No Misclassification Errors Sample Clauses

No Misclassification Errors. To examine the causal effect of WTO membership on MEA participation, the assumption of exogenous selection into the WTO provides a useful starting point for the analysis. It is expressed as P [M (1) = 1, WTO∗] = P [M (1) = 1] which implies P [M (1) = 1, WTO∗ = 1] = P [M (1) = 1, WTO∗ = 0] = P [M (1) = 1]. Accordingly, using (2) implies P [M (1) = 1] = P [M = 1|WTO∗ = 1] P [M (0) = 1] = P [M = 1|WTO∗ = 0] . Finally, with exogenous selection and no misclassification errors, the ATE is nonparametrically identified and given by ATE = P [M (1) = 1] − P [M (0) = 1] (6) = P [M = 1|WTO∗ = 1] − P [M = 1|WTO∗ = 0] .

No Misclassification Errors. In the absence of measurement error, but with no assumptions concerning selection, the only information available concerning the missing counterfactuals are that they lie in the unit interval since they represent probabilities; formally, P [M (1) = 1|WTO∗ = 0], P [M (0) = 1|WTO∗ = 1] ∈ [0, 1]. Accordingly, the individual components of the ATE are bounded as follows P [M = 1, WTO∗ = 1] ≤ P [M (1) = 1] ≤ P [M = 1, WTO∗ = 1] + P (WTO∗ = 0) P [M = 1, WTO∗ = 0] ≤ P [M (0) = 1] ≤ P (WTO∗ = 1) + P [M = 1, WTO∗ = 0] . Note, the width of the bounds on P [M (1) = 1] is the censoring probability, P (WTO∗ = 0), while the width of the bounds on P [M (0) = 1] is the inclusion probability, P (WTO∗ = 1). As a result, although the bounds on the ATE are sharp, the width always equals unity and includes zero (Manski 1995). So, without identifying restrictions on the selection mechanism, it is impossible to sign the ATE. While the sign is unknown, extreme values are excluded from the bounds, thus providing some potentially useful information.