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Section_12

# Section_12 - Section 12 Econ 140 GSI Hedvig Tarso Xiaoyu 1...

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Unformatted text preview: Section 12 - Econ 140 GSI: Hedvig, Tarso, Xiaoyu * 1 Experiments 1.1 Basic model • Y i = β + β 1 X i + u i , where: Y i is the dependent variable X i is a treatment dummy * X i = 1 , if individual i was randomly included in the treatment group * X i = 0 , if individual i was randomly included in the control group u i is the error term * E [ u i | X i ] = 0 : conditional mean-zero assumption • Treatment e ect : E [ Y i | X i = 1]- E [ Y i | X i = 0] = ( β + β 1 )- ( β ) = β 1 The OLS estimator b β 1 from the regression of Y i on X i is then called the di erences estimator . * b β 1 = Y treatment- Y control "Treatment e ect" means the causal e ect of a treatment on some outcome of interest in an ideal randomized controlled experiment. The term "causal e ect" comes from this setting. 1.2 General model • What if the treatment and control groups di er in observable characteristics? • Y i = β + β 1 X i + β 2 W 1 i + ... + β 1+ r W ri + u i W s are additional regressors representing observable characterisitcs of the treated entities. * Example: If the randomization occurs separately for each level of education and for each gender, then we must include these variables in our regression to estimate β 1 . E [ u i | X i ,W 1 i ,...,W ri ] = E [ u i | W 1 i ,...,W ri ] : conditional mean-independence assumption * Interpretation: After conditioning on the observable characteristics, the treatment variable ( X i ) is uncorrelated with the error term ( ⇒ consistency of the estimator de ned below). * If it holds, the coe cient on X i (variable of interest) will have a causal interpretation, but the coe cients on W 1 i ,...,W ri (control variables) will not! • The OLS estimator b β 1 from the regression of Y i on X i ,W 1 i ,...,W ri is then called the di erences estimator with additional regressors . * Many thanks to previous GSIs, Edson Severnini and Raymundo M. Campos-Vazquez, as this note is based on theirs. All errors are ours. 1 1.3 Basic model with panel data • Δ Y i = β + β 1 X i + u i Δ Y i = Y after i- Y before i , which assumes that we have observations on the same subjects before and after the treatment (panel data). • Treatment e ect : E [Δ Y i | X i = 1]- E [Δ Y i | X i = 0] = ( β + β 1 )- ( β ) = β 1 • The OLS estimator b β 1 from the regression of Δ Y i on X i is then called the di erences-in-di erences estimator , because: b β 1 = Δ Y treatment- Δ Y control = Y after- Y before treatment- Y after- Y before control • We also can estimate the treatment e ect in this setting using an OLS regression on the following model: Y it = β + β 1 X i + β 2 After t + β 3 ( X i * After t ) + u it * X...
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