The dynamics of health in the British Household Panel Survey

2.1. Sample and Variables

In estimating the empirical models for self-assessed health we exploit the panel data available in the first eight waves (1991–1998) of the British Household Panel Survey (BHPS).3 This includes rich information on occupational, sociodemographic and health variables. The BHPS is a longitudinal survey of private households in Great Britain (England, Wales and Scotland south of the Caledonian Canal), and was designed as an annual survey of each adult (16+) member of a nationally representative sample of more that 5000 households, with a total of approximately 10,000 individual interviews. The first wave of the survey was conducted between 1st September 1990 and 30th April 1991. The initial selection of households for inclusion in the survey was performed using a two-stage stratified systematic sampling procedure designed to give each address an approximately equal probability of selection.4 The same individuals are re-interviewed in successive waves and, if they split off from their original households are also re-interviewed along with all adult members of their new households. In this analysis we use both balanced samples of respondents, for whom information on all the required variables is reported at each wave, and unbalanced samples that exploit all available observations. The unbalanced sample does not include new entrants but tracks all of those who are observed at wave 1. The issue of sample attrition is discussed below.

2.2. Data Description

Figure 1 describes the distribution of SAH across all eight waves.6 The distributions show a long right tail, with the majority of observations being either excellent or good. The figure shows an interesting pattern; although the interviewee is asked to report their self-assessed health relative to a representative person of their own age, there is a trend for the distribution of health to become worse over time. This can be seen by the steady increase in the proportions of observations in the fair, poor and very poor categories, while there is a gradual decrease in the proportion reporting excellent health. The age profile of SAH is reinforced by Figure 2, which shows the distribution by age groups at wave 1.

Figure 3 displays the relationship between health and income by showing the distribution of self-assessed health, pooled over eight waves, by quintiles of mean equivalized real household income. The figure shows that the distribution of SAH improves as household income increases; as we move up the income distribution the proportions of observations in the excellent and good categories increase while those in the fair, poor and very poor categories decrease. An alternative way of considering this pattern is to consider the empirical CDF of household income by self-assessed health status; where the y-axis shows the proportion of observations with household income below the values given on the x-axis. If a positive relationship exists between health and income we would expect the proportion of observations with income below a particular level to be higher when self-assessed health is poorer; we would expect the CDFs for successively lower levels of health to be successively above those for higher levels of health. This is observable in Plate 1, which plots the empirical distribution functions of equivalized real income for the different categories of SAH. Moving from left to right to compare the distribution of income across increasing levels of SAH, these show evidence of stochastic dominance.

Educational status is often considered as a determinant of health and source of health inequalities, and it is of interest to consider how health is related to educational attainment in our data. Figure 4 shows the distribution of SAH, pooled over all eight waves, by maximum educational attainment. This reveals a positive gradient between education and health for both men and women.

State Dependence

The main focus of this paper is on health dynamics: how does health status in the previous period affect the probability distribution of current health status? While the model-based approaches allow us to condition on other variables when ascertaining the effect of previous health status, Figure 5 and Table II inform this aspect of health dynamics without conditioning on other variables. Figure 5 shows the distribution of SAH at wave 2 by SAH at wave 1. Persistence in health outcomes is observable from the figure. For the most extreme cases, it is clear that the probabilities of transitions to very poor health from excellent health, or the reverse, are almost zero. This observation can be generalized; individuals are far more likely to remain close to their initial state than move far away from it. An alternative way of seeing this is by considering the transition matrices in Table II. Here the rows indicate previous health state while the columns indicate the current state; e.g. the elements of the first row provide information on the conditional distribution of SAH at t given excellent SAH at t − 1. The elements of the table can be interpreted as the conditional probabilities under a Markov model. Persistence is again observable by considering the relative magnitudes of the diagonal elements and those close to them compared to those far from the diagonal.

Table II. Transition matrices, balanced panel

(a) Men
SAH	EX	GOOD	FAIR	POOR	VERY POOR	N
EX	0.600	0.342	0.046	0.010	0.002	5485
GOOD	0.184	0.651	0.142	0.019	0.004	9263
FAIR	0.055	0.361	0.471	0.100	0.012	3433
POOR	0.029	0.120	0.340	0.418	0.093	1031
VERY POOR	0.032	0.073	0.133	0.423	0.339	248
N	5231	9287	3565	1111	266	19 460

(b) Women
SAH	EX	GOOD	FAIR	POOR	VERY POOR	N
EX	0.572	0.353	0.059	0.013	0.004	5164
GOOD	0.150	0.657	0.162	0.026	0.005	11 306
FAIR	0.040	0.362	0.465	0.116	0.017	4928
POOR	0.021	0.156	0.360	0.365	0.098	1587
VERY POOR	0.014	0.106	0.192	0.326	0.362	423
N	4884	11 329	5082	1649	464	23 408

3.1. Correlated Effects and Initial Conditions

To allow for the possibility that the observed regressors may be correlated with the individual effect we parameterize the individual effect (Mundlak, 1978; Chamberlain, 1984; Wooldridge, 2002a). This allows for correlation between the individual effects and the means of the regressors. In addition, because we are estimating dynamic models we need to take account of the problem of initial conditions. Heckman (1981) describes two assumptions that are typically invoked concerning a discrete time stochastic process with binary outcomes. The same issues arise with an ordered categorical variable. The first assumption is that the initial observations are exogenous variables. This is invalid when the error process is not serially independent and the first observation is not the true initial outcome of the process. In our case, the latter condition is violated, while the former is unlikely to be correct. Treating the lagged dependent variables as exogenous when these assumptions are incorrect leads to inconsistent estimators. The second assumption often invoked is that the process is in equilibrium such that the marginal probabilities have approached their limiting values and can therefore be assumed time-invariant. This assumption is untenable when non-stationary variables such as age and time trends are included in the model as we do here.

Wooldridge (2002a) has suggested an approach to deal with the initial conditions problem in non-linear dynamic random effects models by modelling the distribution of the unobserved effect conditional on the initial value and any exogenous explanatory variables. This conditional maximum likelihood (CML) approach results in a likelihood function based on the joint distribution of the observations conditional on the initial observations. Parameterizing the distribution of the unobserved effects leads to a likelihood function that is easily maximized using pre-programmed commands with standard software (e.g. STATA). However it should be noted that the CML approach does specify a complete model for the unobserved effects and may therefore be sensitive to misspecification.

We implement this approach by parameterizing the distribution of the individual effects as:

(5)

where x̄_i is the average over the sample period of the observations on the exogenous variables. u_i is assumed to be distributed and independent of the x variables, the initial conditions, and the idiosyncratic error term (ε_it). Substituting equation (5) into equation (1) gives a model that has a random effects structure, with the regressors at time t augmented to include h_i1 (a vector of dummy variables) and x̄_i. Three features should be noted. First, this specification implies that the identified coefficients of any time-invariant regressors are composite effects of the relevant elements of β and α₂.16 Secondly, all time dummies must be dropped from x̄_i to avoid perfect collinearity. Thirdly, the estimates of α₁ are also of interest as they are informative about the relationship between the individual effect and initial health. We would expect there to be a positive gradient in the coefficient estimates as we move from very poor to excellent health.

3.2. Attrition Bias

Tables V–VIII have shown evidence of systematic patterns of attrition by socioeconomic characteristics and previous levels of health, but it remains to be seen whether this will lead to attrition bias in our empirical models of SAH. To test for attrition bias we use simple variable addition tests as proposed by Verbeek and Nijman (1992, p. 688). The test variables we use are (i) an indicator for whether the individual responds in the subsequent wave (NEXT WAVE), (ii) an indicator of whether the individual responds in all eight waves and, hence, is in the balanced sample (ALL WAVES) and (iii) a count of the number of waves that are observed for the individual (NUMBER OF WAVES). Each of these are added to our dynamic correlated effects ordered probit model, given by equations (1) and (5) and estimated with the unbalanced sample. This gives three separate tests for attrition bias. These tests may have low power and do not correct the estimates for attrition bias (Verbeek, 2000). Additional evidence can be provided by Hausman-type tests that compare estimates from the balanced and unbalanced samples. However, in the context of ordered probit models, this is complicated by the fact that the coefficients from the two specifications do not have a common scale. Instead we rely on a comparison of average partial effects, which do share the same scale.

To allow for attrition we adopt an inverse probability weighted (IPW) estimator and apply it to the pooled ordered probit model (Wooldridge, 2002b, c).17 To implement this estimator we estimate (probit) equations for response (d_it = 1) versus non-response (d_it = 0) at each wave, t = 2, …, 8, conditional on a set of characteristics (z_i1) that are measured for all individuals at the first wave. This relies on ‘selection on observables’ and implies that attrition can be treated as ignorable non-response, conditional on z_i1 (Fitzgerald et al., 1998; Wooldridge, 2002c, p. 588).18 In practice z_i1 includes the initial values of all of the regressors, including initial health states. Also it includes initial values of other indicators of morbidity (whether the individual reports a limiting health problem, whether they report a disability, and their GHQ-12 score which indicates their psychological well-being as measured by the general health questionnaire) along with initial values of their activity status (self-employed, unemployed, retired, maternity leave, caring for the family, student and long-term sick, with employed as the reference category). The probits for response/non-response are estimated at each wave of the panel, from wave 2 to wave 8, using the full sample of individuals who are observed at wave 1. The inverse of the fitted probabilities from these models, 1/p̂_it, are then used to weight observations in the ML estimation of the pooled ordered probit model using:19

(6)

Wooldridge (2002b) shows that, under the ignorability assumption:

(7)

the IPW estimator is consistent and asymptotically normal. Wooldridge also shows that using the estimated p̂_it rather than the true p_it and ignoring the implied adjustment to the estimated standard errors leads to ‘conservative inference’ so that the standard errors are larger than they would be with an adjustment for the use of fitted rather than true probabilities. Therefore we do not adjust the standard errors.20

The IPW estimator can be adapted to allow the elements of z to be updated and change across time, for example adding z variables measured at t − 1 to predict response at t. This should improve the power of the probit models to predict non-response and hence make the ignorability assumption more plausible. In this case the probit model for attrition at wave t is estimated relative to the sample that is observed at wave t − 1. This relies on attrition being an absorbing state and is therefore confined to ‘monotone attrition’ where respondents never re-enter the panel. Also, because estimation at each wave is based on the selected sample observed at the previous wave, the construction of inverse probability weights has to be adapted. The predicted probability weights are constructed cumulatively using , where the denote the fitted selection probabilities from each wave. In this version of the estimator the ignorability condition has to be extended to include future values of h and x (see Wooldridge, 2002c, p. 589). Once again Wooldridge shows that omitting a correction to the asymptotic variance estimator leads to conservative inference so we do not adjust the standard errors.

4.1. Tests for Attrition

Table IX presents the variable addition tests for attrition bias estimated using the dynamic correlated effects ordered probit model. All three test variables show evidence of attrition for men and also suggest attrition among women, although the p-values on the statistics are higher. The positive coefficients on the test variables are consistent with the fact that response rates are positively associated with health. Adding these test variables to the model is not intended to ‘correct’ the estimates for attrition, but it is informative to compare the estimates with the baseline model that does not include the test variables. It is striking that, for key variables such as income and lagged health state, the differences between the estimated coefficients are negligible. This suggests that attrition may not bias the estimates of these effects, a result that is reinforced below.

4.2. Estimates of Dynamic Ordered Probits

Tables X and XI present the coefficient estimates for the ordered probit models based on pooled and random effects specifications. In all cases we estimated a correlated effects version of the models by parameterizing the distribution of the unobserved individual effects as in equation (5). Both the pooled and random effects specifications were estimated on the balanced and unbalanced samples. The estimates for the pooled ordered probit models allow for serial correlation in the errors by using a robust estimator of the covariance matrix. In addition we estimated the pooled model using IPW to adjust for attrition. Both variants of the IPW estimator are presented: IPW-1 uses wave 1 regressors to predict non-response, IPW-2 also includes values from the previous wave as well as the initial wave and the sample is restricted by excluding observations that exhibit non-monotone attrition. For the random effects specifications we incorporated unobserved heterogeneity explicitly by including Gaussian random effects. These models were estimated by maximum likelihood using Gauss–Hermite quadrature with 12 evaluation points using reoprob.ado.21

The estimated coefficients for the random effects model are not directly comparable to those reported for the pooled models due to different scaling of the error variance. The pooled ordered probit assumes that the error term as a whole is distributed N(0,1) for identification of β. The random effects ordered probit restricts ε_it to be N(0,1), so that the overall error variance equals . This implies different scaling of the estimated coefficients in the two models. However we can compare the relative effects of pairs of variables across the two models. Also, the magnitudes of the effects of individual variables can be compared by computing their average partial effects. These average partial effects are defined and tabulated below for the key variables of interest.

To formally test for state dependence we estimated dynamic models which included dummy variables representing one-period lags of the categories of the dependent variable (SAHEX(t − 1) − SAHVPOOR(t − 1)). Including state dependence is important. For the pooled models in columns (1)–(4), the estimated coefficients on the lagged categories of the dependent variable are large and highly statistically significant. There is a gradient across the estimated effects of previous health status as one moves from previous health status of very poor to excellent (the baseline category is lagged good health). Most of the coefficients, on the lagged variables and socioeconomic characteristics, are stable across the balanced and unbalanced samples and when the inverse probability weights are used to adjust for attrition.

Columns (5) and (6) introduce explicit unobserved individual heterogeneity into the dynamic model by specifying random effects. Again, these models were estimated by maximum likelihood. Allowing for heterogeneity substantially improves the fit of the model as evidenced by the change in log-likelihood. For men approximately 33% and for women approximately 31% of the latent error variance is attributable to unobserved heterogeneity, as measured by the intra-class correlation coefficient (ICC).

All of the models presented in Tables X and XI parameterize the unobserved individual effect as a function of the within-individual averages of the time-varying regressors and a vector of dummy variables to represent the first-period observations on the dependent variable.22 The estimated coefficients for the initial period observations are reported as SAHEX(1) − SAHVPOOR(1). There is a positive gradient in the estimated effects as we move from very poor to excellent initial period health. This implies that there exists a positive correlation between the initial period observations and unobserved latent health. It is striking that, comparing the random effects specifications to the pooled models, the coefficients on lagged health states are smaller and coefficients on the initial health states are larger. In fact the relative magnitudes of the effects of lagged health relative to initial health are reversed in the random effects models compared to the pooled models.

For men, educational attainment is not statistically significant but it is highly significant for women. Conditioning on the within-individual average of income (mean ln(INCOME)) renders the current income variable (ln(INCOME)) non-significant except for the random effects specifications for men and for the unbalanced sample of women. Caution should be used in interpreting the results for mean income as, in general, it is not possible to separate a causal effect of long-term economic status on health and the correlation between mean income and the unobservable individual effect. However one way of interpreting the results is to regard current income as a measure of transitory income shocks and mean income as a measure of long-term or ‘permanent’ income (see e.g. Contoyannis et al., 2004; Frijters et al., 2003 who adopt this interpretation).23

4.3. Average Partial Effects

The scaling of the ordered probit coefficients is arbitrary. To provide an indication of the magnitude of the associations between SAH and the regressors we present average partial effects (APEs). For continuous regressors, such as income, these are obtained by taking the derivative of the ordered probit probabilities with respect to the variable in question. For discrete regressors, such as lagged health state, they are obtained by taking differences. In both cases the partial effects are functions of h_it−1, x_it, and also the individual effect α_i. One option is to compute these at E(α_i) = α₀ + α′₁h̄_i1 + α′₂x̄_i. The alternative, that we adopt, is to compute average partial effects (see e.g. Wooldridge, 2002a, p. 22). In this case the partial effects are averaged over the population distribution of heterogeneity and computed using the population averaged parameters β_α. In the random effects specifications these are given by .24 Wooldridge (2002a) shows that computing the partial effect at the observed values of the regressors for each observation and averaging the estimates over the observations provides a consistent estimate of the APE.25

In the ordered probit model it is possible to compute APEs for each of the five categories of self-assessed health. For parsimony, Table XII summarizes the APEs of income, educational attainment and lagged health state on the probability of reporting excellent health. In this case the sign of the APE has a clear qualitative interpretation, with a positive sign implying a positive association with health and vice versa. These are presented for all six versions of the model, along with the sample standard deviations of the partial effects. Comparing the balanced and unbalanced samples gives very similar results within the pooled and random effects models, suggesting that non-response does not lead to differences in the estimated APEs. This is reinforced by the fact that the pooled models with and without inverse probability weights are virtually identical.

The income effects are larger for mean income (mean ln(INCOME)) than for current income (ln(INCOME)) and are a little larger in the random effects models than the pooled models. The APEs for lagged health state are noticeably lower for the random effects specifications, particularly for excellent and good previous health. This is driven by the fact that the relative magnitudes of the effects of lagged health relative to initial health are reversed in the random effects models compared to the pooled models. The APEs of educational attainment are larger in the random effects models and are larger for women than for men.

4.4. Subsample Analysis

As noted in the descriptive analysis, it appears that the dynamics of health may be influenced by age, income and educational status. To investigate this further we split the sample of males and females into subsamples based on age (≤45 and >45) at the first wave, income quartile at the first wave, and highest attained educational qualification. For each subsample we estimated a dynamic random effects ordered probit model controlling for the initial conditions and correlated effects. The results are presented in Tables XIII–XV. For both men and women the proportion of latent error variance attributable to unobserved heterogeneity is approximately the same for the age (0.32–0.34 for men and 0.31–0.33 for women), education (0.29–0.34 for men and 0.27–0.33 for women) and income (0.29–0.36 for men and 0.29–0.32 for women) subsamples compared to the full model (0.33 for men and 0.31 for women). This implies that conditional on age, education and income effects, the composition of the error variance for latent health is approximately the same across age, education and income groups.

Table XIII shows that the APEs for mean income are stable for women but larger for younger men than older men. The APEs for education are larger for younger people than older. The magnitude of the state dependence effects are a little smaller for older people. Tables XIV does not exhibit strong patterns in the APEs by education, although the effect of state dependence tends to be lower among those with no formal qualifications. Table XV does not reveal any clear patterns in the APEs when the sample is split into income quartiles.