fitLCmulti: Function to fit multi-population mortality models

Description

R function for fitting additive, multiplicative, common-factor (CFM), augmented-common-factor (ACFM), or joint-k multi-population mortality model developed by: Debon et al. (2011), Russolillo et al. (2011), Carter and Lee (1992), LI and Lee (2005), and Carter and Lee (1992), respectively. These models follow the structure of the well-known Lee-Carter model (Lee and Carter, 1992) but include different parameter(s) to capture the behavior of each population considered in different ways. In case, you want to understand in depth each model, please see Villegas et al. (2017). It should be mentioned that this function is developed for fitting several populations. However, in case you only consider one population, the function will fit the single population version of the Lee-Carter model, the classical one.

Usage

fitLCmulti(model = c("additive"), qxt, periods, ages, nPop, lxt = NULL)

Value

A list with class LCmulti including different components of the fitting process:

ax parameter that captures the average shape of the mortality curve in all considered populations.
bx parameter that explains the age effect x with respect to the general trend kt in the mortality rates of all considered populations.
kt represent the national tendency of multi-mortality populations during the period.
Ii gives an idea of the differences in the pattern of mortality in any region i with respect to Region 1.
formula additive multi-population mortality formula used to fit the mortality rates.
model provided the model selected in every case.
data.used mortality rates used to fit the data.
qxt.crude corresponds to the crude mortality rates. These crude rate are directly obtained by dividing the number of registered deaths by the number of those initially exposed to the risk for age x, period t and in each region i.
qxt.fitted fitted mortality rates using the additive multi-population mortality model.
logit.qxt.fitted fitted mortality rates in logit way.
Ages provided ages to fit the data.
Periods provided periods to fit the periods.
nPop provided number of populations to fit the periods.
warn_msgs vector with the populations where the model has not converged.

Arguments

model: multi-population mortality model chosen to fit the mortality rates c("additive", "multiplicative", "CFM","ACFM", "joint-K"). In case you do not provide any value, the function will apply the "additive" option.
qxt: mortality rates used to fit the additive multipopulation mortality model. These rates can be provided in the matrix or in a data.frame.
periods: number of years considered in the fitting in a vector way c(minyear:maxyear).
ages: vector with the ages considered in the fitting. If the mortality rates provide from an abridged life tables, it is necessary to provide a vector with the ages, see the example.
nPop: number of population considered for fitting. If you consider 1 the model selected will be the singel version of the Lee-Carter model.
lxt: survivor function considered for every population, not necessary to provide.

References

Carter, L.R. and Lee, R.D. (1992). Modeling and forecasting US sex differentials in mortality. International Journal of Forecasting, 8(3), 393–411.

Debon, A., Montes, F., and Martinez-Ruiz, F. (2011). Statistical methods to compare mortality for a group with non-divergent populations: an application to Spanish regions. European Actuarial Journal, 1, 291-308.

Lee, R.D. and Carter, L.R. (1992). Modeling and forecasting US mortality. Journal of the American Statistical Association, 87(419), 659–671.

Li, N. and Lee, R.D. (2005). Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method. Demography, 42(3), 575–594.

Russolillo, M., Giordano, G., & Haberman, S. (2011). Extending the Lee–Carter model: a three-way decomposition. Scandinavian Actuarial Journal, 2011(2), 96-117.

Villegas, A. M., Haberman, S., Kaishev, V. K., & Millossovich, P. (2017). A comparative study of two-population models for the assessment of basis risk in longevity hedges. ASTIN Bulletin, 47(3), 631-679.

Examples

Run this code

#The example takes more than 5 seconds because it includes
#several fitting and forecasting process and hence all
#the process is included in donttest
# \donttest{
#First, we present the data that we are going to use
SpainRegions
ages <- c(0, 1, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90)

#Charge some libraries needed for executing the function
library(gnm)
library(forecast)
#There are some model that we can fit using this function:
#1. ADDITIVE MULTI-POPULATION MORTALITY MODEL
#In the case, the user wants to fit the additive multi-population mortality model
additive_Spainmales <- fitLCmulti(model = "additive",
                              qxt = SpainRegions$qx_male,
                              periods = c(1991:2020),
                              ages = c(ages),
                              nPop = 18,
                              lxt = SpainRegions$lx_male)

additive_Spainmales

#If the user does not provide the model inside the function fitLCmult()
#the multi-population mortality model applied will be additive one.

#Once, we have fit the data, it is possible to see the ax, bx, kt, and Ii
#provided parameters for the fitting.
plot(additive_Spainmales)

#2. MULTIPLICATIVE MULTI-POPULATION MORTALITY MODEL
#In the case, the user wants to fit the multiplicative multi-population mortality model
multiplicative_Spainmales <- fitLCmulti(model = "multiplicative",
                              qxt = SpainRegions$qx_male,
                              periods = c(1991:2020),
                              ages = c(ages),
                              nPop = 18,
                              lxt = SpainRegions$lx_male)

multiplicative_Spainmales

#Once, we have fit the data, it is possible to see the ax, bx, kt, and It
#provided parameters for the fitting.
plot(multiplicative_Spainmales)

#3. COMMON-FACTOR MULTI-POPULATION MORTALITY MODEL
#In the case, the user wants to fit the common-factor multi-population mortality model
cfm_Spainmales <- fitLCmulti(model = "CFM",
                             qxt = SpainRegions$qx_male,
                             periods = c(1991:2020),
                             ages = c(ages),
                             nPop = 18,
                             lxt = SpainRegions$lx_male)

cfm_Spainmales

#Once, we have fit the data, it is possible to see the ax, bx, kt, and It
#provided parameters for the fitting.
plot(cfm_Spainmales)

#4. JOINT-K MULTI-POPULATION MORTALITY MODEL
#In the case, the user wants to fit the augmented-common-factor multi-population mortality model
jointk_Spainmales <- fitLCmulti(model = "joint-K",
                                qxt = SpainRegions$qx_male,
                                periods = c(1991:2020),
                                ages = c(ages),
                                nPop = 18,
                                lxt = SpainRegions$lx_male)

jointk_Spainmales

#Once, we have fit the data, it is possible to see the ax, bx, kt, and It
#provided parameters for the fitting.
plot(jointk_Spainmales)

#5. AUGMENTED-COMMON-FACTOR MULTI-POPULATION MORTALITY MODEL
#In the case, the user wants to fit the augmented-common-factor multi-population mortality model
acfm_Spainmales <- fitLCmulti(model = "ACFM",
                              qxt = SpainRegions$qx_male,
                              periods = c(1991:2020),
                              ages = c(ages),
                              nPop = 18,
                              lxt = SpainRegions$lx_male)

acfm_Spainmales

#Once, we have fit the data, it is possible to see the ax, bx, kt, and It
#provided parameters for the fitting.
plot(acfm_Spainmales)

#6. LEE-CARTER FOR SINGLE-POPULATION
#As we mentioned in the details of the function, if we only provide the data
#from one-population the function fitLCmulti()
#will fit the Lee-Carter model for single populations.
LC_Spainmales <- fitLCmulti(qxt = SpainNat$qx_male,
                              periods = c(1991:2020),
                              ages = ages,
                              model = "additive",
                              nPop = 1)

LC_Spainmales

#Once, we have fit the data, it is possible to see the ax, bx, and kt
#parameters provided for the single version of the LC.
plot(LC_Spainmales)
# }