Mortality Projection

Main Demographical Trends

Expansion over time
Regularization overtime
Increasing trend over time in life expectancy
Downward trend over time in death rates

Dynamic Life Table

Consider the mortality rates depending on both age and calender year \(q_x(t)\).
Can read dynamic life tables in different ways:
- Diagonal gives information about a cohort’s progression

Columns, known as a period, refers to all people living in year \(t\)

Rows, shows the mortality trend at age \(x\), aka the mortality profile

Extrapolation Methods

Utilise the graduation of past rates to predict the future
- Need to consider how far back you want to use
- Need to consider what ages you want to extrapolate
Traditional Methods
- Data are simple numbers
- Does not allow for statistical features
- Point estimates
Statistical Methods
- Data are outcomes of random variables
- Probabilistic model linking observations and future outcomes
- Both point and interval estimates

Traditional Methods

Reduction Factors

Considering a base year \(t'\) where \(t>t'\)

\(q_x(t) = \underbrace{q_x(t')}_{\text{Base}}*\underbrace{R_x(t',t)}_{\text{Reduction Factor}}\)

Often simplified:
- Only dependent on the length of the period

\[ R_x(t',t) = R_x(t-t')\] * We often assume an exponential structure of the observed \(q_x(t)\)

\[ \frac{q_x(t_{h+1})}{q_x(t_h)} \approx e^{-\delta_x(t_{h+1}-t_h)}\]

We estimate \(\delta_x\) via a least squares procedure
- Typically with the constraint \(\hat{q}_x(t_n)=q_x(t_n)\)
  - Constrained to pass through the last point
Can also use weighted average or a simple single reduction formula:

\[ r_x = \left[\frac{q_x(t_n)}{q_x(t_1)}\right]^{\frac{1}{t_n-t_1}}\]

Therefore we can think of a reduction factor that only depends on the length of forecast:

\[ R_x(t',t) = R_x(t-t') = r_x^{t-t'} = e^{-\delta(t-t')}\]

To avoid \(q(\inf) = 0\), a limit scenario is often added to the formula:

\[ q_x(t) = q_x(t')[\alpha_x + (1-\alpha_x)r_x^{t-t'}]\]

Stochastic Mortality Models

Stochastic projections have central/point estimates as well as interval estimates (explicitly allowing for uncertainty)

Lee Carter Model

\(\alpha_x\) captures the age specific pattern of mortality
\(\kappa_t\) is dependent on time and captures the decline in mortality over time
\(\beta_x\) captures the differences in improvements in different ages

\[ log(m_{xt}) = \alpha_x + \beta_x\kappa_t\]

Parameter estimation is done with least squares based on matrix of available past death rates, with parameter constraints:

\[ \sum_x \beta_x = 1,\quad\sum_t\kappa_t = 0\]

\(\alpha_x\) is the aveage of log-central death rates over calender years
\(\kappa_t\) is modelled as a random walk with drift
- This can be simulated to create trajectories for the confidence intervals
Forecasted mortality rates are computed as follows:

\[ m_{xt}=\exp[\hat{\alpha}_x+\hat{\beta}_x\underbrace{\kappa_t}_{\text{Projected}}] = m_{x,t'}\exp[\hat{\beta}_x(\kappa_t-\kappa_{t'})]\]

Lee Carter: Poisson Log-Bilinear

LC implicitly assumes that errprs are homoskedastic, which is often unrealistic due to low estimates at high ages.
Can reformulate the hypothesis in terms of \(\mu_{xt}\) and maximise the poisson likelihood:

\[ D_{xt}\sim P(E^c_{xt}\mu_{xt})\]

\[ \log(\mu_{xt}) = \alpha_x+\beta_x\kappa_t\]

Cairns-Blake-Dowd Model

Carines-Blake-Dowd fits a linear regression for each calender year, getting the intercept and slope from each to create a kappa time series
- Works mainly for older ages as they tend to be linear, while younger ages do not
- Allows to capture mortality decline as well as the different rate of improvements between ages

\[ logit(q_{xt}) = \kappa^{(1)}_t+(x-\bar{x})\kappa^{(2)}_t \]

\(\kappa^1\) is a stochastic process for the intercept. This is generally declining as mortality rates have been decreasing over time
\(\kappa^2\) is a stochastic process for the slope. This is generally increasing as mortality improvements have been happening at a greater rate for lower ages than at higher ages.
Estimate the parameters through least squares or MLE using the glm framework, and forecase using a multivariate random walk with drift.

Cohort Effects

Statistical evidence that there is cohort effect on the improvement of mortality
Period effects approximate contemporary factors
- General health status of population
- Healthcare services
- Critical weather conditions
Cohort effects approximate historical factors
- World War II
- Diet
- Welfare status
- Smoking habits

Age-Period-Cohort Model

The classic APC model considers cohort and period however doesn’t consider age appropriately:
- \(\gamma_{t-x}\) captures the cohort effect

\[ \log(\mu_{xt})=\alpha_x+\kappa_t+\gamma_{t-x}\]

Estimation using possion MLE or weighted least squares.
Impose constraints

\[ \sum_t\kappa_t = 0,\quad\sum_c\gamma_c = 0,\quad\sum_c c\gamma_c =0\]

Many extensions to existing models have been proposed.
- Some of these are able to appropriately capture age, period and cohort, which results in a very strong model.

There is also a class of generalised APC models, analogous to GLMs:

Goodness of Fit - Residuals

The deviance residuals of the model are analysed, typically graphically:

\[ r_{xt} = sign(d_{xt}-\hat{d}_{xt})\sqrt{\frac{dev(x,t)}{\hat{\phi}}}\]

Graphically we can observe patterns
Can observe the residual sign plot:
- Note dark grey are negative residuals
- We look for:
  - Diagonal patters - cohort
  - Horizontal patterns - age
  - Vertical patterns - period
- Less pattern = better fit

Can also observe the residual scatterplot
- Plotted against age, calender year and year of birth for age, period and cohort patterns respectively