Second-Order Random Walk Prior with Seasonal Effect

Use a second-oder random walk with seasonal effects as a model for a main effect, or use multiple second-order random walks, each with their own seasonal effects, as a model for an interaction. Typically used with temrs that involve time.

Usage

RW2_Seas(
  n_seas,
  s = 1,
  sd = 1,
  sd_slope = 1,
  s_seas = 0,
  sd_seas = 1,
  along = NULL,
  con = c("none", "by")
)

Arguments

n_seas: Number of seasons
s: Scale for prior for innovations in random walk. Default is 1.
sd: Standard deviation of initial value. Default is 1. Can be 0.
sd_slope: Standard deviation for initial slope of random walk. Default is 1.
s_seas: Scale for innovations in seasonal effects. Default is 0.
sd_seas: Standard deviation for initial values of seasonal effects. Default is 1.
along: Name of the variable to be used as the 'along' variable. Only used with interactions.
con: Constraints on parameters. Current choices are "none" and "by". Default is "none". See below for details.

Value

Object of class "bage_prior_rw2randomseasvary", "bage_prior_rw2randomseasfix", "bage_prior_rw2zeroseasvary", or "bage_prior_rw2zeroseasfix".

Details

If RW2_Seas() is used with an interaction, a separate series is constructed within each combination of the 'by' variables.

Argument s controls the size of innovations in the random walk. Smaller values for s tend to produce smoother series.

Argument n_seas controls the number of seasons. When using quarterly data, for instance, n_seas should be 4.

By default, the magnitude of seasonal effects is fixed. However, setting s_seas to a value greater than zero produces seasonal effects that evolve over time.

Mathematical details

When RW2_Seas() is used with a main effect,

$$\beta_j = \alpha_j + \lambda_j, \quad j = 1, \cdots, J$$ $$\alpha_1 \sim \text{N}(0, \mathtt{sd}^2)$$ $$\alpha_2 \sim \text{N}(0, \mathtt{sd\_slope}^2)$$ $$\alpha_j \sim \text{N}(2 \alpha_{j-1} - \alpha_{j-2}, \tau^2), \quad j = 3, \cdots, J$$ $$\lambda_j \sim \text{N}(0, \mathtt{sd\_seas}^2), \quad j = 1, \cdots, \mathtt{n\_seas} - 1$$ $$\lambda_j = -\sum_{s=1}^{\mathtt{n\_seas} - 1} \lambda_{j - s}, \quad j = \mathtt{n\_seas}, 2 \mathtt{n\_seas}, \cdots$$ $$\lambda_j \sim \text{N}(\lambda_{j-\mathtt{n\_seas}}, \omega^2), \quad \text{otherwise},$$

and when it is used with an interaction,

$$\beta_{u,v} = \alpha_{u,v} + \lambda_{u,v}, \quad v = 1, \cdots, V$$ $$\alpha_{u,1} \sim \text{N}(0, \mathtt{sd}^2)$$ $$\alpha_{u,2} \sim \text{N}(0, \mathtt{sd\_slope}^2)$$ $$\alpha_{u,v} \sim \text{N}(2 \alpha_{u,v-1} - \alpha_{u,v-2}, \tau^2), \quad v = 3, \cdots, V$$ $$\lambda_{u,v} \sim \text{N}(0, \mathtt{sd\_seas}^2), \quad v = 1, \cdots, \mathtt{n\_seas} - 1$$ $$\lambda_{u,v} = -\sum_{s=1}^{\mathtt{n\_seas} - 1} \lambda_{u,v - s}, \quad v = \mathtt{n\_seas}, 2 \mathtt{n\_seas}, \cdots$$ $$\lambda_{u,v} \sim \text{N}(\lambda_{u,v-\mathtt{n\_seas}}, \omega^2), \quad \text{otherwise},$$

where

$\pmb{\beta}$ is the main effect or interaction;
$\alpha_j$ or $\alpha_{u,v}$ is an element of the random walk;
$\lambda_j$ or $\lambda_{u,v}$ is an element of the seasonal effect;
$j$ denotes position within the main effect;
$v$ denotes position within the 'along' variable of the interaction; and
$u$ denotes position within the 'by' variable(s) of the interaction.

Parameter $\omega$ has a half-normal prior $$\omega \sim \text{N}^+(0, \mathtt{s\_seas}^2)$$. If s_seas is set to 0, then $\omega$ is 0, and the seasonal effects are fixed over time.

Parameter $\tau$ has a half-normal prior $$\tau \sim \text{N}^+(0, \mathtt{s}^2)$$.

Constraints

With some combinations of terms and priors, the values of the intercept, main effects, and interactions are are only weakly identified. For instance, it may be possible to increase the value of the intercept and reduce the value of the remaining terms in the model with no effect on predicted rates and only a tiny effect on prior probabilities. This weak identifiability is typically harmless. However, in some applications, such as forecasting, or when trying to obtain interpretable values for main effects and interactions, it can be helpful to increase identifiability through the use of constraints.

Current options for constraints are:

"none" No constraints. The default.
"by" Only used in interaction terms that include 'along' and 'by' dimensions. Within each value of the 'along' dimension, terms across each 'by' dimension are constrained to sum to 0.

Examples

RW2_Seas(n_seas = 4)               ## seasonal effects fixed
#>   RW2_Seas(n_seas=4) 
#>     n_seas: 4
#>          s: 1
#>         sd: 1
#>   sd_slope: 1
#>     s_seas: 0
#>    sd_seas: 1
#>      along: NULL
#>        con: none
RW2_Seas(n_seas = 4, s_seas = 0.5) ## seasonal effects evolve
#>   RW2_Seas(n_seas=4,s_seas=0.5) 
#>     n_seas: 4
#>          s: 1
#>         sd: 1
#>   sd_slope: 1
#>     s_seas: 0.5
#>    sd_seas: 1
#>      along: NULL
#>        con: none
RW2_Seas(n_seas = 4, sd = 0)       ## first term in random walk fixed at 0
#>   RW2_Seas(n_seas=4,sd=0) 
#>     n_seas: 4
#>          s: 1
#>         sd: 0
#>   sd_slope: 1
#>     s_seas: NULL
#>    sd_seas: 1
#>      along: NULL
#>        con: none