29.4 Propensity Scores
Even though I mention the propensity scores matching method here, it is no longer recommended to use such method in research and publication (G. King and Nielsen 2019) because it increases
imbalance
inefficiency
model dependence: small changes in the model specification lead to big changes in model results
bias
(Abadie and Imbens 2016)note
The initial estimation of the propensity score influences the large sample distribution of the estimators.
Adjustments are made to the large sample variances of these estimators for both ATE and ATT.
The adjustment for the ATE estimator is either negative or zero, indicating greater efficiency when matching on an estimated propensity score versus the true score in large samples.
For the ATET estimator, the sign of the adjustment depends on the data generating process. Neglecting the estimation error in the propensity score can lead to inaccurate confidence intervals for the ATT estimator, making them either too large or too small.
PSM tries to accomplish complete randomization while other methods try to achieve fully blocked. Hence, you probably better off use any other methods.
Propensity is “the probability of receiving the treatment given the observed covariates.” (Rosenbaum and Rubin 1985)
Equivalently, it can to understood as the probability of being treated.
\[ e_i (X_i) = P(T_i = 1 | X_i) \]
Estimation using
logistic regression
Non parametric methods:
boosted CART
generalized boosted models (gbm)
Steps by Gary King’s slides
reduce k elements of X to scalar
\(\pi_i \equiv P(T_i = 1|X) = \frac{1}{1+e^{X_i \beta}}\)
Distance (\(X_c, X_t\)) = \(|\pi_c - \pi_t|\)
match each treated unit to the nearest control unit
control units: not reused; pruned if unused
prune matches if distances > caliper
In the best case scenario, you randomly prune, which increases imbalance
Other methods dominate because they try to match exactly hence
\(X_c = X_t \to \pi_c = \pi_t\) (exact match leads to equal propensity scores) but
\(\pi_c = \pi_t \nrightarrow X_c = X_t\) (equal propensity scores do not necessarily lead to exact match)
Notes:
Do not include/control for irrelevant covariates because it leads your PSM to be more random, hence more imbalance
Do not include for (Bhattacharya and Vogt 2007) instrumental variable in the predictor set of a propensity score matching estimator. More generally, using variables that do not control for potential confounders, even if they are predictive of the treatment, can result in biased estimates
What you left with after pruning is more important than what you start with then throw out.
Diagnostics:
balance of the covariates
no need to concern about collinearity
can’t use c-stat or stepwise because those model fit stat do not apply