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## Appendix 5: Implementation Mortality Rate Dependence

Mortality rates used in the PWBMsim are benchmarked to the Social Security Administrations’ age- and gender-specific projections of mortality rates through the year 2090. (The SSA rates are consistent with the Social Security Trustees’ annual report for 2016.) This Appendix describes how those rates are further distinguished by categories of ethnicity, $r$, education, $e$, and marital status, $m$, and re-benchmarked to SSA mortality rates by age and gender so that PWBMsim’s overall age-gender mortality rates remain equal to those SSA rates.

PWBMsim’s year-specific mortality rates are derived by decomposing SSA mortality rates by gender and age, $m^\ast [g,a]$, by ethnicity, $r$. The decomposed rates by ethnicity, gender, and age are denoted below as $m^\ast [r \mid g, a]$. The decomposition is implemented as shown in equations 4.1 and 4.2:

\begin{align} m^\ast [r \mid g, a] &= m^{\text{SSA}} [g, a] \times \rho[r \mid g, a] \times s[g, a, r] \text{(A4.1)} \\ s[g, a, r] &= \frac{1}{\Sigma_r \rho (r \mid g, a) \times d(r \mid g, a)} \text{(A4.2)}. \end{align}

Here, the term $\rho(r\mid g, a)$ refers to the mortality rate differential of those with ethnicity r relative to those with $r=white$ for given age and gender and $d(e\mid g,a)$ represents the population share of such individuals. According to the first equation above, mortality differentials by education attainment, $\rho[r\mid g,a]$, are applied to $m^{SSA} [g,a]$ along with a shift parameter, $s[g,a,r]$. (‘a’ denotes age groups (0:0-24, 1:25-44, 2:45-64, 3:65-84, 4:85 and over) and ‘e’ denotes education attainment groups based on completed years of education (0:0-11, 1:12, 2:13-15, 4:16, 5:17 and over). $\rho[e\mid g,r,a]=1$ for $e=0$.) The latter term adjusts mortality rates for all education levels, so that average mortality rates for each subgroup are maintained at SSA’s benchmark mortality rates by age and gender. This shift term is computed in each year as shown in equation A4.2 and the result is used in the equation A4.1 when PWBMsim is executed.

The decomposition by education is implemented analogously as shown via equations A4.3 and A4.4:

\begin{align} m^\ast [e\mid g,a,r] &= m^\ast [r\mid g,a]\times\rho[e\mid g,a,r]\times s[g,a,r,e]\text{(A4.3)} \\ s[g,a,r,e] &= \frac{1}{\Sigma_e\rho(e\mid g,a,r)\times d(e\mid g,a,r)}\text{(A4.4)} \end{align}

Here, the term $\rho(e│g,a,r)$ refers to the mortality rate differential of those with education level $e$ relative to those with $e = \text{high-school diploma}$ for a given age and gender and $d(e│g,a,r)$ represents the population share of such individuals. According to the first equation above, mortality differentials by education attainment, $\rho[e\mid g,a,r]$, are applied to $m^\ast [r\mid g,a]$ along with a shift parameter $s[g,a,r,e]$. The latter term adjusts mortality rates for all education levels, so that average mortality rates for each subgroup are maintained at SSA’s benchmark mortality rates by age and gender. This shift term is computed in each year as shown in equation A4.4 and the result is used in the equation A4.3 when PWBMsim is executed.

The marital status differentials are applied in an analogous way, leading to the final computation of mortality rates, $m^\ast [m│g,a,r,e]$, conditional on marital status along with other demographic attributes - gender, age, ethnicity, and education. (‘m’ denotes marital status (0:single, 1:married, 2:divorced/separated, 3:widowed). The analogous remark as in the earlier footnote applies.)

\begin{align} m^\ast [m \mid g,a,r,e] &= m^\ast [e \mid g,a,r]\times\rho[m \mid g,a,r,e]\times{s[g,a,r,e,m]} \text{(A4.5)} \\ s[g,a.r,e,m] &= \frac{1}{\Sigma_e\rho(m\mid g,a.r,e)\times{d(m\mid g,a,r,e)}} \text{(A4.6)} \end{align}

A similar interpretation of the terms in equations 4.5 and 4.6 applies as described earlier for equations A4.1-A4.4. If information for a particular ethnicity is absent because of small populations in micro-survey data (married educated females aged 45 of “other” ethnicity, for example), those ethnicities are merged with others: in particular, Asian ethnicity category is merged with the white ethnicity category, and “other” and Hispanic ethnicities are merged with the black ethnicity category. When education attainment indicators are inconsistent between Census micro-data and other information sources on relative mortality differentials, education categories are collapsed (for example, education level below 8th grade is not distinguished from the secondary high-school education level). Finally, if estimated relative education differentials are not significant or are negative, the differentials are set equal to the normalized value for the reference education group, namely 1.0. For data sources on relative differentials, see notes to Tables 1 and 2 in the main text.