IV Diagnostics¶

Quick Reference

Applies to: PanelIV results Import: from panelbox.models.iv import PanelIV Stata equivalent: estat firststage, estat overid, estat endogenous R equivalent: ivreg::summary.ivreg(diagnostics=TRUE)

Overview¶

Instrumental Variables estimation requires careful diagnostic checking. Unlike OLS, where the main concerns are specification and heteroskedasticity, IV estimation can fail silently if instruments are weak, invalid, or unnecessary. Three fundamental questions must be addressed:

Are the instruments strong enough? (Relevance -- testable)
Are the instruments valid? (Exogeneity -- partially testable when overidentified)
Is IV actually needed? (Is the regressor truly endogenous?)

This page covers the diagnostic tools available in PanelBox for answering these questions and provides practical guidance for interpreting results.

Three Requirements for Valid Instruments¶

Requirement	Formal condition	Testable?	Key diagnostic
Relevance	\(\text{Cov}(Z, X) \neq 0\)	Yes	First-stage F-statistic
Exogeneity	\(\text{Cov}(Z, \varepsilon) = 0\)	Partially (if overidentified)	Hansen J / Sargan test
Exclusion	\(Z\) affects \(y\) only through \(X\)	No (maintained assumption)	Economic reasoning

Weak Instruments Test¶

First-Stage F-Statistic¶

The most important IV diagnostic. The first-stage regression tests whether the instruments have sufficient predictive power for the endogenous variable.

from panelbox.models.iv import PanelIV

model = PanelIV(
    formula="invest ~ capital + value | capital + lag_value + lag2_value",
    data=df,
    entity_col="firm",
    time_col="year",
    model_type="fe",
)
results = model.fit(cov_type="clustered")

# Check first-stage F for each endogenous variable
for var, fs in results.first_stage_results.items():
    f_stat = fs["f_statistic"]
    r2 = fs["rsquared"]
    print(f"Endogenous variable: {var}")
    print(f"  First-stage F-statistic: {f_stat:.2f}")
    print(f"  First-stage R-squared:   {r2:.4f}")
    if f_stat < 10:
        print("  WARNING: Weak instruments (F < 10)")
    else:
        print("  OK: Instruments are sufficiently strong")

Interpreting the F-Statistic¶

The Stock & Yogo (2005) rule of thumb:

F-statistic	Interpretation	Action
\(F > 10\)	Strong instruments	Proceed with 2SLS
\(F \in [5, 10]\)	Borderline	Results should be interpreted with caution
\(F < 5\)	Weak instruments	Do not trust 2SLS; find better instruments

Automatic Weak Instruments Warning¶

PanelBox automatically flags weak instruments:

if results.weak_instruments:
    print("At least one endogenous variable has a weak first stage (F < 10)")

Consequences of Weak Instruments¶

When instruments are weak:

2SLS estimates are biased toward OLS (the very estimator you are trying to improve upon)
Standard errors are unreliable (often understated)
Confidence intervals have poor coverage
Test statistics have incorrect size (too many rejections)

What to Do About Weak Instruments¶

Find better instruments: the best solution is stronger instruments
Use more instruments: additional relevant instruments increase F (but beware overfitting)
Consider LIML: Limited Information Maximum Likelihood is less biased than 2SLS with weak instruments
Use GMM: if the endogeneity comes from dynamic structure, Arellano-Bond GMM generates instruments internally
Report reduced-form: regress \(y\) directly on \(Z\) to show the instrument has an effect

First-Stage Diagnostics¶

Partial R-Squared¶

The first-stage \(R^2\) measures how much variation in the endogenous variable is explained by all instruments plus exogenous controls:

for var, fs in results.first_stage_results.items():
    print(f"{var}: R² = {fs['rsquared']:.4f}")

A high \(R^2\) is necessary but not sufficient -- what matters is the marginal contribution of the excluded instruments (captured by the F-statistic).

Multiple Endogenous Variables¶

With multiple endogenous variables, check each first-stage separately:

# Each endogenous variable has its own first-stage regression
for var in results.endogenous_vars:
    fs = results.first_stage_results[var]
    print(f"\n--- First stage: {var} ---")
    print(f"  F-stat:    {fs['f_statistic']:.2f}")
    print(f"  R-squared: {fs['rsquared']:.4f}")

Multiple endogenous variables

With multiple endogenous variables, each needs its own strong first stage. A high F for one variable does not help if another has a weak first stage. In this case, Shea's partial \(R^2\) (comparing the full and restricted first-stage regressions) provides a more informative diagnostic.

Overidentification Test¶

When you have more instruments than endogenous variables (overidentified model), you can test whether all instruments are jointly valid.

Hansen J / Sargan Test¶

\(H_0\): All instruments are exogenous (\(\text{Cov}(Z_j, \varepsilon) = 0\) for all \(j\))
\(H_1\): At least one instrument is endogenous

# The overidentification test is available when:
#   n_excluded_instruments > n_endogenous
n_excluded = results.n_instruments - len(results.exogenous_vars)
n_endog = results.n_endogenous
print(f"Excluded instruments: {n_excluded}")
print(f"Endogenous variables: {n_endog}")

if n_excluded > n_endog:
    print("Model is overidentified -- can test instrument validity")
else:
    print("Model is exactly identified -- cannot test instrument validity")

Interpreting the Test¶

Test result	Interpretation	Action
Fail to reject (\(p > 0.05\))	Instruments appear valid	Proceed
Reject (\(p < 0.05\))	At least one instrument may be invalid	Investigate which instruments are problematic

Limitations

The overidentification test requires that at least one instrument is valid. It tests validity of the "extra" instruments relative to the just-identified case. If all instruments are invalid, the test may not reject. It is a necessary but not sufficient check.

Endogeneity Test¶

Is IV Actually Needed?¶

Before using IV, test whether the regressor is actually endogenous. If it is not, OLS is more efficient.

Durbin-Wu-Hausman Test¶

Compare OLS and 2SLS estimates. Under the null of exogeneity, both are consistent but OLS is more efficient. Under the alternative, only 2SLS is consistent.

The procedure:

Estimate the first stage and obtain residuals \(\hat{\nu}\)
Include \(\hat{\nu}\) in the outcome equation alongside \(X\)
Test whether the coefficient on \(\hat{\nu}\) is zero

If the coefficient on \(\hat{\nu}\) is significant, the regressor is endogenous and IV is needed.

Practical Comparison¶

# Fit both OLS and IV
from panelbox import FixedEffects

# Standard FE (assumes exogeneity)
ols_model = FixedEffects("invest ~ capital + value", df, "firm", "year")
ols_results = ols_model.fit(cov_type="clustered")

# IV FE (allows endogeneity)
iv_model = PanelIV(
    "invest ~ capital + value | capital + lag_value + lag2_value",
    df, "firm", "year", model_type="fe",
)
iv_results = iv_model.fit(cov_type="clustered")

# Compare coefficients
print("Variable     | OLS      | IV       | Difference")
print("-" * 55)
for var in ols_results.params.index:
    if var in iv_results.params.index:
        ols_coef = ols_results.params[var]
        iv_coef = iv_results.params[var]
        print(f"{var:<12} | {ols_coef:>8.4f} | {iv_coef:>8.4f} | {iv_coef - ols_coef:>8.4f}")

If the coefficients are similar, endogeneity may not be a practical concern.

Diagnostic Checklist¶

Use this checklist when reporting IV results:

1. Instrument Relevance¶

# First-stage F > 10?
for var, fs in results.first_stage_results.items():
    assert fs["f_statistic"] > 10, f"Weak instruments for {var}"
print("PASS: Instruments are strong")

2. Instrument Validity (if overidentified)¶

# Hansen J test not rejected?
n_excluded = results.n_instruments - len(results.exogenous_vars)
if n_excluded > results.n_endogenous:
    # Overidentification test available
    print("Check: overidentification test p-value > 0.05")
else:
    print("Exactly identified: cannot test overidentification")

3. Endogeneity Justified¶

# Hausman test rejects OLS?
print("Check: OLS and IV estimates differ significantly")
print("If similar, OLS is more efficient -- use OLS")

4. Results Plausible¶

# Coefficients have expected signs and reasonable magnitudes?
print(results.params)
print("Check: signs and magnitudes are economically plausible")

Summary Table¶

Step	Test	Threshold	Result
1	First-stage F	\(> 10\)	Instruments strong?
2	Hansen J (if overidentified)	\(p > 0.05\)	Instruments valid?
3	Hausman (OLS vs IV)	\(p < 0.05\)	Endogeneity confirmed?
4	Economic plausibility	--	Signs and magnitudes sensible?

If all four pass, the IV results are credible. If any fail, investigate further.

Common Pitfalls¶

Too Many Instruments¶

Using many instruments can:

Overfit the first stage (high F but biased 2SLS)
Make the overidentification test lose power
Bias 2SLS toward OLS in finite samples

Rule: keep the number of instruments modest relative to the sample size.

Instruments That Are "Too Good"¶

If the first-stage \(R^2\) is very close to 1, the instruments may be essentially the same as the endogenous variable. This can happen with lagged values when the variable is highly persistent.

Testing Exclusion Restrictions¶

The exclusion restriction -- that instruments affect \(y\) only through \(X\) -- cannot be tested statistically. It must be justified by economic reasoning. The overidentification test is a necessary but not sufficient check (it assumes at least one instrument is valid).

Tutorials¶

Tutorial	Description	Link
IV Diagnostics	Complete diagnostic walkthrough	Static Models Tutorial

References¶

Stock, J. H., & Yogo, M. (2005). Testing for weak instruments in linear IV regression. In D. W. K. Andrews & J. H. Stock (Eds.), Identification and Inference for Econometric Models (pp. 80-108). Cambridge University Press.
Sargan, J. D. (1958). The estimation of economic relationships using instrumental variables. Econometrica, 26(3), 393-415.
Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica, 50(4), 1029-1054.
Hausman, J. A. (1978). Specification tests in econometrics. Econometrica, 46(6), 1251-1271.
Angrist, J. D., & Pischke, J.-S. (2009). Mostly Harmless Econometrics. Princeton University Press. Chapter 4.
Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data (2^nd ed.). MIT Press. Chapters 5-6.