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c7.tex
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\documentclass{beamer}
%\usepackage[table]{xcolor}
\mode<presentation> {
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text}{fg=red}
\newtheorem{assumption}{Assumption}
\setbeamertemplate{caption}[numbered]\newcounter{mylastframe}
%\usepackage{color}
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\usetikzlibrary{arrows}
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%\usepackage[all]{xy}
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\usepackage{lipsum}
\newenvironment{changemargin}[3]{%
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\usetikzlibrary{arrows}
%\usepackage{palatino}
%\usepackage{eulervm}
\usecolortheme{lily}
\newtheorem{com}{Comment}
\newtheorem{lem} {Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{thm}{Theorem}
\newtheorem{defn}{Definition}
\newtheorem{cor}{Corollary}
\newtheorem{obs}{Observation}
\numberwithin{equation}{section}
\newtheorem{iass}{Identification Assumption}
\newtheorem{ires}{Identfication Result}
\newtheorem{estm}{Estimand}
\newtheorem{esti}{Estimator}
\newcommand{\indep}{{\bot\negthickspace\negthickspace\bot}}
%Box Types
\title[Causal Inference] % (optional, nur bei langen Titeln nötig)
{Causal Inference}
\author{Justin Grimmer}
\institute[University of Chicago]{Associate Professor\\Department of Political Science \\ University of Chicago}
\vspace{0.3in}
\date{April 25th, 2018}
\begin{document}
\begin{frame}
\titlepage
\end{frame}
\begin{frame}
\frametitle{Selection on Unobservables}
\begin{Problem}
Often there are reasons to believe that treated and untreated units differ in unobservable
characteristics that are associated with potential outcomes even
after controlling for differences in observed characteristics.\\\bigskip In
such cases, treated and untreated units are not directly comparable. What can we
do then?
\end{Problem}
\end{frame}
%\section{Motivating Example: The Mariel Boatlift}
\begin{frame}
\frametitle{Example: Minimum wage laws and employment}
\begin{itemize}
\item Do higher minimum wages decrease low-wage employment?\medskip
\item Card and Krueger (1994) consider impact of New Jersey's 1992 minimum wage increase from \$4.25 to \$5.05 per hour\medskip
\item Compare employment in 410 fast-food restaurants in
New Jersey and eastern Pennsylvania before and after the rise\medskip
\item Survey data on wages and employment from two waves:
\begin{itemize}
\item Wave 1: March 1992, one month before the minimum wage increase
\item Wave 2: December 1992, eight months after increase
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Locations of Restaurants (Card and Krueger 2000)}
\begin{center}
\includegraphics[height=3in,keepaspectratio=1]{CK1.pdf}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Wages Before Rise in Minimum Wage}
\begin{center}
\includegraphics[height=3in,keepaspectratio=1]{CKbefore.pdf}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Wages After Rise in Minimum Wage}
\begin{center}
\includegraphics[height=3in,keepaspectratio=1]{CKafter.pdf}
\end{center}
\end{frame}
%
%\begin{frame}
% \frametitle{Motivating Example: The Mariel Boatlift}
%\begin{itemize}
%\item How do inflows of immigrants affect the wages and employment of natives in local labor markets?\medskip
% \item Card (1990) uses the Mariel Boatlift of 1980 as a natural experiment
%to measure the effect of a sudden influx of immigrants on unemployment among less-skilled natives
%\end{itemize}
%\begin{overprint}
%\onslide<1>
%\begin{center}
% \includegraphics[height=1.8in,keepaspectratio=1]{Mariel1a.pdf}
%\end{center}
%\onslide<2>
%\begin{itemize}
%\medskip
%\item The Mariel Boatlift increased the Miami labor force by 7\%\medskip
%\item Individual-level data on unemployment
%from the Current Population Survey (CPS) for Miami and four
%comparison cities (Atlanta, Los Angeles, Houston and Tampa-St. Petersburg)
%\end{itemize}
%\end{overprint}
%\end{frame}
%
%\begin{frame}
% \frametitle{Motivating Example: The Mariel Boatlift}
%\begin{itemize}
%\item How do inflows of immigrants affect the wages and employment of natives in local labour markets?\bigskip
% \item Card (1990) uses the Mariel Boatlift of 1980 as a natural experiment
%to measure the effect of a sudden influx of immigrants on unemployment among less-skilled natives\bigskip
%\item The Mariel Boatlift increased the Miami labor force by 7\%\bigskip
%\item Individual-level data on unemployment
%from the Current Population Survey (CPS) for Miami and four
%comparison cities (Atlanta, Los Angeles, Houston and Tampa-St. Petersburg)
%\end{itemize}
%\end{frame}
%\begin{frame}
% \frametitle{The Mariel Boatlift}
%\begin{center}
% \includegraphics[height=2.9in,keepaspectratio=1]{Mariel1b.pdf}
%\end{center}
%\end{frame}
%
%\begin{frame}
% \frametitle{The Mariel Boatlift}
%\begin{center}
% \includegraphics[height=2.9in,keepaspectratio=1]{Mariel1a.pdf}
%\end{center}
%\end{frame}
\section{Difference-in-Differences: Setup}
\begin{frame}
\frametitle{Two Groups and Two Periods}
\begin{Definition}
Two groups:
\begin{itemize}
\item $D=1$ Treated units
\item $D=0$ Control units
\end{itemize}\medskip
Two periods:
\begin{itemize}
\item $T=0$ Pre-Treatment period
\item $T=1$ Post-Treatment period
\end{itemize}\medskip
Potential outcomes $Y_d(t)$:
\begin{itemize}
\item $Y_{1i}(t)$ potential outcome unit $i$ attains in period $t$ when treated between $t$ and $t-1$
\item $Y_{0i}(t)$ potential outcome unit $i$ attains in period $t$ with control between $t$ and $t-1$
\end{itemize}
\end{Definition}
\end{frame}
\begin{frame}
\frametitle{Two Groups and Two Periods}
\begin{Definition}
Causal effect for unit $i$ at time $t$ is
\begin{itemize}
\item $\tau_{it}=Y_{1i}(t)-Y_{0i}(t)$
\end{itemize}\medskip
Observed outcomes $Y_i(t)$ are realized as
\begin{itemize}
\item $Y_i(t)=Y_{0i}(t)\cdot (1-D_i(t))+Y_{1i}(t)\cdot D_i(t)$
\end{itemize}\medskip
Fundamental problem of causal inference:
\begin{itemize}
\item If $D$ occurs only after $t=0$ ($D_i=D_i(1)$ and $Y_i(0)=Y_{0i}(0)$) we have: $Y_i(1)=Y_{0i}(1)\cdot (1-D_i)+Y_{1i}(1)\cdot
D_i$
\end{itemize}
\end{Definition}
\begin{estm}[ATT]
Focus on estimating the average
effect of the treatment on the treated: $\tau_{ATT}=E[Y_{1}(1)-Y_{0}(1)|D=1]$
\end{estm}
\end{frame}
\begin{frame}
\frametitle{Two Groups and Two Periods}
\begin{estm}[ATT]
$\tau_{ATT}=E[Y_{1}(1)-Y_{0}(1)|D=1]$
\end{estm}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
{\bf } & {\bf Post-Period (T=1)} & {\bf Pre-Period (T=0)} \\
\hline
\multirow{2}{*}{Treated D=1} & \multirow{2}{*}{$E[Y_1(1)|D=1]$} & \multirow{2}{*}{$E[Y_0(0)|D=1]$} \\
& & \\
\hline
\multirow{2}{*}{Control D=0} & \multirow{2}{*}{$E[Y_0(1)|D=0]$} & \multirow{2}{*}{$E[Y_0(0)|D=0]$} \\
& & \\
\hline
\end{tabular}
\end{center}
\begin{overprint}
\onslide<1|handout:1>
\begin{problem}
Missing potential outcome: \textcolor{red}{$E[Y_0(1)|D=1]$}, ie. what is the average post-period outcome for the treated in the absence of the treatment?
\end{problem}
\onslide<2|handout:2>
Control Strategy: Before vs. After\begin{itemize}
\item Use: $E[Y(1)|D=1]-E[Y(0)|D=1]$
%\item Assumes: \textcolor{red}{$E[Y_0(1)|D=1]$}$=E[Y_0(0)|D=1]$
\end{itemize}
\onslide<3|handout:3>
Control Strategy: Before-After Comparison\begin{itemize}
\item Use: $E[Y(1)|D=1]-E[Y(0)|D=1]$
\item Assumes: \textcolor{red}{$E[Y_0(1)|D=1]$}$=E[Y_0(0)|D=1]$
\end{itemize}
\onslide<4|handout:4>
Control Strategy: Treated-Control Comparison in Post-Period\begin{itemize}
\item Use: $E[Y(1)|D=1]-E[Y(1)|D=0]$
%\item Assumes: \textcolor{red}{$E[Y_0(1)|D=1]$}$=E[Y_0(1)|D=0]$
\end{itemize}
\onslide<5|handout:5>
Control Strategy: Treated-Control Comparison in Post-Period\begin{itemize}
\item Use: $E[Y(1)|D=1]-E[Y(1)|D=0]$
\item Assumes: \textcolor{red}{$E[Y_0(1)|D=1]$}$=E[Y_0(1)|D=0]$
\end{itemize}
\onslide<6|handout:6>
Control Strategy: Difference-in-Differences (DD)\begin{itemize}\small
\item Use:\\$\Bigl\{ E[Y(1)|D=1]-E[Y(1)|D=0] \Bigr\}-$\\
$\Bigl\{ E[Y(0)|D=1]-E[Y(0)|D=0] \Bigr\}$
%\item Assumes: \textcolor{red}{$E[Y_0(1)-Y_0(0)|D=1]\,$}$=E[Y_0(1)-Y_0(0)|D=0]$
\end{itemize}
\onslide<7|handout:7>
Control Strategy: Difference-in-Differences (DD)\begin{itemize}\small
\item Use:\\$\Bigl\{ E[Y(1)|D=1]-E[Y(1)|D=0] \Bigr\}-$\\
$\Bigl\{ E[Y(0)|D=1]-E[Y(0)|D=0] \Bigr\}$
\item Assumes: \textcolor{red}{$E[Y_0(1)-Y_0(0)|D=1]\,$}$=E[Y_0(1)-Y_0(0)|D=0]$
\end{itemize}
\end{overprint}
\end{frame}
%\begin{frame}
% \frametitle{Graphical Representation: Difference-in-Differences}
%\begin{center}
% \includegraphics[height=2.7in,keepaspectratio=1]{did1.pdf}
%\end{center}
%\end{frame}
\begin{frame}
\frametitle{Graphical Representation: Difference-in-Differences}
\setlength{\unitlength}{1cm}
\hspace*{-1.5cm}\begin{picture}(8,6)(-4,-0.5)
\linethickness{0.5pt}
\thicklines
\put(0,0){\vector(1,0){7}}
\put(0,0){\vector(0,1){5.5}}
\put(1,0.5){\line(3,1){3.5}}
\put(1,2){\line(5,3){3.5}}
\put(1,0.5){\circle*{0.1}}
\put(1,2){\circle*{0.1}}
\put(4.5,1.663){\circle*{0.1}}
\put(4.5,4.1){\circle*{0.1}}
\put(1,-0.1){\line(0,1){0.2}}
\put(4.5,-0.1){\line(0,1){0.2}}
\put(0.6,-0.5){$t=0$}
\put(4.1,-0.5){$t=1$}
\thinlines
\multiput(0,0.5)(0.2,0){5}{\line(1,0){0.1}}
\multiput(0,2)(0.2,0){5}{\line(1,0){0.1}}
\multiput(0,1.663)(0.2,0){22}{\line(1,0){0.1}}
\multiput(0,4.1)(0.2,0){22}{\line(1,0){0.1}}
\put(-2.5,0.4){\small$E[Y(0)|D=0]$}
\put(-2.5,2){\small$E[Y(0)|D=1]$}
\put(-2.5,1.45){\small$E[Y(1)|D=0]$}
\put(-2.5,4){\small$E[Y(1)|D=1]$}
\onslide<2->\multiput(1,2)(0.6,0.2){6}{\line(3,1){0.4}}
\put(4.5,3.163){\circle{0.1}}
\onslide<3->
\multiput(0,3.163)(0.2,0){22}{\line(1,0){0.1}}
\put(-2.5,3.1){\small$E[Y_0(1)|D=1]$}
\onslide<4-> \thicklines
\put(4.5,3.25){\vector(0,1){0.8}}
\put(4.5,4){\vector(0,-1){0.75}}
\put(4.7,3.5){\small $E[Y_1(1)-Y_0(1)|D=1]$}
\end{picture}
\end{frame}
\section{Difference-in-Differences: Identification}
\begin{frame}
\frametitle{Identification with Difference-in-Differences}
\begin{iass}[parallel trends]
\textcolor{blue}{$E[Y_0(1)-Y_0(0)|D=1]=E[Y_0(1)-Y_0(0)|D=0]$}
\end{iass}
\begin{overprint}
\onslide<1|handout:1>
\begin{ires}
Given parallel trends the ATT is identified as:\\
\begin{eqnarray*}
E[Y_1(1)-Y_0(1)|D=1]&=&\Bigl\{ E[Y(1)|D=1]-E[Y(1)|D=0] \Bigr\}\\
&-&\Bigl\{ E[Y(0)|D=1]-E[Y(0)|D=0] \Bigr\}
\end{eqnarray*}
\end{ires}
\onslide<2|handout:2>
\scriptsize
\begin{Proof}
Note that the identification assumption implies
$\alert{E[Y_0(1)|D=0]} = E[Y_0(1)|D=1] - E[Y_0(0)|D=1] + E[Y_0(0)|D=0]$\\
plugging in we get
\begin{eqnarray*}
% \nonumber to remove numbering (before each equation)
& & \left\{ E[Y(1)|D=1]-E[Y(1)|D=0] \right\}
-\left\{ E[Y(0)|D=1]-E[Y(0)|D=0] \right\}\\
&=& \left\{ E[Y_1(1)|D=1]-\alert{E[Y_0(1)|D=0]} \right\}
-\left\{ E[Y_0(0)|D=1]-E[Y_0(0)|D=0] \right\} \\
&=& \left\{ E[Y_1(1)|D=1]- \left( E[Y_0(1)|D=1] - E[Y_0(0)|D=1] + E[Y_0(0)|D=0] \right) \right\} \\
&-& \left\{ E[Y_0(0)|D=1]-E[Y_0(0)|D=0] \right\}\\
&= & E[Y_1(1)- Y_0(1)|D=1] + \left\{ E[Y_0(0)|D=1] - E[Y_0(0)|D=0] \right\} \\
&-&\left\{ E[Y_0(0)|D=1]-E[Y_0(0)|D=0] \right\} \\
&=&E[Y_1(1)-Y_0(1)|D=1]
\end{eqnarray*}
\end{Proof}
\end{overprint}
\end{frame}
%\begin{footnotesize}
%\begin{eqnarray*}
%% \nonumber to remove numbering (before each equation)
% & & \left\{ E[Y(1)|D=1]-E[Y(1)|D=0] \right\}
% -\left\{ E[Y(0)|D=1]-E[Y(0)|D=0] \right\}\\
% &=& \left\{ E[Y_1(1)|D=1]-E[Y_0(1)|D=0] \right\}
% -\left\{ E[Y_0(0)|D=1]-E[Y_0(0)|D=0] \right\} \\
% & & \\
% & & \mbox{Now notice that Assumption \ref{assumption:id} implies:} \\
% & & E[Y_0(1)|D=0] = E[Y_0(1)|D=1] - E[Y_0(0)|D=1] + E[Y_0(0)|D=0]\\
% & & \mbox{plug into above and get:} \\
% & & \\
% &=& \left\{ E[Y_1(1)|D=1]- \left( E[Y_0(1)|D=1] - E[Y_0(0)|D=1] + E[Y_0(0)|D=0] \right) \right\}
% -\left\{ E[Y_0(0)|D=1]-E[Y_0(0)|D=0] \right\}\\
% &= & E[Y_1(1)- Y_0(1)|D=1] + \left\{ E[Y_0(0)|D=1] - E[Y_0(0)|D=0] \right\}
% -\left\{ E[Y_0(0)|D=1]-E[Y_0(0)|D=0] \right\} \\
% &=& E[Y_1(1)-Y_0(1)|D=1]
%\end{eqnarray*}
%\end{footnotesize}
\section{Difference-in-Differences: Estimation}
\begin{frame}
\frametitle{Difference-in-Differences: Estimators}
\begin{estm}[ATT]\vspace{-.15in}
\begin{eqnarray*}\small
E[Y_1(1)-Y_0(1)|D=1]&=&\Bigl\{ E[Y(1)|D=1]-E[Y(1)|D=0] \Bigr\}\\
&-&\Bigl\{ E[Y(0)|D=1]-E[Y(0)|D=0] \Bigr\}
\end{eqnarray*}
\end{estm}\vspace{-.05in}
%\pause
\begin{esti}[Sample Means: Panel]\small
\[
\left\{\frac{1}{N_1}\sum_{D_i=1} Y_i(1) -
\frac{1}{N_0}\sum_{D_i=0} Y_i(1)\right\} -
\left\{\frac{1}{N_1}\sum_{D_i=1} Y_i(0) -
\frac{1}{N_0}\sum_{D_i=0} Y_i(0)\right\}
\]
\[
=\left\{\frac{1}{N_1}\sum_{D_i=1} \{Y_i(1)-Y_i(0)\} -
\frac{1}{N_0}\sum_{D_i=0} \{Y_i(1)-Y_i(0)\}\right\},
\]
where $N_1$ and $N_0$ are the number of treated and control units respectively.
\end{esti}
\end{frame}
\begin{frame}
\frametitle{Sample Means: Minimum wage laws and employment}
\begin{center}
\includegraphics[height=2.4in,keepaspectratio=1]{CKresults1.pdf}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Difference-in-Differences: Estimators}
\begin{esti}[Sample Means: Repeated Cross-Sections]\small
Let
$\{Y_i, D_i, T_i \}_{i=1}^n$ be the
pooled sample (the two different cross-sections merged) where $T$
is a random variable that indicates the period (0 or 1) in which the
individual is observed.\\\bigskip The difference-in-differences estimator is given by:
\begin{multline}
\left\{\frac{\sum D_i\cdot T_i\cdot Y_i}{\sum D_i\cdot T_i} -
\frac{\sum (1-D_i)\cdot T_i\cdot Y_i}{\sum (1-D_i)\cdot T_i}\right\}\\ -
\left\{\frac{\sum D_i\cdot(1-T_i)\cdot Y_i}{\sum D_i\cdot(1-T_i)} -
\frac{\sum (1-D_i)\cdot (1-T_i)\cdot Y_i}{\sum (1-D_i)\cdot (1-T_i)}\right\}\nonumber
%\label{equation:sdid}
\end{multline}
\end{esti}
\end{frame}
\begin{frame}
\frametitle{Difference-in-Differences: Estimators}
\begin{esti}[Regression: Repeated Cross-Sections]\small
Alternatively, the same estimator can be obtained using regression techniques.
Consider the linear model:
\[
Y = \mu + \gamma\cdot D + \delta\cdot T + \tau \cdot (D\cdot T) + \varepsilon,
\]
where $E[\varepsilon|D,T]=0$.
\end{esti}\medskip
\begin{overprint}
\onslide<1|handout:1>
Easy to show that $\tau$ estimates the DD effect:
\begin{eqnarray*}
\tau =&
\left\{
E[Y|D=1, T=1] - E[Y|D=0, T=1]
\right\}\\
-&
\left\{
E[Y|D=1, T=0] - E[Y|D=0, T=0]
\right\}
\end{eqnarray*}
\onslide<2|handout:2>
\begin{center}
\small
\begin{tabular}{|c|c|c|c|}
\hline
{\bf } & {\bf After (T=1)} & {\bf Before (T=0)} & {\bf After - Before} \\
\hline
\multirow{2}{*}{{\bf Treated D=1}} & \multirow{2}{*}{$\mu + \gamma + \delta + \tau$} & \multirow{2}{*}{$\mu + \gamma$} & \multirow{2}{*}{$\delta + \tau$} \\
& & & \\
\hline
\multirow{2}{*}{{\bf Control D=0}} & \multirow{2}{*}{$\mu + \delta$} & \multirow{2}{*}{$\mu$} & \multirow{2}{*}{$\delta$} \\
& & & \\
\hline
\multirow{2}{*}{{\bf Treated - Control}} & \multirow{2}{*}{$\gamma + \tau$} & \multirow{2}{*}{$\gamma$} & \multirow{2}{*}{$\tau$} \\
& & & \\
\hline
\end{tabular}
\end{center}
%Correct standard errors to account for temporal dependence!
\end{overprint}
\end{frame}
\begin{frame}[fragile]
\frametitle{Regression: Minimum wage laws and employment}
\footnotesize
\begin{lstlisting}[language=R, basicstyle=\ttfamily]
> d <- read.dta("CK1994_longformat.dta",convert.factors = FALSE)
> head(d[, c('ID', 'nj', 'postperiod', 'emptot')])
ID nj postperiod emptot
1 1 0 0 40.50
2 1 0 1 24.00
3 2 0 0 13.75
4 2 0 1 11.50
5 3 0 0 8.50
6 3 0 1 10.50
\end{lstlisting}
\end{frame}
\begin{frame}[fragile]
\frametitle{Regression: Minimum wage laws and employment}
\footnotesize
\begin{lstlisting}[language=R, basicstyle=\ttfamily]
with(d,
(
mean(emptot[nj == 1 & postperiod == 1], na.rm = TRUE) -
mean(emptot[nj == 1 & postperiod == 0], na.rm = TRUE)
) -
(mean(emptot[nj == 0 & postperiod == 1], na.rm = TRUE) -
mean(emptot[nj == 0 & postperiod == 0], na.rm = TRUE)
)
)
[1] 2.753606
\end{lstlisting}
\end{frame}
\begin{frame}[fragile]
\frametitle{Regression: Minimum wage laws and employment}
\footnotesize
\begin{verbatim}
> ols <- lm(emptot ~ postperiod * nj, data = d)
> coeftest(ols)
t test of coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 23.3312 1.0719 21.7668 < 2e-16 ***
postperiod -2.1656 1.5159 -1.4286 0.15351
nj -2.8918 1.1935 -2.4229 0.01562 *
postperiod:nj 2.7536 1.6884 1.6309 0.10331
\end{verbatim}
Note: Should adjust standard errors to account for temporal dependence
\end{frame}
%\begin{frame}[fragile]
% \frametitle{Regression: Minimum wage laws and employment}
%
%\footnotesize
%\begin{verbatim}
%
%library(plm)
%library(lmtest)
%> # define panel data
%> d <- plm.data(d, indexes = c("ID", "postperiod"))
%
%> # run DID regression
%> did.reg <- plm(emptot ~ postperiod * nj, data = d,
% model = "pooling")
%
%> # get clustered SEs
%> coeftest(did.reg, vcov=function(x)
% vcovHC(x, cluster="group", type="HC1"))
%
%t test of coefficients:
%
% Estimate Std. Error t value Pr(>|t|)
%(Intercept) 23.3312 1.3457 17.3370 < 2e-16 ***
%postperiod1 -2.1656 1.2173 -1.7790 0.07562 .
%nj -2.8918 1.4387 -2.0100 0.04477 *
%postperiod1:nj 2.7536 1.3058 2.1087 0.03529 *
%\end{verbatim}
%
%\end{frame}
\begin{frame}
\frametitle{Difference-in-Differences: Estimators}
\begin{esti}[Regression: Repeated Cross-Sections]\small
Can use regression version of the DD estimator
to include covariates:
\[
Y = \mu + \gamma\cdot D + \delta\cdot T + \tau \cdot (D\cdot T) + X'\beta + \varepsilon.
\]\vspace*{-0.5cm}
\begin{itemize}
\item introducing time-invariant $X$'s is not helpful (they get differenced-out)
\item be careful with time-varying $X$'s: they are often affected by the treatment and may introduce endogeneity (e.g. price of meal)
\item always correct standard errors to account for temporal dependence
\end{itemize}
Can interact time-invariant
covariates with the time indicator:
\begin{equation}
Y = \mu + \gamma\cdot D + \delta\cdot T + \alpha\cdot (D\cdot T) + X'\beta_0 + (T\cdot X')\beta_1 +\varepsilon \nonumber
%\label{equation:tvdid}
\end{equation}
\end{esti}
$\Rightarrow$ $X$ is used to explain differences in trends.
\end{frame}
\begin{frame}
\frametitle{Difference-in-Differences: Estimators}
\begin{esti}[Regression: Panel Data]\small
With panel data we can estimate the difference-in-differences effect using a fixed effects regression with unit and period fixed effects:
\[
Y_{it} = \mu + \gamma_i + \delta T + \tau D_{it} + X'_{it}\beta + \varepsilon_{it}
\]\vspace*{-0.5cm}
\begin{itemize}
\item One intercept for each unit $\gamma_i$
\item $D_{it}$ coded as 1 for treated in post-period and 0 otherwise
%\item Correct standard errors to account for temporal dependence
\end{itemize}
Or equivalently we can use regression with the dependent variable in first differences:
\[
\Delta Y_i = \delta + \tau \cdot D_i + u_i,
\]
where $\Delta Y_i = Y_i(1)-Y_i(0)$ and $u_i=\Delta \varepsilon_i$.\\\bigskip
%\begin{itemize}
%\item Often helps with temporal dependence.
%\item With two periods this gives the same result as other regressions
%\end{itemize}
%$\Rightarrow$ Can also add $X$ to explain differences in trends.
\end{esti}
\end{frame}
\begin{frame}[fragile]
\frametitle{Regression: Minimum wage laws and employment}
\footnotesize
\begin{verbatim}
library(plm)
library(lmtest)
> d$Dit <- d$nj * d$postperiod
> d <- plm.data(d, indexes = c("ID", "postperiod"))
> did.reg <- plm(emptot ~ postperiod + Dit, data = d,
model = "within")
> coeftest(did.reg, vcov=function(x)
vcovHC(x, cluster="group", type="HC1"))
t test of coefficients:
Estimate Std. Error t value Pr(>|t|)
postperiod1 -2.2833 1.2465 -1.8319 0.06775 .
Dit 2.7500 1.3359 2.0585 0.04022 *
\end{verbatim}
\end{frame}
%\begin{frame}[fragile]
%\begin{verbatim}
%library(plm)
%library(lmtest)
%> ck_data <- plm.data(ck_data, indexes = c("ID", "postperiod"))
%> fixed.mod <- (plm(emptot ~ postperiod * nj, data = ck_data,
% model = "within"))
%> coeftest(fixed.mod, vcov=function(x)
% vcovHC(x, cluster="group",
% type="HC1"))
%
%t test of coefficients:
%
% Estimate Std. Error t value Pr(>|t|)
%postperiod1 -2.2833 1.2465 -1.8319 0.06775 .
%postperiod1:nj 2.7500 1.3359 2.0585 0.04022 *
%---
%Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
%\end{verbatim}
%
%\end{frame}
\begin{frame}[fragile]
\frametitle{Regression: Minimum wage laws and employment}
\footnotesize
\begin{verbatim}
> firstdiff.mod <- plm(emptot ~ postperiod * nj,
data = d, model = "fd")
> coeftest(firstdiff.mod, vcov=function(x) vcovHC(x, type="HC0"))
t test of coefficients:
Estimate Std. Error t value Pr(>|t|)
postperiod1 -2.2833 1.2465 -1.8319 0.06775 .
postperiod1:nj 2.7500 1.3359 2.0585 0.04022 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
\end{verbatim}
\end{frame}
\section{Difference-in-Differences: Threats to Validity}
\begin{frame}
\frametitle{Difference-in-Differences: Threats to Validity}
\begin{enumerate}
% \item \emph{Compositional differences}: In repeated cross-sections
% we do not want that the composition
% of the sample changes between periods.
% \begin{itemize}
% \item Falsification Test: Distribution of $(D,X)$ should be similar for the pre-treatment and
% post-treatment periods.
% \end{itemize}\bigskip
\item Non-parallel dynamics\bigskip
\item Compositional differences\bigskip
\item Long-term effects versus reliability\bigskip
\item Functional form dependence\bigskip
\end{enumerate}
Bias is a matter of degree. Small violations of the identification assumptions may not matter much as the bias may be rather small. However, biases can sometimes be so large that the estimates we get are completely wrong, even of the opposite sign of the true treatment effect.\\\bigskip
Helpful to avoid overly strong causal claims for difference-in-differences estimates.
\end{frame}
%\subsection{Non-parallel dynamics}
\begin{frame}
\frametitle{Difference-in-Differences: Threats to Validity}
\begin{enumerate}
% \item \emph{Compositional differences}: In repeated cross-sections
% we do not want that the composition
% of the sample changes between periods.
% \begin{itemize}
% \item Falsification Test: Distribution of $(D,X)$ should be similar for the pre-treatment and
% post-treatment periods.
% \end{itemize}\bigskip
\item \emph{Non-parallel dynamics}: Often treatments/programs are targeted based on pre-existing differences in outcomes.\medskip
\begin{itemize}
\item ``Ashenfelter dip'': participants in training programs often experience a dip in earnings
just before they enter the program (that may be \emph{why} they participate). Since wages have a natural tendency to mean reversion, comparing wages of participants and non-participants using DD leads to an upward biased estimate of the program effect\medskip
\item Regional targeting: NGOs may target villages that appear most promising (or worst off)\medskip
% \item In repeated cross-sections, we do not want that the composition of the sample changes between periods.
\end{itemize}
% \begin{itemize}
% \item Falsification Test: Under the parallel trends assumption during the periods $t=-1,0,1$, we have:
% \[
% E[Y(0)-Y(-1)|D=1]- E[Y(0)-Y(-1)|D=0]=0
% \]
% \item That is, apply DID estimator to $t=-1,0$ and test if $\alpha=0$
% % \item Can use similar placebo test with second control group or other placebo outcomes that
% % are known to be unaffected
% \end{itemize}
%\item \emph{Long-term effects vs. reliability}:
%\begin{itemize}
% \item Parallel trends assumption for DD is more likely to hold over
%a shorter time-window. In the long-run, many other things may happen and confound the effect of the treatment.
%\item Should be cautious to extrapolate short-term effects to long-term effects
%\end{itemize}
\end{enumerate}
\end{frame}
%\begin{frame}
% \frametitle{Checks for Difference-in-Differences Design}
%\begin{enumerate}
% \item Falsification test using data for prior periods
% \begin{itemize}
%\item Given parallel trends during periods $t=-1,0,1$, we have:
% \[
% E[Y(0)-Y(-1)|D=1]- E[Y(0)-Y(-1)|D=0]=0
% \]
%\item run placebo DD on data from $t=-1,0$ and test if $\alpha=0$. If not, your
% estimate comparing $t=0$ and $t=1$ may be biased
% \end{itemize}\medskip \pause
%\item Falsification test using data for alternative control group
% \begin{itemize}
%\item if the placebo DD with the alternative control is different from the DD with the original control, then the original DD may be biased
% \end{itemize}\medskip \pause
%\item Falsification test using alternative placebo outcome that is not supposed to be affected by the treatment
% \begin{itemize}
% \item if DD from placebo outcome is non-zero, then the DD estimate for original outcome may be biased
% \end{itemize}
%\end{enumerate}
%\end{frame}
\begin{frame}
\frametitle{Checks for Difference-in-Differences Design}
\begin{enumerate}
\item Falsification test using data for prior periods \bigskip
% \begin{itemize}
%\item Given parallel trends during periods $t=-1,0,1$, we have:
% \[
% E[Y(0)-Y(-1)|D=1]- E[Y(0)-Y(-1)|D=0]=0
% \]
%\item run placebo DD on data from $t=-1,0$ and test if $\alpha=0$. If not, your
% estimate comparing $t=0$ and $t=1$ may be biased
% \end{itemize}\medskip \pause
\item Falsification test using data for alternative control group \bigskip
% \begin{itemize}
%\item if the placebo DD with the alternative control is different from the DD with the original control, then the original DD may be biased
% \end{itemize}\medskip \pause
\item Falsification test using alternative placebo outcome that is not supposed to be affected by the treatment
% \begin{itemize}
% \item if DD from placebo outcome is non-zero, then the DD estimate for original outcome may be biased
% \end{itemize}
\end{enumerate}
\end{frame}
\begin{frame}
\frametitle{Falsification test: Data for prior periods}
\vspace{-.2in}
\begin{center}
\includegraphics[height=3.2in,keepaspectratio=1]{KCfalsification1.pdf}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Falsification test: Data for prior periods}
\vspace{-.2in}
\begin{center}
\includegraphics[height=3.2in,keepaspectratio=1]{KCfalsification2.pdf}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Falsification test: Data for prior periods}
\vspace{-.2in}
\begin{center}
\includegraphics[height=3.2in,keepaspectratio=1]{KCfalsification3.pdf}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Falsification test: Data for prior periods}
\vspace{-.2in}
\begin{center}
\includegraphics[height=3.2in,keepaspectratio=1]{KCfalsification4.pdf}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Longer Trends in Employment (Card and Krueger 2000)}
\vspace{-.1in}
\begin{center}
\includegraphics[height=3in,keepaspectratio=1]{PreTrend.pdf}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Falsification test: Alternative control group}
\vspace{-.2in}
\begin{center}
\includegraphics[height=1.3in,keepaspectratio=1]{CKresults4.pdf}
\end{center}
% \begin{itemize}
%\item
If placebo DD between original and alternative control group is not zero, then the original DD may be biased
% \end{itemize}\medskip \pause
\end{frame}
\begin{frame}
\frametitle{Triple DDD: Mandated Maternity Benefits (Gruber, 1994)}
\hspace*{0.5cm}\begin{overprint}
\onslide<1|handout:1>\includegraphics[height=3.2in,keepaspectratio=1]{DDD1markup.pdf}
\onslide<2|handout:2>\includegraphics[height=3.2in,keepaspectratio=1]{DDD2markup.pdf}
\onslide<3|handout:3>\includegraphics[height=3.2in,keepaspectratio=1]{DDDmarkup.pdf}
\end{overprint}
\end{frame}
\begin{frame}
\frametitle{How useful is the Triple DDD?}
\begin{itemize}
\item The DDD estimate is the difference between the DD of interest and the placebo DD (that is supposed to be zero)\bigskip
\begin{itemize}
\item If the placebo DD is non zero, it might be difficult to convince reviewers that the DDD removes all the bias\medskip
\item If the placebo DD is zero, then DD and DDD give the same results but DD is preferable because standard errors are smaller for DD than for DDD
\end{itemize}
\end{itemize}
\end{frame}
%\section{Example: Gerhard Schr\"oder's Flood}
%
%\begin{frame}
% \frametitle{The Elbe Valley}
% \vspace{-.06in}
%\begin{center}
% \includegraphics[height=3.1in,keepaspectratio=1]{elbe1.pdf}
%\end{center}
%\end{frame}
%
%
%%\begin{frame}
%% \frametitle{The Elbe Valley: 2001 and 2002}
%% \vspace{-.06in}
%%\begin{center}
%% \includegraphics[height=3.1in,keepaspectratio=1]{NASA.pdf}
%%\end{center}
%%\end{frame}
%
%\begin{frame}
% \frametitle{Schr\"oder Sends the Troops}
%\begin{columns}
% \begin{column}{0.5\textwidth}
% \centerline{\includegraphics[height=1.2in,keepaspectratio=1]{flood1.pdf}}
% \centerline{\includegraphics[height=1.2in,keepaspectratio=1]{flood2.pdf}}
%
%
% \end{column}
%
% \begin{column}{0.3\textwidth}
% \centerline{\includegraphics[height=1.2in,keepaspectratio=1]{flood3.pdf}}
% \centerline{\includegraphics[height=1.2in,keepaspectratio=1]{flood4.pdf}}
% \end{column}
%\end{columns}
%
%\end{frame}
%
%%\begin{frame}
%% \frametitle{Gerhard Schr\"oder's Flood}
%%
%% \begin{itemize}
%% \item Panel data for Germany's 299 electoral districts. $Y$: PR-vote share of Social Democratic Party SPD\bigskip
%% \item Two periods
%% \begin{itemize}
%% \item $T=0$: Federal election 1998
%% \item $T=1$: Federal election 2002
%% \item The Elbe flood occurred in August of 2002 and the election was held in September of 2002
%% \end{itemize}\bigskip
%% \item Treatment: Flooded Districts
%% \begin{itemize}
%% \item $D=1$ Flooded districts (all of which received federal disaster aid)
%% \item $D=0$ Unaffected districts
%% \end{itemize}
%% \end{itemize}
%%\end{frame}
%
%\subsection{Long-term effects versus reliability}
\begin{frame}
\frametitle{Difference-in-Differences: Further Threats to Validity}
\begin{enumerate}
\setcounter{enumi}{1}
\emph{\item Compositional differences}\bigskip