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<title>Rubin Causal Model</title>
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<h1 class="title toc-ignore">Rubin Causal Model</h1>
</div>
<p>Rubin Causal Model (RCM) is made of three distinct building blocks: a treatment allocation rule, that decides who receives the treatment; potential outcomes, that measure how each individual reacts to the treatment; the switching equation that relates potential outcomes to observed outcomes through the allocation rule.</p>
<div id="the-treatment-allocation-rule" class="section level1">
<h1>The treatment allocation rule</h1>
<p>The first building block of the RCM is the treatment allocation rule. Throughout this class, we are going to be interested in inferring the causal effect of only one treatment with respect to a control condition. Extensions to multi-valued treatments are in general self-explanatory.</p>
<p>In the RCM, treatment allocation is captured by the variable <span class="math inline">\(D_i\)</span>. <span class="math inline">\(D_i=1\)</span> if unit <span class="math inline">\(i\)</span> receives the treatment and <span class="math inline">\(D_i=0\)</span> if unit <span class="math inline">\(i\)</span> does not receive the treatment and thus remains in the control condition.</p>
<p>The treatment allocation rule is critical for several reasons. First, because it switches the treatment on or off for each unit, it is going to be at the source of the FPCI. Second, the specific properties of the treatment allocatoin rule are going to matter for the feasibility and bias of the various econometric methods that we are going to study.</p>
<p>Let’s take a few examples of allocation rules. First, let’s imagine a treatment that is given to individuals. Whether each individual receives the treatment partly depends on the level of her outcome before receiving the treatment. Let’s denote this variable <span class="math inline">\(Y^B_i\)</span>, with <span class="math inline">\(B\)</span> standing for “Before”. It can be the health status assessed by a professional before deciding to give a drug to a patient. It can be the poverty level of a household used to assess its eligibilty to a cash transfer program.</p>
<div id="the-sharp-cutoff-rule" class="section level2">
<h2>The sharp cutoff rule</h2>
<p>The sharp cutoff rule means that everyone below some threshold <span class="math inline">\(\bar{Y}\)</span> is going to receive the treatment. Everyone whose outcome before the treatment lies above <span class="math inline">\(\bar{Y}\)</span> does not receive the treatment. Such rules can be found in reality in a lot of situations. They might be generated by administrative rules. One very simple way to model this rule is as follows: <span class="math display">\[
D_i = \uns{Y_i^B\leq\bar{Y}},
\]</span> where <span class="math inline">\(\uns{A}\)</span> is the indicator function, taking value <span class="math inline">\(1\)</span> when <span class="math inline">\(A\)</span> is true and <span class="math inline">\(0\)</span> otherwise.</p>
<!-- \begin{numexample} -->
<p>Imagine for example that <span class="math inline">\(Y_i^B=\exp(y_i^B)\)</span>, with <span class="math inline">\(y_i^B=\mu_i+U_i^B\)</span>, <span class="math inline">\(\mu_i\sim\mathcal{N}(\bar{\mu},\sigma^2_{\mu})\)</span> and <span class="math inline">\(U_i^B\sim\mathcal{N}(0,\sigma^2_{U})\)</span>. Now, let’s choose some values for these parameters so that we can generate a sample of individuals and allocate the treatment among them. I’m going to switch to R for that.</p>
<pre class="r"><code>param <- c(8,.5,.28,1500)
names(param) <- c("barmu","sigma2mu","sigma2U","barY")
param</code></pre>
<pre><code>## barmu sigma2mu sigma2U barY
## 8.00 0.50 0.28 1500.00</code></pre>
<p>Now, I have choosen values for the parameters in my model. For example, <span class="math inline">\(\bar{\mu}=\)</span> and <span class="math inline">\(\bar{Y}=\)</span>. What remains to be done is to generate <span class="math inline">\(Y_i^B\)</span> and then <span class="math inline">\(D_i\)</span>. For this, I have to choose a sample size (<span class="math inline">\(N=1000\)</span>) and then generate the shocks from a normal.</p>
<pre class="r"><code># for reproducibility, I choose a seed that will give me the same random sample each time I run the program
set.seed(1234)
N <-1000
mu <- rnorm(N,param["barmu"],sqrt(param["sigma2mu"]))
UB <- rnorm(N,0,sqrt(param["sigma2U"]))
yB <- mu + UB
YB <- exp(yB)
Ds <- ifelse(YB<=param["barY"],1,0) </code></pre>
<p>Let’s now build a histogram of the data that we have just generated.</p>
<pre class="r"><code># building histogram of yB with cutoff point at ybar
# Number of steps
Nsteps.1 <- 15
#step width
step.1 <- (log(param["barY"])-min(yB[Ds==1]))/Nsteps.1
Nsteps.0 <- (-log(param["barY"])+max(yB[Ds==0]))/step.1
breaks <- cumsum(c(min(yB[Ds==1]),c(rep(step.1,Nsteps.1+Nsteps.0+1))))
hist(yB,breaks=breaks,main="")
abline(v=log(param["barY"]),col="red")</code></pre>
<div class="figure" style="text-align: center">
<img src="RCM_files/figure-html/histyb-1.png" alt="Histogram of $y_B$" width=".5\textwidth" />
<p class="caption">
Histogram of <span class="math inline">\(y_B\)</span>
</p>
</div>
<p>You can see on Figure~ a histogram of <span class="math inline">\(y_i^B\)</span> with the red line indicating the cutoff point: <span class="math inline">\(\bar{y}=\ln(\bar{Y})=\)</span>. All the observations below the red line are treated according to the sharp rule while all the one located above are not. In order to see how many observations eventually receive the treatment with this allocation rule, let’s build a contingency table.</p>
<pre class="r"><code>table.D.sharp <- table(Ds)
xtable(table.D.sharp,caption='Treatment allocation with sharp cutoff rule',label='tab:table.D.sharp')</code></pre>
% latex table generated in R 3.3.3 by xtable 1.8-2 package % Tue Oct 24 18:57:14 2017
<p>We can see on Table~ that there are treated observations. \end{numexample}</p>
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