In This Set of Notes We’re Going to
Causality has nothing to do with linear regression. In the typical undergraduate economic class, causality is often only discussed in relation to linear regressions which I think muddies the waters. We don’t need to muddy the waters. So let’s be clear: causality has nothing to do with linear regression.
We intuitively understand the idea of causality. When you walk into a room and hit the light switch, the lights come on. When you walk out of the room and you hit the switch again, the lights go off. The switch causes the lights to go on or off. That’s the notion of causality that we’re all familiar with and the one that we’ll use in this class.
Instead of light switches, though, we’ll be interested in specific policies like housing vouchers, zoning regulation, and housing first initiatives — to list a few. We’ll want to know, for instance, the average impact of providing a family with a housing voucher on the children’s incomes when they are adults.
Intuitively, the average effect of a voucher on earnings relies on two pieces of information. First, for each person we’d like to compare their earnings when their family has a voucher to when they do not have a voucher. And second, we’d like to average these individual level differences across the population.
Now, we can’t actually do that first step. For each person, we can’t see their earnings when their family was given a housing voucher and compare their earnings to when their family didn’t have a housing voucher. Holding time fixed, each family either has a housing voucher or doesn’t. So a central problem in estimating the causal effects from data is overcoming this issue.
While we have an intuitive and conceptual understanding of causality and can articulate a central challenge in observing causal effects (in that a person is either treated or not treated) it will be helpful to express these ideas in mathematical notation. This will both be more convenient and allow us to be more precise in our language. In this class we’ll do so using the potential outcome function.
The Potential Outcome Function is, as the name suggests, a function. A function, at a high level, transforms inputs into outputs. Formally, we can represent this as follows.
$$ x \overset{f}{\longrightarrow} f(x) $$