• First partition the population on the basis of covariate values into \(G\) strata, i.e. if the covariate space is \(\mathbb{W}\), partition \(\mathbb{W}\) into \(\mathbb{W}_1, \cdots, \mathbb{W}_G\), so that \(\bigcup_g \mathbb{W}_g = \mathbb{W}\) and \(\mathbb{W}_g \cap \mathbb{W}_{g'} = \emptyset\) if \(g \neq g'\).
• Let \(G_{ig} = 1_{W_i \in \mathbf{W}_g}\) and \(N_g\) be the number of units in cluster \(g\).
• Let \(G_t\) be the number of clusters to treat.
• Let \(\overline{Z}_g \equiv 1/N_g \sum_{i: G_{ig} = 1} Z_i\).
• The assignment probability is: \[ \mathbb{P}[\mathbf{Z}|\mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0)] = \begin{pmatrix} G \\ G_t \end{pmatrix}^{-1}, \] if \(\forall g, \overline{Z}_g = \{0, 1\}, \sum_{g = 1}^G \overline{Z}_g = G_t\) and \(0\) otherwise.

• The clusters may be villages, states, and other geographical entities.
• Given a fixed sample size, this design is in general not as efficient as completely randomized or stratified randomized experiment.
• One motivation is that there may be interference between units in the same cluster but not across different clusters.
• Another motivation is the convenience and practical limitations.

• The finite-sample average treatment effect: \[ \tau_{fs} \equiv \frac{1}{N} \sum_{i = 1}^N [Y_i^\ast(1) - Y_i^\ast(0)]. \]
• The unweighted average of the within-cluster average effects. \[ \tau_C \equiv \frac{1}{G} \sum_{g = 1}^G \tau_g, \tau_g \equiv \frac{1}{N_g} \sum_{i: G_{ig} = 1} [Y_i^\ast(1) - Y_i^\ast(0)]. \]

• \(\tau_{fs}\) is usually more relevant for policies.
• However, \(\tau_{C}\) is easier to analyze: Once we aggregate the data at the cluster level, we can use the inference methods for completely randomized experiments by regarding the clusters as the units.
• Moreover, the estimates of \(\tau_C\) is often more precise than that of \(\tau_{fs}\) by a clustered randomized experiment.

• We can use the inference methods for completely randomized experiments.
• The estimator is: \[ \hat{\tau}_C \equiv \frac{1}{G_t} \sum_{g: \overline{Z}_g = 1} \overline{Y}_g^{obs} - \frac{1}{G_c} \sum_{g: \overline{Z}_g = 0} \overline{Y}_g^{obs}. \]
• We can estimate the variance of \(\hat{\tau}_C\) by Neyman’s variance: \[ \widehat{\mathbb{V}}_{Neyman}(\hat{\tau}_C) \equiv \frac{s_{Cc}^2}{G_c} + \frac{s_{Ct}^2}{G_t}. \] where \[ s_{Ct}^2 \equiv \frac{1}{G_t - 1} \sum_{g: \overline{Z}_g = 1}(\overline{Y}_g^{obs} - \frac{1}{G_t} \sum_{g': \overline{Z}_{g'} = 1} \overline{Y}_{g'}^{obs}). \]

• We obtain the same estimate by using the ordinary least squares method for a linear regression function: \[ \overline{Y}_g^{obs} = \alpha + \tau \cdot \overline{Z}_g + \eta_g. \]
• If we consider a unit-level linear regression function: \[ Y_i^{obs} = \alpha + \tau \cdot Z_i + \epsilon_i, \] we obtain an estimator identical to \(\hat{\tau}_C\) if we run a weighted least squares method where unit \(i\) is weighted by \(1/N_{g(i)}\).

• We can estimate \(\tau_{fs}\) by running an (unweighted) ordinary least squares method for the unit-level linear regression function: \[ Y_i^{obs} = \alpha + \tau \cdot Z_i + \epsilon_i. \]

• Because the assignment is perfectly correlated between units in the same cluster, we use the Liang-Zeger cluster-robust covariance estimates for \((\hat{\alpha}_{ols}, \hat{\tau}_{ols})\):

\[ \begin{split} \Bigg( \sum_{i = 1}^N \begin{pmatrix} 1 & Z_i \\ Z_i & Z_i \end{pmatrix} \Bigg)^{-1} \\ \times \Bigg( \sum_{g = 1}^G \sum_{i: G_{ig} = 1} \begin{pmatrix} \hat{\epsilon}_i\\ Z_i \cdot \hat{\epsilon}_i \end{pmatrix} \sum_{i: G_{ig} = 1} \begin{pmatrix} \hat{\epsilon}_i\\ Z_i \cdot \hat{\epsilon}_i \end{pmatrix}' \Bigg) \\ \times \Bigg( \sum_{i = 1}^N \begin{pmatrix} 1 & Z_i \\ Z_i & Z_i \end{pmatrix} \Bigg)^{-1}. \end{split} \]

• Note that this is different from the Eicker-Huber-White heteroskedasticity robust covariance estimates.

• Section 8, Athey, Susan, and Guido Imbens. 2016. “The Econometrics of Randomized Experiments.” arXiv [stat.ME]. arXiv. http://arxiv.org/abs/1607.00698.