demofit (version 0.1.4)

CFMS: Common factor model

Description

Fits and forecasts mortality rates of two populations using common factor model.

Usage

CFMS(
  x,
  M1,
  M2,
  curve = c("gompertz", "makeham", "oppermann", "thiele", "wittsteinbumsted", "perks",
    "weibull", "vandermaen", "beard", "heligmanpollard", "rogersplanck", "siler",
    "martinelle", "thatcher", "gompertz2", "makeham2", "oppermann2", "thiele2",
    "wittsteinbumsted2", "perks2", "weibull2", "vandermaen2", "beard2",
    "heligmanpollard2", "rogersplanck2", "siler2", "martinelle2", "thatcher2"),
  h = 10,
  jumpoff = 1
)

Value

An object of class CFMS with associated S3 methods coef, forecast (which = 1 for smoothed (default); which = 2 for raw), plot (which = 1 gives parameter estimates (default); which = 2 gives residuals and forecasts), and residuals.

Arguments

x

vector of ages.

M1

matrix of mortality rates of population 1 (rows as years and columns as ages).

M2

matrix of mortality rates of population 2 (rows as years and columns as ages).

curve

name of mortality curve for smoothing forecasted mortality rates (including gompertz, makeham, oppermann, thiele, wittsteinbumsted, perks, weibull, vandermaen, beard, heligmanpollard, rogersplanck, siler, martinelle, thatcher, gompertz2, makeham2, oppermann2, thiele2, wittsteinbumsted2, perks2, weibull2, vandermaen2, beard2, heligmanpollard2, rogersplanck2, siler2, martinelle2, thatcher2, where first 14 curves' parameters are unconstrained and last 14 curves' parameters are generally restricted to be positive).

h

forecast horizon (default = 10).

jumpoff

if 1, forecasts are based on estimated parameters only; if 2, forecasts are anchored to observed mortality rates in final year (default = 1).

Details

The common factor model (CFM) is specified as

\(ln(m_{x,t,i}) = \alpha_{x,i} + B_x K_t + \beta_{x,i} \kappa_{t,i} + \epsilon_{x,t,i}\).

The model is estimated by Newton updating scheme and is forecasted by ARIMA applied to \(K_t\) and \(\kappa_{t,i}\). Constraints include sum of \(B_x\) is one, sum of \(K_t\) is zero, sum of \(\beta_{x,i}\) is one, and sum of \(\kappa_{t,i}\) is zero. It can be applied to whole age range.

References

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

Examples

x <- 60:89
a1 <- c(-5.18,-5.12,-4.98,-4.92,-4.82,-4.73,-4.66,-4.53,-4.45,-4.35,
-4.26,-4.17,-4.05,-3.95,-3.84,-3.73,-3.65,-3.52,-3.40,-3.29,
-3.14,-3.02,-2.88,-2.76,-2.64,-2.49,-2.37,-2.25,-2.12,-2.00)
a2 <- c(-4.78,-4.68,-4.57,-4.49,-4.39,-4.29,-4.19,-4.10,-4.00,-3.89,
-3.80,-3.69,-3.60,-3.49,-3.39,-3.29,-3.17,-3.07,-2.96,-2.85,
-2.71,-2.62,-2.49,-2.37,-2.26,-2.14,-2.04,-1.91,-1.82,-1.72)
B <- c(0.0381,0.0340,0.0420,0.0389,0.0423,0.0414,0.0406,0.0393,0.0415,0.0400,
0.0411,0.0362,0.0387,0.0381,0.0384,0.0385,0.0356,0.0314,0.0317,0.0337,
0.0316,0.0298,0.0284,0.0270,0.0248,0.0262,0.0205,0.0215,0.0142,0.0145)
K <- c(9.66,9.89,10.66,9.83,9.52,7.39,7.64,6.36,2.32,4.18,
2.91,-0.61,0.28,-0.38,-1.79,-3.34,-1.74,-3.50,-4.28,-4.77,
-4.98,-7.13,-5.09,-6.41,-5.56,-5.65,-6.12,-5.64,-7.35,-6.28)
b1 <- c(0.0012,-0.0033,0.0523,0.0161,0.0529,0.0220,0.0312,0.0437,0.0709,0.0444,
0.0398,0.0361,0.0403,0.0396,0.0506,0.0315,0.0428,0.0261,0.0384,0.0388,
0.0300,0.0269,0.0275,0.0256,0.0239,0.0421,0.0314,0.0284,0.0174,0.0314)
k1 <- c(-1.24,-1.38,-3.48,-2.51,-1.32,-1.90,-3.42,-0.94,0.24,-0.48,
-0.26,2.70,1.39,-0.46,1.74,2.53,0.90,1.43,0.76,2.48,
0.74,2.32,0.42,1.69,-0.64,1.30,0.19,-0.69,-1.11,-1.01)
b2 <- c(-0.0014,0.0272,0.0083,0.0273,0.0209,0.0253,0.0144,0.0333,0.0460,0.0439,
0.0439,0.0674,0.0331,0.0443,0.0312,0.0240,0.0570,0.0312,0.0403,0.0376,
0.0500,0.0289,0.0466,0.0418,0.0349,0.0149,0.0366,0.0178,0.0361,0.0372)
k2 <- c(2.35,0.62,-0.38,0.12,0.00,0.80,-1.39,0.38,2.47,0.40,
0.76,3.06,1.42,-0.73,0.79,1.94,0.12,0.60,-0.43,0.29,
0.17,0.98,-1.01,-0.13,-2.46,-1.24,-1.65,-2.48,-2.32,-3.06)
set.seed(123)
M1 <- exp(outer(k1,b1)+outer(K,B)+matrix(a1,nrow=30,ncol=30,byrow=TRUE)+rnorm(900,0,0.07))
M2 <- exp(outer(k2,b2)+outer(K,B)+matrix(a2,nrow=30,ncol=30,byrow=TRUE)+rnorm(900,0,0.07))
fit <- CFMS(x=x,M1=M1,M2=M2,curve="makeham",h=30,jumpoff=2)
coef(fit)
forecast::forecast(fit)
plot(fit)
residuals(fit)