clustering, the numerator will be overwhelmed by the denominator and the estimator $\tau^\mathrm{gcr}_\mathrm{ITR}(1,0)$ will be a dramatic underestimate of the true average treatment effect, and it’s clear that $\tau^\mathrm{ind}_\mathrm{ITR}(1,0)$ would be even worse. In order for meaningful relative bias reduction to occur, the clustering must capture the structure of the dependence between units specified by the matrix of coefficients ${\mathbf{B}}$.

In Appendix \[appendix:balanced\_linear\_case\], we derive similar intuitions from an alternative graph cluster randomization that preserves balance between the sizes of the treatment and control group. There graph cluster randomization no longer always achieves bias reduction for every clustering over independent assignment, but meaningful bias reduction is again possible and depends on how the clustering captures ${\mathbf{B}}$ in an identical way.

This linear outcome model has as special cases some other models of interest. In particular, it has as a special case the linear-in-means model, which is widely studied and used in econometrics [e.g., @manski_identification_1993_reflection; @lee_identification_2007; @bramoulle_identification_2009; @goldsmith_social_2013]. Consider $a(x) = x$ in Eq. \[probit\_model\]. Then for $t \geq 1$ the quantity $E^U[Y_{i,t}(z)]$ is $$\begin{aligned}
{\mathrm{E}}^U[Y_{i,t}(z)] = \alpha +  \beta z_i+ \gamma \frac{{A}_i^{'} {\mathrm{E}}^U[Y_{t - 1}(z)]}{k_i}.\end{aligned}$$ The closed form solution for $E^U[Y_{t}(z)]$ for any $t \ge 0$ is then given by 