# Statistical Tools for Causal Inference

*The SKY Community*

*2019-04-30*

# Introduction

Tools of causal inference are the basic statistical building block behind most scientific results. It is thus extremely useful to have an open source collectively aggreed upon resource presenting and assessing them, as well as listing the current unresolved issues. The content of this book covers the basic theoretical knowledge and technical skills required for implementing staistical methods of causal inference. This means:

- Understanding of the basic language to encode causality,
- Knowledge of the fundamental problems of inference and the biases of intuitive estimators,
- Understanding of how econometric methods recover treatment effects,
- Ability to compute these estimators along with an estimate of their precision using the statistical software R combined with latex using Rmarkdown.

All the notions and estimators are introduced using a numerical example and simulations.

All the code behind this book is written in Rmarkdown and is publically available on GitHub. Feel free to propose corrections and updates.

In its most basic form, the FPCI states that our causal parameter of interest (\(TT\), short for Treatment on the Treated, that we will define shortly) is fundamentally unobservable, even when the sample size is infinite. The main reason for that is that one component of \(TT\), the outcome of the treated had they not received the program, remains unobservable. We call this outcome a counterfactual outcome. The FPCI is a very dispiriting result, and is actually the basis for all of the statistical methods of causal inference. All of these methods try to find ways to estimate the counterfactual by using observable quantities that hopefully approximate it as well as possible. Most people, including us but also policymakers, generally rely on intuitive quantities in order to generate the counterfactual (the individuals without the program or the individuals before the program was implemented). Unfortunately, these approximations are generally very crude, and the resulting estimators of \(TT\) are generally biased, sometimes severely.

The Fundamental Problem of Statistical Inference (FPSI) states that, even if we have an estimator \(E\) that identifies \(TT\) in the population, we cannot observe \(E\) because we only have access to a finite sample of the population. The only thing that we can form from the sample is a sample equivalent \(\hat{E}\) to the population quantity \(E\), and \(\hat{E}\neq E\). Why is \(\hat{E}\neq E\)? Because a finite sample is never perfectly representative of the population. What can we do to deal with the FPSI? I am going to argue that there are mainly two things that we might want to do: estimating the extent of sampling noise and decreasing sampling noise.