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Causality has nothing to do with linear regression. In the typical undergraduate Economics class, causality is often discussed only in relation to linear regression which muddies the waters. We don’t need to muddy the water. So let’s be clear: causality has nothing to do with linear regression.
Thinking of causality from only a linear model perspective muddies the water.
We intuitively understand the idea of causality. When you walk into a room and hit the light switch, the light(s) come on. When you walk out of the room and you hit the switch again, the light(s) 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 policies like housing vouchers, zoning regulation, and housing first initiatives — to name a few.
When examining the effectiveness of a policy, we’ll typically be interested in something called the average treatment effect. That is, the effect the policy has on average across the entire population of people. For example, under the voucher policy some people might see really large changes in earnings. Perhaps a housing voucher allows them to move to a better neighborhood with better schools and better jobs so that their lifetime income is greatly improved. Perhaps, others though, face challenges from a move. They experience discrimination, have a weaker social network, have trouble maintaining housing. The average treatment effect gives us a simple indication of how people tend to fare under the policy.
The challenge in all of this, though, is that we can’t measure individual level treatment effects. In the housing voucher example, we can’t measure how much person 1 benefits from a housing voucher because holding time fixed, either one has a housing voucher or one doesn’t. A person can’t both have the treatment and not have the treatment at the same time. So at its core, causal inference is a missing data problem.
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.
A function, at a high level, transforms inputs into outputs. Formally, we can represent this as follows.
$$ x \overset{f}{\longrightarrow} f(x) $$