Econometrics & Causal Inference
Weak-Instrument First-Stage F and Stock-Yogo 2SLS Bias Formula
Compute the excluded-instrument first-stage partial F-statistic, concentration parameter mu^2 = L_excl * F, and approximate 2SLS relative bias toward OLS 1 / (F + 1).
Why weak instruments pull finite-sample 2SLS straight back toward biased OLS
When excluded instruments Z have tiny partial explanatory power R^2_p in the first stage after netting out K_exog included controls, fitted values X_hat overfit first-stage sampling noise correlated with structural errors! Staiger and Stock (1997) and Stock and Yogo (2005) showed that finite-sample 2SLS bias relative to OLS bias scales inversely with the first-stage partial F-statistic: Relative_Bias approx 1 / (F_first + 1), governed by concentration parameter mu^2 = L_excl * F_first.
Test two regimes with N = 95 observations (B2), K_exog = 2 included controls (B3), and L_excl = 3 excluded instruments (B4, giving residual df_2 = 95 - 2 - 3 = 90): when partial R^2_p = 0.0625 (B5), F_first = (0.0625 / 3) / (0.9375 / 90) = 2.0000, concentration mu^2 = 6.0000, and 1 / (F + 1) = 0.3333 (33.3% of OLS bias survives!). Raising partial R^2_p to 0.2500 lifts F_first to 10.0000, mu^2 to 30.0000, and trims 1 / (F + 1) below 10% to 0.0909!
| First-Stage Partial R^2_p (df_2 = 90, L = 3) | First-Stage Partial F | Approx 2SLS Relative Bias 1/(F+1) |
|---|---|---|
| Weak Instruments (R^2_p = 0.0625) | F = 2.0000 (mu^2 = 6) | 0.3333 (Severe Weak-IV Bias) |
| Staiger-Stock Threshold (R^2_p = 0.2500) | F = 10.0000 (mu^2 = 30) | 0.0909 (< 10% Relative Bias) |
| Strong Instruments (R^2_p = 0.5000) | F = 30.0000 (mu^2 = 90) | 0.0323 (~3.2% Relative Bias) |
Excluded-Instrument First-Stage Partial F-Statistic Formula
Always compute F_first using the PARTIAL R^2_p of the excluded instruments (not the full first-stage R^2 inflated by exogenous controls!) and residual degrees of freedom df_2 = N - K_exog - L_excl.
=LET(n_obs, B2, k_ex, B3, l_iv, B4, r2_p, B5, df_2, n_obs - k_ex - l_iv, IF(OR(l_iv < 1, k_ex < 0, df_2 <= 0, r2_p < 0, r2_p >= 1), "Invalid First-Stage Design", ROUND((r2_p / l_iv) / ((1 - r2_p) / df_2), 4)))Approximate 2SLS-to-OLS Relative Bias Ratio 1 / (F_first + 1) Formula
Combine F_first >= 10 with 1 / (F_first + 1) so collaborators see why weak instruments require Anderson-Rubin weak-IV-robust confidence intervals rather than naive 2SLS Wald z-tests.
=IF(ISTEXT(E2), E2, ROUND(1 / (E2 + 1), 4))Try the example
Use the original values or change them to test your understanding.