SDQ) scores – multi-dimensional parent-reported score on child’s difficulties, $c{d}_{i,age}$ – an age-specific outcome of whether child develops a conduct disorder, and $co{g}_{i,age}$ – an age-specific child’s cognitive skills measure. Finally, let $Y_{i,age}$ be the age-specific vector of life-cycle outcomes (further, outcomes) for child $i$ . These outcomes can be further classified as social, health and economic outcomes, i.e. $Y_{i,age}\equiv \{S_{i,age},H_{i,age},E_{i,age}\}$ , where $S_{i,age},H_{i,age},E_{i,age}$ are the vectors of social, health and economic outcomes respectively. It is allowed for the vector $X_i$ to also contain elements of $\{S_i,H_i,E_i\}$ .

At each age the individual probability of dying $pr.dea{d}_{i,age}$ is modelled and defined over the closed interval from zero to one, i.e. $pr.dea{d}_{i,age}(E_{i,age},S_{i,age},H_{i,age})\in [0,1]$ , which then determines the discrete outcome $dea{d}_{i,age}$ – whether the individual at a certain age is dead ( $dea{d}_{i,age}=1$ ) or alive ${\displaystyle (dea{d}_{i,age}=0).}$ More specifically, we can represent the outcome ­‘dead or alive’ by a function $l(.)$ such that if in the previous year individual was alive then they can be either alive or dead in the following year, i.e. ${\displaystyle dea{d}_{i,age}=l(pr.dea{d}_{i,age},{\zeta }_{i,age}|dea{d}_{i,age-1}=0)\in \{0,1\}}$ (where ${\zeta }_{i,age}$ represents stochasticity); and if in the previous year individual was dead then, because death is an ‘absorbing state’, they can be only dead in the following year, i.e. ${\displaystyle dea{d}_{i,age}=l(.|dea{d}_{i,age-1}=1)=1}$ . Individual life span is then $T_i={\sum }_{age=0}^T\left(1-dea{d}_{i,age}\right)$ .

To describe the initial conditions (in vector $X_i$ ), we draw observations on each child, i.e. 100,000 times in total to represent a cohort of 100,000 individuals, using re-sampling with replacement from the initial sweeps of MCS – a longitudinal survey of English children born in 2000-2001 (see panels A-C of Table 1).

Similarly, we use MCS data from all of the sweeps (up to age 14) to collect data for the vector ${\theta }_{i,age}$ – information on child SDQ conduct problem subscale score and a further parent-reported “behavioural impact” score (see panel D of Table 1). Both of these scores range from 0-10, with a higher score representing more conduct problems and a more severe impact of difficulties in child’s life. MCS data are reported at sweeps every 2 to 4 years, so we use the most recent MCS sweep data available to fill in the missing values in the time gaps, and for age 15-18.

We then use the reported SDQ score components to model whether or not a child develops conduct disorder, using a previously developed algorithm which predicts a child’s probability of developing conduct disorder as a function of SDQ score components – the SDQ conduct problem score and the “behavioural impact” score. This provides a specific probability of conduct disorder based on a classification as either “possible” or “probable” (Goodman et al., 2003; Goodman et al., 2000).⁴ This modelled probability is then combined with a random draw from a uniform distribution over 0-1, which allows us to simulate the discrete outcome of whether or not a child develops conduct disorder. Formally, the age-specific conduct disorder outcome $c{d}_{i,age}$ can be represented using a function g(.) as:

where ${\xi }_{i,age}$ represents stochasticity.

We also use the MCS data from later sweeps (up to age 14) to build $co{g}_{i,age}$ – a single measure of each child’s cognitive skills at each age throughout their childhood up to age 18 (see panel E of Table 1). More specifically, our cognitive skills measure is an age-specific common factor extracted from the cognitive skills measures available in MCS, including the British Ability Scales II (for ages 3, 5, 7, 11), Bracken School Readiness Assessment (for age 3), National Foundation for Educational Research Progress in Maths (for age 7), Cambridge Neuropsychological Test Automated Battery tests (for ages 11 and 14) and Applied Psychology Unit (for age 14). We extract a common factor for each age where test results are available using principal component analysis, and standardise it to be with a mean of 1.00 and standard deviation of 0.15 (following Jones and Schoon (2008)). Similar to the SDQ score data, we use the most recent MCS sweep data available to fill in the missing values in the time gaps, and for age 15-18.

During adulthood, child’s SDQ scores, conduct disorder outcomes and cognitive skills are assumed to stay fixed at the level achieved by the end of childhood, i.e. ${\theta }_{i,age}={\theta }_{i,18}$ , $c{d}_{i,age}=c{d}_{i,18}$ and $co{g}_{i,age}=co{g}_{i,18}$ for $age=19..T_i$ .

Over the life-cycle ( $age=\mathrm{0..18}..T_i$ ), the vector of other life-cycle outcomes $Y_{i,age}$ evolves as:

where ${\eta }_{i,age}$ represents stochasticity. It should be noted that separate outcomes in the vector $Y_{i,age}$ can depend on a subset (and not necessarily all) of the outcomes in the vectors ${\displaystyle Y_{i,age-1},Y_{i,age},{\theta }_{i,18},c{d}_{i,18},co{g}_{i,18},X_i,}$ which can be achieved by restricting coefficients. Also, a period-specific outcome in the vector $Y_{i,age}$ will generally not depend on itself, but can depend on other outcomes at that time period included in the vector $Y_{i,age}$ .

The model structure specified by $k_{age}(.)$ changes as individuals progress through key life stages. In each life stage, the dependencies between the initial conditions and the life-course outcomes are represented by model structure diagrams in Figures 2 and 3, and are also summarised in Table 2. In the model structure diagram each solid arrow is modelled using equations (as we will explain in more detail in Section 2.3).

Figure 2

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Figure 3

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In choosing the model outcomes and formulating the model structure, we consulted with experts in childhood development and childhood policy, demography, epidemiology, human capital economics, and labour economics (see list of advisory group members in the acknowledgements) and were also guided by inter-disciplinary theory on human capital formation in childhood and how this influences educational attainment, earnings, physical illness, mental illness, mortality, and other outcomes with important impacts on individual wellbeing and public cost (Almond et al., 2018; Goodman et al., 2015; Nelson et al., 2020; Cunha and Heckman, 2010; Adler et al., 2010; O’Donnell et al., 2015;; Layard et al., 2014; Shonkoff, 2010; Black et al., 2017).

LifeSim also models variables relevant to the public budget (Figure 4). This includes modelling the public costs over time associated with certain life outcomes, such as conduct disorder, being in prison, mental illness, coronary heart disease, as well as cash benefits paid to people who are in poverty and/or unemployed. This also includes modelling the taxes paid over time on individual earnings and financial gains. These can be aggregated, to assess the overall impact on the public budget as well as cost savings under different policy scenarios and over various time spans. Details of the evidence and assumptions about the unit costs of public services and our simple approach to modelling long-run taxes and benefits are found in Appendix A.

Figure 4

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To model later life outcomes, we use equations parameterised using (i) cross-sectional target data which describe expected levels of and associations between variables at a point in time, usually based on up-to-date survey or administrative data, and (ii) effect estimates which attempt to draw inferences about the effect of one variable on another variable, either at the same time or a future point in time, usually based on statistical analysis of longitudinal data on historical cohorts.

Our target data comes from recent and nationally representative available surveys and administrative records in England. Our effect estimates come from studies based on longitudinal data in a UK context, unless robust estimates are only available from other high-income countries.

Our effect estimates come from studies of longitudinal data which control for observed confounding factors and focus on plausible causal relationships for which there is a large body of theoretical and empirical evidence. Nevertheless, our estimates are subject to potential omitted variable bias and cohort bias. For example, we take the estimated effect of childhood SDQ score on earnings in young adulthood from a study of longitudinal data on children born in 1970, which controls for observed child-level, family-level, and neighbourhood-level factors. We interpret this as a causal estimate i.e. if you increase SDQ score you will increase adult earnings by this amount. However, this estimate may be too low or too high if there are unobserved variables which influence both SDQ score and earnings (”omitted variable bias”). It may also be biased if the underlying stochastic processes have changed since 1970, such that SDQ score is now a more or less powerful determinant of adult earnings (”cohort bias”). Using estimates based on past cohorts of individuals thus relies on the assumption that micro-level causal effects do not change much over many decades (e.g. the proportional effect of social skills on earnings for an individual), even though the macro-level prevalence of each outcome within society may change dramatically (e.g. the average levels of social skills and earnings).

Table 2 summarises the dependencies between the modelled outcomes together with parameter sources for effects estimates, if applicable, as well as the dependencies of the modelled outcomes on the target datasets and the variables from the MCS childhood dataset. More details, as well as the full description of the target datasets are found in Appendix A.

Most of the equations modelling the outcomes can be described as one of the following: (i) simple level equations based on target data only; (ii) complex level equations based on target data supplemented with effect estimates; (iii) simple difference equations based on age associations observed in cross-sectional target data; (iv) complex difference equations based on age associations observed in cross-sectional target data supplemented with effect estimates. We illustrate each below in turn with a simple example. We also use equations that do not fit this taxonomy to model specific variables, such as savings behaviour and wealth accumulation over time, as well as public costs (more on this can be found in Appendix A).

Conventional methods of unweighted benefit-cost analysis do not provide direct information about impacts on wellbeing and can be criticised on two important grounds. First, by focusing on unweighted consumption they ignore the well-established concept in economics of diminishing marginal value of consumption; second, they provide no information about the social distribution of costs and benefits and their impact on inequalities (see discussion in Cookson et al. (2021)). There is a large literature on the theoretical and practical shortcomings of unweighted cost-benefit analysis and the advantages of alternative utilitarian and prioritarian approaches to economic evaluation based on explicit individual wellbeing and social welfare functions (Adler and Fleurbaey, 2016).

Our framework generates individual-level outcomes that could be used in many different ways to create summary indices of wellbeing for use in economic evaluation. In our illustrative evaluation we follow Cookson et al. (2021), who propose a simple approach based on the quality-adjusted life year (QALY) concept in health economics, but adjusting for consumption as well as health-related quality of life. Our approach could be used to construct many other multidimensional measures of wellbeing that have been proposed in the literature, including equivalent income measures and measured based on life satisfaction (Adler and Fleurbaey, 2016). Cookson et al. (2021) refer to their approach as an ”equivalent life” approach (Canning, 2013), and the resulting wellbeing metric as ”years of good life” or ”wellbeing QALYs”. Following them, we represent individual wellbeing in year $t$ by a function $w_t\left(\right)$ increasing in both consumption and health. More specifically, $w(..)=healt{h}_{i,age}+u\left(consumptio{n}_{i,age}\right)$ where $u(.)$ is a standard isoelastic utility of income function defined as $u(.)=A-B\times consumptio{n}_{i,age}^{1-\eta }$ . The parameter $\eta \S gt;1$ captures diminishing marginal value of income, and $A$ and $B$ are constants which depend on normative parameters: $\eta$ (already mentioned), minimal consumption for a life worth living and standard consumption for a good life. In the current application we set minimal consumption at £1,000 (estimated amount required to buy basic food supplies in the UK for a year) and standard consumption at £24,000 (the mean consumption in the LifeSim simulated cohort), and $\eta =1.26$ (see Cookson et al. (2021)).

The interpretation is that a good year is a year lived enjoying full health and consuming the equivalent of the average consumption in a rich country. The good-years measure is more informative than conventional monetary measures because it takes into account the notion that one pound of additional consumption is worth substantially more to a poor individual than a rich individual.

LifeSim is implemented in software R (tested on R version 3.6.2) using object-oriented programming for R (requires R6 and tidyverse packages). The code and related data files are available on GitHub (https://github.com/ievask/lifesim-simulator) and can be run on a high performance computing (HPC) cluster (Slurm Workload Manager).

When we split the simulation into 500 partitions, it takes 28 minutes to run it on the HPC cluster. The simulation can also be run on a standard PC, for any chosen number of individuals.

The code is written using an object-oriented approach built around individuals, capturing their initial endowments and the skills and assets they acquire through life as they undergo various experiences, the probability of which are influenced by their past histories. This allows us to simulate individual life histories in an intuitive manner and easily communicate and validate our modelling assumptions in discussion with domain experts in various stages of the life-course. The code is currently written in R allowing us to elegantly incorporate advanced statistical methods into our modelling. However, R being an interpreted language can be slow to run and if performance was a concern our code could easily be translated into a compiled object oriented programming language such as C++. There are also ways of re-writing the original R code in more compact ways, known as “vectorisation”, which are harder for non-specialists to follow but faster to run because they avoid conventional programming loops that require the same time-consuming interpretation operations to be applied repeatedly.

Table 6 compares the LifeSim predictions with data from the 1970 Birth Cohort Study (BCS70) at ages 26, 29, 42 and 46, as a simple validation check. We list the number of observations, means and standard-deviations of the LifeSim variables for children born in the year 2000 and the BCS70 variables for children born in the year 1970, representing the same outcomes. For each outcome, we quantify the difference between the LifeSim distribution and BCS70 distribution in terms of the absolute difference in their means and standard deviations.

We would expect some adult outcomes to be similar (e.g. health) but others to be substantially different (e.g. earnings, rates of smoking and university education), and so this can be seen as a simple validation check to ensure that our model provides broadly similar findings in the same ballpark where appropriate, and substantially different findings where we know different generations had very different experiences e.g. smoking. Nevertheless, most variables do not deviate substantially from the same quantities characterising the cohort born in 1970.

One exception already mentioned is smoking, which is expected and can be explained by the change in smoking rates over time. Another exception is education – the proportion of people with a degree under 30 years old – which is much higher in the LifeSim cohort. This can be explained by the change in higher education participation rates over time, and increased equality between the genders in the cohort born in 2000. Over time the 1970s cohort partially catches up with the LifeSim cohort by obtaining qualifications at a later age – at the age 46 the proportion of people with a university degree is more similar in both samples than at the age 26. Finally, the LifeSim earnings at all ages on average exceed the 1970s cohort earnings. This can be explained by cohort effects, such as general differences in economy, society, culture and politics experienced by the two cohorts.

To avoid such general cohort effects which arise when comparing two generations born 30 years apart, we also carry out a simple validity check using more recent cross-sectional datasets. More specifically, we compare our age-specific LifeSim outcomes with age-specific outcomes in cross-sectional data.

Figure 6 compares the age-earnings profile for males and females in the LifeSim simulation with our target dataset – ONS Annual Survey of Hours and Earnings in year 2015, and in the Understanding Society survey in year 2015. The concave trend with age, initially increasing and then – decreasing earnings, is very similar in the tree datasets.

Figure 6

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Figure 7 compares the earnings distributions by sex and different age groups in the LifeSim cohort and the Understanding Society data. Both distributions have similar medians for the different sex-age groups, and also become more uniform with increasing age. One issue left to be addressed as part of future work is modelling of the relatively longer right hand side tail which can be observed for the Understanding Society data and not for the LifeSim data. This tail represents the highest-earning people in the distribution. The LifeSim earnings output does not have this tail, as we do not model the outcome of being employed in extremely-high earning jobs. Addressing this feature in LifeSim would require modelling the link with variables in early life that would lead to such extremely-high earning states.

Figure 7

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In Figure 8, we compare the prevalence of the different discrete outcomes in LifeSim cohort, and in our corresponding target datasets, which include Health Survey for England for the health-related outcomes, ONS Labour Force Survey for unemployment and Department for Education estimates for participation in higher education.

Figure 8

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The simulated outcomes matches the target data well, but there is some small discrepancy with the Understanding Society data, which can be explained by differences how data on similar outcomes is collected across different surveys.

Internal validity relates to claims about cause and effect within the study population, whereas external validity relates to how applicable the findings are to real world policy settings.

According to Statistics Canada, they developed a dynamic microsimulation model in the 1990s with a rich set of co-evolving economic, social and health outcomes, called LifePaths (Spelauer et al., 2013), which has subsequently been discontinued. However, this model seems to have had limited detail on developmental outcomes in childhood and we could not find detailed technical information or any published economic evaluations based on it.

Because the MCS data was only available up to age 14 when our model was developed, we approximate the outcomes between ages 15-18 using the MSC data for age 14; this can now be updated with the data from the latest MCS wave which has recently become available and at which children are 17 years old.

More specifically, the algorithm allocates a probability of 0.61 for children with the SDQ conduct problem score of at least 5 combined with the impact score of at least 2; a probability of 0.31 for children with the conduct problem score equal to 4 (irrespective of the impact score) and a probability of 0.06 for all the other children with conduct problem scores below 4.

This equation and other equations in this section are simplified examples of the actual equations that we use; see Appendix A for the full mortality equation and the other equations that we use.

The total residential care cost figure does not include the substantial private costs of residential care, which we assume fall on individuals if they have sufficient savings, nor the public costs of residential care before the age of 60. It may be an underestimate of public costs, because we make simple and conservative assumptions about the need for residential care and eligibility for public funding - for example, we use simple sex-specific rates of care home use in people aged 65 and over (2% for men and 4% for women) but do not model the rapid age-related increase in risk which results in much higher rates for people surviving into their 80s and belyond.

Standard period estimates of gaps in healthy life expectancy by current socioeconomic status are substantially larger than our cohort estimate of gaps by early childhood circumstance, due to dynamic interdependence between health and social status over the lifecourse. Adult-onset illness that is unrelated to early childhood circumstances may cause downward social mobility, and deterioration of social and economic outcomes that is unrelated to early childhood circumstances may cause deteriorating health.

It should be noted that the average marginal effect is not always a good approximation of the true effect, as the actual individual marginal effect is not constant across individuals. So this method is a crude way of modelling the effect.

We assume that whether an individual obtains a university degree is determined at age 19.

See details on UK new State Pension at https://www.gov.uk/new-state-pension.

This threshold is chosen as a maximum, informing from historical UK households savings ratios reported by ONS: https://www.ons.gov.uk/economy/grossdomesticproductgdp/timeseries/dgd8/ukea.

See the paragraph below about “Consumption” in Section A.3.5.

See UK income tax rates at https://www.gov.uk/income-tax-rates. We use the year 2018/19 rates, converted to year 2015/16 prices.

We use the average of the direct and informal care cost, to quantify the annual cost of a person with CHD.

The functional form of each modelling equation is chosen depending on (a) the type of variable that we model (continuous quantity vs. indicator); and (b) the format in which the parameter estimate is reported (e.g. coefficient estimates from a linear regression, odds ratios from a logistic regression, percentage changes, etc.).

Table A1 explains the notation that we use to specify the modelling equations throughout the rest of the Table A2 summarises the target data; Table A3 lists the literature sources of the parameter estimates used in parameterising the modelling equations; Table A4 summarises what other variables these literature sources control for. We then provide full detailed specifications of the modelling equations to model each of the lifecourse outcomes, as well as full details on modelling taxes, cash benefits and costs associated with costly outcomes, in the next subsection.

We present the full specification of the modelling equations which follows the structure outlines in Table 2 in the main text. This material should be used together with Table A1, which clarifies the notation, as well as Table A2 in the main text, which specifies the target data sources, and Table A3 and Table A4, which specify the parameters, and details about their sources.