judge_setup.tex

\input{../Papers/header.tex}

\begin{document}

Consider a sample of $N$ individuals, with $J$ judges. Each judge is assigned $n_{j}$ individuals, with $\sum_{j}n_{j} = N$.

Define the first-stage outcome (dismissal) as $X_{i}$ for person
$i$. Let $\text{Judge}_{i}$ denote the judge that is assigned to
person $i$.

Then, let $1(\text{Judge}_{i} = j)$ be an indicator for person $i$'s
judge equal to $j$. Hence, mechanically,
$\sum_{j=1}^{J}1(\text{Judge}_{i} = j) = 1$.

Consider that the judges are randomly assinged. We want to use this in a first-stage regression of the following:

\begin{equation}
  X_{i} = \pi_{0} + \sum_{j=1}^{J-1}\pi_{j}1(\text{Judge}_{i} = j)
\end{equation}

Note that mechanically, $\pi_{0}$ will be EXACTLY equal
$N^{-1}\sum_{i}X_{i}1(\text{Judge}_{i} = J)$, or the average
(non-leave-one-out) mean leniency. Similarly, $pi_{j}$ is just
$N^{-1}\sum_{i}X_{i}1(\text{Judge}_{i} = j) -
N^{-1}\sum_{i}X_{i}1(\text{Judge}_{i} = J)$, or the relative leniency.

This is $J-1$ instruments!

Our predicted values for $X_{i}$ is just $\bar{X}(j)$, the simple average for a given judge $j$ who was assigned to $i$.

Now consider that we are overidentified. If we implelemented jackknife IV (Imbens Angrist Krueger), our predicted values would be our predicted values for $X_{i}$ still uses the SAME approach, but now ``leave-outs'' our own observation.

Finally, in Dobbie Goldsmith-PInkham and Yang + other leniency appraoches, they diference out their ``own-location'', because the first stage result controls for fixed effects. E.g., consider $\alpha_{l}$, a location fixed effect that is necessary for identification. Consider the following first stage:

\begin{equation}
  X_{i} = \pi_{0} + \sum_{j=1}^{J-1}\pi_{j}1(\text{Judge}_{i} = j) + \alpha_{l}
\end{equation}

Now, consider the residual regression -- projecting onto $\alpha_{l}$
will subtract the own location averages (across all judges within that location).


\end{document}