Lesson M10.L04: Weak Instruments

Field	Detail
Module	M10 — Instrumental Variables
Level	Graduate (ECMT5001 S1 2027)
Duration	30–35 min
Learning Objective	Diagnose weak instruments using the first-stage F-statistic; understand why weak instruments cause bias and poor inference; apply the Stock-Yogo rule of thumb and interpret the Sargan-Hansen J-test in overidentified models
Provenance	Wooldridge, Introductory Econometrics, Ch 15.5

Explanation

The Weak Instrument Problem

Even if an instrument \(z\) is technically relevant (\(\text{Cov}(z,x) \neq 0\)) and exogenous (\(\text{Cov}(z,u) = 0\)), it may be weakly correlated with the endogenous variable. Weak instruments cause severe problems in finite samples:

Problem 1: Bias Toward OLS

In finite samples, the IV estimator is biased. The direction and magnitude of the finite-sample bias depends on the correlation between the instrument and the endogenous variable. When the instrument is weak:

\[\text{Bias}(\hat{\beta}^{\text{IV}}) \approx \frac{1}{F_1 - 1} \cdot \text{Bias}(\hat{\beta}^{\text{OLS}})\]

where \(F_1\) is the first-stage \(F\)-statistic. As \(F_1 \to 1\) (essentially no first stage), IV bias \(\to\) OLS bias — the IV estimator provides no improvement over the biased OLS.

Interpretation: A weak instrument yields an IV estimate that is contaminated by the same endogeneity it is supposed to cure. You've gone to the trouble of finding an instrument and the inference is nearly as bad as OLS.

Problem 2: Large Standard Errors

From M10.L02, \(\text{Var}(\hat{\beta}^{\text{2SLS}}) \propto 1/R^2_1\) where \(R^2_1\) is the first-stage R². A weak instrument means small \(R^2_1\) → enormous SEs → very wide confidence intervals.

Problem 3: Size Distortions

Standard 2SLS \(t\)-tests dramatically over-reject the null when instruments are weak. Staiger and Stock (1997) showed that even when the null is true, \(t\)-tests at the 5% level can reject at rates of 30–40% when \(F_1 \approx 3\). Nominal 5% tests become 40% tests — a catastrophic failure.

First-Stage F-Statistic: The Rule of Thumb

The Stock-Yogo (2005) rule of thumb provides a simple diagnostic:

\[\boxed{F_1 > 10 \quad \Rightarrow \quad \text{instrument is sufficiently strong}}\]

where \(F_1\) is the \(F\)-statistic for the joint hypothesis that all instruments are excluded from the first stage (\(H_0: \pi_2 = \pi_3 = \cdots = 0\)).

Derivation context: Stock & Yogo derived critical values such that, if \(F > c^*\), the worst-case relative bias of 2SLS over OLS is less than 10%. The rule \(F > 10\) corresponds approximately to this 10% maximal bias criterion.

The Staiger-Stock Rule (One Instrument)

For the special case of one instrument and one endogenous variable, the first-stage is a simple regression with \(t\)-statistic \(t_z\) on the instrument coefficient. The rule:

\[|t_z| > 3.2 \quad \Leftrightarrow \quad F_1 = t_z^2 > 10.24 \approx 10\]

A \(t\)-statistic of 2.0 (significant at 5% in a standard test) implies \(F_1 = 4 < 10\) — the instrument is weak even though the first-stage coefficient is conventionally significant!

The Stock-Yogo Weak Instrument Test

More formally, Stock & Yogo (2005) tabulate critical values \(c^*(\ell, m, b)\) where: - \(\ell\) = number of instruments
- \(m\) = number of endogenous variables
- \(b\) = maximal acceptable relative bias (typically 10%)

\(\ell\) instruments (one endogenous)	\(c^*\) (10% max bias)
1	16.38
2	19.93
3	22.30

With one instrument, the formal threshold is actually \(F > 16.38\) for 10% maximal relative bias — the rule of thumb \(F > 10\) is a lower bound. Many practitioners use \(F > 10\) as a screening threshold and \(F > 16\) as a stronger standard.

The Sargan-Hansen J-Test (Revisited)

In the over-identified case (\(\ell > m\)), the Sargan-Hansen J-test tests instrument exogeneity:

\[H_0: \text{Cov}(\mathbf{z}, u) = 0 \quad \text{(all instruments are exogenous)}\]

Procedure: 1. Estimate by 2SLS; save residuals \(\hat{u}_i^{\text{2SLS}}\)
2. Regress \(\hat{u}_i^{\text{2SLS}}\) on all instruments \(\mathbf{z}\) and exogenous controls \(\mathbf{w}\)
3. \(J = n \cdot R^2_{\text{aux}} \sim \chi^2_{\ell - m}\) under \(H_0\)
4. Reject \(H_0\) if \(J > \chi^2_{\ell-m,\alpha}\)

Important caveats: - J-test is not available in exactly identified models (\(\ell = m\))
- Rejecting \(H_0\) means at least one instrument is endogenous — but the test doesn't tell you which
- Failing to reject is not confirmation of exogeneity — just insufficient evidence against it
- J-test has low power when instruments are weak

Remedies for Weak Instruments

Remedy	Description
Find stronger instruments	Return to theory/institutional knowledge for better \(z\)
Anderson-Rubin confidence sets	Weak-instrument robust inference (AR intervals)
Conditional Likelihood Ratio (CLR)	Moreira (2003); optimal weak-instrument-robust test
LIML (Limited Info Maximum Likelihood)	Less biased than 2SLS in weak instrument settings
Report OLS as benchmark	When instruments are too weak, OLS bias may be preferable to IV variance

Worked Example

Setup: Wage regression with quarter of birth (Q1–Q3) as instruments for education (Angrist-Krueger design).

\[\log(\text{wage}_i) = \beta_0 + \beta_1 \text{educ}_i + \beta_2 \text{age}_i + u_i\]

Three instruments: \(z_1 = \mathbf{1}[\text{Q1 birth}]\), \(z_2 = \mathbf{1}[\text{Q2 birth}]\), \(z_3 = \mathbf{1}[\text{Q3 birth}]\) (Q4 is the omitted category).

So \(\ell = 3\), \(m = 1\) → over-identified (J-test has 2 df).

First-stage results:

\[\text{educ}_i = \pi_0 + \pi_1 \text{age}_i + \pi_2 z_{1i} + \pi_3 z_{2i} + \pi_4 z_{3i} + v_i\]

Variable	\(\hat{\pi}\)	SE	\(t\)
Intercept	16.72	0.41	40.8
age	−0.082	0.008	−10.3
Q1 birth (\(z_1\))	−0.091	0.032	−2.84
Q2 birth (\(z_2\))	−0.052	0.032	−1.63
Q3 birth (\(z_3\))	−0.024	0.032	−0.75

Joint F-test (H₀: \(\pi_2 = \pi_3 = \pi_4 = 0\)): \(F = 3.4\)

Weak instrument diagnosis: \(F = 3.4 < 10\) → instruments are weak. Despite Q1 birth having a borderline significant first-stage coefficient (\(t = -2.84\)), the joint \(F\) falls well below the Stock-Yogo threshold.

Consequences: - 2SLS estimates will be biased toward OLS estimates
- Standard errors are unreliable; confidence intervals are misleading
- Nominal 5% tests may reject at rates far above 5%

Comparing weak vs strong instrument estimates:

Method	\(\hat{\beta}_1\) (educ)	SE	\(t\)
OLS	0.067	0.003	22.3
2SLS (weak, \(F=3.4\))	0.074	0.052	1.42
2SLS (strong, \(F=12.4\))	0.129	0.047	2.74

The weak-instrument 2SLS estimate (0.074) is close to the OLS estimate (0.067) — consistent with bias toward OLS. The strong-instrument estimate (0.129, from the proximity-to-college design) differs substantially. Standard errors for weak IV are enormous (0.052 vs 0.047 for strong IV).

Sargan-Hansen J-test (over-identified, \(\ell - m = 2\)):

\(J = 4.12\). Critical value \(\chi^2_{2,0.05} = 5.99\). Fail to reject → over-identifying restrictions not violated.

Common Misconception

"A significant first-stage coefficient guarantees a strong instrument — if the \(t\)-stat on \(z\) is above 2, the instrument is fine."

This is false. Statistical significance at the 5% level (\(|t| > 1.96\) or equivalently \(F > 3.84\)) does not imply instrument strength. The Stock-Yogo rule of thumb requires \(F > 10\) (equivalently, \(|t| > 3.16\) for a single instrument). A \(t\)-statistic of 2.5 is statistically significant but implies \(F = 6.25 < 10\) — the instrument may still be "weak" in the sense that 2SLS suffers substantial bias and size distortions.

The rule to remember: - Conventional significance test: \(F > 3.84\) (5% critical value for 1 df)
- Stock-Yogo weak instrument threshold: \(F > 10\) (rule of thumb) or \(F > 16\) (10% max bias criterion)

These are very different thresholds. Always report the first-stage F-statistic in any IV paper or assignment.

Practice Prompts

A researcher uses a single instrument \(z\) with a first-stage coefficient estimate of \(\hat{\pi} = 0.18\) and standard error \(0.08\). Compute the first-stage \(t\)-statistic and \(F\)-statistic. Is the instrument strong by the Stock-Yogo rule of thumb? What maximal relative bias does Stock-Yogo associate with \(F \approx 5\)?
In the Angrist-Krueger design above, all three quarter-of-birth indicators have first-stage \(t\)-statistics below 3.2. A student argues: "We should drop Q2 and Q3, since they're individually insignificant, and keep only Q1 as the instrument — that solves the weak instrument problem." Critique this selection strategy. Does dropping weak instruments necessarily solve the problem?
A researcher finds \(J = 6.8\) with two over-identifying restrictions (df = 2). At the 5% level, do they reject the null of instrument exogeneity? (Recall \(\chi^2_{2,0.05} = 5.99\).) What does rejection imply about their instrument set? Can the researcher determine which instrument is endogenous from this test alone?

Further Resources

Wooldridge, Introductory Econometrics (7th ed.), Ch 15.5
Staiger, D. & Stock, J.H. (1997), "Instrumental Variables Regression with Weak Instruments," Econometrica, 65(3), 557–586
Stock, J.H. & Yogo, M. (2005), "Testing for Weak Instruments in Linear IV Regression," in Identification and Inference for Econometric Models (Andrews & Stock, eds.)
Moreira, M.J. (2003), "A Conditional Likelihood Ratio Test for Structural Models," Econometrica
Python: linearmodels.iv.model.IV2SLS, check .first_stage attribute; Stata: ivregress 2sls ..., first