Longitudinal causal variance decomposition — causal_decompose

Decomposes the change in treatment effect on inequality from a reference period to each subsequent period into behavioral, compositional, and pre-treatment components, using a unified sequential parameter switching scheme that tracks each switch's contribution to both \(\tau_B\) and \(\tau_W\) simultaneously.

Usage

causal_decompose_longit(
  params,
  order = c("behavioral", "compositional", "pretreatment"),
  ref = NULL
)

Arguments

params: An ineqx_params object with multiple time periods
order: Character vector of length 3, a permutation of c("behavioral", "compositional", "pretreatment") specifying the order in which parameter groups are switched from baseline to time-t values. The mapping to the underlying 5-parameter sequence is: behavioral -> (beta, lambda), compositional -> (pi), pretreatment -> (mu, sigma), with \(\beta\) switched before \(\lambda\) and \(\mu\) switched before \(\sigma\) within their meta-levels. Use order = "shapley" to average over all 6 meta-orderings. Default: c("behavioral", "compositional", "pretreatment").

Value

An object of class "ineqx_causal_longit" containing:

results: List keyed by time period, each containing the 6 components plus 3 combined components and the total
order: The ordering used
ystat: The inequality measure
params: The input ineqx_params object

Details

For each parameter \(p \in \{\beta, \lambda, \pi, \mu, \sigma\}\), the result reports two split components:

Delta_<p>_B: Contribution of switching p (from t0 to t) to the change in \(\tau_B\)
Delta_<p>_W: Contribution of switching p (from t0 to t) to the change in \(\tau_W\)

For the variance (ystat = "Var"), \(\tau_B\) depends only on \((\pi, \mu, \beta)\) and \(\tau_W\) only on \((\pi, \sigma, \lambda)\), so the off-diagonal split parts (Delta_beta_W, Delta_lambda_B, Delta_mu_W, Delta_sigma_B) are exactly zero. For CV\(^2\), both \(\tau_B\) and \(\tau_W\) share the grand-mean denominator, so all five parameters can contribute to both sides; the split components capture this coupling exactly.

By construction, the split components sum to the cross-sectional change: \(\sum_p \text{Delta\_<p>\_B} = \tau_B(t) - \tau_B(t_0)\) and similarly for the W side, so Delta_total equals \((\tau_B(t) + \tau_W(t)) - (\tau_B(t_0) + \tau_W(t_0))\).

Aggregate parameter components (Delta_beta, Delta_lambda, Delta_mu, Delta_sigma) are reported as the sum of their B and W parts; for V they equal the previous (single-side) values exactly. Compositional effects remain split as Delta_pi_B and Delta_pi_W.