Treats each sector's share as a separate instrument and tests whether the just-identified estimates \(\hat\beta_n\) are mutually consistent, using a precision-weighted Cochran's Q statistic \(Q=\sum_n (\hat\beta_n-\bar\beta)^2/\widehat{\mathrm{Var}}(\hat\beta_n)\) referred to a \(\chi^2_{K-1}\) distribution. Rejection points to a failure of shares/shocks exogeneity **or** to treatment-effect heterogeneity across instruments (Goldsmith-Pinkham, Sorkin & Swift 2020). Very weak instruments are down-weighted automatically; use `min_F` to drop near-dead instruments entirely.
Value
A list (class `ssb_overid`) with `Q`, `df`, `p`, `I2`, `beta_bar`, `n_instruments`, `n_dropped`.
Details
**Caveat.** The \(\hat\beta_n\) are estimated from the *same* sample and are therefore mutually correlated; the \(\chi^2_{K-1}\) reference treats them as independent and ignores that covariance. Read the p-value as a heuristic screen for gross cross-instrument disagreement, not as a formal overidentification test — for the latter, use a J-type test with an estimator robust to many instruments (e.g. the HFUL-based test in Goldsmith-Pinkham, Sorkin & Swift 2020).