The Two Core Requirements of Instrumental Variables: How to Argue Relevance and Exogeneity? What Tests Are Needed? From What Angles Should the Argument Be Constructed?

I. Introduction: You Found an Instrumental Variable—Then the Reviewer Asked Three Questions, and You Could Only Answer One

You are studying the effect of years of education on income. You know that OLS has a classic endogeneity problem—ability ($u_i$) affects both educational choices and income levels, and omitting ability biases the coefficient on years of education. You need an instrumental variable.

You read a classic paper—Angrist and Krueger (1991) used quarter of birth as an instrument for education. The logic is: compulsory schooling laws require students to enter school in September of the year they turn 6 and allow them to drop out after turning 16—children born early in the year reach the dropout age earlier than those born late in the year, so they complete fewer years of schooling. Quarter of birth affects years of education (relevance)—but quarter of birth should not seem to directly affect income (exogeneity).

You decide to follow suit. You find a set of instrumental variables and run two-stage least squares (2SLS):

ivreg2 income (education = quarter_of_birth), first

The output shows: first-stage F = 18.7 (> 10, decent), first-stage coefficient = 0.082 (quarter of birth and education are positively correlated). You use this result to write the first draft of your paper.

Then the reviewer asks three questions:

1. "Your instrument is relevant—the first-stage F is indeed greater than 10. But is greater than 10 enough? How large is 'strong enough'?"

2. "Is your instrument exogenous? You say quarter of birth 'should not seem to' directly affect income—but can you prove it? Are there possible channels—such as quarter of birth affecting maturity advantages at school entry, which in turn affect long-run cognitive development—such that quarter of birth affects income through channels other than education?"

3. "How many instruments do you have? Only one? If you only have one, how do you test exogeneity? If you have multiple—did the overidentification test pass? Does passing it mean all instruments are exogenous?"

You can still answer the first question using the Stock-Yogo critical value table. You are stumped by the latter two—because exogeneity, in essence, cannot be fully proven by any statistical test.

Core Message: The argument for instrumental variables consists of two asymmetric parts. Relevance can be statistically tested—the first-stage F statistic (> 10 is the old standard; > 104.7 is the new threshold for 5% worst-case bias), the Stock-Yogo critical values for weak instruments, and limited information maximum likelihood (LIML) as a robust alternative under weak IVs—these tools provide clear, quantifiable criteria. Exogeneity cannot be formally tested—it is essentially an argument about the "exclusion restriction," which can only be supported through theoretical and institutional reasoning, falsification tests, overidentification tests (when multiple IVs are available), and sensitivity analyses that "accumulate evidence" rather than "provide proof." The most powerful angles for arguing exogeneity come from three paths: (1) institutional/natural experiment narratives—why can the source of variation in this IV be reasonably viewed as "as-if random"? (2) falsification tests—in samples where no causal effect should theoretically exist, is the reduced-form effect of the IV on Y insignificant? (3) ruling out alternative channels—arguing one by one why the IV cannot affect Y through paths other than X. A good IV paper does not merely report first-stage F > 10 and an overidentification test p > 0.05—it tells a complete story: where this instrument comes from, why it affects Y only through X, and what efforts you made to convince the reader of this.

II. The Two Necessary Conditions for Instrumental Variables—What Are Relevance and Exogeneity?

2.1 What Problem Do Instrumental Variables Solve?

The standard regression model:

$Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i$

If $\text{Cov}(X_i, \varepsilon_i) \neq 0$ (X is endogenous—due to omitted variables, measurement error, or reverse causality), the OLS estimator $\hat{\beta}_1$ is inconsistent.

The instrumental variable Z is introduced to extract the variation in X that is uncorrelated with $\varepsilon$—using only this clean variation to identify $\beta_1$. This is two-stage least squares (2SLS):

• First stage: Use Z (and all exogenous control variables) to predict X, obtaining $\hat{X}$.
• Second stage: Replace X with $\hat{X}$ and regress Y on it.

2.2 Two Necessary Conditions—If Either Is Missing, IV Fails

The instrumental variable Z must simultaneously satisfy:

Condition One: Relevance

$\text{Cov}(Z_i, X_i) \neq 0$

Z must have a sufficiently strong partial correlation with the endogenous variable X (after controlling for other exogenous variables). If the correlation between Z and X is too weak—the "weak instrument problem"—2SLS is not only biased (biased toward the OLS direction), but standard errors are underestimated, making inference unreliable.

Condition Two: Exogeneity / Exclusion Restriction

$\text{Cov}(Z_i, \varepsilon_i) = 0$

Z is uncorrelated with the error term in the structural equation. Equivalently—Z affects Y only through the single path of X. Z cannot enter the equation for Y directly, nor can it indirectly affect Y through any uncontrolled intermediate variable.

Together, the two conditions mean: Z is an "exogenous shifter" of X—it pushes X to change, but it does not itself appear in the equation for Y. Like an instrument in a laboratory—it indirectly affects Y by manipulating X, but it never touches Y directly.

2.3 Why Is Exogeneity Harder to Argue Than Relevance?

Relevance is a statistical property—you can quantify it with data and tests. Exogeneity is a causal statement—"Z does not affect Y, except through the path of X"—which is logically impossible to test directly with data. This is because what you need to test is $\text{Cov}(Z, \varepsilon) = 0$—but $\varepsilon$ is unobservable. All you can observe is $\hat{\varepsilon} = Y - \hat{\beta}_0 - \hat{\beta}_1 X$—and these residuals are themselves computed under the assumption that the IV is valid.

This is a fundamental asymmetry: you can use data to quantify how "strong" your instrument is—but you cannot use data to prove that your instrument is "clean." The latter requires not more tests, but a better story, deeper institutional understanding, and more creative falsification designs.

III. Arguing Relevance—From "First-Stage F > 10" to a Complete Weak Instrument Diagnostic

3.1 Step One: The First-Stage Regression—Look at the F Statistic

The form of the first-stage regression (assuming one endogenous variable X, one instrument Z, and K control variables W):

$X_i = \pi_0 + \pi_1 Z_i + \pi_2 W_{1i} + \dots + \pi_{K+1} W_{Ki} + u_i$

The core question is testing $H_0: \pi_1 = 0$ (Z and X are unrelated—the instrument is irrelevant).

The first-stage F statistic is the standard test of this null hypothesis. The larger the F statistic → the stronger the relationship between Z and X → the more "relevant" the instrument.

* Report first-stage F and coefficients
ivreg2 y (x = z), first
* Output includes first-stage F, partial R², etc.

3.2 Step Two: Understanding the F Threshold—From "F > 10" to Stock-Yogo Critical Values

The old standard: F > 10 (Staiger and Stock, 1997)

Staiger and Stock's rule of thumb—if the first-stage F < 10, your instrument is "weak." Under weak instruments, the finite-sample bias of 2SLS can be substantial (biased toward the inconsistent OLS estimator), and standard errors are underestimated, so confidence interval coverage is far below nominal levels.

But "F > 10" is only a rule of thumb—it does not account for the number of endogenous variables, the number of instruments, or how much worst-case bias you are willing to tolerate.

The new standard: Stock-Yogo (2005) critical values

Stock and Yogo provide a more precise diagnostic framework. They compute the minimum first-stage F required for a given tolerance of "worst-case relative bias":

Recommendation: In modern empirical papers, cite Stock-Yogo critical values rather than simply saying "F > 10." If your F = 18.7, you should write: "The first-stage F statistic is 18.7, exceeding the Stock-Yogo (2005) 10% worst-case bias critical value of 16.38—indicating that weak IV is not a serious concern in this paper."

Recent reassessment: Lee, McCrary, Moreira, and Porter (2022) suggest a more conservative standard—controlling worst-case over-rejection at the 5% significance level requires F statistics far larger than the old critical values (for a single instrument and a single endogenous variable, they suggest F > 104.7). In practice, this is a very high bar—most IV studies have first-stage F statistics well below this. Therefore, in practice, one should not mechanically apply any single threshold—instead, one should combine the F statistic, the robustness of LIML (limited information maximum likelihood), and the Anderson-Rubin test to assess the severity of weak IV from multiple dimensions.

3.3 Step Three: Multiple Endogenous Variables and Multiple IVs—More Than One F

With multiple endogenous variables, looking only at the F for a single variable is insufficient—you need to examine the partial F statistics and the Cragg-Donald Wald F statistic (the multivariate generalization of the first-stage F).

ivreg2 y (x1 x2 = z1 z2 z3), first
* Focus on the Cragg-Donald Wald F statistic
* And the Kleibergen-Paap rk Wald F (when using clustered or heteroskedasticity-robust standard errors)

If the Cragg-Donald F (or Kleibergen-Paap F) is below the Stock-Yogo critical value → you may have a weak instrument problem.

3.4 Step Four: What If the Instruments Are Truly Weak?

If the first-stage F is below the Stock-Yogo critical value:

1. Use LIML (limited information maximum likelihood) instead of 2SLS. LIML has far smaller finite-sample bias than 2SLS under weak instruments and is the default choice in weak-instrument settings. The drawback of LIML is that standard errors are slightly larger than those of 2SLS (moderately so).

ivreg2 y (x = z), liml

2. Report the Anderson-Rubin test. The AR test directly tests "$\beta_1$ = some candidate value"—it is robust to weak instruments—and does not depend on a judgment of "weak vs. strong IV."

ivreg2 y (x = z), first
* The AR test appears in the first-stage output (Weak identification robust tests)

3. Use the Fuller corrected estimator (Fuller-k estimator)—k = 1 corresponds to LIML, k = 4 corresponds to Fuller's suggested adjustment, which reduces standard errors while preserving LIML's bias properties.

4. Look for stronger instruments. If none of the above suffices—the most honest approach is to acknowledge that "the instrument in this paper suffers from a weak IV problem; conclusions should be based on Anderson-Rubin confidence intervals, and future research should seek stronger instruments."

3.5 Complete Checklist for Arguing Relevance

• Report the first-stage F statistic (or Cragg-Donald / Kleibergen-Paap F).
• Compare against the Stock-Yogo critical value table—state at what bias tolerance your instrument passes the weak IV test.
• Report the full coefficient table from the first-stage regression—not just F, but also whether the magnitude and sign of $\pi_1$ align with theoretical expectations.
• If more than one IV → report the overidentification test (note: this does not directly test relevance, but too many weak IVs can amplify 2SLS bias—using LIML rather than 2SLS can reduce this bias).
• If the first-stage F is not high enough → use LIML as the main specification and report Anderson-Rubin weak-IV-robust confidence intervals.

IV. Arguing Exogeneity—Cannot Be Directly Tested, but Can Be "Besieged"

4.1 The Fundamental Dilemma: The Exclusion Restriction Cannot Be Statistically Tested

The exogeneity condition $\text{Cov}(Z, \varepsilon) = 0$—Z affects Y only through X—is logically untestable. Because $\varepsilon$ contains all unobserved factors that affect Y and are not mediated by X—you cannot observe $\varepsilon$, and therefore cannot test whether it is correlated with Z.

This means: any claim that "I used some test to prove the exogeneity of my instrument" is wrong. What you can do is not "prove" exogeneity—but accumulate evidence from various sources that makes the exogeneity assumption "more credible." This requires not p-values, but a layered progression of narrative, institutions, and falsification.

4.2 Argument Angle One: Institutional and Theoretical Narrative—"Why Is Z As-If Random?"

The most powerful exogeneity arguments typically come from institutional and natural experiment narratives—not from statistical tests.

The core question is: In your research setting, what is the source of variation in Z? Why can this variation be reasonably viewed as an "exogenous shifter"—that is, why is it uncorrelated with other factors that determine Y but are not controlled in the model?

Three classic narrative patterns:

Pattern A: Natural experiment / institutional shock

"Federal law required all states to raise the minimum legal dropout age from 16 to 17—this law was implemented uniformly nationwide, not directly influenced by each state's own economic conditions or educational attainment. Whether a student is affected by this law depends on their year of birth—whether it falls before or after the policy implementation year. The interaction of birth year and policy creates an exogenous source of variation in years of education."

→ Core argument: The design and implementation of the policy is exogenous (not determined by the behavior of affected individuals) + the policy's impact on individuals depends on a characteristic that is exogenous to individual control (year of birth, region of residence, etc.).

Pattern B: Distance / geographic characteristics

"The distance from each county to the nearest two-year college—college location decisions were typically determined by historical factors decades earlier (such as railroad lines, land grants), not by current local income levels or employment rates."

→ Core argument: Historical or geographic factors are fixed in the short run—they are not influenced by current individual behavior.

Pattern C: Random / quasi-random assignment

"The draft lottery—the random drawing of birth dates (1–365) is a completely random process. Birth date determines whether an individual is drafted into military service (X—whether one serves), but the birth date itself is uncorrelated with any pre-conscription individual characteristics (family background, education, ability)."

→ Core argument: The data-generating process of Z is a known, auditable random process.

The common thread across these three narratives: They are not saying "tests show that Z and ε are uncorrelated"—they are saying "the data-generating process of Z makes the correlation between Z and ε logically implausible." This is the gold standard of exogeneity arguments.

4.3 Argument Angle Two: Overidentification Tests (When You Have Multiple Instruments)

If you have at least two instruments—and at least one of them is valid (exogenous)—you can use an overidentification test to examine:

$H_0: \text{All instruments are exogenous}$

Intuition: If all IVs are exogenous, then the $\hat{\beta}$ estimated using any subset of instruments should be "close" (within sampling error) to the $\hat{\beta}$ estimated using another subset. If some IVs are endogenous, different subsets of IVs will give systematically different $\hat{\beta}$ estimates—the overidentification test detects this inconsistency.

• Sargan test: Used under conditional homoskedasticity.
• Hansen J test: Used under heteroskedasticity or clustering (but may lack power with few instruments and many endogenous variables).

ivreg2 y (x = z1 z2 z3), robust
* Hansen J statistic: p > 0.05 → cannot reject "all IVs are exogenous"

Three key limitations of overidentification tests (this is what you need to know to avoid over-relying on them):

1. "Failure to reject the null" ≠ "evidence that all IVs are exogenous." Insufficient sample size, collinearity among IVs, or lack of power in the test itself can all produce p > 0.05 even when the IVs are in fact endogenous. This is a "silent defense"—not affirmative evidence.

2. The reliability of overidentification tests depends on the premise that "at least one IV is valid." If all IVs are endogenous—and their endogeneity biases happen to point in the same direction (e.g., all IVs are positively correlated with the same omitted variable)—the overidentification test may still be insignificant. This premise itself is untestable.

3. When you have many IVs (e.g., > 5), the power of overidentification tests declines, while the bias of 2SLS under weak IVs increases. Ironically: more IVs give you overidentifying freedom—but may also amplify weak-IV bias.

Practical advice on overidentification tests: If you have multiple IVs → report the Hansen J p-value, but do not write "the J test proves the exogeneity of the IVs." The correct phrasing is "the overidentification test cannot reject the null hypothesis that the instruments are exogenous (Hansen J p = 0.42), which is consistent with the exogeneity argument based on institutional analysis in this paper, but does not constitute independent proof of exogeneity."

4.4 Argument Angle Three: Falsification Tests—"If the IV Should Be Ineffective Somewhere, Is It Indeed Ineffective?"

Falsification tests are among the most powerful ways to argue for exogeneity—they do not attempt to directly prove "Cov(Z, ε) = 0," but rather attempt to test whether the IV is indeed ineffective in settings where it should be ineffective.

Common strategies:

Strategy 1: Test the reduced form in samples that should theoretically be unaffected

If your IV is "changes in local education spending policy"—this policy should only affect educational outcomes of school-age children, and should not affect the income of adults who have already graduated. Run a reduced-form regression (Y on Z) on the subsample of individuals older than 25 who have completed their education—if Z still significantly predicts Y in this group → Z may affect Y through channels other than X (exogeneity is called into question).

In short: $Y_i = \pi_0 + \pi_1 Z_i + \varepsilon_i$ (reduced form) in a subsample where the "pathway is blocked"—$\pi_1$ should ≈ 0. If $\pi_1$ is significantly nonzero → exogeneity is doubtful.

Strategy 2: Test whether the IV predicts predetermined variables

If Z is exogenous—it should not be systematically correlated with any individual characteristics determined before treatment. Select a set of important predetermined variables determined before Z is realized (such as gender, year of birth, pre-treatment income/education levels), and regress each variable on Z—check whether the coefficient on Z is significant.

This approach is analogous to the balance test in RCTs: in a randomized experiment, treatment and control groups should be balanced on pre-treatment observed characteristics. Similarly—an exogenous Z should be "balanced" on pre-treatment characteristics (Z should not predict these characteristics).

Strategy 3: Use a "placebo outcome"

Find a variable that is theoretically unrelated to X but may share common confounders with Y—if Z affects Y through channels other than X, this placebo outcome should also be predicted by Z. If Z does not predict the placebo outcome but does predict Y → supports exogeneity.

The logical essence of falsification tests: You fire various "attacks" at the wall of exogeneity—if the wall withstands all attacks (all falsification tests are insignificant), you are accumulating evidence that "this wall might be real." But you cannot say this wall is impossible to breach—you simply have not found the attack that breaches it.

4.5 Argument Angle Four: Ruling Out Alternative Channels—Closing Each Alternative Path from Z to Y One by One

This is the part that requires the most "thinking" rather than "computing." For your instrument, draw a diagram of all logically possible causal paths from Z to Y—then for each path other than through X, argue one by one why, in your research setting, this path is closed or already controlled.

Using Angrist and Krueger (1991)—"quarter of birth"—as an example:

• Path 1 (main path): Quarter of birth → years of education (due to compulsory schooling laws) → income. ✅

• Alternative channel 1: Quarter of birth → relative age at school entry (maturity advantage) → cognitive development → income (not through education). Angrist and Krueger's response: if such a "relative age effect" exists, it should be strongest in early grades and diminish as grade level increases—but they do not observe this pattern of decay in the data.

• Alternative channel 2: Quarter of birth → season of birth (nutrition, sunlight, maternal health) → long-run health and cognition → income. Response: if seasonal birth effects exist, they should be consistent across regions and time periods—empirically, such effects are very small and inconsistent.

• Alternative channel 3: Parental fertility planning—certain types of parents are more likely to have children in particular seasons (e.g., high-SES parents tend to have children in the spring). Response: if such selection effects exist, quarter of birth should be correlated with parental education/income—but balance tests show no such correlation.

How to write this in your paper: In your instrumental variables section, add a subsection on "ruling out alternative channels"—"As shown in Table X, the IV is not systematically correlated with pre-treatment observable characteristics (balance), and the reduced form is insignificant in subsamples that should theoretically be unaffected (falsification tests)—these findings do not prove exogeneity, but are consistent with the institutional argument above and support the validity of the IV."

4.6 Argument Angle Five: Sensitivity Analysis—"Even If Exogeneity Does Not Hold Perfectly, Would My Conclusions Flip?"

In recent years, an increasing number of empirical papers have begun to include sensitivity analysis—not claiming that "the IV is perfectly exogenous," but asking: "If the IV is not perfectly exogenous—that is, if it has a small, direct effect independent of X—would my conclusions still hold?"

Conley, Hansen, and Rossi (2012) provide a framework: allow the IV to have a "plausibly exogenous" bias, then re-estimate the confidence interval for β when the correlation between Z and ε lies within some range (the "support interval").

Intuition: Instead of insisting on $\text{Cov}(Z, \varepsilon) = 0$ (perfect exogeneity) → assume $\text{Cov}(Z, \varepsilon)$ lies within $[-\delta, +\delta]$ (plausible exogeneity) → see whether the confidence interval for $\hat{\beta}$ still excludes zero over this range. If your conclusions still hold over a reasonable range of $\delta$—then even if the IV is not perfectly exogenous, your main inferences may still be robust.

This type of analysis is not yet standard practice in empirical work—but it represents a more mature and honest attitude toward exogeneity: not merely claiming "Z is exogenous"—but showing readers whether, in the worst reasonable case, your conclusions would flip.

V. A Complete Framework for the Argument—A Systematic Path from Theory to Tests

5.1 The Complete Path for Arguing Relevance

├── 1. Theoretical level: Why should Z affect X? What is the economic theory/institutional mechanism?
│
├── 2. First-stage regression: Report the full coefficient table (not just F, but also coefficient magnitude and sign)
│   ├── Is the sign consistent with theoretical expectations?
│   └── Is the magnitude of the coefficient reasonable?
│
├── 3. First-stage F statistic
│   ├── F > Stock-Yogo critical value (specify bias tolerance)
│   ├── If multiple endogenous variables → Cragg-Donald / Kleibergen-Paap F
│   └── If F is not high enough → use LIML (main specification) + Anderson-Rubin robust confidence intervals
│
└── 4. Weak IV robustness: LIML + AR test + Fuller correction

5.2 The Complete Path for Arguing Exogeneity

├── 1. Institutional/theoretical narrative: What is the source of variation in Z? Why can it be viewed as an "exogenous shifter"?
│   ├── Natural experiment/institutional shock?
│   ├── Distance/geographic characteristics/historical factors?
│   └── Random/quasi-random assignment?
│
├── 2. Statistical falsification tests
│   ├── Balance tests: Does Z systematically predict pre-treatment individual characteristics?
│   ├── Falsification tests: In subsamples that should theoretically be unaffected, is the reduced form insignificant?
│   └── Placebo outcomes: Does Z fail to predict "fake outcomes"?
│
├── 3. Ruling out alternative channels
│   └── List all possible paths from Z to Y (other than through X) one by one → close or control each one
│
├── 4. If multiple IVs → overidentification test (Hansen J)
│   └── Correct phrasing: "cannot reject the null hypothesis," not "proved exogeneity"
│
└── 5. Sensitivity analysis (optional but recommended)
    └── Conley-Hansen-Rossi: If Z is allowed to have "plausibly exogenous" bias, are the conclusions robust?

5.3 Order of Presentation in the Paper

A good IV empirical paper typically presents the instrumental variable argument in the following order:

1. Institutional background and IV construction (Section 3)—"Where does my IV come from, and why can it be reasonably viewed as exogenous"—this is the most important part of the entire paper and usually appears before the data and baseline results.
2. First-stage regression results (Section 4 or the first part of Section 5)—report the first-stage coefficient, sign, and F statistic.
3. Balance and falsification tests (subsequent parts of the same section)—present in table form that "the IV does not predict predetermined characteristics" and "the reduced form is insignificant in irrelevant subsamples."
4. Overidentification test (if applicable, reported alongside the 2SLS output).
5. Weak IV robustness (LIML + AR + Fuller)—presented in the robustness checks tables.

VI. Common Misconceptions

6.1 Misconception One: "Overidentification Test p > 0.05 → My IVs Are All Exogenous"

An insignificant overidentification test means you cannot reject the joint null hypothesis that "all IVs are exogenous." But "cannot reject" ≠ "confirmed as true." There are three reasons: (1) the test has limited power—it may fail to detect systematic violations of exogeneity due to noise; (2) the overidentification test is only meaningful if at least one IV is valid; (3) if all IVs have endogeneity bias in the same direction—the overidentification test may still be insignificant. An overidentification test p > 0.05 provides "no signal of exogeneity violation"—not "exogeneity is confirmed." These are two fundamentally different conclusions.

6.2 Misconception Two: "First-Stage F > 10 → Weak IV Is Not a Problem"

F > 10 is the Staiger-Stock 1997 rule of thumb. Stock-Yogo 2005 research shows that even at a 10% worst-case bias tolerance, the F critical value exceeds 10 in most cases (16.38 is needed for a single IV). And the alternative F thresholds suggested by Lee, McCrary, Moreira, and Porter (2022) are even higher. In your paper, you should cite Stock-Yogo critical values rather than simply saying "F > 10"—and certainly should not abandon LIML or AR tests merely because F > 10.

6.3 Misconception Three: "The Exogeneity of an Instrument Is a Statistical Hypothesis—It Can Be Tested with Data"

Exogeneity is the most central part of the instrumental variable argument—and it is precisely the part that cannot be fully tested with data. What you can do is not "find the right test and then say, 'look, it's exogenous'"—but accumulate evidence from multiple sources: institutional narrative + falsification tests + ruling out alternative channels + overidentification (if available) + sensitivity analysis. This is a comprehensive treatment, not a single test bullet.

6.4 Misconception Four: "I Have Multiple Instruments → One Passed the Overidentification Test → All IVs Can Be Used"

The performance of 2SLS with multiple IVs depends on the weakest IV. If one IV is endogenous, the entire 2SLS estimate may be contaminated (although the direction depends on the nature of the endogeneity and the relative strength of the IVs). Passing the overidentification test ≠ all IVs are safe—again, it may miss same-direction endogeneity violations. Rather than adding more IVs, it is better to carefully select a few that are institutionally clean and sufficiently strong. More IVs do not necessarily bring greater safety—they often bring worse weak-IV bias (because collinearity among weak IVs is amplified by 2SLS bias).

6.5 Misconception Five: "Falsification Tests Passed → Exogeneity Is Proven"

An insignificant falsification test means you found no evidence of exogeneity violation—not that you found evidence that exogeneity holds. "No evidence of guilt" ≠ "proof of innocence." This is the same fundamental asymmetry between "cannot reject the null" and "confirming the null is true." Your phrasing should reflect this humility: "The falsification tests fail to provide evidence against the exogeneity of the IV"—rather than "the falsification tests prove the exogeneity of the IV."

6.6 Misconception Six: "As Long as the First-Stage F Is Large, the Weak IV Problem Is Solved"

A large first-stage F is good—but if your sample size is very large, a large F may be driven by N rather than by the substantive strength of the relationship between Z and X—and this kind of "large F" amplifies 2SLS bias when there is even slight IV endogeneity (Bound, Jaeger, and Baker, 1995). Therefore, you should not rely solely on the numerical value of the F statistic—you should also report whether the magnitude and sign of $\pi_1$ align with theoretical expectations, and the partial R² (the additional explanatory power of Z for X, after controlling for other variables) to assess whether relevance is not merely "statistically significant with a large F"—but also "economically relevant."

VII. Summary

Seven core takeaways for arguing IV relevance and exogeneity:

1. Relevance is testable—exogeneity is not. This is the most fundamental asymmetry in the IV argument. Relevance has F statistics, Stock-Yogo critical values, LIML, and AR tests—you can quantify the strength of your instrument. Exogeneity has no such ultimate test—it requires an accumulation of theory, institutional narrative, and falsification tests.

2. Relevance is not "F > 10"—it is "F > Stock-Yogo critical value." Specify the worst-case relative bias you can tolerate (e.g., 10%) → look up the corresponding Stock-Yogo critical value → report that your F exceeds it. If F is not high enough → LIML + Anderson-Rubin.

3. The starting point of the exogeneity argument is "why is Z as-if random?" The best IV papers find a credible "exogenous shifter" in institutions, history, law, or geography—and only then use data and tests to reinforce it. Narrative is the skeleton; tests are the flesh—flesh without a skeleton is weak.

4. Falsification tests are the pillar of the exogeneity argument. Test the reduced form in subsamples where no effect should theoretically exist—if Z is "quiet" in these places, its credibility is higher where it should "speak." Balance tests show that Z is uncorrelated with pre-treatment characteristics. Placebo outcomes show that Z does not predict "fake outcomes."

5. Overidentification tests are defensive—not offensive. They cannot "prove" exogeneity—they can only "find no evidence to refute" it. If significant → at least one IV is endogenous (but not which one). If insignificant → either all IVs are exogenous, or there is insufficient power, or the biases point in the same direction.

6. Ruling out alternative channels requires "thinking," not "computing." Draw a diagram of all causal paths from Z to Y → for every path other than the channel of interest, explain one by one why it is blocked or already controlled in your setting.

7. Sensitivity analysis is the future. Rather than claiming the IV is perfectly exogenous—show "even if the IV has a reasonable bias, do the conclusions still hold?" This is more honest and more persuasive than merely reporting a J-test p-value.

One-sentence conclusion:

"For the relevance of your instrument, you can quantify with the F statistic—to what extent Z pushes X. You can protect yourself against weak-IV bias with LIML, Anderson-Rubin, and Fuller corrections. But for the exogeneity of your instrument—whether Z affects Y only through X—you can never answer with a single test. You can show that the overidentification test p-value is 0.42, you can show that in falsification tests Z is indeed ineffective in samples where it should be ineffective, you can show that Z is uncorrelated with twenty pre-treatment characteristics—but ultimately, deep in the castle of the exclusion restriction, you need a story. This story is not 'statistical tests prove exogeneity'—this story is 'history, institutions, and logic make the variation in Z close to random assignment in a natural experiment.' And this story—not your tests—is the argument through which your instrument truly 'passes' in the minds of reviewers and readers."

VIII. Presentation Suggestions for Bilibili/WeChat Official Account

• Bilibili video: Suggest using "two doors" as the core visual metaphor. Opening: one door labeled "Relevance," with a precision instrument panel beside the door frame (F statistic, Stock-Yogo critical values, various numbers flashing). Voiceover: "The key to the first door is statistics. First-stage F = 18.7—exceeding the Stock-Yogo threshold of 16.38. This door has a precision lock—you can open it with data and tests." The camera moves to the second door—"Exogeneity." This door has no instrument panel, no numbers—only a story. On the door frame are written: "Why is quarter of birth like random assignment?" "Why is river density unrelated to policy preferences?" "Why is the timing of the law's implementation unaffected by local economic conditions?" Voiceover: "The second door has no key—you can only persuade it to open itself. It does not ask 'what is the p-value of your test'—it asks 'in what institutional history did you find this instrument? Why should I believe your story?'" Act One, "The Staircase of Relevance": progressively show F → Stock-Yogo → LIML → AR. F