Propensity score matching difference-in-differences

PSM DiD is an extension to the standard DiD approach. Using this approach, outcomes between treatment and control groups are compared, after matching them with similar observable factors, followed by estimation by DiD [40–42]. Combining the PSM approach with DiD allows further elimination of any time-invariant differences between the treatment and control groups, and allows selection on observables and unobservables which are constant over time [40, 43]. Additionally, matching on the propensity score accounts for imbalances in the distribution of the covariates between the treatment and control groups [40] ⁴. We present this model as follows [40],

Where and is the outcome in the post-intervention and pre-intervention period for individual patient episode i respectively, indicates individual patient episode i is in the treatment group, indicates individual patient episode i is in the control group, represents the probability of treatment assignment conditional on observed characteristics in the pre-intervention period.

In our final PSM DiD estimation model we estimate the average treatment effect on the treated (ATT) using nearest neighbour matching propensity scores, by selecting the one comparison unit i.e. patient episode whose propensity score is nearest to the treated unit in question. We present our estimation model as follows:

Where and represent the treatment and control groups respectively, the nearest neighbour matching weights, and S is the area of common covariate support⁵.

Additionally, PSM makes the parallel trends assumption more plausible as the control groups are based on similar propensity scores in the PSM DiD approach. PSM forms statistical twin pairs before conducting DiD estimation, thus increasing the credibility of the identification of the treatment effect [40]. Instead, PSM relies on the conditional independence assumption (CIA). This assumption states that, in absence of the intervention, the expected outcomes for the treated and control groups would have been the same, conditional on their past outcomes and observed characteristics pre-intervention [40, 44]. However, it is also important to note, that even if covariate balance is achieved in PSM DiD, this does not necessarily mean that there will be balance across variables that were not used to build the propensity score [40, 44]. It is for this reason that the CIA assumption is still required.

Furthermore, recent developments of the DiD approach have highlighted that additional assumptions are necessary to ensure the estimated treatment effects are unbiased [45]. It is proposed that estimates will remain consistent after conditioning on a vector of pre-treatment covariates [45]. This was our motivation for employing the PSM DiD approach, as it accounts for pre-intervention characteristics, which allow to further minimise estimation bias. PSM DiD achieves this by properly applied propensity scores, based on matched pre-intervention characteristics, thus eliminating observations that are not similar between treatment and control groups [41]. Further developments have been made to account for multiple treatment groups, which receive treatment at various time periods i.e. differential timing DiD [46]. However, this does not affect our analysis, as the introduction of ABF in our empirical example took place at one time.