Model structure

This page sketches the methodology behind ineqx. Three equations carry the intuition; for the full derivations see Rosche (2026).

1. Descriptive within/between decomposition

Take the vector $Y_t$ of individual incomes at time $t$, and the vector $G_t$ of group memberships ($G_t = j$ for $j = 1,\dots,J$ mutually exclusive groups). The variance of $Y_t$ splits cleanly into a within- and a between-group part:

$$\begin{equation} V_t \;=\; \underbrace{\sum_j \pi_{jt}\,\sigma_{jt}^2}_{\text{within } W_t} \;+\; \underbrace{\sum_j \pi_{jt}\,\big(\mu_{jt} - \bar\mu_t\big)^2}_{\text{between } B_t} \tag{1} \end{equation}$$

with $\pi_{jt}$ the share of group $j$, $\mu_{jt}$ its mean, $\sigma_{jt}^2$ its variance, and $\bar\mu_t = \sum_j \pi_{jt}\mu_{jt}$ the overall mean. Cross sections collapse to a single $t$. With repeated cross-sections, the change $V_t - V_{t_0}$ further decomposes into contributions from changing means $(\Delta_\mu)$, changing dispersions $(\Delta_\sigma)$, and changing group shares $(\Delta_\pi)$, with each component split into its between- and within-group share. This is what ineqx() returns when no treat is supplied — it is essentially the Western & Bloome (2009) decomposition with a Shapley-averaged path.

2. Causal cross-sectional decomposition

Let $D \in \{0,1\}$ be a binary treatment, $\beta_j$ its causal effect on group $j$’s mean, and $\lambda_j$ its causal effect on group $j$’s log-SD (so the treated SD is $\sigma_j(0)\cdot e^{\lambda_j}$). The treatment effect on inequality, $\tau = V[Y(1)\mid D] - V[Y(0)\mid D]$, splits into a between- and a within-group part, and each of those further splits into a heterogeneity and a covariance term:

$$\begin{equation} \boxed{\;\tau_B \;=\; \underbrace{\mathrm{Var}_\pi(\beta)}_{\text{het}_B} \;+\; \underbrace{2\,\mathrm{Cov}_\pi(\mu(0),\,\beta)}_{\text{cov}_B} \;} \qquad \boxed{\;\tau_W \;\approx\; \underbrace{2\,\bar\sigma^2(0)\,\bar\lambda}_{\text{het}_W} \;+\; \underbrace{2\,\mathrm{Cov}_\pi(\sigma^2(0),\,\lambda)}_{\text{cov}_W}\;} \tag{2} \end{equation}$$

The four sub-components have direct substantive readings:

• $\text{het}_B = \mathrm{Var}_\pi(\beta) \ge 0$. Heterogeneity in the treatment effect on the mean always widens between-group inequality. Equal $\beta$ across groups means $\text{het}_B = 0$.
• $\text{cov}_B = 2\,\mathrm{Cov}_\pi(\mu(0),\beta)$. Sorting of gains. If positive, larger gains accrue to already-advantaged groups (reinforces the hierarchy). If negative, treatment compresses the hierarchy.
• $\text{het}_W \propto \bar\lambda$. The average effect on within-group dispersion. Sign tracks $\bar\lambda$: treatment can stretch or compress within-group spread.
• $\text{cov}_W = 2\,\mathrm{Cov}_\pi(\sigma^2(0),\lambda)$. Sorting of variance effects. If positive, treatment amplifies dispersion most in already-disperse groups.

plot(<causal>, type = "wibe", stats = c("het_b","cov_b","het_w","cov_w")) shows these four series directly.

The package supports both $V$ (variance) and $CV^2 = \sigma^2/\mu^2$ (squared coefficient of variation) as the underlying inequality measure.

3. Longitudinal three-channel decomposition

With repeated cross-sections, the change in the treatment effect on inequality, $\Delta\tau = \tau_t - \tau_{t_0}$, decomposes into three interpretable channels:

$$\begin{equation} \Delta\tau \;=\; \underbrace{(\Delta_\beta + \Delta_\lambda)}_{\text{behavioral}} \;+\; \underbrace{(\Delta_{\pi,B} + \Delta_{\pi,W})}_{\text{compositional}} \;+\; \underbrace{(\Delta_\mu + \Delta_\sigma)}_{\text{pre-treatment}} \tag{3} \end{equation}$$

• Behavioral — changes in how treatment affects each group’s mean and variance ($\Delta_\beta$, $\Delta_\lambda$). The Kitagawa-Blinder-Oaxaca “coefficient effect.”
• Compositional — changes in who gets treated across groups ($\Delta_{\pi,B}, \Delta_{\pi,W}$). The KBO “endowment effect.”
• Pre-treatment — changes in the baseline distribution that determine how much a fixed treatment effect translates into inequality ($\Delta_\mu, \Delta_\sigma$). Has no KBO analogue — this is what the variance-decomposition framing adds.

Sequential decompositions of this kind are path-dependent: the size of $\Delta_\beta$ depends on whether you switch $\beta$ before or after $\pi$. ineqx() defaults to order = "shapley", which averages across all $3! = 6$ orderings to give an order-invariant decomposition. plot(<causal>, type = "shapley") displays the Shapley point estimate and the range across orderings.

4. A note on the variance of logs

ineqx() accepts ystat = "VL" for the descriptive case (it log-transforms $y$ and runs the standard variance decomposition), but the paper recommends against $V_L$ for substantive decomposition work. The reason: because the log transformation is non-linear, the within- and between-group components of $V_L$ can move in the opposite direction from within- and between-group changes on the income scale. A treatment that unambiguously raises within-group dispersion in dollars can register as compressing it in $V_L$. $V$ and $CV^2$ do not have this pathology and decompose into components that remain interpretable on the original income scale. ineqx() issues a runtime warning whenever ystat = "VL" is used — see the FAQ for details.