week_10_recap.tex

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\title[Introduction]{Econometrics 2 (Part 2)}

\author[Lychagin  \& Mu\c co]{Arieda Mu\c co}
\institute[CEU]{Central European University}
\date{Spring 2020}


\AtBeginSection[] {
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    \frametitle{TOC}
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\pgfdeclareimage[height=.7cm]{logo}{rgs2}
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\begin{document}

\frame{\titlepage}


\begin{frame}
\frametitle{Contact}
\begin{itemize}
\item Sergey Lychagin: \href{mailto:LychaginS@ceu.edu}{LychaginS@ceu.edu}\\
\sout{Office: Budapest campus, Nador 13, 513}
\item Arieda Mu\c co: \href{mailto:MucoA@ceu.edu}{MucoA@ceu.edu}\\
\sout{Office:  Budapest campus, Nador 13, 507}
\item Boldizs\'ar Juh\'asz (TA): \href{mailto:juhasz_boldizsar@phd.ceu.edu}{juhasz\_boldizsar@phd.ceu.edu}
\end{itemize}

\end{frame}

\begin{frame}
\frametitle{Textbooks}
\begin{itemize}
\item Introductory Econometrics: A Modern Approach by Wooldridge
\item Mostly Harmless Econometrics by Angrist and Pischke
\item \textcolor{blue}{Mastering Metrics by Angrist and Pischke}
\item \textcolor{blue}{Casual Inference: The Mixtape by Cunningham}
\item Reading list of applied papers
\end{itemize}
\end{frame}


\begin{frame}
\frametitle{Outline: Tentative Schedule}
\begin{enumerate}
\item Economic research questions: causality
\item The experimental ideal
\item Linear regression
\item Instrumental Variables
\item Panel Data, Fixed Effects, Differences-in-Differences
\item Matching
\item Program Evaluation: Nonparametric Methods
\item \textcolor{blue}{Regression Discontinuity}
\end{enumerate}
\end{frame}


 \frame{ \frametitle{Della Vigna and Card}
\begin{center}
\begin{figure}
\includegraphics[width=1\linewidth]{graphs/what_economist_do.png}
\end{figure}
\end{center}
 }
 
 \frame{ \frametitle{}
\begin{center}
\begin{figure}
\includegraphics[width=1\linewidth]{graphs/what_economist_do1.png}
\end{figure}
\end{center}
 }

 \frame{ \frametitle{Applied Microeconometrics Papers}
\begin{center}
\begin{figure}
\includegraphics[width=1\linewidth]{graphs/what_economist_do2.png}
\end{figure}
\end{center}
 }

\frame{ \frametitle{}
\begin{center}
\begin{figure}
\includegraphics[width=0.8\linewidth]{graphs/detective.png}
\end{figure}
\end{center}
}
 
\begin{frame}
\frametitle{Statistical Models of Shoe and Leather}
Freedman, David A. (1991) "Statistical Models and Shoe Leather", Sociological Methodology, 21, 291-313
\bigskip

John Snow studies of the cholera epidemics in Europe in the 19th century and proves that cholera is a waterborne infectious disease

\begin{itemize}
  \item In the $19^{th}$ century no microbiology, limited microscopes
  \item Theory: diseases result from "poison in the air" - miasma
  \item Cholera Europe in epidemic waves
  \item Snow studied spatial pattern of epidemics along tracks of human commerce
  \item Influence of water supply on incidence of Cholera?
\end{itemize}
\end{frame}


\begin{frame}
\frametitle{Is cholera a waterborne or an airborne disease?}

London in the 1800's: different water companies serve different areas

\begin{itemize}
  \item Some companies take water from the Thames polluted by sewage
  \item 2 companies
  \begin{itemize}
  \item Southwark \& Vauxhall: downstream from sewage discharges
  \item Lambeth: intake point upstream
  \end{itemize}

  \item Both companies served the same parts of London during the 1853-54 cholera epidemic
  \item Sometimes houses next to each other in the same street were served by the 2 different companies
    \begin{itemize}

  \item Each company supplies rich and poor, large and small houses, no difference in condition or occupation
  \end{itemize}
  \item Idea: compare number of cholera victims
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Method of Shoe and Leather}

\begin{itemize}
  \item Snow surveyed houses in large parts of London
  \item Water company
  \item Cholera victims
  \item 300,000 households involved
  \item Reward: clear result
\end{itemize}
\end{frame}

 \frame{ \frametitle{}
\begin{center}
\begin{figure}
\includegraphics[width=1\linewidth]{graphs/SnowMap_Points.png}
\end{figure}
\end{center}
 }
 

 

 
\frame{ \frametitle{}
\begin{center}
\begin{figure}[t]
\includegraphics[width=1.1\linewidth]{graphs/freedman_tab1.pdf}
\end{figure}
\end{center}
 }

\begin{frame}
\frametitle{The Experimental Ideal}

Social experiment is the most influential research design. Why?
\bigskip

Solves Selection Problem
\end{frame}




   \begin{frame}
\frametitle{Potential Outcome Model}
$D_i=\{0,1 \}$ treatment variable (hospital care)

For each population unit $i$ we consider two \emph{potential outcomes} (health status)
\[\begin{array}{cl}
    Y_{1i} & \text{outcome with treatment} \\
    Y_{0i} & \text{outcome without treatment}
  \end{array}
\]
The gain from treatment or \emph{causal effect} for unit $i$ is
\[ Y_{i1}-Y_{i0} \]
Problem: For each $i$, only one of  $ Y_{i1}$ or $Y_{i0}$ is observed.

%\emph{Missing data problem}

\end{frame}

\begin{frame}
\frametitle{Observed outcome}

We observe
\begin{eqnarray*}
Y_i  &=& \left\{ \begin{array}{cc}
                                                Y_{1i} & \text{if}\;\; D_i=1 \\
                                                Y_{0i} & \text{if}\;\; D_i=0
                            \end{array}
                            \right. = Y_{0i}+ (Y_{1i}-Y_{0i}) D_i
\end{eqnarray*}


In the population distribution of $Y_{1i}$ and $Y_{0i}$, we can compare the average health of \emph{treated} and \emph{non-treated}


\begin{eqnarray*}
  \underset{\text{observed difference}}{\underbrace{E[Y_i|D_i=1]-E[Y_i|D_i=0]}} &=&  \\
  \underset{\text{average treatment effect}}{\underbrace{E[Y_{1i}|D_i=1]-E[Y_{0i}|D_i=1]}}  &+& \underset{\text{selection bias}}{\underbrace{E[Y_{0i}|D_i=1]-E[Y_{0i}|D_i=0] }}
\end{eqnarray*}

\end{frame}


\begin{frame}\tiny

\frametitle{Observed outcome}

Remind that the observed difference is $E[Y_{i}|D_i=1]-E[Y_{i}|D_i=0]$\\
Which can be rewritten as:
\[
E[\underbrace{Y_{0i}+ (Y_{1i}-Y_{0i}) D_i}_{\text{$Y_i$ }}|D_i=1]-E[\underbrace{Y_{0i}+ (Y_{1i}-Y_{0i}) D_i}_{\text{$Y_i$ }}|D_i=0]
\]
From the properties of the conditional expectation we can rearrange the above equation as:
\begin{equation*}
E[Y_{0i}|D_i=1]+E [(Y_{1i}-Y_{0i}) D_i|D_i=1]-E[Y_{0i}|D_i=0]-E[(Y_{1i}-Y_{0i}) D_i|D_i=0]
\end{equation*}

Which can be rewritten as:
\[
E[Y_{0i}|D_i=1]+E [(Y_{1i}-Y_{0i})|D_i=1]-E[Y_{0i}|D_i=0]
\]
And is equivalent to:

\[
E[Y_{0i}|D_i=1]+E [Y_{1i}| D_i=1]-E[Y_{0i}|D_i=1]-E[Y_{0i}|D_i=0]
\]
Rearranging we get:

\begin{eqnarray*}
  \underset{\text{average treatment effect}}{\underbrace{E[Y_{1i}|D_i=1]-E[Y_{0i}|D_i=1]}}  &+& \underset{\text{selection bias}}{\underbrace{E[Y_{0i}|D_i=1]-E[Y_{0i}|D_i=0] }}
\end{eqnarray*}
\end{frame}




\begin{frame}
\frametitle{Random assignment as a solution}

Random assignment makes $D_i$ independent of the potential outcome. If $D_i$ is independent of $Y_i$ then $E[Y_i|D_i]=E[Y_i|D_i=1]=E[Y_i|D_i=0]=E[Y_i]$ 
\begin{eqnarray*}
E[Y_i|D_i=1]-E[Y_i|D_i=0]  &=& E[Y_{1i}|D_i=1]-E[Y_{0i}|D_i=0] \\
        &=& E[Y_{1i}|D_i=1]-E[Y_{0i}|D_i=1]  \\
         &=& E[Y_{1i}-Y_{0i}|D_i=1]=E[Y_{1i}-Y_{0i}] \\
\end{eqnarray*}
The observed difference in mean outcomes equals the \emph{average treatment effect}.
Examples:
\begin{itemize}
  \item health treatments
  \item government sponsored training programs
  \item education production: effect of class size, teacher quality, etc. on  student achievement
\end{itemize}

\end{frame}

 
 \begin{frame}
\frametitle{Regression Analysis of Experiments}
Assume $Y_{i1}-Y_{i0} = \rho$ \emph{constant} treatment effect
\begin{eqnarray*}
 Y_i &=&  \alpha + \rho D_i + \eta_i
\end{eqnarray*}

\begin{eqnarray*}
 E[Y_i|D_i=1] &=&  \alpha + \rho  +  E[\eta_i|D_i=1] \\
 E[Y_i|D_i=0] &=&  \alpha  +  E[\eta_i|D_i=0]\\
 E[Y_i|D_i=1]-E[Y_i|D_i=0]&=&    \rho +  \underset{\text{selection bias}}{\underbrace{E[\eta_i|D_i=1]-E[\eta_i|D_i=0]}}
\end{eqnarray*}
Selection bias amounts to correlation between regression error $\eta_i$ and $D_i$.
\end{frame}


 \begin{frame}
\frametitle{Regression Analysis of Experiments}
We know about the selection bias
\begin{eqnarray*}
 E[\eta_i|D_i=1]-E[\eta_i|D_i=0]= E[Y_{0i}|D_i=1]-E[Y_{0i}|D_i=0]
\end{eqnarray*}
If $D_i$ is randomly assigned, the selection bias is equal to zero.

Thus estimating the regression model results in the \emph{causal effect} $\rho$.
\bigskip

Regression model with covariates
\begin{eqnarray*}
 Y_i &=&  \alpha + \rho D_i + \beta X_i+ \eta_i
\end{eqnarray*}
If $X_i$ uncorrelated with $D_i$, including them will not affect estimate of $\rho$, but increase precision.

\end{frame}

\end{document}