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Endogenous Health Groups and Heterogeneous Dynamics of the Elderly

We propose a novel methodology to classify individuals into groups of health and characterize their transition across these groups as they age. We use MCMC techniques to estimate a panel Markov switching model that exploits information from both the crosssectional and time series dimensions. Using the Health and Retirement Study, we identify four clearly differentiated and persistent health groups, depending on individual’s physical and mental disabilities, with heterogeneous transitions across gender and education. Our classification outperforms existing measures of health used in the literature at explaining entry in nursing homes, home health care, out-of-pocket medical expenses and mortality.


Introduction
Life-cycle models with heterogeneous agents are becoming increasingly popular among scholars as a tool for designing optimal policies related to Social Security, inequality, insurance markets or health care among others. In order to use these models as accurate laboratory economies, it is therefore crucial to appropriately capture health and earnings risk in order to understand individuals' decisions. As a result, macro models are benefiting from a recent and influential strand of the empirical literature estimating richer earning dynamics (see Guvenen et al. 2015;Arellano et al. 2017) and analysing their macroeconomic implications (De Nardi et al. 2018;Gorea and Midrigan 2017, among others). Different from earnings, health is a multidimensional attribute hard to measure and summarize. This paper proposes a dynamic latent variable model for jointly estimating a parsimonious health classification and the associated process for health transitions. Health dynamics are allowed to differ across gender and education types to capture heterogeneity in health risks across the population. The methodology exploits both the cross-sectional and the time-series dimension of panel data sets based on detailed surveys, which contain a wide array of variables about different aspects of elderly's health. If we restrict to the cross-sectional dimension, our method mimics the latent class model (Lazarsfeld, 1950); that is, it allocates individuals to latent groups to fit the joint distribution of all the observed health variables considered. On the other side, along the time-series dimension, our strategy emulates Hamilton (1989)'s model inasmuch as it infers the health status of an individual at a point in time using her whole timeseries through the auto-correlation structure. Altogether, we assign each individual at each point in time to a given group using her health information and that of every individual in past, current and future periods. We thus reduce the dimensionality of the data to a discrete variable which corresponds to the endogenous groups.
We apply the methodology to allocate individuals in the Health and Retirement Study (HRS) into four groups according to their difficulty with Activities of Daily Living (ADLs) and Instrumental Activities of Daily Living (IADLs). Additionally, we characterize health dynamics and survival as a hidden Markov chain which incorporates heterogeneity across age, gender, and education. Precisely, we model transitions across health groups as logistic functions of the aforementioned attributes whose parameters change depending on the current health status.
Our modelling approach presents three desirable features. First, it considers the classification of individuals and groups' dynamics jointly. This way the health classification is not based solely on the information of the current period but on all the observations including death events. Moreover, potential misreporting is smoothed out by the algorithm which reduces possible biases affecting groups' dynamics. Secondly, even though the resulting health measure is discrete, we also obtain as a by-product the probability of belonging to each group conditional on the whole sample, which enables to weight observations according to their representativeness of each group using a continuous measure. Third, the latent nature of our groups allows classifying an individual's health even in the case of missing information as long as we have past or future information.
The empirical strategy requires the estimation of thousands of hidden Markov chains, one per individual, together with hundreds of parameters. For that reason, we resort to Markov chain Monte Carlo methods. In particular, we rely on a Metropolis-within-Gibbs algorithm which involves two main blocks. First, given the health group of each individual, it is straightforward to sample the parameters driving the I-ADLs binary processes through a Metropolis step; and likewise, the parameters ruling the dynamics. 1 Then, conditional on these parameters we obtain, for each individual, a realization of the latent health group using Kim (1994)'s smoother algorithm. To save the computational burden to future researchers, the probabilities for each individual and time are available at the authors' website. Further, based on our results we suggest an estimation-free classification that improves currently used ones, although it performs worse than the endogenous one. 2 Four groups which divide individuals into physically frail, mentally frail, impaired, and healthy represent health suitably. The impaired have both types of limitations, physical and cognitive, while the healthy have no or light difficulties with I-ADLs. 3 In turn, the physically frail have limited mobility, while the mentally frail have difficulties with more cognitive tasks such as managing money. Importantly, and in line with gerontology literature (e.g. Morris et al., 2013), not all the I-ADLs are equally informative for classifying individuals in health 1 Along the paper we use I-ADLs to denote the set of both ADLs and IADLs; likewise, I-ADL refers to one of these variables. 2 Bueren's webpage provides the probabilities of each individual in the HRS identified by hhidpn and wave, and the parameters of the model. 3 Along the paper, we use italics to refer to our states, hence a healthy individual is a member of the group we label healthy.
groups. For example, if a person has difficulties with getting in or out of bed, she belongs to the physically frail group with a probability higher than one third but to the mentally frail with a probability lower than 5%. In contrast, an individual incapable of taking medications is much more likely to belong to the mentally rather than the physically frail group.
Groups' dynamics features stylized facts previously documented in the literature of aging (Manton and Soldo, 1985): older individuals have relatively worse health, health deteriorates with age, individuals in worse health have larger chances of dying, and females live longer than males. Furthermore, in line with Brown (2002) and Meara et al. (2008), we find a large educational gradient in life expectancy. Nonetheless, despite living longer, educated individuals spend, on average, less time impaired, consistent with Pijoan-Mas and Ríos-Rull (2014).
Even though any health classification reveals the protective effect of education, they lead to very different magnitudes. Precisely, while high-school graduates live on average around 30% less time in our unhealthiest group and 40% more in the healthiest one, these gradients equal 55% and 140% if we rely on self-reported health.
Aside from education, current health status constitutes an important source of heterogeneity because of the groups' persistence. For instance, a 75-years-old impaired respondent has a probability of remaining impaired of 60%; thus she faces a health risk different from a healthy respondent who stays healthy with 80% probability. This feature is consistent with our groups being closely linked to long-term care (LTC) needs and it is less pronounced in the case of self-reported heath.
We then compare access to medical and care services across health groups based on the estimated probabilities. On average, impaired (healthy) individuals spend around $10,043 ($2,310) per year in out-of-pocket medical spending. Likewise, mentally frail individuals spend $1,343 more than physically frail ones, who employ $3,565. The use of LTC services also presents large differences across groups. While 9% of the individuals mentally frail live in a nursing home at the time of the interview, only 1.6% of the physically frail do so. This disparity widens between members of the healthy group, who avoid the nursing home almost surely, and those of the impaired, out of which 33.6% reside in these facilities. A similar pattern arises if we compare the received professional care of these two extreme groups. Nonetheless, mentally and physically frail individuals need a medical-trained person to look after them at home with the same probability.
Finally, we contrast our estimated health groups with other commonly used health classi-fications, namely, five different levels of self-reported health, whether the individual reports difficulty with any ADL, and the division of a frailty index into five equally sized groups. 4 To do so, we consider three main variables associated with health-related spending, particularly, out-of-pocket medical expenditures, and indicators of residing in a nursing home and receiving care which the macro literature has identified as crucial drivers of savings (De Nardi et al. 2010; Barczyk and Kredler 2018;Ameriks et al. 2015). Our four groups classification generates more differentiated groups; furthermore, it explains about three times more variance than self-reported health and twice as much as the use of an ADL indicator. These results resemble an out-of-sample exercise since these variables do not enter the classification model.
Additionally, we analyze the ability of the different classification to predict mortality and find that our four groups dominate the alternatives.
Our paper complements the literature analyzing the effect of health on economic decisions.
This literature relies on dynamic structural models to quantify the importance of mechanisms or to derive implications for policymaking. Due to the curse of dimensionality, researchers undertake an ad-hoc decision over which of all the possible health variables from the available surveys to use as a state variable. Van der Klaauw and Wolpin (2008) and French and Jones (2011) divide individuals into two groups of self-reported health to analyze how health affects the retirement decision. De Nardi et al. (2010) use the same strategy to quantify the effect of health-related expenses on the savings decision of the elderly. With a similar objective, Ameriks et al. (2015) and Barczyk and Kredler (2018) classify individuals as unhealthy if they report a difficulty with ADLs or require care, respectively. Alternatively,  splits a frailty index into five quintiles to introduce health in their insurance demand model. This paper relates to an extensive literature which proposes econometric methods to analyze different issues in health economics (see Jones, 2000, for a survey). Closely related to our paper is Deb and Trivedi (1997) who show that a finite mixture of negative binomials, characterizing "healthy" and "ill" individuals, explains counts of medical care utilization by the elderly in the U.S. better than previously proposed specifications. They, however, do not classify individuals into the aforementioned categories. Moreover, they disregard health dynamics which is of first-order relevance: Contoyannis et al. (2004) stress the importance of health persistence using a dynamic panel ordered probit model for self-reported health.
We also contribute to a growing literature that summarizes health variables into a single index that explains most of the variation related to health (see Searle et al., 2008). Regarding HRS, Yang and Lee (2009) compute a frailty index based on chronic conditions, ADLs, IADLs, depressing symptoms, self-reported health, and obesity. Nonetheless, its continuous nature prevents researchers to include it in structural models. One exception is Bound et al. (2010) who considers health as a continuous latent variable and include it into a structural model to analyze retirement. To be able to solve the model; though, they assume that individuals are completely unable to self-insure against medical expenses.
The rest of the paper is structured as follows. We briefly describe the HRS data in Section 2. Then, the econometric model and the estimation strategy are presented in Section 3. Next, we present the main results in Section 4 and we compare our proposed classification with alternative ones in Section 5. Finally, Section 6 concludes.

HRS and I-ADLs
Our data comes from the RAND HRS dataset which comprises a cleaned version of the 6 probabilities. While the median age is 72 years, the share of individuals is decreasing in age as they die. Likewise, females account for 58% of the sample as their life expectancy is higher than the males' one. In terms of education, 72% of individuals completed high school which constitutes 74% of the sample due to its superior life expectancy.
[ Figure 1 about here.] The HRS provides dozens of health-related variables, but we restrict to individual's ability to perform Activities of Daily Living (ADLs) and Instrumental Activities of Daily Living (IADLs) to infer the health status. ADLs were proposed by Katz et al. (1963) as a measure of how independent a patient is, and consequently, they include very basic activities such as if they can walk or dress. IADLs, in contrast, consist of activities more closely related with cognition as the possibility of using a phone or controlling her medication. Accordingly, these variables relate to the need for LTC which is the dimension of health we aim to identify.
Although our model could incorporate more information, reducing the set of variables eases the interpretation of the groups. Besides, by excluding other variables, we can use them to compare the performance of our classification against other alternatives.
Precisely, we utilize twelve binary variables, denoted as I-ADLs, which include six ADLs and six IADLs that describe whether individuals have any difficulty to perform these types of basic tasks. We extract this information from the HRS questionnaire to which respondents select one out of six possible answers: Yes and Can't Do that we label as 1, No to which we assign a value of 0, and Don't Do, Don't Know, and Refuse to answer, which are recorded as missing. Altogether, 30% of respondents battle with at least one I-ADL. These probabilities, nevertheless, change substantially across demographic groups and age as Figure 2 shows. When they are 60 years old, more than 40% of the individuals who drop out high school already report 7 difficulties with at least one I-ADL. On the other hand, only one high-school graduate out of five struggle with daily activities. Regarding gender, these proportions are also heterogeneous since 22% of females present some type of difficulty compared to 19% in the case of males.
The differences across gender shrink as people age; while at the same time, the share of them facing troubles with an ADL or IADL increases for all groups systematically.
[ Table 1  The HRS also includes a question to qualify respondent's self-reported health (SRH). Since another strand of the literature hinges on subjective measures of health to classify individuals, in the last five columns of Table 1 we compare this measure with the answers related to ADLs and IADLs. Not surprisingly, we observe that as people report worse health, they are more likely to present problems with I-ADLs, nonetheless, the importance of each activity differs.
In particular, individuals reporting poor health are not able to walk, dress or bath with probabilities around 40%, while for the remaining three ADLs the corresponding figures barely surpass 30%. Similarly, difficulties with IADLs are also diverse within the worst self-reported health groups since 50% of individuals endeavor to shop but only 20% encounter complications to take their medications.

Econometric model
We have an unbalanced panel of individuals i = 1, . . . , N followed for t i = 1, . . . , T i periods which correspond from age a i 1 to age a i T i where a ∈ (a, a). For each individual, we observe K dummy variables corresponding to each I-ADL across time (x 1,i,t , x 2,i,t , . . . , x K,i,t ), provided the individual is alive and interviewed. All or some of the variables for a given individual who is alive can also be missing for some period t i . Although we take missing observations into account under the assumption that they occur completely at random, we abstract from them in the model description to simplify the exposition.
We assume that the main source of heterogeneity in the population is represented by a finite number of possible health groups or clusters which are not observed by the researcher.
Conditioning on education, e; age, a; and gender, s; the current health cluster of individual i is independent of previous health clusters except for the most recent one (Markov first-order property). Besides transiting across health groups, individuals may also die which is represented by an observable and absorbing state labeled as D.
Specifically, we consider that individual i at time t belongs to a health group h i,t out of H possible ones. Given her group is g, the probability of facing difficulties with the k'th I-ADL, say x i,k,t = 1, is µ k,g . Under the assumption that I-ADLs are independently distributed conditional on the health status, the joint distribution of where µ g = (µ 1,g , µ 2,g , . . . , µ K,g ) . Therefore, individuals within the same health group have the same probabilities of experiencing problems with an I-ADL whereas these probabilities might vary if individuals do not belong to the same group. Similarly, the same individual might face a different likelihood regarding I-ADLs if she changes groups during her life.
In favor of parsimony, we model health outcomes as independent across time and individuals conditional on the health group. In the case of I-ADLs, it seems plausible that their persistent component is only due to health, nonetheless, the model can accommodate other types of persistence if the researcher wants to extend the set of conditioning variables. We take into account health dynamics by explicitly modeling the transition probabilities across groups. In particular, an individual i at time t who belongs to group g transits to group c with probability where H is the set that contains the H health groups. The remaining possible event is that the individual dies, which is an observable state that occurs with probability .
This specification allows health groups to own distinct dynamics as parameters differ according to the current health group. Moreover, to capture within-group heterogeneity, transition probabilities can depend on age, gender and education level through the function f g,c (a, s, e) whose parametric specification is given by f g,c (a, s, e) = β 1,g,c + β 2,g,c a + β 3,g,c s + β 4,g,c e + β 5,g,c (a × s) + β 6,g,c (a × e).

Posterior simulation
We aim to recover the posterior of all the parameters and the latent variables that classify the health group to which each individual belongs at each point in time. To do so, we use a Gibbs sampling procedure to estimate the models for different choices of the number of health groups H. In essence, this amounts to reducing a complex problem, that is, sampling from the joint posterior distribution of both parameters and state variables, into a sequence of tractable ones, i.e., sampling from conditional distributions for a subset of the parameters conditional on all the other parameters, for which the literature already provides a solution.
, as the collection of all health groups, and µ and β as the vectors stacking the parameters of the I-ADLs process and the transition probabilities, respectively. In addition, we include in X the data we observe; that is, age, gender, education, if the individual is death or alive, and her situation in terms of ADLs and IADLs. The Metropolis-within-Gibbs algorithm involves sampling sequentially from several blocks. Specifically, iteration m involves: The empirical results shown in the next sections are based on 40,000 draws. The first 2,000,000 draws are disregarded as burn-in and of the remaining 4,000,000, one every 100 draws is retained.

Sampling the states: Kim's Smoother
To sample the states, we apply the methodology developed by Kim (1994): 1. Using the filter proposed in Hamilton (1989) we obtain p(h i,T = g|β, µ, X) for all g ∈ H.
3. Similarly, we sample h i,t conditional on β,µ, X and h i,t+1 , using the following result: As a result, each individual has a different probability of belonging to a given group depending on her past, current and future answers regarding I-ADLs. Moreover, this probability also incorporates information about the individuals' death wave, as well as her age, gender, and education.
To form a complete likelihood, we need to know the unconditional distribution of h i,1 for each i, p(h i,1 |β). Since the model is non-stationary due to its dependence on age, we cannot compute the unconditional distribution without further assumptions. In particular, we consider the unconditional distribution at the age of 60 coincides with the stationary distribution given by the parameters of the first transition (from 60 to 62).

Sampling the transition probabilities and the Bernoulli parameters
In this step, we sample from the posterior of the parameters of the Bernoulli distributions and the ones governing the health dynamics (µ, β) conditional on the health groups, H, and the data, X.
Regarding priors, we consider a uniform on [0, 1] for the elements of µ and a diffuse Gaussian prior centered at 0 and covariance matrix 100 · I for β. Hence, the posterior of the parameters governing the health dynamics and the one driving the Bernoulli distributions are independent conditional on the latent health group. Precisely, their posterior distributions are given by

Starting the algorithm
To obtain the starting set of parameters µ 0 and β 0 for the algorithm, we sample from an approximate model in two steps. First, we obtain µ 0 as the mode of the posterior described in equation (1) under the assumption that h i,t are independent across both dimensions. 6 Second, we use the same model to simulate h i,t from the posterior probability p(h i,t |µ, x i,t ). Given a sample of health groups, we get the mode of the posterior of β, β 0 , under the assumption that groups follow the same multinomial logit specification as in the baseline model.

Obtaining moments
In most applications, as in the following sections, researchers aim to compute several sample moments conditional on a given health level. Our model, however, results in a probability of being in each group even if one fixes the parameters. While we can impute individuals to their most likely groups, using these probabilities to weight observations enhances our measure without losing the discrete nature of the variable.
For instance, assume the researcher wants to obtain the expectation of several outputs, say where M denotes the specific structural economic model in hand and A are the quantities of interest. In our context, the dimension of the state space is greater than 2 K , thus she must discretize X intoX = {x 1 , ...,x b } and then the final result equals First, our procedure provides a natural way of obtainingX. Second, the proposed methodology also determines the probabilities of eachx given the sample such that we can obtain Thus, even though we can only compute A at some points, we can weight each observation by its representativeness of each group.

Health groups
We first describe how the algorithm classifies individuals into groups and then how health evolves as individuals age taking into account differences in education and gender. To define the groups, the model identifies those that explain the joint distribution of difficulties with I-ADLs the best, taking into account the dynamics. In this context, the only parameter that is not endogenous is the total number of clusters, whose value we vary from two to five to discern what is the contribution of each successive cluster.
In what follows, we report the median of the posterior distribution of the parameters -or relevant functions of them. Figure 3 reports the probability of reporting difficulties with each I-ADL conditional on being in each cluster, that is µ k,g in equation (1). Each panel corresponds to a different number of clusters H. Meanwhile, each marker symbol represents a cluster and each tick in the horizontal axis refers to an ADL (the first six) or an IADL (the remaining ones). The higher the marker is, the more likely is that an individual in that specific group struggles with the corresponding I-ADL.

Endogenous classification
[ Figure 3 about here.] If we set H = 2, the algorithm divides individuals into one group whose probability of declaring problems with an I-ADL is close to 0 for every I-ADL and another one which owns a higher likelihood of facing problems with every I-ADLs. We label the former group as healthy (circumferences) and the latter as impaired (triangles). We also find large differences in the probabilities across I-ADLs within the impaired group which suggests that activities differ in their importance for categorizing individuals. For example, as regards the impaired group these probabilities range from 31% in the case of eating to 77% in the case of shopping.
The upper right panel of Figure 3 presents the same graph but with H = 3. There is still one group with almost zero probability to face difficulties with any I-ADL and another with again the highest probabilities of struggling with all I-ADL. Nevertheless, the probabilities of this group are slightly higher than when we consider only two groups as some individuals previously classified as impaired belong to the new group whose probabilities lie between the other two.
When we allow for four groups, the impaired and the healthy groups become more distant.
In addition, the middle group splits into two very different ones. One group with moderate probabilities to suffer difficulties with an ADL but low probabilities to have problems with IADLs, reflecting that those individuals are physically frail; and another one which consists of mentally frail elderly in the sense that they are mostly dependent in terms of IADLs but not as much in terms of ADLs.
Lastly, we consider H = 5 in the lower right panel. In that case, the previous groups remain almost unchanged and the new group that emerges is extremely similar to the healthy one, with the exception that individuals struggle reading a map. As one adds more groups, their connection to health is even weaker; therefore, in the remaining of the paper, we focus on the case of four groups.
While Figure 3 characterizes individual's health in each cluster, it is silent about the meaningfulness of each I-ADLs for classifying individuals. For instance, in the case of H = 2, the elderly in the impaired group present a much higher probability of facing difficulties reading a map than eating. This comparison, however, disregards that unconditionally only 5% of individuals struggle to eat but 16% are not able to read a map.
To overcome this issue, Figure 4 plots the probability of belonging to group g given that the individual faces difficulties with I-ADL k, that is, where the relative size of the bars indicates which I-ADL is more informative.
[ Figure 4 about here.] Following the same example, if a person has difficulties to eat, she belongs to the impaired group with probability 90%, according to the upper left panel. Meanwhile, individuals incapable of reading a map have almost the same likelihood to be part of the impaired or healthy group; thus, MAP is uninformative. The pattern of these two I-ADLs remains unchanged when H = 3 and H = 4; MAP is never informative while EAT is the best indicator to classify individuals into the impaired group. This evidence is in line with previous evidence in the medical literature (see Morris et al., 2013, and references therein) which argues that difficulties with eating are the best predictor of full dependence. however, the joint structure of these variables also contributes significantly to identification.
To see this, in the third and fourth columns in Table 2 we provide the proportion of respondents who report difficulties with at least one ADL or IADL. Consistent with the previous discussion, individuals in the impaired group are the ones more likely to present difficulties with an I-ADL; actually, they face problems with one I-ADL almost surely. The other side of the coin is the healthy group which probability of reporting troubles with ADLs varies between around 4% and 9% depending on the number of groups. In the third panel (four groups), the distinction between physically frail and the mentally frail becomes salient. While in the former 80% of respondents struggle with ADLs and 61% with I-ADLs, the latter faces more problems with IADLs (100%) and less with ADLs (55%).
[ Groups are not only different in terms of I-ADLs but also in terms of demographics. For instance, if our classification correctly identifies the health status of individuals we expect members of the impaired group to be older than those of the other groups. In that regard, Table 2 shows they are indeed on average nine years older than the ones in the healthy cluster and six years older than those physically frail. Additionally, the difference between mentally frail and impaired is smaller which is consistent with mental conditions caused by aging.
Next, in terms of education, high school graduates are overrepresented in the healthy group which is in line with previous literature on health inequality such as Mackenbach et al. (2008).
Another interesting pattern is that worse health groups contain a significantly higher proportion of women. These differences lead us to study pattern of heterogeneity of health dynamics across gender and education groups.

Heterogeneous health dynamics
The distribution of elderly into health groups changes with age, gender and education. Figure 5 plots the probability of being in each group through age. The left panels correspond to dropouts whereas the right ones present the results for high-school graduates; meanwhile, the upper graphs refer to males and the lower ones to females. The most common health status is healthy at early ages but starting at age 90, impaired becomes the predominant group. Further, the physically and mentally frail have very different dynamics. The former is stable throughout life while the latter increases steeply as elderly age. These patterns are very similar across education and gender, although the initial composition of individuals varies with demographic characteristics.
[ Figure 5 about here.] Since, in the estimation, mortality and health deterioration is allowed to vary by education group, we find that dropouts and high-school graduates encounter a very distinct health risks. [ Table 3 about here.] Individuals' incentives might also change across health groups since their expected health path might differ. Figure 6 displays the transition probabilities according to age and current health status. For example, a healthy elderly owns a very low probability to become impaired, thus a low health risk, everything else equal. In contrast, once an individual enters the impaired group, she is very likely to stay in that group; hence, her expected future medical spending is very high. In general, groups are very persistent and health is more likely to worsen than to improve, in line with our interpretation of the endogenous groups as different levels of LTC needs. Although mentally frail and impaired individuals do not recover, their large mortality rates limit the time spent in high levels of need.

Comparison with alternative indices
The need for a discrete measure of health has led researchers to use ad-hoc classifications.
In this section, we compare our endogenous classification with the main three alternatives: self-reported health, if the individual struggles with an ADL, and the quintiles of a frailty index. In addition, we also consider the Cartesian product of whether the individuals report difficulty with i) at least one ADL and ii) IADLs (excluding MAP) as an unsophisticated proxy of our endogenous classification. To perform the comparison, we focus on mortality and three variables related to the financial risk due to health: OOP medical expenditures, and indicators of receiving home-care and residing in a nursing home. OOP medical spending is a direct measure of the economic consequences of health. It includes the costs -in constant 2000 US dollars-of hospital and nursing home stays, doctor visits, dental treatments, outpatient surgery, prescription drugs, home health care, and special facilities. Received home care equals 1 if a medically-trained person has come to the respondent's home to help her, and nursing home resident takes value 1 for those individuals who live in a nursing home at the time of the interview.
[ Table 4 about here.] The health classification most widely used in the literature relies on an individuals selfassessment on their health status which can take 5 different values between excellent and poor. The self-reporting nature of the answer induces two opposing effects. On the one hand, individuals might know more about their health than researchers can ever measure. On the other hand, respondents might misjudge their health condition, incorporate other information as mood or consider different benchmarks of being good. Previous literature has analyzed the net effect of these two channels and establishes that the disadvantages often offset any benefit.
For instance, Crossley and Kennedy (2002) directly checks the reliability of self-assessment and finds that 28% of individuals change their answer from the beginning to the end of the survey. Moreover, this measurement error correlates with important socioeconomic variables; hence, it raises concerns about the validity of self-reported health (see Currie and Madrian, 1999, for a survey).
Nevertheless, the first panel of Table 4 confirms that self-reported health has information about the financial risks. Those respondents reporting worse health spend more on medical consumption and care, and are more likely to reside in a nursing home than those who claim to be healthy. The difference between the five groups varies though. In particular, answering excellent, very good, and good relates to almost the same risk, whilst fair and poor correspond to much more spending. Previous literature, thus, merges the three healthiest and the two worst groups. We denote this latter classification as self-reported health (2 groups).
Grouping individuals according to if they have an ADL or not is similar to our approach, specifically to identify the healthy respondents; hence the proportions of healthy and No-ADL almost coincide. This classification, however, considers every ADL equally important and disregards the number of ADLs, as well as difficulties with IADLs. Actually, elderly who struggle to eat are usually more dependent than those who are unable to dress themselves (Williams et al., 1994). and alcohol consumption to create a frailty index. Although the inclusion of more information improves the measure of health and allows to create more groups, the relevance of each variable is still assumed to be the same. Additionally, the resulting index is continuous which forces them to allocate individuals into five equally sized groups according to the quintiles of the index. As a result, the healthiest groups are very similar among themselves and the worst group present the same features as those who have an ADL in the Yes/No classification.
Finally, classifying individuals regarding whether they struggle with at least one ADL, IADL, both or none, which we denote as 4-I-ADL, can be understood as a simple approximation to our four groups. In contrast to the frailty index by  who effectively separates individuals without problems with any ADL in four groups, this method divides respondents who recognize problems to perform an ADL into three groups. Since these individuals are more heterogeneous, the resulting groups become more differentiated in all the variables considered.
Even if the four aforementioned alternative classifications are highly correlated with the health outcomes that we use, our estimated groups seem to be more differentiated across them.
For instance, using our methodology, the average difference in terms of OOP between healthy and impaired elderly is $7,751. According to self-reported health, however, an individual belonging to the worst group only expends $3,333 more than one in the best group. Similarly, the fact that you report an ADL implies that your average OOP medical spending is $2,648 higher; meanwhile, being a part of the worst, rather than the best, frailty quintile costs $3,305.
Not surprisingly, 4-I-ADL is the closest to our classification but the distance between the best and worst groups hardly surpasses $4,000. As for the intermediate groups, they are again less distinct in the case of the alternative classifications as their increment in spending is below $1,000 except from the two worst groups, compared to $1,281 which is the minimum difference between our groups.
Regarding the probability of residing in a nursing home, a similar pattern arises and the difference between the best and worst of our health groups at least duplicates the same difference using the alternative methods. The same holds true for home care when we look at self-reported health or struggling with at least one ADL but, in this case, our four groups outranks 4-I-ADL just mildly.
In line with the previous discussion, our classification also identifies future death events more accurately. In particular, an impaired individual dies with 40% probability whereas only 3.4 out of 100 healthy ones do not survive to the next wave. Instead, the difference between the healthiest and unhealthiest groups does not reach 25 percentage points with alternative classifications. Though relevant, this result might follow from the inclusion of death in the classification algorithm. 18

A horse race
Most of the time, the researcher's concern might not be to classify individuals into distant groups but to create a categorical index that captures most of the variation coming from health.
To assess the performance of the grouping methods in that context, Table 5 displays the R 2 of the following regression: where y i,t is the variable used as a reference, z i,t includes gender and education, and d i,t is a vector of dummy variables indicating to which group the individual belongs. 7 In the case of our classification, we use two alternative approaches. First, we substitute d i,t by a vector containing the probability of individual i at time t of belonging to each cluster (we label it Probs). Secondly, we assign each individual to her most likely state (which we label as Mode).
[ Nursing home residency, by virtue of being binary, contains a lower measurement error; nevertheless, the same ranking persists. Any measure that includes ADLs beats self-reported health by at least 5 percentage points, which doubles if we consider our unsophisticated method. Further, weighting each I-ADL, our health groups enhance the naive 4-I-ADL by 65% because it identifies the extreme dependent individuals better. In sum, our proposed classification explains almost 4 times more variance than self-reported health and 2.5 times more than ADL:Yes/No.
In contrast to nursing home residents, most elderly who need home care preserve a high degree of independence. As a consequence, the weighting of I-ADLs loses importance and our 7 The exclusion of the covariates does not modify our results. It just changes the level of the R 2 for all classifications measure, although remains to be the optimal, barely improves classifications based on ADLs.
Nonetheless, it explains 50% more variance than self-reported health.
Regarding mortality, we have constructed a division that performs better than self-reported health. This contribution is relevant because most of the literature (see Idler and Benyamini, 1997, for a survey) shows that subjective measures of health usually predict mortality beyond objective indicators. Notably, the R 2 using 4-I-ADL exceeds by 0.7 percentage points that of self-reported health which indicates that part of the improvement on the mortality prediction relies on the incorporation of I-ADLs, and not on the use of death in the classification algorithm.

Dynamics: self-reported health versus endogenous classification
The comparison regarding groups' dynamics generates new insights about the differences between grouping methods. To obtain smooth dynamics, we assume that the transition probabilities of self-reported health follow a logistic specification as described by Equation (2).
Furthermore, to ease the comparison we focus on the best and worse groups of each method, that is, we compare healthy according to our method with excellent as reported by individuals and impaired with poor. For completeness, we also include the two groups of self-reported health in the comparison. 8 There are two main risks associated with health transitions which increase the incentives to save. The first one is survival risk. Individuals optimally want to consume everything but the bequest they desire to leave before their death day. In reality, however, this day is not known, hence they have to save in case they live more than expected. The second risk relates to the direct costs of health. Under the fear of entering into a health status with high medical costs, individuals increase their savings. Figure 7 reports the median probability of dying. The left panel corresponds to the healthiest groups, whereas the right panel presents the results for the most unhealthy ones. Up to the age of 80, individuals who report an excellent health, as well as those classified as healthy own very small probabilities of dying. After this age, elderly with a low survival probability still assess their health as excellent. On the other hand, age is not as important for the healthy group as mortality less than doubles between age 80 and 98. One possible explanation is that individuals compare themselves with relatives and friends of the same age to assess their health 8 One groups includes excellent, very good, and good; while the other comprises fair and poor. 20 status; thus, respondents of age 65 and 90 have a different benchmark. Furthermore, while the difference between the mortality rates of healthy and impaired are sizable, this is not the case for the groups based on self-reported health, which suggests that this method does not predict mortality at older ages. In addition, impaired individuals feature a higher death probability than those who assess themselves as in poor health at any age.
[ Figure 7 about here.] The second relevant element of health risk is persistence. If the process is not persistent, health today would contain relatively little information on tomorrows health and survival probabilities thus affecting individuals' saving behavior. Additionally, the persistence of each classification sheds some light on the type of health process. In particular, we aim to create an indicator of LTC needs which is by definition persistent in contrast to others such as the flu or a sprained ankle. Figure 8 depicts the probability of remaining in the same group conditional on the group you are at a given age. We find that individuals that report excellent in one wave have less than 40% of probability to provide the same answer in the following wave, whereas respondents classified as healthy are extremely likely to remain in that state. This fact indicates that some non-persistent factors might drive self-reported health. If we focus on individuals in bad health, our classification displays a larger persistence as individuals age which is in line with the idea that as you become older the harder it is to recover. In contrast for fair and/or poor self-reported health, individuals are more likely to report improvements in their health status as people age which points towards changes in their health benchmark.
[ Figure 8 about here.] Lower persistence and a worse ability to predict mortality indicate that self-reported health overestimates the uncertainty faced by individuals. The effect of this bias on individuals' decisions depends on its severity across socio-economic groups and the specific structural model. 9 To shed some light on the former, Figure 9 plots the additional percentage of time than a high-school graduate spends in the healthiest state (left-hand panel) and the unhealthiest state (right-hand side) in expectation. While our classification indicates that high school graduates spend around 40% more time in the healthy state and 30% less in the impaired state, using selfreported, these differences at least double. Given that our classification was able to explain a larger fraction of the variance of different health outcomes, these results suggest that selfreported health contains a measurement error correlated with education. More precisely, low educated individuals tend to report worse health status or high-school graduates overestimate their wellness, or both.

Conclusion
As retirees age, they face large risks of requiring persistent and expensive care. The macroeconomic literature underlines the importance of this uncertainty to explain the dissaving pattern of the elderly and the labor supply decisions of the individuals close to retirement.
They face, however, an important empirical challenge: summarizing the information content of several health variables into a few groups, which is a requirement for quantitative models to be computationally feasible.
This paper develops a methodology to classify individuals, into a reduced number of categories, exploiting the richness of the health information available in panel surveys. In addition, by profiting from the panel dimension of the data we estimate transitions across groups conditioning on current health, age, education, and gender, which are of paramount importance when calibrating macroeconomic models.
Individuals LTC needs can be parsimoniously represented with four different groups, namely, healthy, impaired, physically and mentally frail. While healthy and impaired have the usual extreme interpretation, the distinction between physically and mentally frail arises from the different pattern of respondents struggling with ADLs and IADLs. Moreover, and in line with the previous literature, health status is highly persistent over time, but with significant differences in the dynamics of health across demographic groups.
We then assess our proposed classification against other commonly used measures. Our comparison exercises show that previous health indices are weakly related to health outcomes and medical utilization rates. In contrast, our health groups explain a significant fraction of the variance in the use of nursing homes, home health care, out of pocket medical expenses, and mortality.   Notes: RAND HRS Data; sample from 1996 to 2014 (10 waves). We select individuals over 60 years old and we drop individuals whose education, gender or age are missing (<0.1% of observations). The final sample consists of 159,025 interviews (including exit waves) which correspond to 27,369 individuals followed 6 waves (12 years) on average. The units of the x-axis are percentage points. Notes: RAND HRS Data; sample from 1996 to 2014 (10 waves). We select individuals over 60 years old and we drop individuals whose education, gender or age are missing (< 0.1% of observations). The final sample consists of 159,025 interviews (including exit waves) which correspond to 27,369 individuals followed on average 6 waves (12 years). The column All indicate the percentage of observations who have problems with a given I-ADL. The last five columns present the same percentage by group of self-reported health (excellent (Exc.), very good (Very), good, fair and poor). Notes: RAND HRS Data; sample from 1996 to 2014 (10 waves). We select individuals over 60 years old and we drop individuals whose education, gender or age are missing (<0.1% of observations). The final sample consists of 159,025 interviews (including exit waves) which correspond to 27,369 individuals followed 6 waves (12 years) on average. Results reported in percentage points. See Section 3 for details about the econometric model and the estimation procedure. Notes: RAND HRS Data; sample from 1996 to 2014 (10 waves). We select individuals over 60 years old and we drop individuals whose education, gender or age are missing (<0.1% of observations). The final sample consists of 159,025 interviews (including exit waves) which correspond to 27,369 individuals followed 6 waves (12 years) on average. Results reported in years. In parentheses we report the 95% high-density intervals. See Section 3 for details about the econometric model and the estimation procedure.  Notes: RAND HRS Data; sample from 1996 to 2014 (10 waves). We select individuals over 60 years old and we drop individuals whose education, gender or age are missing (<0.1% of observations). The final sample consists of 159,025 interviews (including exit waves) which correspond to 27,369 individuals followed 6 waves (12 years) on average. Then, we restrict the sample to those observations that can be classified according to all criteria. Results reported in percentage points. Numbers correspond to the R 2 of the following regression:

Tables
where y i,t is the variable used as a reference, z i,t includes gender and education, and d i,t is a vector of dummy variables indicating to which group the individual belongs.    Notes: This table presents the summary statistics of the two samples used in the paper. The estimation sample corresponds to the observations that we incorporate into the estimation procedure. Due to missing data, we might not be able to classify individuals in this sample; hence we use a restricted sample in order to compare across classification methods. The summary statistics of this latter sample are included in the second panel.  Notes: Numbers correspond to the estimates and standard errors (in parenthesis) of the following regression: where y i,t is the variable used as a reference, z includes gender and education, and d i,t is a vector of dummy variables indicating to which group the individual belongs. * * * , * * , * indicate significance at the 99%, 95% and 90% confidence level.  Notes: Numbers correspond to the estimates and standard errors (in parenthesis) of the following regression: where y i,t is the variable used as a reference, z includes gender and education. d i,t is a vector that includes the probabilities of being physically frail, mentally frail, or impaired. Healthy is the excluded category. * * * , * * , * indicate significance at the 99%, 95% and 90% confidence level.   Notes: Numbers correspond to the estimates and standard errors (in parenthesis) of the following regression: where y i,t is the variable used as a reference, z includes gender and education. d i,t is a vector that includes 3 dummy variables that take value one if the most likely health group is physically frail, mentally frail, or impaired. Healthy is the excluded category. * * * , * * , * indicate significance at the 99%, 95% and 90% confidence level.  -0.027 * * -0.019 -0.001 * * * 0.001 * * * 0.003 * * * 0.005 * * * 0.005 * * * (0.008) (0.010) (0.000) (0.000) (0.000) (0.000) (0.000) HS -2.868 * * * -3.515 * * * -0.029 * * * -0.012 0.031 0.058 * * 0.037 * (0.541) (0.674) (0.008) (0.010) (0.016) (0.020) (0.016) HS×Age 0.047 * * * 0.056 * * * 0.000 * * * 0.000 -0.000 * -0.001 * * * -0.001 * * * (0.008) (0.009) (0.000) (0.000) (0.000) (0.000) (0.000) Notes: Numbers correspond to the estimates and standard errors (in parenthesis) of the following regression: where y i,t is the variable used as a reference, z includes gender and education. d i,t is a vector that includes 3 dummy variables that take value one if the individual presents difficulties with an ADL but no IADL (1,0), if she struggles with an IADL but no ADL (0,1) and if she has difficulties with at least one of each (1,1). Individuals without difficulties compose the excluded category. * * * , * * , * indicate significance at the 99%, 95% and 90% confidence level.  Notes: Numbers correspond to the estimates and standard errors (in parenthesis) of the following regression: where y i,t is the variable used as a reference, z includes gender and education. d i,t is a vector that includes 4 dummy variables that take value one if the individual belongs to the second, third, fourth of fifth quantile of the frailty index proposed by . Individuals in the quintile with the lowest frailness compose the excluded category. * * * , * * , * indicate significance at the 99%, 95% and 90% confidence level. 8 Table S.VIII: Parameters of the regression in Wave Current Next Current Next Current Next Next Age -0.025 * * -0.005 0.000 0.002 * * * 0.003 * * * 0.005 * * * 0.006 * * * (0.008) (0.010) (0.000) (0.000) (0.000) (0.000) (0.000) HS -3.096 * * * -3.400 * * * 0.002 0.029 * * 0.062 * * * 0.083 * * * 0.074 * * * Notes: Numbers correspond to the estimates and standard errors (in parenthesis) of the following regression: where y i,t is the variable used as a reference, z includes gender and education. d i,t is a dummy variable that takes value one if the individual presents difficulties with an ADL. Individuals without difficulties compose the excluded category. * * * , * * , * indicate significance at the 99%, 95% and 90% confidence level.  Notes: Numbers correspond to the estimates and standard errors (in parenthesis) of the following regression: where y i,t is the variable used as a reference, z includes gender and education. d i,t includes four dummy variables that takes value one if the individual report very good, good, poor or very poor health. Individuals reporting excellent health compose the excluded category. * * * , * * , * indicate significance at the 99%, 95% and 90% confidence level.  Notes: Numbers correspond to the estimates and standard errors (in parenthesis) of the following regression: where y i,t is the variable used as a reference, z includes gender and education. d i,t is a dummy variable that takes value one if the individual reports poor or very poor health (Bad). Individuals reporting excellent, very good, good compose the excluded category. * * * , * * , * indicate significance at the 99%, 95% and 90% confidence level. Notes: Numbers correspond to the R 2 of the following regression: where y i,t is the variable used as a reference and d i,t is a vector of dummy variables indicating to which group the individual belongs. In the case of our classification, we use two alternative approaches. First, we substitute d i,t by a vector containing the probability of individual i at time t of belonging to each cluster (we label it Probs). Secondly, we assign each individual to her most likely health group (which we label as Mode). Notes: Parameter estimates and standard errors (in parenthesis) of the logit estimation using two groups of self-reported health. The most healthy group is composed for those individuals who report excellent, very good, or good health; meanwhile, the least healthy one includes those respondents who report poor or very poor health. The second column corresponds to the parameter of an individual who is currently in the healthiest group while the fourth column refers to unhealthy individuals. The first panel shows the estimation results for the transitions to the healthy group whereas the second includes those of the unhealthiest group. Notes: Parameter estimates and standard errors (in parenthesis) of the logit estimation using the five groups of self-reported health. Each column refers to the current health group of the individual while each panel presents the parameters of the transition to a different health group. For instance the fourth column of the third row of the first panel (-2.279) indicates that high school graduates who currently report very good health are less likely to report excellent in the next wave compared with dropouts.          Notes: Median and standard deviation (in parenthesis) of the posterior of each parameter in the estimation with two groups. estimates and standard errors. Each column refers to the current health group of the individual while each of the first two panels presents the parameters of the transition to a different health group. The last panel gathers the estimation results of the Bernouilli process that drives I-ADLs.   Notes: Median and standard deviation (in parenthesis) of the posterior of each parameter in the estimation with four groups. estimates and standard errors. Each column refers to the current health group of the individual while each of the first four panels presents the parameters of the transition to a different health group. The last panel gathers the estimation results of the Bernouilli process that drives I-ADLs. Notes: Median and standard deviation (in parenthesis) of the posterior of each parameter in the estimation with five groups. estimates and standard errors. Each column refers to the current health group of the individual while each of the first five panels presents the parameters of the transition to a different health group. The last panel gathers the estimation results of the Bernouilli process that drives I-ADLs.