simulate.fitiMoMo: Simulate future sample paths from an improvement Model
In amvillegas/iMoMo: Mortality Improvement Rate Modelling

Description Usage Arguments Details Value

Simulate mortaility improvment rates and mortality rates using the fit from a mortality improvement rate model. The period indexes are κ_t^{(i)}, i = 1,..N, are forecasted using integrated vector autoregressive model. The cohort index γ_{t-x} is forecasted using an ARIMA(p, d, q). By default an ARIMA(1, 1, 0) with a constant is used.

## S3 method for class 'fitiMoMo'
simulate(
  object,
  nsim = 1000,
  seed = NULL,
  h = 50,
  kt.order = c(1, 0),
  kt.include.constant = TRUE,
  kt.include.trend = FALSE,
  gc.order = c(1, 0, 0),
  gc.include.constant = TRUE,
  jumpRates = NULL,
  kt.lookback = NULL,
  gc.lookback = NULL,
  ...
)

`object`	an object of class `"fitiMoMo"` with the fitted parameters of an improvement rate mortality model.
`nsim`	number of sample paths to simulate.
`seed`	either `NULL` or an integer that will be used in a call to `set.seed` before simulating the time series. The default, `NULL` will not change the random generator state.
`h`	number of years ahead to forecast.
`kt.order`	an optional vector indicating the order of autorregression and of differetiation of the VARI model. The two components (p, d) are the AR order and the degree of differencing.
`kt.include.constant`	a logical value indicating if the VARI model should include a constant value. The default is `TRUE`.
`kt.include.trend`	a logical value indicating if the VARI model should have a linear trend. The default is `FALSE`.
`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, 0, 0).
`gc.include.constant`	a logical value indicating if the ARIMA model should include a constant value. The default is `TRUE`.
`jumpRates`	optional vector of moratlity rates for the last year used as starting rates. for the projection. If it is not provided the rates from the the actual rates from the final year are used.
`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 κ_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 γ_{t-x} are used.
`...`	other arguments.

The modelling of the period indexes kappa_t is done using a integrated vector autoregressive model of differencing order d and autorregressive order p:

Δ^d k_t = C+Dt+∑_{i=1}^p A_i Δ^d k_{t-i} + ε_t

where C and D are N-dimensional vectors for parameters and A_1,...,A_p are N\times N matrices of autoregressive parampeters. If kt.include.constant is TRUE then C is included in the equation. Similarly, if kt.include.trend is TRUE then D is included in the equation.

Fitting and simulating of the VAR model is done using the fucntion simpleVAR2.

Fitting and simulating 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.

A list of class "simiMoMo" with components:

`improvements`	a three dimensional array with the future simulated improvement rates.
`rates`	a three dimensional array with the future simulated mortality rates.
`ages`	vector of ages corresponding to the first dimension of `improvements`.
`years`	vector of years for which a forecast has been produced. This corresponds to the second dimension of `improvements`.

`kt.s`	information on the simulated paths of the period indexes of the model. This is a list with the `model` fitted to κ_t; the simulated paths `sim`; 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.s`	information on the simulated paths of the cohort index of the model. This is a list with the `model` fitted to γ_c; the simulated paths (`sim`); and the `cohorts` for which simulations were produced. If the mortality model does not have a cohort effect this is set to `NULL`.
`fittedImprovements`	a three dimensional array with the in-sample improvements of the model for the years for which the improvement rate model was fitted.