project_parameters: Estimate the time series of the time dependent parameters in...

In RobbenJ/MultiMoMo: Multi-population mortality models

Estimate the time series of the time dependent parameters in the calibrated Li-Lee model and make projections

Description

This function estimates the parameters in the time series specifications for the time dependent parameters in the calibrated Li-Lee model, namely \mjeqnK_tASCII representation and \mjeqn\kappa_tASCII representation, jointly for males and females.

Usage

project_parameters(fit_M, fit_F, n_ahead, n_sim, arima_spec, est_method)

Arguments

fit_M	The calibrated Li-Lee model for males.
fit_F	The calibrated Li-Lee model for females.
n_ahead	The number of years to project in the future.
n_sim	The number of simulations/projections.
arima_spec	The ARIMA time series specifications for \mjeqn(K_t^M, \kappa_t^M, K_t^F, \kappa_t^F)ASCII representation.
est_method	The estimation method of the parameters in the time series specifications.

Details

\loadmathjax

The arguments fit_M and fit_F are the outputs of the function fit_li_lee.

The argument arima_spec is a list specifying the type of time series used for each time dependent parameter in the Li-Lee model and is of the form:

list("K.t_M" = , "k.t_M" = , "K.t_F" = , "k.t_F" = ).

The capital K.t refers to the time dependent parameter in the Lee-Carter model for the common trend, i.e. \mjeqnK_tASCII representation. The small k.t refers to the time dependent parameter \mjeqn\kappa_tASCII representation in the Lee-Carter model for the country-specific deviation from the common trend. We jointly model the time series dynamics for men and women by assuming a multivariate Gaussian distribution with mean \mjeqn(0,0,0,0)ASCII represenation and covariance matrix \mjeqnCASCII represenation for the error terms \mjeqn(\epsilon_t^M, \delta_t^M, \epsilon_t^F, \delta_t^F)ASCII representation of the multivariate time series \mjeqn(K_t^M, \kappa_t^M, K_t^F, \kappa_t^F)ASCII representation. Possible choices for the time series dynamics are RWD (e.g. for \mjeqnK_tASCII representation) or ARk.j, where k refers to an AR(k) process, an auto-regressive process with lag k. Further \mjeqnj \in {0,1}ASCII representation and refers to an AR process without or with an intercept respectively.

The argument est_method specifies the estimation method of the parameters in the time series specifications, given in arima_spec. A first option is SUR, referring to seemingly unrelated regression, that uses the function systemfit. This method only works for a limited amount of cases. In particular, all time series must jointly start at the same time t. For example, this means that in the example below, we start the joint time series estimation at year 1975 (since an AR5.0 is used). A second option is PORT. This is a self-written objective function that maximizes the log-likelihood of the multivariate Gaussian distribution for \mjeqn(\epsilon_t^M, \delta_t^M, \epsilon_t^F, \delta_t^F)ASCII represenation using the function nlminb in the stats-package. In the example below, the joint estimation consists of three parts. For t = 1971-1972 only the RWD processes are considered leading to a bi-variate normal log-likelihood. Then for t = 1973-1974 the two RWD processes together with k.t_M are used, leading to a 3-variate normal log-likelihood. From 1975 on, we deal with the four processes (4-variate normal log-likelihood). The sum of these three likelihoods is maximized at once using PORT routines.

Value

A list containing the following objects

• The estimated parameters in the time series for \mjeqnK_t^MASCII representation: $coef_KtM

• The estimated parameters in the time series for \mjeqn\kappa_t^MASCII representation: $coef_ktM

• The estimated parameters in the time series for \mjeqnK_t^FASCII representation: $coef_KtF

• The estimatedparameters in the time series for \mjeqn\kappa_t^FASCII representation: $coef_ktF

• The estimated covariance matrix \mjeqnCASCII representation: $cov_mat

• The \mjeqnK_t^MASCII representation simulations: $K.t_M

• The \mjeqn\kappa_t^MASCII representation simulations: $k.t_M

• The \mjeqnK_t^FASCII representation simulations: $K.t_F

• The \mjeqn\kappa_t^FASCII representation simulations: $k.t_F

Examples

lst   <- MultiMoMo::european_mortality_data
dat_M <- lst$M
dat_F <- lst$F
xv    <- 0:90
yv = yvSPEC <- 1970:2018
Countries   <- names(dat_M$UNI)
CountrySPEC <- "BE"
fit_M <- fit_li_lee(xv, yv, yvSPEC, CountrySPEC, dat_M, "NR", TRUE, FALSE)
fit_F <- fit_li_lee(xv, yv, yvSPEC, CountrySPEC, dat_F, "NR", TRUE, FALSE)

arima_spec <- list(K.t_M = "RWD", k.t_M = "AR3.1", K.t_F = "RWD", k.t_F = "AR5.0")
n_ahead    <- 50
n_sim      <- 10000
est_method <- "PORT"
proj       <- project_parameters(fit_M, fit_F, n_ahead, n_sim, arima_spec, est_method)

RobbenJ/MultiMoMo documentation built on June 28, 2022, 9:29 p.m.