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REGRESSION, IV, MATCHING Treatment effect Boualem RABTA Center for World Food Studies (SOW-VU) Vrije Universiteit - Amsterdam

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The problem Consider a dependent variable y, a treatment variable u and a vector of characteristics x (observables). The issue is to estimate correctly how for given observed x, variation in u impacts y. Examples: - Effect of school feeding programs (On: Enrollment, school attendance, School dropout, Child Nutritional Status, Learning..) - Impact of policy response to poverty patterns. - Failing to properly estimate the treatment effect could result in the costly implementation of ineffective programs or faulty public policies that block the development of real solutions for the problem that was meant to be addressed.

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The main difficulty is due to the fact that variations in unobserved factors q, to the extent they correlate with variation in u might explain the effect on y, rather than variation in u itself. The main challenge is to disentangle the effect of the treatment u from that of unobserved q. In experimental studies : design the variation in u so as to break any correlation with q (e.g., randomization). In non-experimental studies : take u and q as given but try to account for their relation explicitly, on an a priori basis. Difficulties

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OLS regression Consider the simple model The OLS estimator for is Under some assumptions OLS estimator is BLUE. BUT Omitted variable bias: OLS regression should include all influential variables. If we miss out an important variable it not only means our model is poorly specified it also means that any estimated parameters are likely to be biased. Systematic errors : Bias can also occur from error in measurements Correlation X with the errors: If X is correlated with the errors then the OLS estimator will be biased. … etc.

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IV regression If X is correlated with the errors then the OLS estimator will be biased. We still can obtain an unbiased estimator by using instrumental variables (IV). The idea is to find a variable Z that is highly correlated with X (relevance) and independent of the errors (validity). (2 stages least squares) Problems with IV regression - How to find good instruments? - How many instruments?

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Matching The method compares the outcomes of treated individuals with those of matched non-treated, where matches are chosen on the basis of similarity in observed characteristics. (counterfactual)

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u is a binary variable, i.e., u = 1 if the subject has received the treatment, 0 otherwise. Let y 1 (resp. y 0 ) the outcome with (resp. without) treatment.

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The treatment effect is the difference y 1 - y 0 We define the average treatment effect as ATE = E(y 1 -y 0 ) the average treatment effect for the treated ATT = E(y 1 -y 0 | T=1) and conditionally to x ATE x = E(y 1 -y 0 | x) ATT x = E(y 1 -y 0 | T=1, x)

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In reality we may observe y 0 or y 1 but not both. Counterfactual: what would have happened to the treated subjects, had they not received treatment? Also, in observational studies it is not possible to randomize. We still be able to estimate ATE.

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Matching The method compares the outcomes on treated with those on matched non treated, where matches are chosen on the basis of similarity in observed characteristics (x). It is motivated by the assumption that the only source of omitted variables or selection bias is the set of observed covariates. The conditional independence assumption becomes E(y i | x, u) = E(y i | x), i=0,1 which implies ATE = E(y 1 - y 0 ) = E(E(y 1 | x, T=0) - E(y 0 | x, T=1))

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Other matching procedures Propensity score matching (estimate the probability of treatment) Nearest neighbor matching (compare with the most similar/nearest individual) Kernel matching (more general definition of proximity)

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