ilc (version 1.0)

lca.rh: A class of generalised Lee-Carter models

Description

A purpose-built regression routine to fit any of the six variants of the class of Lee-Carter model structures using an iterative Newton-Raphson fitting method.

Usage

lca.rh(dat, year = dat$year, age = dat$age, series = 1, max.age = 100, 
	   dec.conv = 6, clip = 3, error = c("poisson", "gaussian"), 
	   model = c("m", "h0", "h1", "h2", "ac", "lc"),  	   restype = c("logrates", "rates", "deaths", "deviance"), scale = F,  	   interpolate = F, verbose = T, spar = NULL)

Arguments

dat

source data object of demogdata class

year

vector of years to be included in the regression (all available years by default)

age

vector of ages to be included in the regression (all available ages by default)

series

numerical index corresponding to the target series to be used from the source data

max.age

highest age to be used in the regression

dec.conv

number of decimal places used to achieve convergence. The lower the value the faster the convergence of the fitting algorithm.

clip

number of marginal birth cohorts to exclude from the regression (i.e., give 0 weights). It is only applicable to the first 5 models (see below)

error

type of error structure of the model choice (Poisson distribution of the errors by default)

model

a character (see usage) or a numeric value (1-6) to specify the model choice

restype

types of residuals, which also controls the type of the fitted value. Thus, in the cases of logrates and rates the function returns as fitted values the log and untransformed mortality rates, respectively. Likewise, the choices of deaths and deviance correspond to the fitted number of deaths

scale

logical, if TRUE, re-scale the interaction parameters so that the $k_t$ has drift parameter equal to 1 (see also lca)

interpolate

logical, if TRUE, replace before regression all zero or missing values in the mortality rates of dat argument by interpolation across calendar years (see also smooth.demogdata)

verbose

logical, if TRUE, the program prints out the updated deviance values along with the starting and final parameter estimates

spar

numerical smoothing spline parameter in the interval (0,1] (with a recommended value of 0.6). If it is not NULL, the interaction effects (i.e. $\beta_x^{(0,1)}$) are smoothed out after the initial regression. Consequently, the period and/or cohort effects are adjusted (smoothed out) accordingly.

Value

A Lee-Carter type fitted object with the following components:

label

data label

age

vector of fitted ages

year

vector of fitted fitted years

matrix of observed (source) mortality rates used for fitting. It is named the same way as the chosen series

ax

parameter estimates of (mean) age-specific mortality rates across the entire fitting period

bx

parameter estimates of age-specific interaction effect between age and period

kt

parameter estimates of year-specific period trend of mortality rates

df

degree of freedom of the fitted GLM model

residuals

residuals of the fitted model in the form of a functional time series object

fitted

fitted values of the fitted model in the form of a functional time series object

varprop

percent of variance

y

source mortality data in the form of a functional time series object

mdev

mean deviance of total and base lack of fit (see also lca)

model

string expression of the fitted model

adjust

type of error structure (e.g. "poisson" or "gaussian")

call

copy of the R call to the model

conv.iter

number of iterations used to reach convergence

bx0

parameter estimates of age-specific interaction effect between age and cohort (only applies to the age-period-cohort model)

itx

parameter estimates of year-specific cohort trend of mortality rates (only applies to the age-period-cohort model)

Details

Implements the modelling approach proposed in Renshaw and Haberman (2006), which extends the basic Lee-Carter model within the GLM framework. The function makes use of tailored iterative Newton-Raphson fitting algorithms to estimate the graduation parameters of the six variants within this class of extended Lee-Carter models.

References

Renshaw, A. E. and Haberman, S. (2003a), ``Lee-Carter mortality forecasting: a parallel generalised linear modelling approach for England and Wales mortality projections", Journal of the Royal Statistical Society, Series C, 52(1), 119-137.

Renshaw, A. E. and Haberman, S. (2003b), ``Lee-Carter mortality forecasting with age specific enhancement", Insurance: Mathematics and Economics, 33, 255-272.

Renshaw, A. E. and Haberman, S. (2006), ``A cohort-based extension to the Lee-Carter model for mortality reduction factors", Insurance: Mathematics and Economics, 38, 556-570.

Renshaw, A. E. and Haberman, S. (2008), ``On simulation-based approaches to risk measurement in mortality with specific reference to Poisson Lee-Carter modelling", Insurance: Mathematics and Economics, 42(2), 797-816.

Renshaw, A. E. and Haberman, S. (2009), ``On age-period-cohort parametric mortality rate projections", Insurance: Mathematics and Economics, 45(2), 255-270.

See Also

dd.rfp, elca.rh, lca

Examples

# standard LC model with Gaussian errors (corresponding to SVD graduation):
#   correct 0 or missing mortality rates before graduation
mod6g <- lca.rh(dd.cmi.pens, mod='lc', error='gauss', max=110, interpolate=TRUE)
# AP LC model with Poisson errors
mod6p <- lca.rh(dd.cmi.pens, mod='lc', error='pois', interpolate=TRUE)
# Model Summary, Coefficients and Plotting:
mod6p; coef(mod6p); plot(mod6p)

# Comparison with standard fitting method
# Standard LC model (with Gaussian errors) - SVD fit (demography package)
modlc <- lca(dd.cmi.pens, interp=TRUE, adjust='none')
# Gaussian (SVD) - Gaussian (iterative)
round(modlc$ax-mod6g$ax, 4)
round(modlc$bx-mod6g$bx, 4)
round(modlc$kt-mod6g$kt, 4)
# -------------------------------------------------- #

# APC LC model fitted to restricted age range with 'deviance' residuals 
#  the remaining 0/NA values reestimated:
# WARNING: for proper fit recommend dec=6, but it can lead to slow convergence!
mod1 <- lca.rh(dd.cmi.pens, age=60:100, mod='m', interpolate=TRUE, res='dev', dec=1)