GAPC: Generalized APC stochastic mortality model

In vital: Tidy Analysis Tools for Mortality, Fertility, Migration and Population Data

Generalized APC stochastic mortality model

Description

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 call genWeightMat with arguments clip and zeroCohorts.
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.

Value

A model specification.

Author(s)

Rob J Hyndman

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

See Also

LC()

Examples

# Fit the same CBD model using GAPC() and CBD()
gapc <- norway_mortality |>
  dplyr::filter(Sex == "Female", Age > 50, Year > 2000) |>
  model(
    cbd1 = GAPC(Mortality,
      link = "log",
      staticAgeFun = FALSE,
      periodAgeFun = c("1", function(x, ages) x - mean(ages))
    ),
    cbd2 = CBD(Mortality)
  )
glance(gapc)
gapc |>
  dplyr::select(cbd2) |>
  report()

vital documentation built on Aug. 21, 2025, 5:34 p.m.