Model structure

1 Descriptive variance decomposition

Take the vector $Y_t$ to be individual incomes at time $t$, and the vector $G_t$ to be the group to which the individuals belong, where $G_t=j$ is a categorical variable with $j=1,…,J$ categories that represent mutually exclusive and exhaustive groups. The variance $V$ in income at time $t$ can then be expressed as the sum of the variance within and between groups:

$$\begin{equation} \begin{split} V_t(Y_t) &= E(V(Y_t |G_t))+V(E(Y_t |G_t )) \\[10pt] &= W_t + B_t \\[10pt] &= \sum_j\pi_{jt}\sigma_{jt}^2+\sum_j\pi_{jt}\big(\mu_{jt}-\sum_j\pi_{jt}\mu_{jt}\big)^2 \end{split} \tag{1} \end{equation}$$

where $\pi_{jt}$ is the proportion of individuals in group $j$ at time $t$, $\mu_{jt}$ is the mean income in group $j$ at time $t$, and $\sigma_{jt}^2$ is the variance around this mean in group $j$ at time $t$.

With repeated cross-sectional or panel data, the change in variance from $t_0$ (baseline) to $t$ (any timepoint post baseline) can then be decomposed into the sum of a within-group effect ($\delta_W$), a between-group effect ($\delta_B$), and a compositional effect ($\delta_C$). That is, $$\begin{equation} V_t-V_{t_0} = \delta_W^t + \delta_B^t + \delta_C^t \\[10pt] \text{where} \\[10pt] \begin{split} \delta_W^t &= \sum_j \pi_{jt_0} \big( \sigma_{jt}^2 - \sigma_{jt_0}^2 \big) \\[10pt] \delta_B^t &= \sum_j \pi_{jt_0} \big( (\mu_{jt} - \sum_j\pi_{jt}\mu_{jt})^2 - (\mu_{jt_0} - \sum_j\pi_{jt_0}\mu_{jt_0})^2 \big) \\[10pt] \delta_C^t &= \sum_j \big( \pi_{jt}-\pi_{jt_0} \big) \big( (\mu_{jt} - \sum_j\pi_{jt}\mu_{jt})^2 + \sigma_{jt}^2 \big) \end{split} \tag{2} \end{equation}$$

The between-group effect ($\delta_B^t$) captures the change in total variance induced by changes in the mean of each group. The within-group effect ($\delta_W^t$) captures the change in total variance induced by changes in the variance around the mean of each group. Finally, the compositional effect ($\delta_C^t$) captures the change in total variance induced by changes in the relative size of each group. The superscript $t$ on the $\delta\text{s}$ indicates that the change over time is considered.

2 Causal variance decomposition

I first entertain the treatment effect framework at a single point in time. Let $D\in\{0,1\}$ be a binary treatment, $Y(D)$ be the potential outcome of the same outcome vector as before, and $\tau=Y(1)-Y(0)$ be the intra-individual causal effect of treatment on the outcome.

The group-specific treatment effect on the mean of the treated ($ATT$) then equals the expected value of the differences in the potential outcomes in each group:

$$\begin{equation} ATT_j=E[Y(1)-Y(0)|G=j,D=1] \tag{3} \end{equation}$$

Further, the group-specific treatment effect on the variance of the treated ($VTT$) equals the difference in the variance in the potential outcomes in each group:

$$\begin{equation} VTT_j=V[Y(1)|G=j,D=1]-V[Y(0)|G=j,D=1] \tag{4} \end{equation}$$

2a Decomposing the effect of treatment

The focus here is on the $ATT$ rather than the average treatment effect ($ATE$) as we are interested in the aggregate consequences of treatment, which depends on the actual distribution of treatment across groups, $E[D|G]$, and therefore on the $ATT$. Note that we assume that the treatment effect on the variance is fully described by its effect on the group-specific means and variances. Given these definitions, the effect of treatment on the variance can be decomposed into a within- and between group component, where the variance in the $ATT$ equals the between-group component and the expected value of the $VTT$ equals the within-group component:

$$\begin{equation} V[Y(1)|D]-V[Y(0)|D] = \delta_B^D + \delta_W^D \\[10pt] \text{where} \\[10pt] \begin{split} \delta_B^D &= \sum_j \pi_j \big( (\mu_j+\beta_j- \sum_j\pi_j(\mu_j+\beta_j))^2 - (\mu_j - \sum_j\pi_j\mu_j)^2 \big) \\[10pt] \delta_W^D &= \sum_j \pi_j \big( (\sigma_j+\lambda_j )^2 - \sum_j\pi_j\sigma_j^2 \big) \end{split} \tag{5} \end{equation}$$

Note that the interpretation of some quantities changes as compared to equation (2). In equation (5), $\pi_j$ is the proportion of individuals in each group receiving treatment (i.e., $E[D|G]$), $\mu_j$ is the pre-treatment mean in group $j$, $\sigma_j$ is the pre-treatment standard deviation in group $j$, $\beta_j$ is the causal effect of treatment on $\mu_j$, $\lambda_j$ is the causal effect of treatment on $\sigma_j$. The superscript $D$ on the $\delta\text{s}$ indicates that the change caused by treatment (at a single point in time) is considered. The between-group effect $\delta_B^D$ captures the change in total variance induced by the effect of treatment on the mean of each group. The within-group effect $\delta_W^D$ captures the change in total variance induced by the effect of treatment on the variance of each group.

2b Decomposing the change in the effect of treatment

The treatment effect on the variance depends both on the treatment effects on the group-specific means and variances and on the distribution of treatment across groups and on the level of pre-treatment inequality. Therefore, with repeated cross-sectional or panel data, the change in total variance from $t_0$ (baseline) to $t$ (any timepoint post baseline) due to change in the effect of treatment can be decomposed into the sum of a between-group effect ($\delta_B^{D,t}$), within-group effect ($\delta_W^{D,t}$) a compositional effect ($\delta_C^{D,t}$), and a pre-treatment effect ($\delta_P^{D,t}$):

$$\begin{equation} (V[Y_t(1)|D_t]-V[Y_t(0)|D_t]) - (V[Y_{t_0}(1)|D_{t_0}]-V[Y_{t_0}(0)|D_{t_0}]) = \delta_B^{D,t} + \delta_W^{D,t} + \delta_C^{D,t} + \delta_P^{D,t} \\[10pt] \text{where} \end{equation}$$ $$\begin{equation} \begin{split} \delta_B^{D,t} &= B(\pi_{t_0},\mu_{t_0}+\beta_t) - B(\pi_{t_0},\mu_{t_0}+\beta_0) \\ &= \sum_j\pi_{j,t_0} \left( \Big(\mu_{j,t_0} + \beta_{j,t} - \sum_j\pi_{j,t_0}(\mu_{j,t_0}+\beta_{j,t}) \Big)^2 - \Big(\mu_{j,t_0} + \beta_{j,t_0} - \sum_j\pi_{j,t_0}(\mu_{j,t_0}+\beta_{j,t_0} ) \Big)^2 \right) \\[10pt] \delta_W^{D,t} &= W(\pi_{t_0},\sigma_{t_0}+\lambda_t) - W(\pi_{t_0},\sigma_{t_0}+\lambda_{t_0}) \\ &= \sum_j\pi_{j,t_0} \left( \Big(\sigma_{j,t_0}+\lambda_{j,t}\Big)^2 - \Big(\sigma_{j,t_0}+\lambda_{j,t_0}\Big)^2 \right) \\[10pt] \delta_C^{D,t} &= \Big( B(\pi_t,\mu_{t_0}+\beta_t) - B(\pi_{t_0},\mu_{t_0}+\beta_t ) \Big) - \Big( B(\pi_t,\mu_t ) - B(\pi_{t_0},\mu_t ) \Big) + \Big( W(\pi_t,\sigma_{t_0}+\lambda_t ) - W(\pi_{t_0},\sigma_{t_0}+\lambda_t ) \Big) - \Big( W(\pi_t,\sigma_t ) - W(\pi_{t_0},\sigma_t ) \Big) \\ &\approx \sum_j(\pi_{j,t}-\pi_{j,t_0} ) \left( \Big(\mu_{j,t_0}+\beta_{j,t}-\sum_j\pi_{j,t}(\mu_{j,t_0}+\beta_{j,t})\Big)^2 - \Big(\mu_{j,t}-\sum_j\pi_{j,t}\mu_{j,t}\Big)^2 + \Big(\sigma_{j,t_0}+\lambda_{j,t}\Big)^2-\sigma_{j,t}^2 \right) \\[10pt] \delta_P^{D,t} &= B(\pi_t,\mu_t+\beta_t) - B(\pi_t,\mu_{t_0}+\beta_t) + W(\pi_t,\sigma_t+\lambda_t) - W(\pi_t,\sigma_{t_0}+\lambda_t) - \Big( B(\pi_{t_0},\mu_t) - B(\pi_{t_0},\mu_{t_0}) + W(\pi_{t_0},\sigma_t) - W(\pi_{t_0},\sigma_{t_0}) \Big) \\ &= \sum_j\pi_{j,t} \left( \Big(\mu_{j,t}+\beta_{j,t}-\sum_j\pi_{j,t}(\mu_{j,t}+\beta_{j,t}) \Big)^2 - \Big(\mu_{j,t_0}+\beta_{j,t}-\sum_j\pi_{j,t}(\mu_{j,t_0}+\beta_{j,t})\Big)^2 + \Big(\sigma_{j,t}+\lambda_{j,t} \Big)^2 - \Big(\sigma_{j,t_0}+\lambda_{j,t}\Big)^2 \right) - \sum_j\pi_{j,t_0} \left( \Big(\mu_{j,t}-\sum_j\pi_{j,t_0}\mu_{j,t}\Big)^2 - \Big(\mu_{j,t_0}-\sum_j\pi_{j,t_0}\mu_{j,t_0}\Big)^2 + \sigma_{j,t}^2-\sigma_{j,t_0}^2 \right) \end{split} \tag{6} \end{equation}$$

2c Decomposing the change in the post-treatment variance due to treatment

Rather than decomposing the change in the effect of treatment on the variance, the change in post-treatment variance induced by the change in the effect of treatment, i.e., $V[Y_t(1)|D_t] - V[Y_(t_0 )(1)|D_(t_0 )]$, can also be decomposed.

$$\begin{equation} V[Y_t(1)|D_t] - V[Y_(t_0 )(1)|D_(t_0 )] = \delta_B^{D,t} + \delta_W^{D,t} + \delta_C^{D,t} + \delta_P^{D,t} \\[10pt] \text{where} \end{equation}$$ $$\begin{equation} \begin{split} \delta_B^{D,t} &=\sum_j\pi_{jt_0} \left( \Big(\mu_{jt_0}+\beta_{jt}-\sum_j\pi_{jt_0}(\mu_{jt_0}+\beta_{jt})\Big)^2 - \Big(\mu_{jt_0}+\beta_{jt_0}-\sum_j\pi_{jt_0}(\mu_{jt_0}+\beta_{jt_0})\Big)^2 \right) \\[10pt] \delta_W^{D,t} &= \sum_j\pi_{jt_0}\left( \Big(\sigma_{jt_0}+\lambda_{jt}\Big)^2 - \Big(\sigma_{jt_0}+\lambda_{jt_0}\Big)^2 \right) = \sum_j\pi_{jt_0} \Big( \lambda_{jt}^2-\lambda_{jt_0}^2 + 2\sigma_{jt_0}(\lambda_{jt}-\lambda_{jt_0}) \Big) \\[10pt] \delta_C^{D,t} &= \sum_j\pi_{jt}\left( \Big(\mu_{jt_0}+\beta_{jt}-\sum_j\pi_{jt}(\mu_{jt_0}+\beta_{jt})\Big)^2 + \Big(\sigma_{jt_0}+\lambda{jt}\Big)^2 \right) - \sum_j\pi_{jt_0}\left( \Big(\mu_{jt_0}+\beta_{jt}-\sum_j\pi_{jt_0} (\mu_{jt_0}+\beta_{jt})\Big)^2 + \Big(\sigma_{jt_0}+\lambda_{jt}\Big)^2 \right) \\ &\approx \sum_j(\pi_{jt}-\pi_{jt_0}) \left( \Big(\mu_{jt}+\beta_{jt}-\sum_j\pi_{jt}(\mu_{jt}+\beta_{jt})\Big)^2 + \Big(\sigma_{jt_0}+\lambda_{jt} \Big)^2 \right) \\[10pt] \delta_P^{D,t} &=\sum_j\pi_{jt} \left( \Big(\mu_{jt}+\beta_{jt}-\sum_j\pi_{jt}(\mu_{jt}+\beta_{jt})\Big)^2 - \Big(\mu_{jt_0}+\beta_{jt}-\sum_j\pi_{jt}(\mu_{jt_0}+\beta_{jt})\Big)^2 + \Big(\sigma_{jt}+\lambda_{jt}\Big)^2 - \Big(\sigma_{jt_0}+\lambda_{jt}\Big)^2 \right) \end{split} \tag{7} \end{equation}$$