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I'm reading Clark(2009) paper regarding the impact of education reform on school performance. The rule is that the school initiated a democratic vote at first and if 50% of students vote in favor of change school to a GM status (another form of school, with more school autonomy), the school can send an application to the government to change their school status. Most of the case the government will accept the application, an exception would be those schools that have been marked closure by the local authority. Another variation would be vote loser can still invoke a second round of vote 2 years later. Hence we could only have a fuzzy RD under the framework of IV.

In this case

$$Y_{i,t+s}=\alpha_i+\delta_1GM_{i,t}+f(c)+e_i$$ where $Win=1(c>=50)$ in the first round is used to instrument for GM status.

(Model explanation: the impact of the voting outcome in year t observed in year t+s)

What I don't understand is that to validate the causal interpretation we implicitly assume $f(·)$ only through the GM treatment status (exclusion restriction assumption). So why the functional specification of the running variable (in this case the voting share) could cause inconsistency of estimation of $\delta_1$?

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