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.
rh_stan( death, exposure, forecast, validation = 0, family = c("poisson", "nb"), ... )
| 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. |
| ... | Arguments passed to |
An object of class stanfit returned by rstan::sampling.
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: $$\psi_1,\psi_2 \sim N(0,10),\quad \sigma_{\gamma}\sim Exp(0.1).$$
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.
#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) #> #> SAMPLING FOR MODEL 'RHmodel' NOW (CHAIN 1). #> Chain 1: #> Chain 1: Gradient evaluation took 0 seconds #> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0 seconds. #> Chain 1: Adjust your expectations accordingly! #> Chain 1: #> Chain 1: #> Chain 1: WARNING: There aren't enough warmup iterations to fit the #> Chain 1: three stages of adaptation as currently configured. #> Chain 1: Reducing each adaptation stage to 15%/75%/10% of #> Chain 1: the given number of warmup iterations: #> Chain 1: init_buffer = 3 #> Chain 1: adapt_window = 20 #> Chain 1: term_buffer = 2 #> Chain 1: #> Chain 1: Iteration: 1 / 50 [ 2%] (Warmup) #> Chain 1: Iteration: 5 / 50 [ 10%] (Warmup) #> Chain 1: Iteration: 10 / 50 [ 20%] (Warmup) #> Chain 1: Iteration: 15 / 50 [ 30%] (Warmup) #> Chain 1: Iteration: 20 / 50 [ 40%] (Warmup) #> Chain 1: Iteration: 25 / 50 [ 50%] (Warmup) #> Chain 1: Iteration: 26 / 50 [ 52%] (Sampling) #> Chain 1: Iteration: 30 / 50 [ 60%] (Sampling) #> Chain 1: Iteration: 35 / 50 [ 70%] (Sampling) #> Chain 1: Iteration: 40 / 50 [ 80%] (Sampling) #> Chain 1: Iteration: 45 / 50 [ 90%] (Sampling) #> Chain 1: Iteration: 50 / 50 [100%] (Sampling) #> Chain 1: #> Chain 1: Elapsed Time: 0.034 seconds (Warm-up) #> Chain 1: 0.975 seconds (Sampling) #> Chain 1: 1.009 seconds (Total) #> Chain 1: