Abnormal Returns

In an Event Study, abnormal returns are the difference between actual outcomes and what the pre-event model predicted: $AR_t = y_t - \hat y_t$. They measure the market's (or system's) "surprise" at the event — the portion of the outcome that "business as usual" cannot explain. Cumulative abnormal returns over the event window quantify the total causal effect.

When to use

Compute abnormal returns whenever you've run an event study. The pattern matters: AR ≈ 0 before the event and a discrete shift after = clean policy effect; nonzero AR before the event = anticipation (markets learned the news early); persistent post-event AR ≠ 0 = the model is missing something (or the event had a delayed effect). PS_3 calls the analogue "abnormal pollution" — AP_t = log_nox_t - log_nox_hat_t — and compares the post-period average against zero.