Individual Survival Expectations and Actual Mortality: Evidence from Dutch Survey and Administrative Data

Abstract

Because of the important role that survival expectations play in individual decision making, we investigate the extent to which individual responses to survival probability questions are informative about actual mortality. In contrast to earlier studies, which relied on the Health and Retirement Study (HRS) of US individuals aged 50 and over, we combine household survey data on subjective survival probabilities with administrative data on actual mortality for Dutch respondents aged 25 and over. Our main finding is that in our sample, individual life expectancies (measured as subjective survival probabilities) do predict actual mortality even when we control for a large set of health indicators. Our results further suggest that, on average, women underestimate their remaining life duration, whereas men tend to predict their survival chances more realistically. Both sexes, however, tend to overestimate the age gradient in mortality risk and underestimate the health risks of smoking.

Appendix

1.1 Derivation of the Median Remaining Life Duration

For each individual in our sample, we observe two survival function values, $SSP_{1,i}$ and $SSP_{2,i}$, at two different target ages $t_{1,i}$ and $t_{2,i}$ where $t_{1,i} < t_{2,i}$. $t_{0,i}$ is the baseline age at which the respondent reported the SSPs:

$$S_{1,i} (t_{0,i} ,t_{1,i} ) = \exp \left\{ { - \int\limits_{{t_{0,i} }}^{{t_{1,i} }} {\theta (s)ds} } \right\} = SSP_{1,i}$$

(6)

$$S_{2,i} (t_{0,i} ,t_{2,i} ) = \exp \left\{ { - \int\limits_{{t_{0,i} }}^{{t_{2,i} }} {\theta (s)ds} } \right\} = SSP_{2,i}$$

(7)

with $(t_{1,i} ,t_{2,i} ) \in \left\{ {(75,80),(80,85),(85,90),(90,95),(95,100)} \right\}$ and $(SSP_{1,i} ,SSP_{2,i} ) \in \left\{ {(P75,P80),(P80,P85),(P85,P90),(P90,P95),(P95,P100)} \right\}$, respectively. $P75$ represents the subjective survival probability (SSP) to age 75, $P80$ that to age 80, and so on. The Gompertz hazard rate function is given by $\theta (t) = \lambda_{i} \exp \left\{ {\gamma_{i} t} \right\}$.

After substituting the hazard rate function into Eqs. (6) and (7), we evaluate the integral to find

$$S_{1,i} \left( {\gamma_{i} ,\lambda_{i} \left| {t_{0,i} ,t_{1,i} } \right.} \right) = \exp \left\{ {\frac{{\lambda_{i} }}{{\gamma_{i} }}\left( {\exp \left\{ {\gamma_{i} t_{0,i} } \right\} - \exp \left\{ {\gamma_{i} t_{1,i} } \right\}} \right)} \right\} = SSP_{1,i}$$

(8)

$$S_{2,i} \left( {\gamma_{i} ,\lambda_{i} \left| {t_{0,i} ,t_{2,i} } \right.} \right) = \exp \left\{ {\frac{{\lambda_{i} }}{{\gamma_{i} }}\left( {\exp \left\{ {\gamma_{i} t_{0,i} } \right\} - \exp \left\{ {\gamma_{i} t_{2,i} } \right\}} \right)} \right\} = SSP_{2,i}$$

(9)

Next, following Perozek (2008), we take logarithms of the survival functions [Eq. (10)] and estimate the parameters $\gamma_{i}$ and $\lambda_{i}$ for each individual using nonlinear least squares (NLLS). This procedure requires that we replace the survival probabilities of 0 and 1 with slightly different numbers, namely 0.01 and 0.99, respectively.

$$\ln \left( {SSP_{j,i} } \right) = \ln \left( {S_{j,i} \left( {\gamma_{i} ,\lambda_{i} \left| {t_{0,i} ,t_{j,i} } \right.} \right)} \right) + \varepsilon_{j,i} ,\quad j \in \left\{ {1,2} \right\}$$

(10)

where $\varepsilon_{j,i}$ is the error term, which is assumed to be independent and identically distributed (i.i.d.) and have a zero mean. We obtain the NLLS estimates of $\gamma_{i}^{*}$ and $\lambda_{i}^{*}$ by minimizing the following expression:

$$\mathop {\hbox{min} }\limits_{{\gamma_{i} ,\lambda_{i} }} \sum\nolimits_{j} {\left( {\ln \left( {SSP_{j,i} } \right) - \ln \left( {S_{j,i} \left( {\gamma_{i} ,\lambda_{i} \left| {t_{0,i} ,t_{j,i} } \right.} \right)} \right)} \right)^{2} }$$

with

$$S_{j,i} \left( {\gamma_{i} ,\lambda_{i} \left| {t_{0,i} ,t_{j,i} } \right.} \right) = \exp \left\{ {\frac{{\lambda_{i} }}{{\gamma_{i} }}\left( {\exp \left\{ {\gamma_{i} t_{0,i} } \right\} - \exp \left\{ {\gamma_{i} t_{j,i} } \right\}} \right)} \right\}$$

We then calculate the subjective median remaining life duration conditional on baseline age for each individual denoted by $L_{i}^{R,S}$, where R and S denote ‘remaining’ and ‘subjective,’ respectively, based on the following formula:

$$S(L_{i}^{R,S} \left| {t_{0,i} } \right.) = \exp \left\{ { - \int\limits_{0}^{{L_{i}^{R,S} }} {\theta (s^{\prime} + t_{0,i} )ds^{\prime}} } \right\} = 0.5$$

(11)

The Gompertz hazard function is then $\theta (s^{\prime} + t_{0,i} ) = \lambda_{i} \exp \left\{ {\gamma_{i} (s^{\prime} + t_{0,i} )} \right\}$, so evaluating the integral in Eq. (11) and taking the natural logarithm of both sides yields

$$- \ln (0.5) = \frac{{\lambda_{i} }}{{\gamma_{i} }}\left( {\exp \left\{ {\gamma_{i} (t_{0,i} + L_{i}^{R,S} )} \right\} - \exp \left\{ {\gamma_{i} t_{0,i} } \right\}} \right)$$

$$L_{i}^{R,S} = \frac{1}{{\gamma_{i} }}\ln \left( {\frac{{\gamma_{i} \ln (2)}}{{\lambda_{i} \exp (\gamma_{i} t_{0,i} )}} + 1} \right)$$

(12)

We then replace $\gamma_{i}$ and $\lambda_{i}$ in Eq. (12) with their estimates $\gamma_{i}^{*}$ and $\lambda_{i}^{*}$, respectively. Because the variable $L_{i}^{R,S}$ is created using individual subjective survival probabilities, it represents the subjective median remaining life duration conditional on baseline age for each individual.

1.2 Comparison of Objective and Subjective Predicted Remaining Life Durations (Table 6)

The common assumption in the objective and subjective mortality models is that life duration can be modeled using a Gompertz distribution. Under this assumption, the hazard function can be written as

$$\theta (t\left| {{\mathbf{x}}_{{\mathbf{i}}} } \right.) = \lambda_{i} \exp \left\{ {\gamma_{i} t} \right\}$$

where $\lambda_{i} = \lambda^{obj} = \exp \left\{ {{\mathbf{x}}_{{\mathbf{i}}} {\varvec{\upbeta}}^{obj} } \right\}$, $\gamma_{i} = \gamma^{obj}$ in the objective mortality model, and $\lambda_{i} = \lambda^{subj} = \exp \left\{ {{\mathbf{x}}_{{\mathbf{i}}} {\varvec{\upbeta}}^{subj} } \right\}$, $\gamma_{i} = \gamma^{subj}$ in the subjective mortality model. Hence, we replace $\lambda^{obj}$, $\gamma^{obj}$, $\lambda^{subj}$, and $\gamma^{subj}$ in Eq. (12) with their estimates, $\hat{\lambda }^{obj}$,$\hat{\gamma }^{obj}$, $\hat{\lambda }^{subj}$, and $\hat{\gamma }^{subj}$, respectively:

$$\hat{L}^{R,O} = \frac{1}{{\hat{\gamma }^{obj} }}\ln \left( {\frac{{\hat{\gamma }^{obj} \ln (2)}}{{\exp \left\{ {{\bar{\mathbf{x}}}_{i} {\hat{\mathbf{\beta }}}^{obj} + \hat{\gamma }^{obj} \bar{t}_{0,i} } \right\}}} + 1} \right)$$

(13)

$$\hat{L}^{R,S} = \frac{1}{{\hat{\gamma }^{subj} }}\ln \left( {\frac{{\hat{\gamma }^{subj} \ln (2)}}{{\exp \left\{ {{\bar{\mathbf{x}}}_{i} {\hat{\mathbf{\beta }}}^{subj} + \hat{\gamma }^{subj} \bar{t}_{0,i} } \right\}}} + 1} \right)$$

(14)

where $\hat{L}^{R,O}$ and $\hat{L}^{R,S}$ denote objective and subjective predicted remaining life durations, respectively, and $\bar{t}_{0,i}$ is the baseline age, which is equal to 45. The vector ${\bar{\mathbf{x}}}_{i}$ contains fixed values of the control variables included in the estimation. For example, in the first row of Table 6, all of the dummy variables in both mortality models are equal to zero.