A Generalized Age-Period-Cohort (GAPC) stochastic mortality mode is defined in Villegas et al. (2018). The StMoMo package is used to fit the model. Separate functions are available to fit various special cases of the GAPC model.
Usage
GAPC(formula, use_weights = TRUE, clip = 0, zeroCohorts = NULL, ...)
LC2(
formula,
link = c("log", "logit"),
const = c("sum", "last", "first"),
use_weights = TRUE,
clip = 0,
zeroCohorts = NULL,
...
)
CBD(
formula,
link = c("log", "logit"),
use_weights = TRUE,
clip = 0,
zeroCohorts = NULL,
...
)
RH(
formula,
link = c("log", "logit"),
cohortAgeFun = c("1", "NP"),
use_weights = TRUE,
clip = 0,
zeroCohorts = NULL,
...
)
APC(
formula,
link = c("log", "logit"),
use_weights = TRUE,
clip = 0,
zeroCohorts = NULL,
...
)
M7(
formula,
link = c("log", "logit"),
use_weights = TRUE,
clip = 0,
zeroCohorts = NULL,
...
)
PLAT(
formula,
link = c("log", "logit"),
use_weights = TRUE,
clip = 0,
zeroCohorts = NULL,
...
)Arguments
- formula
Model specification
- use_weights
If
TRUE, will callgenWeightMatwith argumentsclipandzeroCohorts.- clip
Passed to
genWeightMat()- zeroCohorts
Passed to
genWeightMat()- ...
All other arguments passed to
StMoMo()- link
The link function to use. Either "log" or "logit". When using logit, the mortality rates need to be between 0 and 1. If they are not, most likely you need to use initial rather than central population values when computing them.
- const
defines the constraint to impose on the period index of the model ensure identifiability. The alternatives are "sum" (default), "last" and "first" which apply constraints \(\sum_{t=1}^T \kappa_t=0\), \(\kappa_T = 0\) and \(\kappa_1 = 0\) respectively.
- cohortAgeFun
A function defining the cohort age modulating parameter \(\beta_x^{(0)}\). It can take values: "NP" for a non-parametric age term or "1" for \(\beta_x^{(0)} = 1\) (the default).
Details
LC2() provides an alternative implementation of the Lee-Carter model based
on a GAPC specification. The advantage of this approach over LC() is that
it allows for 0 rates in the mortality data. Note that it does not return
identical results to LC() because the model formulation is different.
For LC2(), do not take logs of the mortality rates because this is handled
with the link function. For LC(), you need to take logs of the mortality rates
when calling the function.
The Renshaw-Haberman (RH) model due to Renshaw and Haberman (2006) is another special case of a GAPC model, that can be considered an extension of a Lee-Carter model with an age-specific cohort effect.
The Age-Period-Cohort (APC) model is a special case of a GAPC model discussed by Renshaw and Haberman (2011).
The Cairns-Blake-Dowd (CBD) model due to Cairns et al (2006) can be considered another special case of a GAPC model that is primarily intended for forecasting mortality patterns in older populations.
Cairns et al (2009) extended the CBD model by adding a cohort effect and a quadratic age effect, giving the M7 model.
Plat (2009) combined the CBD model with some features of the Lee-Carter model to produce a model that is suitable for full age ranges and captures the cohort effect.
Each of these functions returns a GAPC model applied to the formula's response
variable as a function of age.
The model will optionally call genWeightMat with arguments clip and zeroCohorts.
All other arguments are passed to StMoMo.
References
Cairns, AJG, Blake, D, and Dowd, K (2006). A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration. Journal of Risk and Insurance, 73(4), 687-718. doi:10.1111/j.1539-6975.2006.00195.x
Cairns AJG, Blake D, Dowd K, Coughlan GD, Epstein D, Ong A, Balevich I (2009). A quantitative comparison of stochastic mortality models using data from England and Wales and the United States. North American Actuarial Journal, 13(1), 1–35. doi:10.1080/10920277.2009.10597538
Lee, RD, and Carter, LR (1992) Modeling and forecasting US mortality. Journal of the American Statistical Association, 87, 659-671. doi:10.1080/01621459.1992.10475265
Plat R (2009). On stochastic mortality modeling. Insurance: Mathematics and Economics, 45(3), 393–404. doi:10.1016/j.insmatheco.2009.08.006
Renshaw, AE, and Haberman, S (2006). A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insurance: Mathematics and Economics, 38(3), 556-570. doi:10.1016/j.insmatheco.2005.12.001
Renshaw, AE, and Haberman, S (2011). A comparative study of parametric mortality projection models. Insurance: Mathematics and Economics, 48(1), 35–55. <doi:10.1016/j. insmatheco.2010.09.003>
Villegas, AM, Millossovich, P, and Kaishev, VK (2018). StMoMo: An R package for stochastic mortality modelling. Journal of Statistical Software, 84(3), 1-38. doi:10.18637/jss.v084.i03
Examples
# Fit the same CBD model using GAPC() and CBD()
gapc <- aus_mortality |>
dplyr::filter(State == "Victoria", Sex == "female", Age > 50) |>
model(
cbd1 = GAPC(Mortality,
link = "log",
staticAgeFun = FALSE,
periodAgeFun = c("1", function(x, ages) x - mean(ages))
),
cbd2 = CBD(Mortality)
)
#> Registered S3 method overwritten by 'quantmod':
#> method from
#> as.zoo.data.frame zoo
glance(gapc)
#> # A tibble: 2 × 8
#> Sex State Code .model loglik deviance nobs npar
#> <chr> <chr> <chr> <chr> <dbl> <dbl> <int> <int>
#> 1 female Victoria VIC cbd1 -26955. 14005. 5930 240
#> 2 female Victoria VIC cbd2 -26955. 14005. 5930 240
gapc |>
dplyr::select(cbd2) |>
report()
#> Series: Mortality
#> Model: GAPC
#> Stochastic Mortality Model fit
#> Call: fit.StMoMo(object = model, Dxt = data2$Dxt, Ext = data2$Ext,
#> Call: ages = data2$ages, years = data2$years, wxt = wxt, verbose = FALSE)
#>
#> Poisson model with predictor: log m[x,t] = k1[t] + f2[x] k2[t]
#>
#> Data: vital
#> Series: female
#> Years in fit: 1901 - 2020
#> Ages in fit: 51 - 100
#>
#> Log-likelihood: -26954.67
#> Deviance: 14005.39
#> Number of parameters: 240