Structural Causal Models and Counterfactuals

A risk manager sees that a fund lost $2m last quarter, and asks: "would it have lost less if we'd cut the position size in half two months ago?" This isn't a question about correlations in historical data — it's a question about one specific fund, on one specific path of events, under a hypothetical action that never actually happened. Answering it needs more than a statistical model of average effects; it needs a mechanical description of how each variable in the system is actually generated, precise enough to "rewind" a single realized history and replay it differently. That mechanical description is a structural causal model, and the rewind-and-replay operation is a counterfactual.

An analogy: rerunning a specific chess game with one move changed

Imagine a recorded chess game and asking "what if White had played a different move on turn 12, everything else about both players' styles and the position held fixed?" You're not asking about White's average performance across many games (that's more like an average treatment effect) — you're asking about this exact game, holding fixed everything that happened before turn 12 and everything about the players that wasn't determined by the move itself, then mechanically working out how the rest would unfold under the new move. That requires a model detailed enough to simulate forward from a changed starting point while keeping every other "random seed" of the game identical to what actually happened.

The model, one symbol at a time

A structural causal model (SCM) specifies each variable $V_i$ as a deterministic function of its direct causes $\text{Pa}_i$ (its "parents" in the DAG) and an independent noise term $U_i$:

$$V_i = f_i(\text{Pa}_i, U_i)$$

In plain English: nothing is left unexplained by "randomness" alone — each variable is generated by a specific mechanism from its direct causes plus some unit-specific background noise that captures everything else about that individual (fund, trader, stock) not otherwise modeled. Given a full SCM, a counterfactual — "what would $Y$ have been for this specific unit, had $D$ been set to $d'$ instead of what actually happened" — is computed in three steps: (1) abduction, infer the noise terms $U_i$ consistent with what was actually observed for this unit; (2) action, replace the equation for $D$ with the fixed value $d'$ (a "do"-intervention, ignoring $D$'s normal causes); (3) prediction, propagate forward through the (modified) equations using the same inferred noise terms from step 1 to get the counterfactual $Y$.

Worked example 1: the risk manager's "what if," made concrete

Suppose the SCM is: $\text{Loss} = 0.02 \times \text{PositionSize} \times \text{MarketShock} + U$, with MarketShock an observed market-wide factor and $U$ fund-specific noise. Actual data: PositionSize = $10m, MarketShock = −8%, observed Loss = $1.8m. Abduction: the modeled channel gives $0.02 \times 10{,}000{,}000 \times 0.08 = 16{,}000$, i.e. $16,000; the rest must be noise, so $U = 1{,}800{,}000 - 16{,}000 = 1{,}784{,}000$, i.e. $1,784,000 (most of this quarter's loss came from fund-specific noise, not the modeled channel — a simplification for illustration). Action: set PositionSize to $5m, keep MarketShock and $U$ fixed at their inferred values. Prediction: new Loss $= 0.02 \times 5{,}000{,}000 \times 0.08 + 1{,}784{,}000 = 1{,}792{,}000$, i.e. $1,792,000. The counterfactual loss barely changes — in this specific quarter, position size wasn't the dominant driver, the noise term was.

Worked example 2: why average effects and counterfactuals can disagree

Across many funds, suppose the average effect of halving position size is a genuine 40% reduction in average loss during shock quarters — a solid, well-estimated ATE (see ATE, ATT and other treatment effect estimands). But for this particular fund, the counterfactual loss reduction was only about 0.4%, because its realized loss was driven almost entirely by unit-specific noise rather than the position-size channel the average effect describes. Average effects and unit-level counterfactuals are genuinely different quantities from the same model, and a population-average recommendation can be nearly irrelevant to any one specific case.

What this means in practice

Structural causal models formalize what "if only we'd done X differently" means precisely enough to compute, and they subsume the potential-outcomes notation from the potential outcomes framework as a special case — $Y_i(d)$ is exactly the SCM's counterfactual prediction for unit $i$ under action $D=d$. They matter most in post-mortems ("would this specific trade have lost money under a different stop-loss rule?"), individualized policy decisions, and mediation questions about which channel drove a specific outcome (see mediation analysis). The catch is that a full SCM requires committing to functional forms $f_i$ for every variable, which is a much stronger assumption than what's needed just to estimate an average effect.