identified and is equal to τ|T=1,X averaged over the distribution of X|Ti=1. In other words, we can measure the average impact of the VRF by taking each borrower, finding an identical non-borrower (conditioned on the _X_ covariates), computing the difference in the outcome variable of interest, and taking its mean.

This procedure would only be straightforward if there were just a few covariates; in practice the problem is more tractable if we can create a summary measure of similarity in the form of a propensity score. Let p( _Xi_ ) be the probability that unit _i_ be assigned to the treatment group, and define _p_ ( _X i_ ) ≡ Pr( _Ti_ = 1 | _X i_ ) = _E_ ( _Ti_ | _X i_ ). (1)

In practice, this probability – the propensity score – could be estimated using a logit or probit equation. Rosenbaum and Rubin (1983) show that conditional independence extends to the propensity score, so that treatment cases may be matched with comparison cases using just the propensity score. Furthermore, the average treatment effect may be obtained by computing the expected value of the difference in the outcome variable between each treated household and the perfectly matched comparison household (as matched using the propensity score). Perfect matching is not possible in reality, so in practice one needs to compute where _Yi_ is the observed outcome for the _i_ th individual who is treated and _Ji_ is the set of comparators for _i_ . The comparators may be chosen with replacement – the approach we take – in which case the bias is lower but the standard error higher than without replacement. We use single nearest neighbor matching, whereby one chooses the closest comparator, although other approaches are possible (Abadie et al. 2001); Dehejia and Wahba (2002) argue that the choice of matching mechanism is not as crucial as the proper estimation of the propensity scores.

Broadly following an algorithm outlined by Dehejia and Wahba (2002), we first estimated propensity scores by applying a probit model to a limited number of covariates. We then sorted the observations by propensity score and divided them into strata sufficiently fine to ensure that there was no statistically significant difference in propensity scores between treated and non-treated households within each stratum. We confined this comparison to the area of “common support” – where the propensity scores of the treated and untreated overlap – and typically needed between 15 and 21 strata. We then checked for the “balancing property,” which means that within each stratum we tested (using a 1% significance level) whether there was a difference in the covariates between the treated and non-treated group. Our initial propensity score models were not well balanced, so we added covariates (including dummy variables for Thailand’s 76 provinces) and we were able to generate models that were adequately balanced. For instance, when we confined our sample to rural areas, the propensity score model had 101 covariates, generated 13 strata, and produced 14 cases where covariates were not balanced. This is acceptable, given that at a one percent level of statistical significance one would expect to find, erroneously, about 13 cases of imbalance (false negatives).

_Village Fund, draft of June 29, 2009_ _Page 10 of 34_