What is the fundamental problem of causal inference in the potential outcomes framework?

## The setup The potential outcomes framework starts by attaching two numbers to every unit (a person, a session, a store) instead of one. For unit `i` and a binary treatment: - `Y_i(1)` is the outcome that unit would show **if treated**. - `Y_i(0)` is the outcome that same unit would show **if not treated**. Both numbers are treated as fixed properties of the unit that exist before anyone decides who gets what. The **individual treatment effect** is their difference: ``` tau_i = Y_i(1) - Y_i(0) ``` This is the quantity a causal question is really about: what did the treatment *do to this unit*. ## The observation rule Let `D_i` be 1 if unit `i` was treated and 0 otherwise. What the data records is ``` Y_i = D_i * Y_i(1) + (1 - D_i) * Y_i(0) ``` One of the two potential outcomes is switched on by the assignment, and the other is switched off forever. You never see both terms for the same unit, so `tau_i` can never be computed for any single unit — not with a bigger sample, not with better instruments, not with more precise measurement. That impossibility is the **fundamental problem of causal inference**. ## A headache pill, one person You have a headache, take a pill, and an hour later rate your pain 3 out of 10. That number is `Y(1)`. The question "did the pill help?" asks about `Y(1) - Y(0)`, where `Y(0)` is the pain you would have reported at that same hour, with that same headache, had you not taken the pill. That hour happened once. `Y(0)` is not an unrecorded number sitting somewhere waiting to be found; there is no world in which it was ever written down. Repeating the experiment tomorrow gives you a different headache, a different unit. Notice which comparisons do **not** rescue you. Pain before the pill is not `Y(0)`: it is the outcome at a different time, and headaches fade on their own. Your friend who skipped the pill is not `Y(0)` either: that is a different person, with a different headache. ## Why "missing data" is more than a metaphor Write the ideal table: one row per unit, one column for `Y(1)`, one column for `Y(0)`. Causal questions are trivial arithmetic on that table. Real data is that same table with exactly one of the two cells blanked out in every row — a missing-data pattern, and a severe one, because half the cells are missing and each row is missing at least one. The mechanism that blanks the cells is the **assignment mechanism**: whatever process decided who got treated. This is the pivot of the whole framework. What you are willing to assume about the assignment mechanism is exactly what you are willing to assume about the missingness, and it is what determines whether the blanks can be filled in on average. If treatment was assigned by a coin flip, the blanked cells are missing in a way unrelated to their values, and the untreated group's observed outcomes are a fair stand-in for the treated group's missing `Y(0)`. If treatment was chosen by the units themselves, or by someone who could guess the outcomes, the blanks are missing precisely because of what they contain, and a naive comparison measures the selection as much as the treatment. ## What survives Individual effects are gone, but averages are not. Population quantities like `E[Y(1) - Y(0)]` are differences of two averages, and an average over a group can be estimated from a *different* set of units than the one that supplied the other average — as long as the two groups are comparable. That is the trade the field makes: abandon the unit-level question, keep the population-level one, and spend all the effort on the comparability argument. Every design you have heard of — randomised experiments, matching, adjustment, natural experiments — is a strategy for making the missing averages recoverable, not for recovering individual counterfactuals. ## Common traps - **"With enough data we could see the individual effect."** Sample size buys precision on averages. It never gives a unit a second history. - **Before-versus-after as the counterfactual.** This substitutes the unit's earlier outcome for `Y(0)`, which is only valid if nothing else changed over the interval — an assumption, not an observation. - **Treating the observed `Y` as if it were both potential outcomes.** Once you write `Y = Y(1)` for a treated unit, remember `Y(0)` for that unit still exists as a quantity and is simply unknown. ## What to say in an interview Define the two potential outcomes, write the observation rule, state that exactly one is realised so the unit-level effect is undefined in the data, then pivot: this makes causal inference a missing-data problem whose missingness mechanism is the assignment mechanism, which is why we estimate averages and why how units came to be treated matters more than which model we fit.