Dynamic SVD-Based Priors for Age Profiles or Age-Sex Profiles

Use components from a Singular Value Decomposition (SVD) to model an interaction involving age and time, or age, sex/gender and time, where the coefficients evolve over time.

Usage

SVD_AR(ssvd, n_comp = NULL, indep = TRUE, n_coef = 2, s = 1)

SVD_AR1(ssvd, n_comp = NULL, indep = TRUE, min = 0.8, max = 0.98, s = 1)

SVD_RW(ssvd, n_comp = NULL, indep = TRUE, s = 1)

SVD_RW2(ssvd, n_comp = NULL, indep = TRUE, s = 1)

Arguments

ssvd: Object of class "bage_ssvd" holding a scaled SVD. See below for scaled SVDs of databases currently available in bage.
n_comp: Number of components from scaled SVD to use in modelling. The default is half the number of components of ssvd.
indep: Whether to use separate or combined SVDs in terms involving sex or gender. Default is TRUE. See below for details.
n_coef: Number of AR coefficients in SVD_RW().
s: Scale for standard deviations terms.
min, max: Minimum and maximum values for autocorrelation coefficient in SVD_AR(). Defaults are 0.8 and 0.98.

Value

An object of class "bage_prior_svd_ar", "bage_prior_svd_rw", or "bage_prior_svd_rw2".

Details

SVD_AR(), SVD_AR1(), SVD_RW(), and SVD_RW2() priors assume that, in any given period, the age profiles or age-sex profiles for the quantity being modelled looks like they were drawn at random from an external demographic database. For instance, the SVD_AR() prior obtained via

SVD_AR(HMD)

assumes that profiles look like they were obtained from the Human Mortality Database.

Mathematical details

When the interaction being modelled only involves age and time, or age, sex/gender, and time

$$\pmb{\beta}_t = \pmb{F} \pmb{\alpha}_t + \pmb{g},$$

and when it involves other variables besides age, sex/gender, and time,

$$\pmb{\beta}_{u,t} = \pmb{F} \pmb{\alpha}_{u,t} + \pmb{g},$$

where

$\pmb{\beta}$ is an interaction involving age, time, possibly sex/gender, and possibly other variables;
$\pmb{\beta}_t$ is a subvector of $\pmb{\beta}$ holding values for period $t$;
$\pmb{\beta}_{u,t}$ is a subvector of $\pmb{\beta}_t$ holding values for the $u$th combination of the non-age, non-time, non-sex/gender variables for period $t$;
$J$ is the number of elements of $\pmb{\beta}_t$;
$V$ is the number of elements of $\pmb{\beta}_{u,t}$;
$n$ is n_coef;
$\pmb{F}$ is a known matrix with dimension $J \times n$ or $V \times n$;
$\pmb{g}$ is a known vector with $J$ or $V$ elements.

$\pmb{F}$ and $\pmb{g}$ are constructed from a large database of age-specific demographic estimates by performing an SVD and standardizing.

With SVD_AR(), the prior for the $k$th element of $\pmb{\alpha}_t$ or $\pmb{\alpha}_{u,t}$ is

$$\alpha_{k,t} = \phi_1 \alpha_{k,t-1} + \cdots + \phi_n \beta_{k,t-n} + \epsilon_{k,t}$$

$$\alpha_{k,u,t} = \phi_1 \alpha_{k,u,t-1} + \cdots + \phi_n \beta_{k,u,t-n} + \epsilon_{k,u,t};$$

with SVD_AR1(), it is

$$\alpha_{k,t} = \phi \alpha_{k,t-1} + \epsilon_{k,t}$$

$$\alpha_{k,u,t} = \phi \alpha_{k,u,t-1} + \epsilon_{k,u,t};$$

with SVD_RW(), it is

$$\alpha_{k,t} = \alpha_{k,t-1} + \epsilon_{k,t}$$

$$\alpha_{k,u,t} = \alpha_{k,u,t-1} + \epsilon_{k,u,t};$$

and with SVD_RW2(), it is

$$\alpha_{k,t} = 2 \alpha_{k,t-1} - \alpha_{k,t-2} + \epsilon_{k,t}$$

$$\alpha_{k,u,t} = 2 \alpha_{k,u,t-1} - \alpha_{k,u,t-2} + \epsilon_{k,u,t}.$$

For more on the $\phi$ and $\epsilon$, see AR(), AR1(), RW(), and RW2().

Scaled SVDs of demographic databases in bage

HMD Mortality rates from the Human Mortality Database.
LFP Labor forcce participation rates from the OECD.

References

For details of the construction of scaled SVDS see the vignette here.

Examples

SVD_AR1(HMD)
#>   SVD_AR1(HMD) 
#>     ssvd: HMD
#>   n_comp: 3
#>      min: 0.8
#>      max: 0.98
#>        s: 1
#>    along: NULL
SVD_RW(HMD, n_comp = 3)
#>   SVD_RW(HMD) 
#>     ssvd: HMD
#>   n_comp: 3
#>        s: 1
SVD_RW2(HMD, indep = FALSE)
#>   SVD_RW2(HMD,indep=FALSE) 
#>     ssvd: HMD
#>   n_comp: 3
#>    indep: FALSE
#>        s: 1