ccxzhang.github.io/blog/CausalInference/RDDs.html at main · ccxzhang/ccxzhang.github.io · GitHub

Name: ccxzhang.github.io/blog/CausalInference/RDDs.html at main · ccxzhang/ccxzhang.github.io · GitHub
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<h1><ahref="https://charliezhang.net/">Charlie Zhang</a></h1>

<p>MS-DSPP 22' @ Georgetown <br> cz317 [at] georgetown.edu</p>

<pclass="view"><ahref="https://www.charliezhang.net"> Home </a></p>

<pclass="view"><ahref="https://charliezhang.net/blog"> Blog </a></p>

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<section>
<small></small>
<h1>Regression Discontinuity Designs</h1>
<pclass="view">by Charlie Zhang</p>
<h2>Sharp RD (SRD)</h2>
<p>Assumptions:</p>
<ol>
<li>\(Y_{i}(0), Y_{i}(1) \perp W_{i} | X_{i}\) (Unconfoundness)</li>
<li>Continuity of Conditional Regression Functions <ahref="https://www.sciencedirect.com/science/article/abs/pii/S0304407607001091">(Imben and Lee, 2008)</a>
 \(E[Y(0)|X]\) and \(E[Y(1)|X]\) are continuous in \(x\) as a result of the violation of the RCM; </li>
<li>Continuity of Conditional Distribution Functions</li>
</ol>

<p>
 Treatment status is a deterministic and discontinuous function of a covariate (MHE, p.189). It is formally expressed as \(P(T = 1 | x <x^∗) = 1\)and\(P(T = 1|x≥x^∗) = 0\)inthecasetheforcingvariableisxandthethresholdis\(x^*\).
</p>
<p>
 Then, the Average causal effect of the treatment at the discontinuity point is:
 $$\tau_{SRD} = E[Y_{1} − Y_{0} | X_{i} = c] = \lim_{x \downarrow c} E[Y_{i} | X_{i} = x] − \lim_{x \uparrow c} E[Y_{i} | X_{i} = x]$$
</p>

<h2>Fuzzy RD (FRD)</h2>
<p>Assumptions:</p>
<p>
 Treatment status is a deterministic and discontinuous function of a covariate (MHE, p.189). It is formally expressed as \(P(T = 1 | x <x^∗) = 1\)and\(P(T = 1|x≥x^∗) = 0\)inthecasetheforcingvariableisxandthethresholdis\(x^*\).
</p>
<p>
 Then, the Average causal effect of the treatment at the discontinuity point is:
 $$\tau_{FRD} = \frac{\lim_{x \downarrow c} E[Y|X=x] - \lim_{x \uparrow c} E[Y|X=x]}{\lim_{x \downarrow c} E[W|X=x] - \lim_{x \uparrow c} E[W|X=x]} \\ = E[Y_{i}(1) - Y_{i}(0) | \text{unit i is a complier and } X_{i} = c]$$
</p>

<h2>Others</h2>
<h3>The problem of external validity</h3>
<ol>
<li>Both SRD and FRD can, at best, provide estimates of the average treatment effect for a subpopulation, namely the subpopulation with covariate value equal to \(X_{i} = c\);</li>
<li>RD designs might have a relatively high degree of internal validity based on unconfoundness.</li>
</ol>
</section>
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<pclass="view"><small> Charlie Zhang on <ahref="https://www.linkedin.com/in/charlie-zhang-b639a1128/"> LinkedIn </a></small></p>

<p><small> This project is maintained by <ahref="https://github.com/ccxzhang">ccxzhang</a></small></p>

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