## 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 because it increases

• imbalance

• inefficiency

• model dependence: small changes in the model specification lead to big changes in model results

• bias

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.”

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 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

### References

Abadie, Alberto, and Guido W Imbens. 2016. “Matching on the Estimated Propensity Score.” Econometrica 84 (2): 781–807.
Bhattacharya, Jay, and William B Vogt. 2007. “Do Instrumental Variables Belong in Propensity Scores?” National Bureau of Economic Research Cambridge, Mass., USA.
King, Gary, and Richard Nielsen. 2019. “Why Propensity Scores Should Not Be Used for Matching.” Political Analysis 27 (4): 435–54.
———. 1985. “The Bias Due to Incomplete Matching.” Biometrics, 103–16.