Matching

Description and Implementation

A natural way of describing matching formally is in the context of the classical potential outcomes model. To describe this model, suppose that a random sample of size n is available from a large population, which is represented by the collection of random variables (Yi, Ti,Xi), i = 1,2,…, n, where Ti ∊ {0,1},

and Xi represents a (possibly high-dimensional) vector of observed characteristics. This model aims to capture the idea that while the set of characteristics Xi is observed for all units, only one of the two random variables (Y0i, Y1i) is observed for each unit, depending on the value of Ti. The underlying random variables Y0i and Y1i are usually referred to as potential outcomes because they represent the two potential states for each unit. [Page 759]For example, this model is routinely used in the program evaluation literature, where Ti represents treatment status and Y0i and Y1i represent outcomes without and with treatment, respectively. In most applications the goal is to establish statistical inference for some characteristic of the distribution of the potential outcomes such as the mean or quantiles. However, using the available sample directly to establish inference may lead to important biases in the estimation whenever units have selected into one of the two possible groups (Ti = 0 or Ti = 1). As a consequence, researchers often assume that the selection process, if present, is based on observable characteristics. This idea is formalized by the so-called conditional independence assumption: conditionally on xi, the random variables (Y0i, Y1i are independent of Ti. In other words, under this assumption, units having the same observable characteristics xi are assigned to each of the two groups (Ti = 0 or Ti = 1) independently of their potential gains, captured by (Y0i, Y1i). Thus, this assumption imposes random treatment assignment conditional on xi. This model also assumes some form of overlap or common support: For some

. In words, this additional assumption ensures that there will be observations in both groups having a common value of observed characteristics if the sample size is large enough. The function is known as the propensity score and plays an important role in the literature. Finally, it is important to note that for many applications of interest, the model described above employs stronger assumptions than needed. For simplicity, however, the following discussion does not address these distinctions.

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