3-Stage-HTE

Note: The model assumes a discrete latent variable (cluster variable) that changes the normal distribution parameters for part of the population.

3 stage algorithm to estimate heterogeneous treatment effects for populations with distinct distributions.

Stage 1 - Parametrization

Sub-stage 1 (ML-Partial):

Initial estimation of normal distribution parameters through maximum-likelihood.

Sub-stage 2 (ML-Partial):

Initial estimation of probability parameters maximum-likelihood.

Sub-stages 1 and 2 are iterated by fixing the parameters from the other sub-stage as "fixed" parameters. This is done for purely practical reasons, since attempting to run the full model from the get-go often resulted in non-convergence.

Substage 3 (ML-Full):

Take in parameters from the last iteration of stages 1 and 2 as the initial values to estimate the full model through maximum-likelihood. $$L(X_N; \{ \hat{p} _ k , \hat { \theta } _k \} _ {k \in K})= \prod _ {i=1} ^N \left[ \sum _ {k \in K}\hat{p_k}f(x_i;\hat{\theta_k})\right]$$

Stage 2 - Prédiction à la Bayes

For every observation, predict the conditional probability of belonging to each cluster through Bayes' theorem with the calibrated density functions. $$\hat{Pr}(i \in C_k) = \frac{\hat{p} _ k f(x_i;\hat {\theta} _ k)}{\sum_{j \in K}\hat{p} _ j f(x_i;\hat {\theta} _ j)}; k \in K$$

Stage 3 - Heterogeneous Treatment Effects (HTE) estimation

Use predicted probabilities to estimate the Heterogeneous Treatment Effect (HTE) through OLS. In the case of two clusters: $$Y_i=\alpha_0 + \alpha_1 \hat{Pr}(i \in C_j) + \beta_0 T_i + \beta_1 T_i \times \hat{Pr}(i \in C_j) + \varepsilon_i$$ Thanks to focusing on conditional probability, this method allows estimation of HTE even when the clusters have the exact same mean (but different variances). By controlling for the probability of belonging to a cluster, the method is also robust to unobserved (and observed, but unused) variables that are affected by cluster and are correlated with the outcome, thus allowing the user to choose only the variables that are fitted best for identification of the cluster.

This is only a proof-of-concept and further advancements will be needed to lift the rather restrictive assumptions.