Understanding Propensity Scores and GMM Estimation in Causal Effect Estimation

Understanding Propensity Scores and GMM Estimation in Causal Effect Estimation

Introduction:
Causal effect estimation is an important concept in econometrics and statistical analysis. It allows researchers to understand the impact of certain interventions or treatments on outcomes of interest. In this article, we will delve into the use of propensity scores and generalized method of moments (GMM) estimation in principal causal effect estimation.

Propensity Scores and Principal Causal Effect Estimation:
Propensity scores offer a way to estimate principal effects by classifying individuals into principal strata. These scores are derived from the probability of being in the treatment group given the observed covariates. By using propensity score-based methods, researchers can identify individuals in the control group who are likely to be compliers. This approach assumes that potential outcomes under control are the same across principal strata, given the observed pre-treatment covariates.

Joint Estimation Methods:
Commonly used joint estimation methods involve simultaneously modeling the membership of principal stratum and the outcome. In these methods, the covariates are first related to the intermediate outcome, such as treatment received, using logistic regression. Propensity score approaches rely on the ability to identify principal strata membership based solely on covariate information. This means that there are pretreatment covariates that can effectively predict principal stratum membership.

Matching and Weighting:
Matching and weighting techniques can be employed to estimate the complier average causal effect (CACE). When estimating the CACE, the treatment group never-takers are not used. In matching, full matching can be utilized, where treated individuals are matched with control individuals based on their covariates. The weights assigned to the control group members reflect the ratio of treated to control in each matched set. This ensures that the control group members resemble the treatment group compliers. Regression adjustment can then be used to estimate the principal effects, with the outcome values regressed on the treatment indicator and covariates, using the weights from full matching.

Generalized Method of Moments (GMM) Estimation:
GMM estimation is another approach used in causal effect estimation. It involves a vector function, observable variables (which may be endogenous or exogenous), and instruments. The optimal weighting matrix for GMM estimation is the inverse of the covariance matrix of the sample moments. This leads to the smallest covariance matrix for the GMM estimator. By obtaining a first consistent estimator, researchers can make reliable causal effect estimates.

Conclusion:
Causal effect estimation is a complex process that requires careful consideration of various factors. Propensity scores and GMM estimation offer valuable methods for estimating principal causal effects. When utilizing propensity scores, it is important to ensure that the assumptions of principal ignorability hold. Additionally, matching and weighting techniques can be employed to improve the estimation of principal effects. Before concluding, here are three actionable pieces of advice for researchers conducting causal effect estimation:

1. Pay close attention to the selection of covariates for propensity score estimation. Ensure that the chosen covariates have a strong predictive power for principal stratum membership.

2. Consider using full matching techniques when estimating the CACE. Full matching allows for a more comprehensive comparison between treated and control individuals, improving the accuracy of the estimated principal effects.

3. Familiarize yourself with the principles and techniques of GMM estimation. GMM estimation offers an alternative approach to causal effect estimation and can be particularly useful in situations where instrumental variables are available.

By combining propensity score methods and GMM estimation, researchers can gain valuable insights into the causal effects of interventions or treatments, contributing to the advancement of econometrics and statistical analysis.