Presentation of Bezy et al.

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Eligible

Age

\mathbb{E}[Y_{i,2016} \vert X_{i,2016}, a_{i,2016} =30] - \mathbb{E}[Y_{i,2016} \vert X_{i,2016}, a_{i,2016}=31]

\mathbb{E}[Y_{i,2016} \vert X_{i,2016}, a_{i,2016} =30] - \mathbb{E}[Y_{i,2016} \vert X_{i,2016}, a_{i,2016}=31]

Idea # 1: Difference-in-Means

Compare outcomes between individuals who are on opposite sides of the age threshold

Idea # 2: Difference-in-Difference

\mathbb{E}[Y_{i,2016} \vert X_{i,2016} , a_{i,2016} =30] - \mathbb{E}[Y_{i,2016} \vert X_{i,2016} , a_{i,2016} =31]

\mathbb{E}[Y_{i,2016} \vert X_{i,2016} , a_{i,2016} =30] - \mathbb{E}[Y_{i,2016} \vert X_{i,2016} , a_{i,2016} =31]

- \mathbb{E}[Y_{i,2015} \vert X_{i,2015} , a_{i,2015} =30] - \mathbb{E}[Y_{i,2015} \vert X_{i,2015} , a_{i,2015} =31]

- \mathbb{E}[Y_{i,2015} \vert X_{i,2015} , a_{i,2015} =30] - \mathbb{E}[Y_{i,2015} \vert X_{i,2015} , a_{i,2015} =31]

Regression Equation # 1

Y_{it} = \alpha _{a(i)} + \delta_t + \beta1_{a(i) \leq 30} \times 1_{t \geq 2016} + X_{it} + \varepsilon_{it}

Y_{it} = \alpha _{a(i)} + \delta_t + \beta1_{a(i) \leq 30} \times 1_{t \geq 2016} + X_{it} + \varepsilon_{it}

Y_{it} = \beta_1 \underbrace{\Big(D_i - \mathbb{E}[D_i \vert a(i), X_{it}, \delta_t]\Big)}_{=0}

Y_{it} = \beta_1 \underbrace{\Big(D_i - \mathbb{E}[D_i \vert a(i), X_{it}, \delta_t]\Big)}_{=0}

Interpretation

\textrm{Let} \ D_i = 1_{a(i) \leq 30} \times 1_{t \geq 16}

\textrm{Let} \ D_i = 1_{a(i) \leq 30} \times 1_{t \geq 16}

Regression Equation # 2

Y_{it} = \alpha _{a(i)} + \delta_t + \sum _{2016-T}^{2016+T}\beta_k 1_{a(i) \leq 30} \times 1_{t \geq 2016} + X_{it} + \varepsilon_{it}

Y_{it} = \alpha _{a(i)} + \delta_t + \sum _{2016-T}^{2016+T}\beta_k 1_{a(i) \leq 30} \times 1_{t \geq 2016} + X_{it} + \varepsilon_{it}

Interpretation

Bezy et al. (2024)