rh_stan {StanMoMo}

Bayesian Renshaw-Haberman model with Stan

Description

Fit and Forecast Bayesian Renshaw-Haberman model (Lee-Carter with cohort effect) introduced in Renshaw and Haberman (2006). The model can be fitted with a Poisson or Negative-Binomial distribution. The function outputs posteriors distributions for each parameter, predicted death rates and log-likelihoods.

Usage

rh_stan(
  death,
  exposure,
  forecast,
  validation = 0,
  family = c("poisson", "nb"),
  ...
)

Arguments

death

Matrix of deaths.

exposure

Matrix of exposures.

forecast

Number of years to forecast.

validation

Number of years for validation.

family

specifies the random component of the mortality model. "Poisson" assumes a Poisson model with log link and "nb" assumes a negative-binomial model with log link and overdispersion parameter \phi.

...

Arguments passed to rstan::sampling (e.g. iter, chains).

Details

The created model is either a log-Poisson or a log-Negative-Binomial version of the Renshaw-Haberman model:

D_{x,t} \sim \mathcal{P}(\mu_{x,t} e_{x,t})

or

D_{x,t}\sim NB\left(\mu_{x,t} e_{x,t},\phi\right)

with

\log \mu_{xt} = \alpha_x + \beta_x\kappa_t+\gamma_{t-x}.

To ensure the identifiability of th model, we impose

\kappa_1=0, \gamma_1=0,\sum gamma_i =0, \gamma_C=0,

where C represents the most recent cohort in the data.

For the priors, the model chooses wide priors:

\alpha_x \sim N(0,100),\beta_{x} \sim Dir(1,\dots,1),\frac{1}{\phi} \sim Half-N(0,1).

For the period term, we consider the standard random walk with drift:

\kappa_{t}=c+\kappa_{t-1}+\epsilon_{t},\epsilon_{t}\sim N(0,\sigma^2)

with c \sim N(0,10),\sigma \sim Exp(0.1).

For the cohort term, we consider a second order autoregressive process (AR(2)):

\gamma_{c}=\psi_1 \gamma_{c-1}+\psi_2 \gamma_{c-2}+\epsilon^{\gamma}_{t},\quad \epsilon^{\gamma}_{t}\sim N(0,\sigma_{\gamma}).

To close the model specification, we impose some vague priors assumptions on the hyperparameters:

\psi_1,\psi_2 \sim N(0,10),\quad \sigma_{\gamma}\sim Exp(0.1).

Value

An object of class stanfit returned by rstan::sampling.

References

Renshaw, A. E., & Haberman, S. (2006). A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insurance: Mathematics and Economics, 38(3), 556-570.

Examples

#10-year forecasts for French data for ages 50-90 and years 1970-2017 with a log-Poisson model
ages.fit<-70:90
years.fit<-1990:2010
deathFR<-FRMaleData$Dxt[formatC(ages.fit),formatC(years.fit)]
exposureFR<-FRMaleData$Ext[formatC(ages.fit),formatC(years.fit)]
iterations<-50 # Toy example, consider at least 2000 iterations
fitRH=rh_stan(death = deathFR,exposure=exposureFR, forecast = 5, family = "poisson",
iter=iterations,chains=1)