StMoMo (version 0.4.1)

forecast.fitStMoMo: Forecast mortality rates using a Stochastic Mortality Model

Description

Forecast mortality rates using a Stochastic Mortality Model fit. The period indexes \(\kappa_t^{(i)}, i = 1,..N,\) are forecasted using ether a Multivariate Random Walk with Drift (MRWD) or \(N\) independent ARIMA\((p, d, q)\) models. The cohort index \(\gamma_{t-x}\) is forecasted using an ARIMA\((p, d, q)\). By default an ARIMA\((1, 1, 0)\) with a constant is used.

Usage

# S3 method for fitStMoMo
forecast(object, h = 50, level = c(80, 95),
  oxt = NULL, gc.order = c(1, 1, 0), gc.include.constant = TRUE,
  jumpchoice = c("fit", "actual"), kt.method = c("mrwd", "iarima"),
  kt.order = NULL, kt.include.constant = TRUE, kt.lookback = NULL,
  gc.lookback = NULL, ...)

Arguments

object

an object of class "fitStMoMo" with the fitted parameters of a stochastic mortality model.

h

number of years ahead to forecast.

level

confidence level for prediction intervals of the period and cohort indices.

oxt

optional matrix/vector or scalar of known offset to be added in the forecasting. This can be used to specify any a priori known component to be added to the forecasted predictor.

gc.order

a specification of the ARIMA model for the cohort effect: the three components \((p, d, q)\) are the AR order, the degree of differencing, and the MA. The default is an ARIMA\((1, 1, 0)\).

gc.include.constant

a logical value indicating if the ARIMA model should include a constant value. The default is TRUE.

jumpchoice

option to select the jump-off rates, i.e. the rates from the final year of observation, to use in projections of mortality rates. "fit"(default) uses the fitted rates and "actual" uses the actual rates from the final year.

kt.method

optional forecasting method for the period index. The alternatives are "mrwd"(default) and "iarima". See details.

kt.order

an optional matrix with one row per period index specifying the ARIMA models: for the ith row (ith period index) the three components \((p, d, q)\) are the AR order, the degree of differencing, and the MA order. If absent the arima models are fitted using auto.arima. This argument is only used when kt.method is "iarima".

kt.include.constant

an optional vector of logical values indicating if the ARIMA model for the ith period index should include a constant value. The default is TRUE. This argument is only used when kt.method is "iarima".

kt.lookback

optional argument to specify the look-back window to use in the estimation of the time series model for the period indexes. By default all the estimated values are used. If kt.lookback is provided then the last kt.lookback years of \(\kappa_t^{(i)}, i = 1,..N,\) are used.

gc.lookback

optional argument to specify the look-back window to use in the estimation of the ARIMA model for the cohort effect. By default all the estimated values are used in estimating the ARIMA model. If gc.lookback is provided then the last gc.lookback years of \(\gamma_{t-x}\) are used.

...

other arguments for iarima.

Value

A list of class "forStMoMo" with components:

rates

a matrix with the point forecast of the rates.

ages

vector of ages corresponding to the rows of rates.

years

vector of years for which a forecast has been produced. This corresponds to the columns of rates.

kt.f

forecasts of period indexes of the model. This is a list with the model fitted to \(\kappa_t\); the mean(central) forecast, the lower and upper limits of the prediction intervals; the confidence level associated with the prediction intervals; and the years for which a forecast was produced. If the model does not have any age-period terms (i.e. \(N=0\)) this is set to NULL.

gc.f

forecasts of cohort index of the model. This is a list with the model fitted to \(\gamma_c\); the mean(point) forecast, the lower and upper limits of the prediction intervals; the confidence level associated with the prediction intervals; and the cohorts for which a forecast was produced. If the mortality model does not have a cohort effect this is set to NULL.

oxt.f

the offset used in the forecast.

fitted

a matrix with the fitted in-sample rates of the model for the years for which the mortality model was fitted.

model

the model fit from which the forecast was produced.

jumpchoice

Jump-off method used in the forecast.

kt.method

method used in the forecast of the period index.

Details

If kt.method is "mrwd", fitting and forecasting of the time series model for the period indexes is done with a Multivariate Random Walk with Drift using the function mrwd.

If kt.method is "iarima", fitting and forecasting of the time series model for the period indexes is done with \(N\) independent arima models using the function iarima. See this latter function for details on input arguments kt.order and kt.include.constant.

Fitting and forecasting of the ARIMA model for the cohort index is done with function Arima from package forecast. See the latter function for further details on input arguments gc.order and gc.include.constant.

Note that in some cases forecast of the cohort effects may be needed for a horizon longer than h. This is the case when in the fitted model the most recent cohorts have been zero weighted. The forecasted cohorts can be seen in gc.f$cohorts.

Examples

# NOT RUN {
#Lee-Carter (random walk with drift)
LCfit <- fit(lc(), data = EWMaleData, ages.fit = 55:89)
LCfor <- forecast(LCfit)
plot(LCfor)

#Lee-Carter (forecast with ARIMA(1,1,2) with drift )
LCfor.iarima1 <- forecast(LCfit, kt.method = "iarima", kt.order = c(1, 1 , 2))
plot(LCfor.iarima1)

#Lee-Carter (forecast with auto.arima)
LCfor.iarima2 <- forecast(LCfit, kt.method = "iarima")
plot(LCfor.iarima2)

#CBD (Multivariate random walk with drift)
CBDfit <- fit(cbd(), data = central2initial(EWMaleData), ages.fit = 55:89)
CBDfor <- forecast(CBDfit)
plot(CBDfor, parametricbx = FALSE)

#CBD (Independent Arima models)
kt.order <- matrix(c(1, 1, 2,  #ARIMA(1, 1, 2) for k[1]
                     0, 1, 1), #ARIMA(0, 1, 1) for k[2]
                   nrow = 2, ncol = 3, byrow = TRUE)
CBDfor.iarima <- forecast(CBDfit, kt.method = "iarima", kt.order = kt.order)
plot(CBDfor.iarima, parametricbx = FALSE)

#APC: Compare forecast with different models for the cohort index
wxt <- genWeightMat(55:89,  EWMaleData$years, clip = 3)
APCfit <- fit(apc(), data = EWMaleData,  ages.fit = 55:89, 
              wxt = wxt)
APCfor1 <- forecast(APCfit)
plot(APCfor1, parametricbx = FALSE, nCol = 3)
APCfor2 <- forecast(APCfit, gc.order = c(0, 2, 2))
plot(APCfor2, only.gc = TRUE)
plot(c(APCfit$years, APCfor1$years), 
     cbind(APCfor1$fitted, APCfor1$rates)["65", ], 
     type = "l", xlab = "year", ylab = "Mortality rate at age 65", 
     main = "Forecasts with different models for gc")
lines(APCfor2$years, APCfor2$rates["65", ], col = "blue")
points(APCfit$years, (APCfit$Dxt / APCfit$Ext)["65", ], pch = 19) 

#Compare Lee-Carter forecast using: 
#  1. Fitted jump-off rates and all history for kt
#  2. Actual jump-off rates and all history for kt
#  3. Fitted jump-off rates and only history for 
#     the past 30 years of kt (i.e 1982-2011)

LCfor1 <- forecast(LCfit)
LCfor2 <- forecast(LCfit, jumpchoice = "actual")
LCfor3 <- forecast(LCfit, kt.lookback = 30) 

plot(LCfit$years, (LCfit$Dxt / LCfit$Ext)["60", ],
     xlim = range(LCfit$years, LCfor1$years),
     ylim = range((LCfit$Dxt / LCfit$Ext)["60", ], LCfor1$rates["60", ],
                  LCfor2$rates["60", ], LCfor3$rates["60", ]),
     type = "p", xlab = "year", ylab = "rate",
     main = "Lee-Carter: Forecast of mortality rates at age 60")
lines(LCfit$years, fitted(LCfit, type = "rates")["60", ])
lines(LCfor1$years, LCfor1$rates["60", ], lty = 2)
lines(LCfor2$years, LCfor2$rates["60", ], lty = 3, col = "blue")
lines(LCfor3$years, LCfor3$rates["60", ], lty = 4, col = "red")
legend("topright",legend = c("Fitted jump-off", "Actual jump-off", 
       "Fitted jump-off, 30 year look-back"), 
       lty = 1:3, col = c("black", "blue", "red"))
# }