Imbens and rubin covers the potential outcomes model
S C O T T C U N N I N G H A M
C AU S A L I N F E R E N C E :
First printing, April 2018
Contents
Potential outcomes causal model 81
Matching and subclassification 105
Synthetic control 287
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1 

2 Wright’s graphical demonstration of the identification problem 21
3 Graphical representation of bivariate regression from y on x 46
and talent (horizontal axis). Top right right figure: Star sample scat
ter plot of beauty and talent. Bottom left figure: Entire (stars and non
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12  first and second grade [Krueger, 1999].  
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14  Smoking and Lung Cancer  
15  Covariate distribution by job trainings and control.  
16  Covariate distribution by job trainings and matched sample.  126  
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19  
20  (using PSID)  

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23  Histogram of propensity score by treatment status 


24  Angrist and Lavy [1999] descriptive statistics  
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26  sample.  
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31  162  
32  Second stage regressions [Angrist and Lavy, 1999].  
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34  165  
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38  Illustration of a boundary problem  
39  Insurance status and age  
40  Card et al. [2008] Table 1  
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Example of outcome plotted against the running variable.  185 
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76 Table 3 from Cornwell and Trumbull [1994] 252
77 Table 2 from Draca et al. [2011] 253
82 Simple DD using sample averages 268
83 DD regression diagram 270
88 Comparison of Internet user and nonuser groups 277
89 Theoretical predictions of abortion legalization on age profiles of gon
92 Subset of coefficients (yearrepeal interactions) for the DDD model,
Table 3 of Cunningham and Cornwell [2013]. 283
97 California cigarette sales vs synthetic California 294
98 Placebo distribution 295
103 Synthetic control graph: Differences between West Germany and Syn
thetic West Germany 298
108 Gap between actual Texas and synthetic Texas 305
109 Histogram of the distribution of ratios of postRMSPE to preRMSPE.
List of Tables
1  Examples of Discrete and Continuous Random Processes. 


2  Total number of ways to get a 7 with two sixsided dice.  25  

3  25  
4  Two way contingency table.  
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48  
7  54  
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9  parity  


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14  Death rates per 1,000 personyears [Cochran, 1968]  
15  Mean ages, years [Cochran, 1968]. 


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17  113  
18  Subclassification example of Titanic survival for large K  118  
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22  Nearest neighbor matched sample  
23  Nearest neighbor matched sample with fitted values for bias correc  
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Completed matching example with single covariate 


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26  Distribution of propensity score for CPS Control group.  142 
29 OLS and 2SLS regressions of Log Quantity on Log Price with wave
height instrument 241
32 Compared to what? Different cities 264
33 Compared to what? Before and After 264
To my son, Miles, one of my favorite people. I
love you. You’ve tagged my head and heart.
I like to think of causal inference as the space between theory and estimation. It’s where we test primarily social scientific hypotheses in the wild. Some date the beginning of modern causal inference with Fisher [1935], Haavelmo [1943], Rubin [1974] or applied labor economics studies; but whenever you consider its start, causal inference is now a distinct field within econometrics. It’s sometimes listed as a lengthy chapter on “program evaluation” [Wooldridge, 2010], or given entire booklength treatments. To name just a few textbooks in the growing area, there’s Angrist and Pischke [2009], Morgan and Winship [2014], Imbens and Rubin [2015] and probably a half dozen others, not to mention numerous, lengthy treatments of specific strategies such as Imbens and Lemieux [2008] and Angrist and Krueger [2001]. The field is crowded and getting more crowded every year.
So why does my book exist? I believe there’s some holes in the market, and this book is an attempt to fill them. For one, none of the materials out there at present are exactly what I need when I teach my own class on causal inference. When I teach that class, I use Morgan and Winship [2014], Angrist and Pischke [2009], and a bunch of other stuff I’ve cobbled together. No single book at present has everything I need or am looking for. Imbens and Rubin [2015] covers the potential outcomes model, experimental design, matching and instrumental variables, but does not contain anything about directed
synthetic control. Angrist and Pischke [2009] is very close, but does not include anything on synthetic control nor the graphical models that I find so useful. But maybe most importantly, Imbens and Rubin [2015], Angrist and Pischke [2009] and Morgan and Winship [2014]
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Finally, this book is written for people very early in their careers, be it undergraduates, graduate students, or newly minted PhDs.
My hope is that this book can give you a jump start so that you don’t have to, like many of us had to, meander through a somewhat labyrinthine path to these methods.
This is probably because I remain deep down a teacher who cares about education. I love helping students discover; I love sharing in that discovery. And if someone is traveling the same windy path that I traveled, then why not help them by sharing what I’ve learned and now believe about this field? I could sell it, and maybe one day I will, but for the moment I’ve decided to give it away – at least, the first few versions.
The second reason, which supports the first, is something that Al Roth once told me. He had done me a favor, which I could never repay, and I told him that. To which he said:


life, and what they want to say about the life they were given to live when they look back on it. Economic models take preferences as given and unchanging [Becker, 1993], but I have found that figuring out one’s preferences is the hard work of being a moral person.
How I got here
“Started from the bottom now we’re here”
– Drake
and working as a qualitative research analyst doing market research
and slowly, stopped writing poetry altogether.4 My job as a qualitative research analyst was eye opening in part because it was my first exposure to empiricism. My job was to do“grounded theory” – a kind of inductive approach to generating explanations of human behavior based on focus groups and indepth interviews, as well as other ethnographic methods. I approached each project as an opportunity to understand why people did the things they did (even if what they did was buy detergent or pick a cable provider). While the job inspired me to develop my own theories 


about human behavior, it didn’t provide me a way of falsifying those theories.
After passing my prelims, I took Mustard’s labor economics field class, and learned about the kinds of topics that occupied the lives of labor economists. These topics included the returns to education, inequality, racial discrimination, crime and many other fascinating and important topics. We read many, many empirical papers in that class, and afterwards I knew that I would need a strong background in econometrics to do the kind of empirical work I desired to do.
And since econometrics was the most important area I could ever learn, I decided to make it my main field of study. This led to me working with Christopher Cornwell, an econometrician at Georgia from whom I learned a lot. He became my most important mentor, as well as a coauthor and friend. Without him, I wouldn’t be where I am today.
Optimization Makes Everything Endogeneous
“I gotta get mine, you gotta get yours”
– MC Breed
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lation with other things. The reason we think is because of what we learn from the potential outcomes model: a correlation, in order to be a measure of a causal effect, must be completely independent of the potential outcomes under consideration. Yet if the person is making some choice based on what she thinks is best, then it necessarily violates this independence condition. Economic theory predicts choices will be endogenous, and thus naive correlations are misleading.



prediction is falsifiable insofar as we can evaluate, and potentially reject the prediction, with data.6The economic model is the framework 

with which we describe the relationships we are interested in, the intuition for our results and the hypotheses we would like to test.7 After we have specified an economic model, we turn it into what is called an econometric model that we can estimate directly with data. One clear issue we immediately face is regarding the functional form 


To illustrate this idea, let’s begin with a basic economic model: supply and demand equilibrium and the problems it creates for estimating the price elasticity of demand. Policymakers and business managers have a natural interest in learning the price elasticity of demand. Knowing this can help firms maximize profits, help governments choose optimal taxes, as well as the conditions under which quantity restrictions are preferred [Becker et al., 2006]. But, the problem is that we do not observe demand curves, because demand curves are theoretical objects. More specifically, a demand curve is a collection of paired potential outcomes of price and quantity. We observe price and quantity equilibrium values, not the potential price and potential quantities along the entire demand curve. Only by tracing out the potential outcomes along a demand curve can we calculate the elasticity.
To see this, consider this graphic from Philip Wright’s Appendix B [Wright, 1928], which we’ll discuss in greater detail later (Figure 2). The price elasticity of demand is the ratio of percentage changes in quantity to price for a single demand curve. Yet, when there are shifts in supply and demand, a sequence of quantity and price pairs emerge in history which reflect neither the demand curve nor the supply