For this study, we use the 1994-2010 waves of the HRS. We start our sample selection by dropping observations of Wave 1 from the full HRS sample to avoid inconsistencies that could arise from the difference in scale between year 1992 and the subsequent years.3 After that, the sample is restricted to individuals aged 51 to 64 years, who were asked about their subjective survival probabilities for two different target ages (75 and 80 or 85), and has non-missing values for these subjective survival probabilities. Finally, we drop the internally inconsistent subjective survival probabilities, that is, when the probability of living up to age 80 or 85 is greater than the probability of living up to age 75 (cases where it is likely that the individual was not able to comprehend the nature of the question).
Figure A1 and A2 on the appendix summarize the sample selection process at individual and observation level.
3As noted in section 2.1, changing the scale causes a difference in the nature of responses in 1992 and 1994 onwards. Indeed, one is a discrete answer on a 0 to 10 scale whereas the other is a continuous answer in terms of probability. Therefore, the reasoning for the answers is different.
3 Methodology
In order to develop a method for estimating the forward-looking age, first we need sub-jective remaining life expectancies expressed in terms of years. As they are not measured directly in the HRS and survival expectations only exist in the form of probability of surviving up to a specified target age, we can use these self-reported subjective survival probabilities to obtain subjective life expectancies.
However, there is a challenge in the use of these subjective survival probabilities due to the structure of the survival questions in the HRS. As indicated by Bissonnette et al.
(2014) among others, these self-reported probabilities are subject to rounding and focal answers. Indeed, when people are asked to choose a real number within a range between 0 and 100, most of them report the nearest multiple of some integer rather than their exact subjective expectations (Dominitz and Manski, 1997). Moreover, a significant fraction of the responses heaps at the end points and in the middle of the given scale. In fact, it is found that subjective survival probabilities at the individual level cluster around some focal responses of 0, 50, and 100, even though they seem reasonable when averaged across respondents (for discussions, see, for example, Hurd and McGarry 2002; Manski 2004, Bissonnette et al. 2014). Particularly, serious bunching at 50 percent is considered to be either non-informative focal answers which do not correspond to respondents’ underlying beliefs (De Bruin et al. 2000; De Bruin and Carman 2012; De Bresser and van Soest 2013;
Hill et al. 2005; Hudomiet and Willis 2013), or an extreme form of rounding (Gan et al.
2005; Kleinjans and Soest 2014; Manski and Molinari 2010). On the other hand, Bisson-nette et al. (2014) find little support for the idea that 50 percent answers are used to avoid answering questions. Therefore, using these subjective survival probabilities in an empirical analysis without correcting for rounding and measurement errors may give us biased results. We propose a three step procedure to calculate forward-looking ages from self-reported survival probabilities:
1. Tackling the focal points problem using random effects ordered probit to obtain refined probabilities which depend on the characteristics of each individual.
2. Non Linear Least Squares estimation of subjective survival functions using these refined probabilities and construction of life tables for groups with various charac-teristics.
3. Using the life tables based on estimated subjective survival curves to apply the ‘Char-acteristic Approach’ proposed by Sanderson and Scherbov (2013, 2014) to calculate forward-looking ages for different groups.
3.1 Tackling the Focal Points Issue in the Data
In the existing literature, various approaches are used to deal with the focal responses (e.g., Gan et al. 2005; Kleinjans and Soest 2014; Bissonnette et al. 2014), but still there is no consensus. Gan et al. (2005) propose a method which takes responses from other subjective probability questions to estimate the probability of giving a focal point answer to the questions about subjective survival probabilities. Therefore, by doing so, they are limiting their analysis to people from whom other information is available related to sub-jective probabilities. Kleinjans and Soest (2014) and later Bissonnette et al. (2014) deal with the focal point problem using an ordered response model to estimate the probability of using a certain rounding rule when giving an answer to the survival probabilities ques-tion. Alternatively, Ludwig and Zimper (2013) propose a method where they model the answering of survival probability questions in a Bayesian update framework. However, as
they point out, the aim of their approach is to explain the individual differences between subjective probabilities and objective data, which is far from our purpose. Moreover, their method is oblivious to individual characteristics, which lies in the core of our approach, and they focus solely on the information update process.
In order to tackle the focal point issue, we take a path different from the literature, and we use random effects ordered probit to estimate the probability that people’s given sub-jective survival probabilities fall in a particular interval. In our method, we do not intend to model the individual reasoning process behind giving a certain probability as an answer.
Instead, we directly estimate the probability of an individual giving an answer according to characteristics. We also include random effects, as individual randomness clearly plays a role in the process of giving a subjective probability of survival. In this way, we attempt to better capture the influence of the characteristics in the survival probabilities, treating other individual effects (such as the rounding process or individual optimism/pessimism) as part of the random term. By using probit, we stand under the assumption that the randomness factor and the disturbances are normally distributed.
Formally, we estimate the probability that a given subjective probability of survival is greater than a particular cut point given all cut points, the individual characteristics and the randomness factor. This probability is given by:
P r(SP Si,a,A> k|κ, Xi,a, vi) = Φ(Xi,aβ+vi−κk)
whereSP Si,a,A is the subjective probability of survival up to target age A for individual iaged a; κ is the vector of cut points - 21 cut points are defined depending on key focal points, vi is the vector of random effects for individual i and Xi,a corresponds to a set of characteristics for individual i at age a including: education dummies (less than high school, high school graduate, more than high school), cohort dummies (cohort 1 if the year of birth is in the interval of 1930-1945; cohort 2 if the year of birth is in the interval of 1946-1959), a dummy for place of birth (created depending on whether the respondent was born in the US), an array of health variables (whether the respondent was diagnosed with some adverse health conditions such as diabetes, cancer, high blood pressure, arthritis, stroke, heart problems, lung problems, psychological problems), and a dummy for smok-ing. Detailed descriptions of the regression variables are presented in Table A1.
Table A2 shows the estimated marginal effects of personal characteristics on the sub-jective probability of survival up to different target ages. Column (I) and Column (III) of the table present the estimates for target age 75, while Column (II) and Column (IV) present the estimates for target age 85. The coefficients are estimated separately for men and women but only the results for white individuals are presented here. We evaluate gender differently, as the story of females is very different from males in terms of actual life expectancy at age 50 (Glei et al., 2010). Thus, the coefficients represent the gender specific effects of the characteristics.
Clearly, subjective survival probabilities increase by age. We also find that the older cohort has higher subjective survival probabilities compared to the younger cohort. This finding is in line with (Bissonnette et al., 2014). We also control for the place of birth by considering that individuals who were born and grew up in a country different from the US may have followed a different ageing path.
The coefficients of education dummies are negative and strongly statistically significant for both men and women at both target ages. This implies that education has a positive effect on subjective survival probabilities and this effect is much larger for women than for men. On the other hand, this effect is smaller for both genders at the older target age.
Hurd and McGarry (1995, 2002) also find a similar effect on subjective survival probabil-ities. This is not surprising, as education contributes to life expectancy in different ways including healthier behavior, higher earnings and higher rates of employment (Hummer and Lariscy, 2011).
The particular causes and the contributing factors of mortality at older age demon-strated by Crimmins et al. (2011) may be the same factors lowering subjective survival probabilities. For the US, Crimmins et al. (2011) show that the prevalence of heart dis-ease, stroke and diabetes is very high and cancer is the most important cause of death.
Thus, our list of health measures roughly corresponds to that used by Crimmins et al.
(2011) and includes high blood pressure, diabetes, cancer, lung condition, heart condition, stroke, arthritis and psychological conditions. Among others, arthritis and psychological conditions may not be life-threatening, but we may still presume that they may reduce the subjective survival probabilities. As expected, we found that the coefficients on these indicators are negative and strongly statistically significant. Also, in line with the results in Hurd and McGarry (1995, 2002), the association between these adverse health condi-tions and survival probabilities is different for men and women. In particular, for both target ages, coefficients on smoking, high blood pressure, diabetes, and arthritis are larger for men than for women, whereas the opposite is true for coefficients on cancer and stroke.
Furthermore, coefficients on adverse lung and heart conditions are found to be larger for women at the younger target age. Similar results are observed for men at the older target age.
As smoking increases the risk of numerous causes of death and people are aware of its associated mortality risk, we can expect smoking to be negatively correlated with sub-jective survival probabilities. Indeed, the coefficient is found to be negative and strongly statistically significant for men and women at each of the target ages, with its magnitude being larger for men at both target ages.
After calculating the probabilities for each of the cut points for each individual, we rebuild the subjective survival probabilities. These ‘refined’ probabilities will be simply the expected value of the subjective survival probabilities given the cut points and the characteristics.