In This Set of Notes We’re Going to
Causal Inference is concerned with the interpretation of statistical results in various context. The key word here is interpretation - the interpretation you use when you’re walking around an art gallery. Causal Inference is a lot like walking around an art gallery, but where the various portraits and paintings correspond to statistical results.
I like this metaphor (simile?) for a number of reasons. It makes clear that causal inference is not concerned with trying to convince someone of something. The “best” you can do is to explain to your audience how you interpret something, and provide the necessary background details so that the audience can form their own interpretation. Maybe they see it like you. Maybe they don’t. Again, it’s a lot like interpreting a piece of art.
Why are there multiple interpretations?
There is a general trade off between questions that are important and questions that can be credibly answered via statistical analysis. The most credible method is a randomized control trial. Yet, as we’ll highlight, few important questions can be answered by only exploiting experimental variation. The challenge, therefore, is to select research projects that have the right balance of importance and credibility.
When you're tackling a question that is somewhat important, you have to make decisions about how to perform the analysis.
You start by defining the parameter of interest — the thing that you are actually interested in. You do so using formal math. Not to highlight that you appear to understand formal math, but to make precise both to yourself (and the reader) what you’re interested in.
You begin by defining the population — the set of people (or firms, etc.) that you’re interested in: $\Omega$. Each entity in the population, $\omega$, will have a treatment effect, $\tau(\omega)$. As the treatment effect is a random variable, it generates a distribution over treatment effects values. We will typically be the average treatment effect value $\mathbb{E}[\tau]$. This note walk one through an example based on Chetty’s Moving to Opportunity paper.
Having defined the thing you’re interested in, you then need to decide on a method for estimating the treatment.