Cognition, Optimism and the Formation of Age-Dependent Survival Beliefs

<b>English Abstract:</b> This paper investigates the roles psychological biases play in deviations between subjective survival beliefs (SSBs) and objective survival probabilities (OSPs). We model deviations between SSBs and OSPs through age-dependent inverse S-shaped probability weighting functions. Our estimates suggest that implied measures for cognitive weakness and relative pessimism increase with age. We document that direct measures of cognitive weakness and motivational attitudes share these trends. Our regression analyses confirm that these factors play strong quantitative roles in the formation of subjective survival beliefs: cognitive weakness rather than optimism is an increasingly important contributor to overestimation of survival chances in old age. <b>German Abstract:</b> Diese Arbeit ist motiviert durch die klassische ökonomische Frage, wie Entscheidungen über den Lebenszyklus von zeitlichen Präferenzen und dem Zeithorizont der Individuen beeinflusst werden. Insbesondere fokussieren wir darauf, wie Individuen ihre Überlebenserwartungen bilden. Zufolge zahlreicher empirischer Studien unterschätzen junge Menschen ihre Überlebenschancen, wohingegen ältere Menschen diese durchschnittlich überschätzen. Was ist die treibende Kraft hinter diesem altersabhängigen Muster der Fehleinschätzungen der ferneren Lebenserwartungen? Gibt es womöglich Effekte, die über einen Lern- und Erfahrungsprozess über Gesundheitsrisiken und -historien hinaus die Einschätzung von Individuen hinsichtlich dieses Risikos beeinflussen? Dieses Paper erörtert, dass psychologische Faktoren von entscheidender Bedeutung zur Beantwortung dieser Frage sind. Wir vergleichen subjektive Überlebenserwartungen aus einer Haushaltsbefragung mit den dazu korrespondierenden objektiven Überlebenswahrscheinlichkeiten, die wir basierend auf individuellen Charakteristika schätzen (z.B. Alter, Geschlecht, Gesundheitshistorie). Wir dokumentieren, dass typische Verzerrungen auftreten: Bis zu einem Alter von 70 Jahren unterschätzen Individuen ihre objektiven Überlebenschancen, ab diesem Alter überschätzen sie diese. Um zu zeigen, dass psychologische Einstellungen wichtige Determinanten dieser Abweichungen zwischen subjektiven Überlebenserwartungen und objektiven Überlebenswahrscheinlichkeiten sind, schätzen wir implizite psychologische Faktoren aus den beobachteten Differenzen. Wir zeigen, dass das implizite Maß sowohl an gemessenem Pessimismus, als auch an Insensitivität zur objektiven Wahrscheinlichkeit, mit dem Alter steigt. Diese Ergebnisse legen also nahe, dass Individuen im Durchschnitt im Alter pessimistischer werden und dass die Fähigkeit, objektive Wahrscheinlichkeiten richtig einzuschätzen, abnimmt. Daraufhin zeigen wir, dass direkte Messungen dieser psychologischen Faktoren diese Trends teilen: Daten Indices zu Pessimismus steigen, zu Optimismus sinken mit dem Alter und ein Index über kognitive Schwäche steigt ebenfalls mit dem Alter. Schlussendlich zeigen wir, dass diese direkten psychologischen Maße tatsächlich wichtige quantitative Rollen in der Bildung subjektiver Überlebenserwartungen spielen. Pessimismus führt zu einer signifikanten Unterschätzung, Optimismus zu einer Überschätzung von Überlebenschancen und eine fehlende Kognition spielt eine zunehmend wichtige Rolle für die beobachtete Überschätzung der Überlebenschancen, wenn Individuen älter werden. Dieser Effekt dominiert, weshalb ältere Menschen ihre Überlebenserwartungen mehr und mehr überschätzen. Wir schließen unsere Analyse mit einem Ausblick, der argumentiert, dass die treibenden Kräfte für Verzerrungen in Wahrscheinlichkeitseinschätzungen auf ökonomische Entscheidungen im Allgemeinen nur mithilfe struktureller Lebenszyklusmodells studiert werden können, welches Wissenschaftlern ermöglicht, simultan multiple Risiken und die Beeinflussung der Erwartungsbildung über diese Risiken durch psychologische Faktoren zu berücksichtigen.


Introduction
Important economic problems, such as the decision about when to retire, how much to save for retirement and whether to purchase life-insurance, depend on the formation of survival beliefs over an individual's life-cycle. A rational individual would be modeled as a statistician whose survival beliefs are given as data-based (Bayesian or frequentist) estimates. For this rational benchmark, any differences between subjective survival beliefs and their objective counterparts can only result from an insufficient amount of data, and biases will decrease when the individual collects more data with age. Empirical studies on subjective survival beliefs, however, do not support this notion of convergence of perceived survival chances to objective survival probabilities. Instead, the literature robustly documents a flatness bias, i.e., respondents of age 50-70 express underestimation, whereas older respondents (older than age 75) express overestimation of survival chances on average. 1 Moreover, we document that these biases are not negligible: on average, 65 year-old respondents underestimate their survival probabilities by roughly 10 percentage points, whereas 85 year-old respondents overestimate them by roughly 15 percentage points. What is driving these age-dependent patterns of survival belief biases on top of any statistical learning process?
In this paper we provide a structural interpretation of these biases by use inverse S-shaped transformations of objective probabilities, as known from experimental prospect theory (PT) (Kahneman and Tversky 1979;Tversky and Kahneman 1992;Wakker 2010), and argue that psychological factors are important drivers of the observed biases. In particular, we show that cognitive weakness/strength and pessimism/optimism play important quantitative roles in the formation of age-dependent subjective survival beliefs.
As our first step, we compare subjective survival beliefs (SSBs) to objective survival probabilities (OSPs) using data from the Health and Retirement Study (HRS). In the HRS, interviewees are asked about their beliefs on whether they will survive from the interview age to some target age that is several years ahead. To compare these individual SSBs with their objective counterparts, we estimate for each interviewee the corresponding individual-level OSP by using the information on actual HRS mortality and several conditioning variables, including mortality trends. We adapt the methods used by Khwaja et al. (2007), Khwaja et al. (2009) and Winter and Wuppermann (2014) to estimate mortality hazard rates at the individual level. Plotting SSBs against OSPs over age, we document the flatness bias in the form of an average underestimation of respondents of age 70 and younger, respectively, an overestimation of respondents of age 75 and older. Within a given age group, we find that respondents with low OSPs express overestimation, whereas respondents with high OSPs express underestimation, resulting in a "flattening out" of SSBs compared to the 45-degree line of OSPs.
To formally describe these biases, we assume that SSBs can be modeled as agedependent Prelec (1998) probability weighting functions (PWFs). In line with the usual interpretation of the parameters of the Prelec function (cf. Wakker (2010)), we assume that the motivational factor of relative pessimism is expressed through the elevation and that the cognitive factor of likelihood insensitivity corresponds to the flatness of the Prelec function. Likelihood insensitivity refers to a cognitive weakness according to which people cannot distinguish well among the respective likelihoods of events that are neither impossible nor certain. An extreme case of flattening-out are fifty-fifty probability judgments, which are well-documented in the psychological literature (Bruine de Bruin et al. 2000). 2 Estimating agespecific Prelec PWFs on our data of SSBs, we find that the elevation of the Prelec function decreases with age, whereas its flatness increases. These findings imply that the implicit measures of the relative pessimism of respondents and their likelihood insensitivity increase with age. 3 Thus, the increasing overestimation with age cannot be explained by an increasing optimism as one may suspect. It is rather a consequence of (increasing) likelihood insensitivity (i.e., the flatness of the PWF) and decreasing objective probabilities to survive.
We next compare the age patterns of these implicit cognitive and motivational factors to directly observable counterparts. Because we do not have individuallevel data on relative pessimism and likelihood insensitivity, we look at proxies for these variables in the HRS. From wave 8 onward, the HRS contains measures on dispositional optimism (pessimism) that are derived from the same statements as in the well-known Life Orientation Test-Revised (LOT-R). 4 We find that dispositional optimism is decreasing with age, whereas dispositional pessimism is increasing, on average. 5 To obtain a good proxy for likelihood insensitivity, we consider HRS measures on the cognitive weakness of the respondent, which is motivated by a cognitive interpretation of likelihood insensitivity (Wakker 2010). This cognitive measure is a version of a composite score taken from RAND, and it combines the results of several cognitive tests. We find that cognitive weakness is strongly increasing with age. Thus, these age patterns are consistent with the age patterns of our implied measures of relative pessimism and likelihood insensitivity that we obtained from the age-dependent Prelec PWFs.
Finally, in order to estimate the quantitative impact of cognitive and motivational factors on subjective survival beliefs, we combine the Prelec transformation of OSPs with the HRS data on direct cognitive and motivational measures. Specifically, we specify both parameters of the Prelec function-relative pessimism and likelihood insensitivity-as linearly dependent on dispositional optimism (pessimism) and cognitive weakness. This linear dependence also features a constant. With our estimate of that constant, we identify a significant base bias in the form of a baseline inverse-S-shaped transformation of the objective survival probabilities. We interpret this base bias as capturing incomplete statistical learning of, respectively, (rational) inattention to, the OSP of the individual. Thus, the base bias reflects that individuals may only partially use their individual-level OSP in their formation of subjective beliefs. The base bias may also reflect rounding of subjective reports on survival. 6 Since the inverse S-shaped transformation of OSPs attributable to this base effect is constant over age by construction, changes in differences between SSBs and OSPs attributable to the base bias reflect movements of the underlying OSPs. For example, because OSPs are relatively high at the age of 65, the base effect induces an underestimation of long-horizon survival chances of approximately 10 percentage points. At age 85, however, OSPs are relatively low and the base bias therefore induces an overestimation of approximately 7 percentage points at that age. In addition, our estimates identify a change of cognition over the life-cycle that describes a clockwise tilting of the probability weighting function in age so that lack of cognition increases with age (the Prelec function becomes flatter). Relative to the base bias, this leads to an additional underestimation at age 65 by minus 5 percentage points and to an additional overestimation at age 85, also by 5 percentage points. In contrast to these dynamic effects of cognition, the effects of the motivational factors pessimism and optimism are roughly constant in age. Pessimism leads to a downward bias by 5 percentage points, and optimism leads to an upward bias by 10 percentage points.
We thus find that cognitive and motivational factors are important drivers of subjective survival beliefs beyond any statistical learning processes that may take place. While the motivational bias measured as optimism and pessimism does not significantly change with age, cognitive weakness, measured as likelihood insensitivity, does. Since objective mortality risk and cognitive weakness increase with age we conclude that cognitive weakness is an increasingly important contributor to the observed overestimation of survival chances in old age.
Relation to the Literature. Our work contributes to the economics literature on subjective expectations (Manski 2004) 7 -, particularly on subjective survival beliefs, which is inspired by Hamermesh (1985). On the one hand, the literature documents that SSBs are broadly consistent with OSPs and co-vary with direct measures of health, such as health behavior (e.g., smoking) or health status, in the same way as OSPs (Hurd and McGarry 1995;Gan et al. 2005) in that probability distributions.
7 Also see the literature concerned with the role of roundoff and focal point answers, cf. Footnote 6.
SSBs serve as predictors of actual mortality (Hurd and McGarry 2002; and that individuals revise their SSBs in response to new adverse (health) shocks ). On the other hand, several authors document important biases in subjective survival beliefs when comparing sample average beliefs to objective survival probabilities (Elder 2013;Ludwig and Zimper 2013;Peracchi and Perotti 2014;Groneck et al. 2016;Bissonnette et al. 2017). We emphasize that motivational (optimism and pessimism) and cognitive factors are important contributors to these biases. 8 In this respect, our work relates to medical studies examining the link between psychosocial dispositions and health shocks (Kim et al. 2011) or subjective body weight (Sutin 2013). Mirowsky and Ross (2000) and Griffin et al. (2013) study how incorporating motivational factors influences subjective life expectancy. We extend their work by controlling for OSPs. D' Uva et al. (2015) investigate the effects of cognition on the accuracy of longevity expectations. We go beyond their analyses by combining motivational and cognitive variables and by focussing on probabilities.
Manifestations of biases driven by motivational factors have also been discussed in the behavioral learning literature in the form of confirmatory biases (Rabin and Schrag 1999), myside biases (Zimper and Ludwig 2009), partisan biases (Jern et al. 2014;Weeks 2015), and irrational belief persistence (Baron 2008) and in the related literature on motivated beliefs (Bénabou and Tirole 2016). Simply speaking, people who are biased by motivational factors 'only see/learn what they want to see/learn' so that any new information tends to confirm already existing beliefs and convictions. One would expect that motivational biases play an important role in the formation of survival beliefs, as most people strongly dislike to die. According to Kastenbaum (2000) "[...], most of us prefer to minimize even our cognitive encounters with death." Under the plausible assumption that "cognitive encounters with death" increase with age, elderly people might express more optimistic attitudes towards their likelihood of surviving. An age-increasing motivational bias in the form of optimism could then explain why elderly people increasingly overestimate their survival chances compared with younger people, for whom the prospect of death is less apparent.
Although our analysis confirms that a motivational (i.e., confirmatory) bias is important for the formation of survival beliefs at all ages, we find that it leads to a roughly constant bias in age, on average. If anything, our descriptive analysis suggests that probability weighting functions express more pessimistic rather than optimistic beliefs as individuals become older. In contrast, both our descriptive and regression analyses suggest that cognitive weakness is an increasingly important quantitative contributor to the overestimation of survival chances over an individual's life-cycle.
To model age-dependent survival beliefs, we employ a Prelec probability weighting function applied to objective survival probabilities, which is a prominent approach in prospect theory (PT). As a generalization of rank-dependent utility theories, pioneered by Quiggin (1981Quiggin ( , 1982, modern PT has developed into a comprehensive decision theoretic framework that combines empirical insightsstarting with Kahneman and Tversky (1979))-with theoretical results about integration with respect to non-additive probability measures; cf. the Choquet expected utility theories of Schmeidler (1989) and Gilboa (1987).
Of the many aspects of PT, our model of age-dependent biases in survival beliefs is thus related to the experimental PT literature, which shows that subjective probability judgments cannot be described as additive probabilities. To be precise, the experimental two-stage PT literature shows that in a first stage, subjective probability judgments (=beliefs) resemble an inverse S-shaped transformation of additive probabilities, while in a second stage, these transformed probability judgments themselves undergo another inverse S-shaped transformation (emphasizing pessimism) to become decision weights reflecting the decision maker's preferences; cf., e.g., Tversky and Fox (1995), Fox et al. (1996), Fox and Tversky (1998), Gonzalez and Wu (1999), Kilka and Weber (2001), and Wakker (2004, 2010. According to this two-stage approach, probability weighting is relevant for the formation of beliefs and decision weights. 9 9 The two-stage approach is not uncontroversial. For example, Barberis (2013) argues that probability weighting is exclusively a feature of preferences (recent contributions in this line of literature are Woodford (2012) and Steiner and Stewart (2016), who postulate that perception biases may arise as a second-best solution if the information processing capacity is limited or Our findings contribute to the literature on the two-stage approach within the special context of age-dependent survival beliefs. While the inverse S-shape of probability judgments has typically been documented in experimental situations, only a very few papers document evidence of inverse S-shaped probability judgments in non-experimental data. 10 In contrast to experimental situations, for which the experimental design can clearly distinguish between risk and uncertainty, this clear-cut distinction does not apply to non-experimental HRS data on survival beliefs: Although there exist OSPs, we cannot know how much the subjects of the HRS are aware of these objective probabilities, so we look at a hybrid situation for which both aspects, risk and uncertainty, are relevant. This is reflected in our estimates of the base bias. Since it is plausible to assume that assessments of long-run survival chances involve ample uncertainty, the strong quantitative role of the base bias we uncover can be interpreted as a confirmation that inverse S-shaped probability weighting is indeed very pronounced in situations with uncertainty.
Importantly, it speaks to the robustness of the experimental PT findings on probability judgments that we can confirm the typical inverse S-shape for the survival belief data at all ages. Moreover, our regression analyses with respect to direct motivational and cognitive measures support the typical interpretations of the PT literature on probability judgments.
The remainder of this paper is organized as follows. Section 2 presents the main stylized facts on survival belief biases. Section 3 provides a structural interpretation of these biases through prospect theory. Section 4 looks at the direct psychological measures elicited in the HRS. Section 5 quantifies the role of cognitive and motivational variables for subjective survival beliefs. Finally, if the processing is noisy). However, this view ignores, e.g., people who judge the chances of A versus NOT A as fifty-fifty, even if they were told that the mutually exclusive events A, B, and C are equally likely. Arguably, many real-life people commit such cognitive errors, "reflecting insufficient sensitivity to changes in likelihood" (Wakker 2004, p. 239). While inverse S-shaped beliefs are thus prevalent even in situations under risk -in which individuals are told objective probabilities-they are apparently even more pronounced in situations under uncertainty, for which no objective probabilities are provided; cf. Wakker (2004).
10 For example, Polkovnichenko and Zhao (2013) and Andrikogiannopoulou and Papakonstantinou (2016) find evidence for inverse S-shaped probability weighting for option prices and betting markets, respectively. Section 6 concludes. Separate appendices contain further information on the data and additional results.

Data Sources
We base our estimates of subjective survival beliefs (SSBs) and of the corresponding objective survival probabilities (OSPs) on the Health and Retirement Study (HRS), which is a national representative panel study of the elderly US population. Individuals are interviewed on a biennial basis. Interviews of the first wave were conducted in 1992. The interviewees are individuals older than 50 and their spouses regardless of age. An overview of the survey, its waves and the interview cohorts is contained in Appendix A.
In our descriptive analyses, we use waves 8-11. In our regression analyses with lagged variables, we use waves 9 − 11 (years 2006 − 2012) because motivational variables are only available for waves 8-12, and our measure of the individual level OSP is dependent on our index of contemporaneous cognitive weakness index which is only available up to wave 11. To estimate the individual-level objective survival probabilities (OSPs), we accordingly use waves 4 − 12 of the HRS as well as data from the Human Mortality Database (HMD). 11 For further details on the sample selection, see again Appendix A.

Subjective Survival Beliefs
In the HRS, an interviewee i of age h is asked about her SSB to live to at least a certain target age m, which we denote as SSB i,h,m . We focus on individuals in the survey who are age 65 and older. This sample restriction is used because the data set does not allow us to estimate OSPs for ages less than 65, with details provided in Subsection 2.3 below. The assignment of target age m(h) to interview age h for our sample is given in Table 1. Observe that the distance between interview age h and target age m(h) is the same across all interview age/target age groups. 11 We explore the index of cognitive weakness up to wave 11 and compute panel mortality

Objective Survival Probabilities
To study survival misconceptions at the individual level, our first objective is to assign to each individual in the sample its respective objective survival probability (OSP). Using aggregate data from (cohort) life-tables for this purpose 12 is ill-suited because individual (objective) survival rates generally deviate from sample averages. To estimate individual-level OSPs, we adapt the methods described in Khwaja et al. (2007), Khwaja et al. (2009), Winter and Wuppermann (2014) and Bissonnette, Hurd, and Michaud (2017). We accordingly employ a duration model to estimate hazard rates conditional on several individual-level characteristics. Among standard variables such as age, socioeconomic status, and health behavior, the set of explanatory variables includes predicted average OSPs (AOSPs) in order to capture time trends of mortality hazards. We extract the time trend by applying the Lee and Carter (1992) procedure; see Appendix A.2. We estimate the relationship between individual-level observable variables and the AOSPs, both collected in x i , and mortality using a hazard function given by where time to failure t is the number of years to death. λ 0 (t) is the baseline hazard, for which we choose the specification given by the Weibull (1951) hazbetween waves 11 and 12. 12 As, e.g., in Perozek (2008), Ludwig and Zimper (2013), Peracchi and Perotti (2014), and Groneck et al. (2016). ard model. 13 This allows us to model duration dependence, i.e., the fact that mortality rates are an increasing function of age. Accordingly, we impose the structure λ 0 (t) = αt α−1 which allows for α > 1 (capturing positive duration dependence). exp(x β) is the proportional hazard. In our estimation, survivors are treated as censored, and we estimate function (1) by maximum likelihood. The objective survival probabilities (OSPs) for all prediction horizons t and each individual i of interview age h are given by (cf., e.g., Wooldridge (2002) and Cameron and Trivedi (2005)): From this, we can also construct the OSP until the target age (with horizon t = m(h) − h), OSP i,h,m(h) , which we assign to the respective SSB i,h,m(h) of individual i.

Biases
Our following descriptive analysis compares individual-specific SSBs from the survey data with our individual-specific measures of OSPs. As a first step, we replicate the results of previous literature-e.g., Hamermesh (1985), Elder (2013), Ludwig and Zimper (2013) and Peracchi and Perotti (2014)-on the age patterns of survival beliefs in Figure 1. As a crucial difference from this literature, we calculate average OSPs from our individual measures instead of using average (cohort) life-tables. The step function in the figure is due to the change in assignment of the interview and target ages, cf. Table 1. Our findings confirm 13 A specification of the hazard function that allows for unobserved heterogeneity may be preferable. However, when we tried to estimate the individual OSPs with a specification of the hazard function that allows for unobserved heterogeneity, we faced serious convergence problems in many of our bootstrap iterations. Thus, we compared the results of the first bootstrap of a specification while allowing for unobserved heterogeneity with our specification in the paper. The coefficient estimates and the estimate for duration dependence are very similar. Additionally, we compared the distributions of the predicted OSPs for both specifications, which are very similar as well. Hence, we are confident that our results are not affected much by ignoring unobserved heterogeneity in our specification of the hazard function. the well-established flatness bias: At ages prior to age 70, individuals, on average, underestimate their probabilities to survive, whereas for ages above 75, they overestimate it.  Next, we take a new perspective for which individual-level data are needed. Instead of computing averages over age, we average over OSPs, i.e., for each OSP, we compute the average SSB. In the upper left panel of Figure 2, we show the corresponding results by plotting the average SSBs against the average OSPs. If the SSBs are aligned along the 45-degree line, then there is no bias. However, we observe a very systematic pattern of misconception: Individuals with low OSPs, on average, overestimate their survival chances, whereas those with high OSPs, on average, underestimate it.
The two perspectives on the data taken in Figure 1 and the upper left panel of Figure 2 suggest a very simple explanation for the observed biases across age. Suppose that individuals were to always resolve any uncertainty about their survival chances in a 50-50 manner (Bruine de Bruin et al. 2000), i.e., their response were a weighted average of a fifty percent chance of survival and the actual OSP. Such a 50-50 heuristic could obviously explain the pattern in the upper left panel of Figure 2. Furthermore, young respondents in our data have OSPs above 50 percent. If they were to apply such a simple heuristic, then they would, on average, underestimate their chances to survive. Old respondents, on the other hand, have long-run OSPs less than 50 percent, on average. 14 Under such a heuristic, they would accordingly overestimate their OSPs, on average.
Hence, such a 50-50 bias could simultaneously explain the patterns of Figure 1 and the upper left panel of Figure 2.
We next argue that there is more information content in the data, giving rise to alternative interpretations. 15 For this purpose, we repeat the previous analysis for different age groups. In the remaining panels of Figure 2, we display SSBs as a function of OSPs for each target age group; cf. Table 1. The figure suggests that the flatness of SSBs against OSPs grows stronger with increasing age-compare, e.g., age group 65-69 with age group 80-84. In addition, the intersection with the 45-degree line moves downward, from approximately 50 percent for age group 65-69 to approximately 40 percent for age group 80-84. Therefore, the average tendency for underestimation increases across age groups.
Hence, it appears that, on average, pessimism and likelihood insensitivity are both increasing with age. The next section provides a structural interpretation of these biases and trends over age.

The Prelec Probability Weighting Function
We interpret these biases in survival beliefs through the lens of prospect theory (PT). 16 More precisely, we take from PT the concept of inverse S-shaped probability weighting functions in order to model probability judgments in the form of SSBs. The use of inverse S-shaped probability weighting functions enables us to model the flatness of SSBs relative to the underlying OSPs shown in Figure 2. We additionally aim at capturing the dynamics of the PWFs across age, i.e., the increasing flatness of SSBs and the decreasing intersection with the 45-degree line 15 The general notion of more information content beyond a mere 50-50 bias is also shared in the earlier work by Hurd and McGarry (1995), Hurd et al. (1999), , , Hurd and McGarry (2002) and Gan et al. (2005). We add to this literature by emphasizing the roles of cognitive and motivational factors.
16 As a generalization of rank dependent utility theories (pioneered by Quiggin (1981Quiggin ( , 1982), modern prospect theory (PT) (Tversky and Kahneman 1992) has developed into a comprehensive decision theoretic framework that combines empirical insights (starting with Kahneman and Tversky (1979)) with theoretical results regarding integration with respect to non-additive probability measures (cf. the Choquet expected utility theories of Schmeidler (1989) and Gilboa (1987)).
we observe in the remaining panels of Figure 2. We approach this by a specific functional form assumption on the probability weighting function using a parsimonious parameterization, which, employing the terminology of Wakker (2010), gives rise to two psychological interpretations of these data features. First, the increasing flatness reflects, along a cognitive dimension, an increasing insensitivity to the objective likelihood of the decision maker (likelihood insensitivity). Second, the decreasing intersection with the 45-degree line reflects increasing pessimism, respectively, decreasing optimism, and hence a motivational interpretation of the data. Our aim is to estimate these implicit cognitive and motivational measures. 17 To simultaneously capture these cognitive and motivational dimensions, we adopt the non-linear probability weighting function (PWF) suggested by Prelec (1998). Consider the mapping of the OSP to the SSB according to for parameters ξ ≥ 0, θ ≥ 0. These two parameters control the elevation and the curvature of the function, which can be interpreted as measures of pessimism/optimism and likelihood insensitivity, respectively. To see this, observe that for ξ = θ = 1, function (4) coincides with the 45degree line. Holding θ constant at one, an increase of ξ above one leads to an S-shaped pattern and a decrease below one leads to an inverse S-shape, whereby the intersection with the 45-degree line is at the objective probability of OSP = exp(−1) ≈ 0.37. 18 This dependency on ξ is illustrated in Panel (a) of Figure 3, where we decrease ξ from one to zero. 19 In the limit where ξ = 0, the curve 17 Our estimates may be biased by two important features of the data. First, for the oldest two age groups, we only cover part of the full support of OSPs, because the long-run objective survival chances do not exceed 70%, respectively, 50%, for these two groups, cf. Figure 2. This censoring of the data may bias our estimates for these groups. Second, survival chances are naturally bounded from below by zero and from above by one so that respondents with very high (low) objective survival probabilities cannot overestimate (underestimate) their survival chances by much, whereas the respective other side is less limited. This may induce a flatness of the PWFs; see also our discussion in Section 5.
18 For ξ = 1, SSB = OSP iff − ln(SSB) = − ln(OSP ) = 1; hence, SSB = OSP = exp(−1) ≈ 0.37. 19 Given the data patterns shown in Figure 2, which resemble an inverse S at all interview ages, a parameterization with ξ > 1 is irrelevant in our context. is flat. Hence, ξ can be interpreted as a measure of likelihood insensitivity: lowering ξ decreases the responsiveness of the SSB in the OSP, i.e., there is a lower likelihood sensitivity, respectively, higher likelihood insensitivity. As we further illustrate in Panel (b) of Figure 3, decreasing θ leads to an upward shift of the PWF, whereas increasing it leads to a downward shift. Accordingly, θ can be interpreted as a measure of relative pessimism whereby a higher value of θ means higher pessimism. Finally, notice that unless θ = 1 (or ξ = 1), these two properties of the PWF interact. This can be seen in Panel (b) of Figure 3, where varying the pessimism parameter θ simultaneously affects the shape (slope) of the probability weighting function. It is instructive to emphasize three different effects by use of Figure 3. Suppose that with age relative pessimism, θ, increases. For all OSPs, this will induce stronger underestimation (Panel (b)). At the same time, however, OSPs decrease with age to the effect that the mass of the population lives to the left of the intersection of the PWF with the 45-degree line, cf. age groups 80-84 and 85-89 in Figure 2. This movement of OSPs induces overestimation in old age. If with age also likelihood sensitivity, ξ, decreases, the corresponding flattening of the PWF leads to additional overestimation of OSPs at low OSP levels (Panel (a)).
Our following quantitative analyses distinguish between these different forces.

Age-Dependent Probability Weighting Functions
We next postulate that relationship (4) is an appropriate model for each individual i's subjective belief of surviving from current (interview) age h to some future age t. Accordingly, we specify for a given OSP i,h,t that Notice that parameters θ h , ξ h are now (interview) age h specific. Through this, we capture the dependency of survival belief formation on the current age, as suggested by the age group-specific bias patterns displayed in Figure 2.
To estimate parameters ξ h , θ h , we further restrict these parameters to be the same for each interview age h assigned with a specific target age m(h), i.e., we let ξ h =ξ m(h) and θ h =θ m(h) . We identify these parameters under this constraint by minimizing the Euclidean distance between the predicted and reported subjective survival beliefs for each individual in group m(h) using the data of Figure 2. Figure 4 shows the predicted probability weighting functions with the corresponding 95% confidence intervals. Standard errors are bootstrapped and confidence intervals are computed using the percentile method. 20,21 For the fitted values of the full sample displayed in the upper left panel, we observe a quite symmetric weighting function intersecting the 45-degree line close to 0.5. As already suggested by the pattern in Figure 2, the age-specific weighting functions depicted in the other panels in Figure 4 reveal two facts: First, the functions become flatter with increasing age, and second, the intersection with the 45-degree 20 Since our data are clustered, we perform a cluster bootstrap that samples the clusters with replacement. Thus, in each bootstrap, we solve The percentile method uses the relevant percentiles of the empirical distribution of our bootstrap estimates of the Prelec parameters. line is at lower values for older ages-it is at approximately 55 percent for age group 65-69 and at approximately 40 percent for age group 80-84.

Figure 5 depicts the parameter estimates
shown in Panel (a) of the figure, are decreasing in h, which according to our interpretation captures increasing likelihood insensitivity. Note, however, that the differences between age groups are not always statistically significant.
Estimates θ h =θ m(h) , shown in Panel (b), show a less clear-cut age pattern. They are increasing between interview age groups 70-74 and 85-89, but the confidence bands for the oldest group are rather large. The estimated pessimism parameters are also decreasing from age group 65-69 to age group 70-74. Overall, we can conclude that pessimism also tends to increase for ages above 70. Combined, these findings suggest that the overestimation of SSBs at ages 75 and older documented in Figure 1 cannot be explained by increasing optimism, as one may suspect. It is rather due to the flatness of the PWF and the reduction of OSPs. In addition, increasing likelihood insensitivity increases flatness of the PWF and thus the overestimation at low OSP levels. Thus, likelihood insensitivity is an increasingly important contributor to the overestimation of survival chances in old age.
In Appendix B, we also investigate the robustness of these results by use of linear probability weighting function and thereby confirm our main findings: implied measures of likelihood insensitivity tend to increase, measures of relative optimism tend to decrease with age. In the next section, we turn to direct measures of cognitive and motivational variables to investigate whether we can observe similar age patterns.

Measures
From wave 8 onward, the HRS contains measures on optimism and pessimism. Measures on dispositional optimism (pessimism) are derived from the same statements as in the well-known Life Orientation Test-Revised (LOT-R). 22 Respondents are given various statements regarding a specific latent variable. For most variables, they were asked "please say how much you agree or disagree with the following statements". Each statement is rated on a scale from one (strongly disagree) to six (strongly agree). Average scores are taken to create indices for each motivational construct. Higher values for the motivational variables imply more-optimistic, respectively, more pessimistic, attitudes. 23 Note that optimism and pessimism are usually measured separately, i.e., respondents are asked questions with negative connotations (pessimism) or positive connotations (optimism). The reason for separate measures is that these two concepts were found to display bi-dimensionality (Herzberg et al. 2006). Figure 6, showing the histograms on both measures in our sample, underscores this aspect. Dispositional pessimism shows a clear focal point at index value 1 (="strongly disagree"), whereas dispositional optimism apparently has focal point answers at values 4, 5 and 6, and the peak is at 5. In our empirical analyses, we therefore use separate variables for each concept, although in our previous descriptive and theoretical analyses, we speak of increasing pessimism and decreasing optimism interchangeably.
For a measure corresponding to likelihood insensitivity, our choice of a proxy variable is motivated by our cognitive interpretation of likelihood insensitivity (Wakker 2010). Thus, we include a variable measuring the cognitive weakness of the respondent. It is a version of a composite score taken from RAND and combines the results of several cognitive tests. For instance, respondents were asked to recall a list of random words, to count backwards and to name the (Vice) President of the United States. In total, there are 35 questions and the results are 22 Such statements are, e.g., "In uncertain times I usually expect the best". 23 The index score is set to missing if responses on more than half of the respective statements are missing. summarized in an ability score. We take RAND's composite score of cognitive ability as given and create our score of cognitive weakness. For this purpose, we subtract the cognitive ability score from the maximal achievable value, i.e., our measure of cognitive weakness is 35 minus cognitive ability. A higher score indicates higher cognitive weakness.
An overview of our three measures of cognitive and motivational variables is given in Table 2.

Age Patterns
Figures 7 and 8 display the age patterns of cognitive weakness and of the two motivational measures, respectively. The average index value of cognitive weakness is quite strongly and statistically significantly increasing from 11.8 to 17.9 between ages 65 and 89. Optimism decreases by 2.9% and pessimism increases by 12.2% from age 65 to 90, but both age patterns are less pronounced. 24 That pessimism increases more strongly than optimism decreases supports the notion of the bi-dimensionality of these two measures.  Notes: Average cognitive weakness score over age with 95% confidence-interval calculated based on a normality assumption. The straight line shows the prediction of a linear regression of cognitive weakness against age. Source: Own calculations, Health and Retirement Study (HRS).
Hence, the age trends of the direct cognitive and motivational measures coincide with the indirect measures we derived from estimating probability weighting functions on the data of subjective survival beliefs. These findings lead us to conjecture that cognition and motivational attitudes play a strong role in the formation of subjective survival beliefs. Our next aim is to investigate this inter- Notes: Average optimism and pessimism scores over age with 95% confidence intervals calculated based on a normality assumption. The straight line shows the prediction of a linear regression of optimism (pessimism) against age. Source: Own calculations, Health and Retirement Study (HRS).
pretation of the data further through regression analyses.

Parameterized Non-linear PWFs
To investigate whether our measures of cognitive and motivational factors play a role in the formation of subjective survival beliefs and to quantify their impact, we consider a parameterized variant of the Prelec (1998) function. Specifically, we postulate that for each individual in the sample i and each age h, the implicit measures of cognition, ξ i,h , and optimism/pessimism, θ i,h , from equation (5) are linearly dependent on the cognitive, respectively, motivational, variables as follows: In the above, c i,h−2 is the lag of our measure of cognitive weakness, and p i,h−2 is the lag of our measure of pessimism, whereas o i,h−2 is the lag of our measure of optimism. Using lags of these measures allows us to treat them as weakly exogenous so that we avoid spurious correlation 25 . It also allows us to interpret our findings on the relationship between cognitive and motivational measures and SSBs, at least tentatively, as causal. 26 Replacing in (5) the age-specific parameters ξ h and θ h with the individual and age-specific parameters ξ i,h , θ i,h and using (6), our specification of survival beliefs reads as which we estimate on the pooled sample of HRS data.
Turning to the parameters of interest in specification (7), we refer back to our analysis of Section 3, in particular to the illustration in Figure 3. In light of our discussion there, parameters ξ 0 and θ 0 capture a base effect in subjective beliefs. With regard to the base effect in cognition, ξ 0 , we conjecture that this base effect indeed exists in form of an inverse S, and therefore, we expect ξ 0 ∈ (0, 1). This may reflect an average degree of cognitive weakness, incomplete statistical learning, (rational) inattention with respect to objective survival probabilities or a statistical artifact from truncation of the data. 27 On the other hand, with regard to a base effect in optimism/pessimism captured θ 0 = 1, we do not have a specific prior expectation. Recall from our discussion in Section 3 that θ 0 < 1 reflects rather optimistic beliefs, whereas θ 0 > 1 reflects rather pessimistic beliefs. Depending on which of these two motivational factors dominate, we would find an average upward or downward shift of the probability weighting function. Furthermore, recalling the illustrative analysis of Figure 3, a lower likelihood sensitivity, ξ h , leads to a flatter PWF. Therefore, if changes in cognitive weakness are relevant to the formation of subjective beliefs, we would find its coefficient to be negative, ξ 1 < 0. Also, since increasing relative pessimism, θ h , leads to a lower elevation of the PWF, we expect that θ 1 > 0. Likewise, since increasing relative optimism leads to a higher elevation, we expect that θ 2 < 0. To summarize, our main hypotheses on the signs of the coefficients are that ξ 0 ∈ (0, 1), ξ 1 < 0, θ 1 > 0 and θ 2 < 0. We do not have a hypothesis on the sign of θ 0 .
Our main results summarized in Table 3 show that there is indeed a significant average inverse-S-shaped transformation of objective survival probabilities (ξ 0 = 0.54 < 1). At the same time, we do not identify an additional baseline shifter of the PWF that would reflect average optimistic or pessimistic beliefs, because θ 0 is not statistically different from 1 and the point estimate is close to one. Thus, the base effect exclusively captures factors that lead to an inverse-S of the PWF relative to the 45-degree line, without additional shifting. Since we cannot distinguish between the alternative explanations for this effect discussed above, we subsequently speak of this average base effect simply as a base bias. Beyond the base bias, increasing lack of cognition leads to increasing likelihood insensitivity (ξ 1 = −0.01), which flattens the non-linear PWF. Furthermore, increasing pessimism leads to a significant downward shift (θ 1 = 0.029) and increasing optimism to a significant upwards shift (θ 2 = −0.05) of the non-linear PWF. Thus, cognitive and motivational factors do have a significant effect on the formation of subjective survival beliefs of the expected sign.
To quantify the impact of these factors, we next compute the predicted subjective beliefs of the probability weighting function and decompose the probability weighting function into the following elements: We first predict the full model and the sub-PWFs defined in (8) for each individual and then take sample averages. In our subsequent description, we denote these respective predicted values by , i.e., SSB stands for the sample average of the predicted PWFs under the "full" model, equation (7), and so forth. The results on the predictions for the full model and its decomposition are displayed in Figure 9. The predicted base bias SSB b ("base bias") displays a pronounced inverse S and intersects with the 45-degree line at OSP 0 = 0.35. Since the estimateθ 0 is close to one, this intersection is close to the theoretical inter-section at exp(−1) ≈ 0.37; cf. Section 3, in particular Figure 3. Predictions for the base bias plus changes in cognitive weakness SSB bc ("base + cogn. weakn.") lead to a clockwise rotation of the PWF, while the intersection with the 45-degree line stays at roughly OSP 0 = 0.35, which again is a consequence ofθ 0 ≈ 1. The predictions from the base model with optimism, SSB bo ("base + opt."), imply a parallel upward shift relative to the base model, which dominates the smaller downward shift of pessimism; see SSB bp ("base + pess."). As a consequence of all these effects, the PWF in the full model ("full") is both flatter and shifted upwards relative to the PWF of the base bias.  Notes: Sample averages of predicted non-linear probability weighting functions according to equations (7) and (8) Figure 1. Panel (a) shows the data on SSBs and OSPs (i.e., the data points of Figure 1) and the predicted values for the full model and for the base bias. Overall, the full model displays a very close match to the average data on SSBs. Consistent with our findings in Figure 9, the base bias leads to a stronger underestimation than in the full model. Panel (b) provides the differences relative to the base bias caused by changes of cognitive weakness with age, the parallel shifts induced by pessimism and optimism, and, finally, the full model. That is, we show the differences between (8b), (8c), respectively, (8d)) to (8a), and we denote these differences by ∆. Due to the increasing cognitive weakness over age, individuals, on average, overestimate their survival chances increasingly more as they grow older: relative to the base bias, cognitive weakness initially leads to a downward bias of approximately -5%p and eventually to an overestimation by slightly more than 5%p. We also observe that pessimism leads to an underestimation of survival chances by roughly -5%p and optimism to a strong overestimation by approximately 10%p for all age groups. Overall, the effects of cognitive and motivational variables on subjective survival beliefs are quite strong, with a net effect of approximately 2%p for age group 65-69 and almost 12%p for age group 85-89. Importantly, the effects of cognitive factors are changing with age, whereas the effects of the two motivational factors optimism and pessimism are roughly constant. We can therefore conclude that lack of cognition plays an increasingly important role in the observed overestimation of survival chances in old age.

Parameterized Linear PWFs
To investigate the sensitivity of our findings with respect to the functional form of the PWF, we repeat our regressions for a neo-additive PWF (Chateauneuf et al. 2007a), which is linear for interior survival probabilities, thereby approximating the non-linear model. The main advantage of the linear model is that we can interpret coefficient estimates directly as marginal effects. Furthermore, it is straightforward to add additional control variables in the linear framework, cf. the following Section 5.3.
However, one drawback is that the neo-additive PWF is only a crude approximation to answering patterns for extreme OSPs close to 0% or 100%. In addition, the structural parameters are only partially identified. In particular, without further assumptions, we cannot identify the base bias and the marginal effect of an increase of cognitive weakness; cf. Appendix B.2. To identify both, we invoke our insights from the estimates of the non-linear PWF and derive in Notes: Sample averages of predicted subjective survival beliefs according to equations (7) and (8)  Appendix B.2 an indirect approach to identify both effects. 28 There, we derive the following reduced form specification of the linear model: The results from estimating (9) are summarized in Table 4. All coefficient estimates on the pure effects are of the expected sign and are significant, whereas the interaction terms between cognition and the motivational variables are small and insignificant. This latter finding is convenient for our decomposition analysis because it means that the marginal effects of an increase in cognitive weakness as defined earlier can be identified by setting these interactions to zero. This marginal effect is accordingly given byβ 2 OSP i,h,m(h) − OSP 0 . Sinceβ 2 < 0, 28 To identify the base bias, we postulate that the base bias plus the effect of cognition, the effect is negative (positive) for OSP i,h,m(h) > OSP 0 (OSP i,h,m(h) < OSP 0 ), which reflects the clockwise tilting of the PWF induced by an increase of cognitive weakness, just as in the non-linear model. 29 To illustrate the effects of cognitive weakness, consider first an individual with an OSP of OSP i,h,m(h) = 0.9. For this person, the marginal effect is −0.0107 · (0.9 − 0.3676) · 100%p = −0.6%p. Likewise, for a person with an OSP of OSP i,h,m(h) = 0.1, the effect is positive at 0.3%p. Our estimates also suggest that a one-point increase of pessimism leads respondents to underestimate survival changes by 1.7%p, and a one-point increase of optimism leads respondents to overestimate them by 2.6%p. 30 In addition to these interpretations of the marginal effects, we decompose the linear probability weighting functions analogously to equation (8) and Figures 9 and 10, with details provided in Appendix B.2. This confirms our main findings for the non-linear model: the quantitative roles of our cognitive measure and of the two motivational variables for the age-specific differences between SSBs and OSPs are very similar. Thus, our findings for the non-linear model are robust to the linear approximation.

The Linear Model and Control Variables
On the basis of formal models of statistical learning, individuals learn their individual OSP by obtaining more information. This suggests that they base their survival beliefs on the OSP and additional variables (e.g., those we use to predict the respective OSPs) as well as (in case of biased beliefs) cognitive and motivational factors. A snapshot of a reduced form learning model, as in Viscusi (1985) and , and for biased beliefs in Ludwig and Zimper (2013) and Groneck et al. (2016), can be approximated as a linear regression by adding controls to (9) so that where x i,h is the vector of control variables, such as age, socio-economic characteristics and various health measures. 31 The results on our main parameters of interest are reported in Table 5, and estimates for the control variables are contained in Table 7 of Appendix B.2 (which are of the expected sign and are in line with findings in the literature). Our main findings from Table 4 are unchanged. The coefficient on the constant decreases and becomes very imprecise (i.e., the constant is basically zero) and the coefficient estimate on the OSP,β 1 , decreases due to the additional explanatory power attributable to the control variables (also leading to a decrease in the AIC, indicating better fit). One interpretation of the finding that additional control variables matter for the formation of subjective survival beliefs and take on explanatory power from the objective survival probabilities is that the base 31 Analogously, we could replace in (7) the variable OSP i,h,m(h) with function f OSP i,h,m(h) , x , for f : OSP i,h,m(h) , x → [0, 1]. E.g., f could be a logistic function or a hazard function. We have experimented with such specifications but faced severe convergence problems. We therefore rely on the linear model to investigate the robustness of our findings w.r.t. the inclusion of control variables. bias indeed partially captures learning mechanisms in the face of uncertain survival beliefs. Otherwise, the parameter estimates are unchanged (the confidence intervals overlap). Thus, our main findings on the effects of cognitive and motivational factors for the formation of subjective survival beliefs are also robust to the inclusion of control variables in the empirical specification.

Quantile Regressions
So far, our analyses have been based on a strong structural interpretation of the data. In particular, we have postulated (and found) that cognitive weakness leads to a clockwise tilting of the PWF and that optimism induces a parallel upward shift of the PWF (respectively, pessimism leads to a parallel downward shift). We now investigate the robustness of these findings by running quantile regressions. This allows us to detect relationships that are not captured by mean effects. In our quantile regressions, we take the difference between SSBs and OSPs as a dependent variable. Additionally, we include the level of the objective survival probability in our set of explanatory variables because the interval of our dependent variable is directly linked to the level of the OSP. We analyze every decile and estimate the results for all deciles jointly. As in our previous OLS regressions, standard errors are bootstrapped. Our regression specification is h,m(h) . (11) By including the OSP on the right-hand-side of the regression, we control for biases induced by truncation and censoring, as underestimators cannot report SSBs less than zero and overestimators cannot report SSBs above one. The clockwise tilting of the PWF from increasing cognitive weakness we identified earlier would be consistent with negative estimates of β 2 in lower percentiles and positive estimates in upper percentiles. This would mean that increasing cognitive weakness leads to a more pronounced underestimation for underestimators (who, on average, have high OSPs) and a more pronounced overestimation for overestimators (who, on average, have low OSPs). Irrespective of the percentiles, we also expect that β 3 < 0 and β 4 > 0.
We report our results in Figure 11, thereby confirming our hypotheses. Interestingly, we also find that the effects of optimism and pessimism are strongest for the intermediate percentiles. This is consistent with the non-linear probability weighting function: in the lowest percentiles, we have individuals with, on average, high OSPs, where the structure of the non-linear PWF forces subjective beliefs to converge to 1; cf. Figure 2. Likewise, in the highest percentile, individuals have, on average, low OSPs, which forces subjective beliefs to converge to 0. Thus, under a non-linear PWF, there is less room for motivational variables to impact the formation of SSBs at extreme OSPs of 0 and 1. This is reflected in our estimates shown in Panels (b) and (c) of Figure 11.
Overall, our results for these less-parametric quantile regressions support our structural interpretation of the data by use of inverse S-shaped probability weighting functions.

Concluding Remarks
This paper compares subjective survival beliefs (SSBs) with objective survival probabilities (OSPs) that we estimate based on individual-level characteristics. We establish a twofold and related strong regularity of survival misperceptions. First, respondents of age 65-75 underestimate their chances of survival, whereas older respondents overestimate them. Second, respondents overestimate survival chances with low objective probabilities and underestimate chances with high objective probabilities. Based on this finding, we estimate inverse S-shaped prob-ability weighting functions on the data and establish a strong age dependency in the shape of these functions. Our coefficient estimates suggest that implied measures of pessimism and of cognitive weaknesses are increasing with age. Direct psychological measures confirm these age patterns.
Motivated by these descriptive findings, we estimate reduced form variants of probability weighting functions. We find that biases induced by the motivational factor optimism leads to an overestimation of subjective survival beliefs by approximately 10 percentage points, on average, whereas the motivational factor pessimism leads to a downward bias by approximately 5 percentage points. Both biases are roughly constant in age. In contrast, increasing cognitive weakness leads to an increasing upward bias in subjective survival beliefs. By thus showing that these factors play an important role in the formation of subjective beliefs, our results support the cognitive and motivational interpretations attached to the parameters of inverse S-shaped probability weighting functions in the theoretical and experimental literature (Gonzalez and Wu 1999;Wakker 2010). Our findings also suggest that (increasing) lack of cognition rather than optimism plays an increasingly important quantitative role in the overestimation of survival beliefs in old age.
What are the economic implications of our findings? This is addressed in our companion work (Grevenbrock et al. 2018), where we show that biased beliefs improve the fit of life-cycle models to data on life-cycle asset holdings, relative to a rational expectations model. However, psychological attitudes may also bias other beliefs-e.g., income expectations (Dominitz and Manski 1997;Rozsypal and Schlafmann 2017)-and may directly affect pure time discounting through cognitive processes (Binswanger and Salm 2017;Gabaix and Laibson 2017). We leave for future research to apply our methods to other expectations data and to explore further the consequences for economic decisions of the multidimensionality of psychologically biased beliefs in calibrated life-cycle models.
Since we also do not have information on SSBs for individuals older than 89, we further restrict to individuals of age less than 90.
The sample for the hazard model includes waves 4-12 (years 1998 − 2014). We exclude waves < 4 due to consistency problems in how some variables were measured. This concerns questions on physical health such as activities of daily living (ADL).
Cross-Sectional Analysis. The HRS contains variables about psychosocial factors from wave 8 (year 2006) onwards. In our analyses we use psychological variables in lags. Our measure of cognitive weakness is not (yet) available in wave 12 (year 2014). Since we use lags, the main cross-sectional analyses of the paper is restricted to waves 9 − 12 (years 2008 − 2014).

A.1.2 Human Mortality Base (HMD)
As we describe in Section A.2, next to a long list of socioeconomic and health variables, we condition OSPs also on average objective survival probabilities (AOSPs), which requires the use of (predicted) cohort life tables in order to capture the time tend of survival risk. Our out of sample predictions of cohort specific survival rates are based on a statistical model, which we estimate on the basis of period life tables taken from the Human Mortality Database (HMD) for years 1993 to 2013. 34

A.2 Estimation of Objective Survival Probabilities (OSP)
We condition the estimation of the hazard model on average objective survival probabilities (AOSPs) and several individual level observable variables.

A.2.1 Average Objective Survival Probabilities
We construct life tables for each cohort c on the basis of a sequence of period t life tables. A period t life table contains average population mortality rates for ages j = 0, 1, 2, ..., J in year t, denoted by δ j,t which is the average probability of an individual aged j born in year c = t − j of dying in year t. A cohort c life table gives the average mortality rate of individuals of a given birth cohort c and in principle are obtained by simple rearrangements of period life tables. If period life tables are available from years t min to t max , then the cohort life table of cohort c is restricted to the age interval {max(t min − c; 0), t max − c}. For our purpose, however, we require for several cohorts mortality rates that exceed the age limit t max − c. 35 Hence, we have to predict future period life-tables from which, by re-arrangement, we can extract the corresponding cohort life tables.
To this purpose, we estimate mortality processes by adopting the Lee and Carter (1992) method and form predicted mortality rates on the basis of these estimates. We accordingly specify that the log mortality rate log(δ j,t ) can be decomposed into a vector of age-specific constants α j and age specific drift terms β j , where the drift is determined by a single index k t according to where ε j,t is some error term that captures age and time specific random deviations from this mortality trend. The single index k t is assumed to obey a unit-root process with drift and error term t ∼ N (0, σ 2 ): We estimate these processes with data from 1950 onward (because of structural breaks in earlier periods). The estimated drift terms areφ = −1.4460 andφ = −1.8114 for men and women, respectively. Based on our estimates we predict mortality rates until 2090 by holding constant the vectorsˆ α,ˆ β and the drift termφ and complete the cohort life tables on the basis of these estimates.

A.2.2 Individual Objective Survival Probabilities
As described in Subsection 2.3 we base our estimates of OSPs on a Weibull (1951) hazard model. We condition the baseline hazard in equation (1) on the AOSPs and on several individual level observable variables, both included in x i . The individual variables are summarized in Table 6.  Figure 13 shows the corresponding distributions of SSBs. Average SSBs decrease as we move up across target age groups, as with OSPs. However, the movement is not as pronounced as for the OSPs and the difference in the averages depicted by the red lines in both figures just reflects the facts shown in Figure 1 of the main text. Second, there are focal point answers at SSBs of 0, 0.5 and 1. Observe that the fraction of individuals providing a focal point answer at 1 decreases whereas the fraction giving answer 0 increases when the target 36 Observe that using AOSPs instead would result in only five different survival probabilities, one for each target age group. age increases. This indicates that focal point answers do have information content that goes beyond simple heuristics that individuals may apply when being confronted with such complicated questions about survival prospects.

A.4 Bootstrap
Standard errors of the parameters of our regressions have to be corrected in order to account for the estimation variance of OSPs. We accommodate this by implementing a two-sample bootstrap procedure with 1, 000 replications to estimate the standard errors of our coefficient estimates. In this procedure we correct for the estimation variance in objective survival probabilities as follows. 37 In each bootstrap replication we (i) draw a sample with replacement from the HRS sample used to estimate OSPs, (ii) estimate the OSPs, (iii) draw a sample with replacement from the cross-sectional sample used for regression analysis, (iv) perform regression analysis. Based on the resulting estimates we compute standard errors using the percentile method.

B.1 The Neo-Additive PWF
As an alternative to non-linear probability weighting functions, we estimate linear approximations in form of neo-additive probability capacities (Chateauneuf et al. 2007b). For OSP i,h,t ∈ (0, 1) 38 the neo-additive capacity is linear and writes as where ξ l h ∈ [0, 1], θ l h ∈ [0, 1] are parameters that are the analogues to parameters ξ h and θ h of the non-linear specification in (5). To see this observe that ξ l h controls the slope of the function, whereby for ξ l h = 1 the line in (14) corresponds with the 45-degree line. Therefore, any ξ l h ∈ [0, 1] can be interpreted as a measure of likelihood insensitivity. Likewise, 1 − θ l h ∈ [0, 1] determines the intersection of (14) with the 45-degree line, whereby the intersection moves down when θ l h increases. Accordingly, θ l h can be interpreted as a measure of pessimism. Figure 14 shows the linearly estimated probability weighting functions and Figure 15 shows the age patterns of the parameter estimates ξ l h =ξ l h , θ l h =θ m(h) , again with the bootstrapped 95% confidence intervals. As for the non-linear specification in the main text, the coefficient estimates ξ h =ξ m(h) , shown in Panel (a) of Figure 15, are decreasing in interview age h up to interview age group 85-89 where the estimates are very imprecise. 39 The point estimates suggest that a one percentage point increase of the OSP for age group 65-69 leads to a 0.6 analyses.
38 Interior OSPs follow from our specification in (3). 39 The imprecision for this age group is much larger than for the corresponding non-linear specification, cf. Figure 5. percentage point increase of the associated SSB, on average. At age group 75-79 the effect is only 0.4 percentage points. As with the non-linear specification, the age pattern of pessimism is less clearcut. Pessimism increases from age group 70-74 to age group 80-84. (Estimates are very imprecise for age group 85-89.) As previously, we also observe that pessimism initially decreases from age group 65-69 to age group 70-74.

B.2.1 The Linear Regression Specification & Identification
We derive the linear specification from using (6) in (14), but also superimpose additional structure based on our insights from the non-linear model. Specifically, we assume that there is no motivational base effect, which in this linear model   Under these assumptions and combining (6) with (14), we get Before turning to the reduced form of (15) and the crucial question of identifica-tion, observe that the isolated effects analogous to (8) are given by From the second line in (16b) we observe that the "pure" (i.e., ignoring the interactions with motivational variables) marginal effect of an increase of cognitive weakness is ξ 1 OSP i,h,m(h) − OSP 0 . For ξ 1 < 0-i.e., increasing cognitive weakness leads to a flattening of the PWF-we find that increasing cognitive weakness gives rise to stronger underestimation for OSP i,h,m(h) > OSP 0 , and to stronger overestimation for OSP i,h,m(h) < OSP 0 , just as in the non-linear model. Likewise, we observe from (16c), that the marginal effect of an increase of pessimism is given by θ 1 (1 − ξ 0 ), respectively the effect of an increase of optimism by θ 2 (1 − ξ 0 ). The reduced form specification follows from rewriting (15) as where β 3 = −β 2 OSP o . The parameters in the second line can be determined by a simple linear regression, where the regression coefficients are related to the structural model parameters by In general, the reduced form does not exactly identify all parameters of the model because there are 8 parameters in the reduced form and 5 in the model. What is identified is the direct marginal effect of an increase in OSPs given by β 1 = ξ 0 , and the marginal effects of pessimism is given by γ 1 = θ 1 (1 − ξ 0 ), respectively of optimism by γ 2 = θ 2 (1 − ξ 1 ). Hence, we expect that β 1 ∈ (0, 1), γ 1 < 0, γ 2 > 0. While the marginal effect of an increase of the interaction between the OSP and cognition, β 2 = ξ 1 , is identified (and while we therefore expect that β 2 < 0), the "pure" marginal effect of an increase in cognitive weakness, ξ 1 OSP i,h,m(h) − OSP 0 , is not identified because OSP 0 is not identified. To see this, observe that using (18b) in (18a) gives OSP 0 = β 0 1−β 1 , whereas using (18c) in (18d) gives OSP 0 = − β 3 β 2 and in general β 0 1−β 1 = − β 3 β 2 . 40 We resolve this issue of non-identification by determining the intersection of the total effect of cognition for each individual i, SSB bc i,h , with the 45-degree line, 40 To illustrate the problem of non-identification, observe that our point estimates reported in Table 4 suggest that OSP 0 = 0.0486 1−0.6316 ≈ 0.1318 or OSP 0 = − 0.0109 −0.0107 ≈ 1.0179 (which violates the bound constraint OSP 0 ∈ [0, 1]).
i.e., we determine OSP 0,i for each individual i from Taking the mean across all individuals gives our (mean group) estimate OSP 0 = 1 n n i=1 OSP 0,i . For our decomposition analysis we further require that the base bias intersects with the 45-degree line in the same point. To this purpose we modify the base bias by an additional additive shifterβ 0 and accordingly write 41 We determineβ 0,i from Again taking the mean across all i we get the (mean group) estimate β 0 = 1 n n i=1β 0,i . To determine the confidence intervals of OSP 0 andβ 0 we repeat these steps for all bootstrap iterations. 42

B.2.2 Decomposition Analyses of Linear Model
The decomposition of the linear probability weighting function is presented in Figure 16. Except for the behavior in the tails, findings are very similar to those from the non-linear model. The more interesting decomposition is the one over age shown in Panels (a)  to those from the non-linear model. The effect of cognitive weakness is basically the same, pessimism leads to a slightly stronger average underestimation and optimism to a stronger average overestimation than for the non-linear model. Notes: Sample averages of predicted non-linear probability weighting functions according to equations (7) and (8)

B.2.3 Statistical Learning and Control Variables
Derivation of Reduced From. To derive the linear reduced form specification, again start from (14) and replace the objective survival rate OSP i,h,t with (1 − w( x i,h )) OSP i,h,t +w( x i,h )π 0 w( x i,h ) where π 0 is some prior belief and w( x i,h ) is a weighting function. This linear approximation to the learning model of Groneck et al. (2016) gives rise to the following specification which nests specification (14). Using (6) in the above, linearly approximating the weighting function w( x i,h ) and ignoring all interactions between x i,h , c i,h , OSP i,h,t Notes: Sample averages of predicted subjective survival beliefs according to equations (7) and (8)  for simplicity 43 results in (10).
Results on Control Variables. Table 7 shows the results of our estimation for the control variables. 43 In a regression where we included these interactions the AIC decreased.

C.1 Focal Point Answers
To investigate the sensitivity of our results with respect to focal point answers, we repeat the estimation of non-linear PWFs by excluding observations with focal point answers at SSBs of 0%, 50% and 100%. Results are presented in Figure 18. In contrast to the corresponding Figure 4, probability weighting functions for the highest target age group are now downward sloping. Since we regard upward sloping PWFs as plausible, this finding is another indication (beyond the histograms shown in Appendix A) that focal point answers do have information content, which justifies including all these observations in our main analyses.

C.2 The Effects of Motivational Factors on Mortality
In our estimation of individual-level OSPs in Section 2, we include cognition as a regressor in the hazard model. 44 Therefore, our estimates in the previous subsections capture the effects of cognitive weakness on the formation of survival beliefs beyond the effects that are channeled through the objective survival probabilities. However, for sample reasons, we do not include motivational variables in this hazard model, because they are only available from wave 8 onwards. If optimists were more likely to survive and if pessimists were more likely to die, then the observed deviations from the SSBs caused by these motivational attitudes we identified would (at least partially) reflect additional information of respondents on their objective mortality risk rather than psychological biases. Notes: Column 2 shows the point estimates. Columns 3 and 4 show the respective bounds of the 95%-confidence intervals (CI-and CI+), which are calculated with the percentile method (1,000 replications).
Source: Own calculations, Health and Retirement Study (HRS), Human Mortality Database (HMD).
We can address this concern in a smaller sub-sample by re-estimating the hazard model with the inclusion of the motivational variables, using HRS data from waves 8-12. The results are shown in Table 8. We find neither optimism nor pessimism to be significant at the 5% significance level. This supports our interpretation of the effects of optimism and pessimism on SSBs as reflecting psychological biases. 44 We find that the lack of cognition is positively correlated with mortality.