Do Empirical Studies Require Rigorous Statistical Tests to Examine Variable Endogeneity?

I. Introduction: The Textbook Says "DWH Test"—But Almost No One Does It in the Papers You Read

You have finished the endogeneity chapter in econometrics. You learned about the Durbin-Wu-Hausman (DWH) test—first run an instrumental variables regression, then compare the difference between the IV and OLS estimates. If the difference is significant → OLS has endogeneity bias → you should use IV. If the difference is not significant → you cannot reject the consistency of OLS → OLS is more efficient, so use OLS.

The logic is clear, and the implementation is simple. You open Stata:

ivreg2 y (x = z), endog(x)
* The p-value of the DWH test appears in the output

You plan to include the DWH test in your empirical paper—as "formal statistical evidence of whether endogeneity exists."

But you have gone through papers published in your target journal over the past three years. Almost no one reports the DWH test. They discuss endogeneity—extensively—but the discussion does not take the form of "the test p < 0.05, therefore endogeneity exists." The discussion takes the form of "the endogeneity challenges facing this paper include: omitted variables (ability, motivation...), reverse causality (high Y may lead to high X...), measurement error (X contains systematic measurement bias...). This paper addresses these through the following strategies..."

You are puzzled: If a mature test exists and is written into textbooks—why does the empirical frontier hardly rely on it for the core endogeneity argument? Should endogeneity be "tested" or "analyzed"?

Core message: Endogeneity cannot, in essence, be fully proven or refuted by statistical tests—because the core definition of endogeneity ($\text{Cov}(X, \varepsilon) \neq 0$) involves an unobservable error term. The question the Durbin-Wu-Hausman test can answer is "whether the OLS and IV estimates differ statistically"—not "whether X is endogenous." The distinction between the two is crucial: if the IV itself is weak or invalid (fails the exclusion restriction), an insignificant DWH result may not signal "no endogeneity"—but rather "the IV is too weak or equally endogenous, and the test lacks power to detect the difference." Conversely, if the instrumental variable is relevant and exogenous, a significant DWH result does suggest a difference between OLS and IV—but this difference may not be driven entirely by endogeneity (e.g., the difference between the local average treatment effect (LATE) and the ATE). Therefore, the empirical frontier does not rely on DWH as the core of the endogeneity argument—not because researchers "do not understand" the test, but because they recognize that the endogeneity argument is a comprehensive project of research design and theoretical reasoning, not a statistical problem that can be resolved by a single p-value. In this paradigm, the way endogeneity is argued shifts from "I tested for endogeneity → it exists/does not exist" to "I analyzed the potential sources of endogeneity → I estimated the direction and magnitude of bias each source could produce → I adopted identification strategies to address the most important sources → I showed that the conclusions remain robust across multiple plausible scenarios."

II. "Testing for Endogeneity"—What Exactly Are You Testing?

2.1 The Definition of Endogeneity Involves an Unobservable Error

The core definition of endogeneity is:

$\text{Cov}(X, \varepsilon) \neq 0$

where $\varepsilon$ is the unobservable error term in the structural equation. You can compute residuals $\hat{\varepsilon} = Y - \hat{\beta}_0 - \hat{\beta}_1 X$—but these residuals are estimated under the assumption that "X is exogenous." You cannot observe the true structural error $\varepsilon$, because you do not know the true $\beta$.

This is a fundamental epistemological dilemma: endogeneity involves an unobservable quantity—and any method that requires you to estimate this unobservable quantity to "test" for endogeneity operates under some assumption. If that assumption itself holds—then the test's conclusion is valid. But if the assumption does not hold in the real data—the test's conclusion may completely mislead you.

2.2 What Is the DWH Test Doing?

The DWH test compares the OLS and IV estimates:

$H = \frac{(\hat{\beta}_{IV} - \hat{\beta}_{OLS})^2}{\text{Var}(\hat{\beta}_{IV}) - \text{Var}(\hat{\beta}_{OLS})}$

• $H_0$: The difference between OLS and IV arises only from sampling error (i.e., both estimators are consistent for $\beta$ → X is exogenous).
• $H_1$: OLS is inconsistent while IV is consistent (i.e., X is endogenous).

$H \sim \chi^2(1) \quad \text{or} \quad H \sim F(1, n-K)$

* Method 1: Directly use the endog option in ivreg2
ivreg2 y x1 x2 (x = z), endog(x)
 
* Method 2: Manual two-step procedure
* Step 1: First-stage regression, compute residuals
reg x z x1 x2
predict v, residual
* Step 2: Add the first-stage residual to the structural equation
reg y x x1 x2 v
* Test whether the coefficient on v is significantly different from 0
test v

The question the DWH test answers is not "whether X is endogenous"—but "whether, under the assumption that the IV is correctly specified (relevant and exogenous), OLS and IV give statistically different answers."

2.3 Three Structural Limitations of the DWH Test

Limitation 1: The test depends on the premise that "the IV is valid." The null hypothesis of the DWH test is "both OLS and IV are consistent"—the alternative hypothesis is "OLS is inconsistent, IV is consistent." But the premise that "IV is consistent" is a maintained assumption—the test itself cannot verify it. If the IV is also endogenous—or if the IV is weak—the DWH conclusion may be completely wrong. A significant DWH result → does not necessarily mean OLS is endogenous; it may mean the IV is endogenous while OLS is actually closer to the truth. Or the IV happens to compensate for the OLS endogeneity bias (opposite biases exactly cancel out), making the two numerically close but with completely different mechanisms.

Limitation 2: Under weak instruments, the DWH test may lack power. If the first-stage F is not high, the variance of the IV estimate is large, and the DWH test may fail to detect OLS bias—even if OLS indeed has serious endogeneity. Under extremely weak IV, the DWH test is essentially comparing a very noisy IV estimate with a precise but inconsistent OLS estimate—the noise makes the difference between the two insignificant, rather than "OLS is exogenous."

Limitation 3: Statistical significance ≠ economic importance. Even if DWH is significant → it only indicates that OLS and IV differ statistically—it does not indicate whether this difference is economically important. With large N, tiny differences that are economically meaningless may be detected as significant. What you need to assess is the direction and magnitude of the bias's impact on your conclusions—not just a p-value.

2.4 The Key Conclusion

In any empirical paper, the DWH test result cannot constitute the core pillar of the endogeneity argument. It can serve as supplementary evidence—"the IV and OLS estimates differ statistically, which is consistent with this paper's endogeneity analysis"—but you should not write "the DWH test p < 0.05, therefore X is endogenous, hence we use IV."

Why? Because DWH tells you "OLS and IV differ"—but you should not need a test to tell you that. You should have already analyzed the potential sources of endogeneity for X (omitted variables, reverse causality, measurement error) at the research design stage, and judged the direction and approximate magnitude of endogeneity based on economic theory. If theory tells you X is almost certainly endogenous (e.g., years of education and ability)—you do not need DWH to tell you that. If theory tells you X is likely exogenous (e.g., a treatment variable in a randomized experiment)—you also do not need DWH to tell you that.

III. If Not by Testing—How Should Endogeneity Be "Analyzed"?

3.1 From "Testing for Endogeneity" to "Analyzing Endogeneity Threats"

The methodological approach to endogeneity at the empirical frontier has undergone a paradigm shift from "detection" to "analysis." Modern empirical papers do not rely on statistical tests to determine whether endogeneity exists—instead, they systematically analyze the sources, direction, and magnitude of endogeneity at the research design stage, and then choose appropriate identification strategies to address it.

The root of this shift is a deeper understanding of endogeneity: endogeneity is not a binary state that "exists or does not exist" in your data—it is a question of degree and source. The covariance between X and ε, $\text{Cov}(X, \varepsilon)$, is not like an on-off switch of "0 or non-zero"—it is a continuous quantity. In some contexts this covariance is very small (close to zero); in others it is large. A test cannot judge the source and magnitude of this covariance for you—but your theoretical analysis and identification strategy can.

This analytical framework consists of three steps:

3.2 Step 1: Classify the Sources of Endogeneity—Three Classical Sources

The analysis of endogeneity begins with "identifying the source." There are three sources of endogeneity, each requiring a different response strategy:

Source 1: Omitted Variables—the Most Common

There is an unobserved variable W that affects both X and Y. W is absorbed into the error term ε → X and ε are correlated.

For example: $u_i$ (innate ability) → affects both education (X) and income (Y).

Response strategies:

• Panel fixed effects (if the omitted variable is time-invariant).
• Instrumental variables (find a Z that affects only X but not Y through W).
• Control for more observable variables (enrich the model as much as possible with observable confounders).
• Exploit exogenous shocks (DID, RDD, natural experiments).

Source 2: Reverse Causality / Simultaneity—X and Y Determine Each Other

X affects Y, but Y also affects X. You cannot distinguish which direction dominates in the data.

For example: high firm profits (Y) → more funds invested in R&D (X) → R&D increases profits (Y).

Response strategies:

• Use lagged independent variables ($X_{t-1}$ rather than $X_t$—but this only addresses contemporaneous reverse causality, not sequential exogeneity issues).
• Instrumental variables (find a Z that affects Y only through X).
• Exploit external shocks (e.g., policy changes that affect only X, not Y).

Source 3: Measurement Error—the True Value of X Differs from What You Measure

You observe $X^* = X + \eta$, where $\eta$ is measurement error. If $\eta$ is classical ($\eta \perp X, \varepsilon$) → OLS produces attenuation bias (coefficient biased toward zero).

If $\eta$ is not classical (e.g., systematic reporting bias—high-income earners tend to underreport income) → the direction of bias is uncertain.

Response strategies:

• Instrumental variables (find another measure of the true X that is uncorrelated with $\eta$).
• Use administrative data or more reliable measurement sources.
• If only classical measurement error exists → you can estimate upper/lower bounds for the attenuation factor.

In your paper, you should clearly distinguish which source (or sources) of endogeneity is primarily present in your data—because different sources correspond to different identification strategies and different judgments about the direction of bias.

3.3 Step 2: Judge the Direction of Endogeneity Bias—Sometimes More Important Than "Whether It Exists"

The sign of endogeneity bias (positive or negative) can often be inferred through economic theory even when the exact magnitude of the bias is unclear. And this directional judgment is sometimes more important than "whether endogeneity exists"—because it tells you whether the OLS estimate is an upper or lower bound.

Three classical judgment rules:

Rule 1 (omitted variables): Bias sign = sign of the correlation between the omitted variable and X × sign of the omitted variable's effect on Y.

• Ability (W) and education (X) are positively correlated (more able people are more likely to obtain more education).
• Ability (W) has a positive effect on income (Y).
• → $\text{Cov}(X, W) > 0$, $\beta_W > 0$ → OLS overestimates the true return to education (upward bias).

Rule 2 (reverse causality): Bias sign = sign of the reverse causal path.

• Firm profits (Y) → firms increase R&D (X) → R&D has a positive effect on profits → the X→Y effect you estimate is confounded by the two-way causal relationship.
• → OLS overestimates the causal effect of R&D on profits (upward bias).

Rule 3 (classical measurement error): The bias direction is always toward zero (attenuation).

• If $X^* = X + \eta$, and $\eta$ is independent of X and Y → the OLS coefficient is biased toward zero.
• This is a special type of "endogeneity"—it causes you to underestimate the true effect.

In your paper, if you can argue the direction of the endogeneity bias ("the endogeneity bias in this paper is likely positive—therefore the OLS estimate should be regarded as an upper bound for the return to education")—this in itself is an extremely valuable and honest analysis of endogeneity. It tells the reader which side of the OLS coefficient the true effect is likely to lie on. Even if you cannot provide a precise unbiased estimator, you can provide a judgment about the direction of bias—this is a substantive response to the endogeneity problem, not merely a statistical test.

3.4 Step 3: Convergence of Multiple Identification Strategies—"They All Point in the Same Direction"

The endogeneity argument in modern empirical papers often no longer relies on a single instrumental variable or natural experiment—instead, it uses multiple identification strategies, starting from different assumptions, and examines whether the conclusions converge.

A typical "multi-strategy" combination:

• Strategy A: OLS (baseline)—contains endogeneity bias, but serves as a starting point.
• Strategy B: Panel fixed effects—controls for time-invariant omitted variables.
• Strategy C: IV 2SLS—based on a different exogeneity assumption.
• Strategy D: DID or natural experiment—if a policy shock or exogenous variation exists.
• Strategy E: Matching or inverse probability weighting—based on conditional independence given observable variables.

If A–E yield the same sign and are reasonably consistent in magnitude → the conclusion is insensitive to the choice of identification assumption. This in itself constitutes a powerful and credible argument—far more persuasive than a single strategy plus a DWH test.

If some strategies yield results that are clearly different from others → this is precisely what you need to analyze and explain in your paper. Such divergence is information, not noise—it tells you where the source of endogeneity may lie. Do not let the DWH test hide this divergence for you.

IV. Supplementary Quantitative Tools—After You Complete the Theoretical Analysis

Although endogeneity cannot be fully resolved by a single test—after you complete the theoretical analysis, the following quantitative tools can serve as supplementary evidence.

4.1 Testing the Severity of Selection Bias: Oster's δ Statistic

Oster (2019) proposed a method to assess the extent to which "selection bias from unobservables" could overturn your conclusions. The core idea is to use the severity of selection on observables as a benchmark for selection on unobservables.

In practice, one computes a quantity called $\delta$—which answers the question: "Relative to the selection bias from observable variables, how much larger would the selection bias from unobservable variables need to be to drive your core coefficient to zero?"

• The larger δ → the less sensitive your conclusion is to omitted unobservables → the more robust.
• Suggested standard: if δ > 1 (i.e., selection on unobservables would need to be more severe than selection on observables to overturn the conclusion) → the conclusion is relatively robust.

* Oster's δ test (using psacalc)
psacalc delta y, mcontrol(x1 x2 x3)

4.2 The Altonji-Elder-Taber (2005) Ratio

Altonji, Elder, and Taber (2005) proposed a method to "calibrate selection on unobservables using selection on observables." Their core ratio is:

$\text{Ratio} = \frac{\hat{\beta}_{\text{full controls}}}{\hat{\beta}_{\text{no controls}} - \hat{\beta}_{\text{full controls}}}$

This ratio measures how much the coefficient attenuates after adding all control variables—relative to the size of the coefficient without controls. If this ratio is large → selection on observables is insufficient to fully explain the effect → selection on unobservables would need to be much larger than selection on observables to overturn the conclusion.

4.3 When an IV Is Available—DWH as Supplementary Rather Than Core

If your paper already uses an IV (based on theoretical and institutional exogeneity arguments), the DWH test can be reported as supplementary evidence—but its position in the paper should not be "the core evidence for the endogeneity argument." The wording in the paper should be:

"The Durbin-Wu-Hausman test rejects the consistency of OLS at the 5% level (p = 0.03), which is consistent with this paper's theoretical analysis—namely, that years of education may be endogenous. However, the primary reason this paper uses IV is not the result of this test, but rather [the institutional argument—why Z is exogenous, and why it provides an independent source of identifying variation]."

V. Endogeneity Argument Strategies in Different Scenarios

5.1 Scenario A: You Only Have OLS (No Panel, No IV, No Natural Experiment)

In this scenario, your endogeneity argument cannot be "I ran the DWH test and it said there is no problem"—nor can it be "I assume X is exogenous" and then move on. You need to do the following:

1. Explicitly list the potential sources of endogeneity: omitted variables (which ones?), reverse causality (why might Y affect X?), measurement error (how might your X be mismeasured?).
2. Judge the direction of bias: based on each source, argue whether the OLS coefficient is upward- or downward-biased. Are you estimating an upper or lower bound of the true effect?
3. Control for confounders as much as possible within the observable range: add all relevant observable control variables you can—show the stability of the core coefficient across different control sets.
4. Use Oster's δ or the Altonji-Elder-Taber ratio: assess how large selection on unobservables would need to be to overturn your conclusion.
5. Be candid: do not claim OLS is an "unbiased causal effect"—instead, position OLS as "the partial correlation between X and Y after controlling for observable confounders." Honestly discuss that "this does not equal causation."

5.2 Scenario B: You Have Panel Data—Fixed Effects Are Your Primary Strategy

Panel fixed effects control for time-invariant omitted variables—but you cannot claim that "FE solves all endogeneity problems." In your paper you should:

1. Clearly distinguish between two sources of endogeneity: FE controls for time-invariant omissions (such as ability, culture, location) → but time-varying omissions may still make X endogenous.
2. Give examples of time-varying confounders: "A firm may encounter an external opportunity in a given year that simultaneously increases productivity and digital investment"—such shocks are time-varying, and FE cannot handle them.
3. If feasible, seek sources of exogenous variation (e.g., average changes of other firms in the industry, policy shocks—i.e., instrumental variables) within the FE model to further address time-varying endogeneity.

5.3 Scenario C: You Have an Instrumental Variable—IV Is Your Primary Strategy

In this scenario, your endogeneity argument is not about "whether endogeneity exists"—but about "whether the IV satisfies exogeneity." The DWH test can be reported as supplementary—but it should not be the core. The focus of the argument shifts to the exclusion restriction (see the article "Relevance and Exogeneity of Instrumental Variables" in this series).

VI. Common Misconceptions

6.1 Misconception 1: "DWH test is insignificant → X is exogenous → just use OLS"

An insignificant DWH test could be because: (a) X is indeed exogenous, (b) the IV is too weak so the test lacks power, or (c) the IV is also endogenous and its bias direction happens to offset the OLS bias. Among these three possibilities, only the first allows you to safely use OLS—but your data cannot distinguish among them. An insignificant DWH cannot rule out (b) or (c)—and both of these possibilities imply that continuing to use OLS is unsafe.

6.2 Misconception 2: "DWH test is significant → IV must be better than OLS → I should only report the IV results"

A significant DWH only indicates that OLS and IV differ—but if the IV itself is also weak or slightly endogenous, the IV may be worse than OLS. Significant DWH + first-stage F not high enough + weak argument for IV exogeneity → IV may not be more credible than OLS. In this case, you should report both OLS and IV results and discuss the source of the difference, rather than simply replacing OLS with IV.

6.3 Misconception 3: "Endogeneity needs to be 'tested'—my paper must have this test to pass peer review"

What you should test is not "whether endogeneity exists"—but whether your identification strategy is credible and whether your conclusions are robust to alternative assumptions. Reviewers will ask you: "What are your identification assumptions? Why are these assumptions reasonable in your research context? If these assumptions do not hold—what analyses have you done to show whether your conclusions still hold under these alternative scenarios?" These questions require not the p-value of a DWH test—but your analysis at the research design stage and your investment in robustness checks.

6.4 Misconception 4: "OLS has endogeneity bias → cannot be used → must find an IV"

Not all research questions require (or can find) a good instrumental variable. If you cannot find a credible IV—you can:

• Use panel FE to control for time-invariant confounders.
• Use matching or regression adjustment to improve balance on observable variables.
• Use a Heckman selection model or similar correction methods.
• Honestly state the implications and limitations of your results—treat the OLS results as "partial correlations" rather than "causal effects."

"Because of endogeneity, I cannot do this research"—this itself is an overreaction. Good empirical research does not perfectly solve all endogeneity problems—rather, it honestly presents the best available evidence while clearly articulating the endogeneity threats.

VII. Summary

Five core paradigm shifts in the endogeneity argument:

1. From "testing for endogeneity → judging by p-value" → to "analyzing the sources of endogeneity → assessing the nature and direction." Endogeneity is not a binary state that can be cleanly adjudicated by a DWH test—it is a data feature that needs to be systematically understood through theoretical analysis at the research design stage. Your argument is not "I proved X is not endogenous"—but "I analyzed the potential sources of endogeneity and adopted targeted strategies."

2. From "DWH test → choose OLS or IV" → to "multiple strategies → see whether they point to the same conclusion." An argument based on a single test plus a single strategy is fragile. The standard for modern empirical papers is: multiple identification strategies starting from different assumptions jointly support the same conclusion. OLS, FE, IV, DID, matching—they identify different parameters under different assumptions—if these parameters are consistent in direction and magnitude, the strength of the argument far exceeds any single test.

3. From "default to OLS until overturned by a test" → to "explicitly state endogeneity threats and adopt the most reasonable response strategy, regardless of test results." If your theoretical analysis indicates that X is almost certainly endogenous—you should not wait for DWH to approve your use of IV. Conversely, if your theoretical analysis indicates that X is exogenous in a specific context—you should not let DWH prevent you from using OLS.

4. From "focusing only on whether endogeneity exists" → to "also focusing on the direction of endogeneity bias." Judging the direction of bias is often more informative than judging whether bias exists—is OLS an upper or lower bound of the true effect? This directional judgment is feasible in your data and theory, and it is extremely useful to readers.

5. From "endogeneity → the research is invalid" → to "endogeneity → but after discussing this threat, the conclusions are honest and still valuable." Any observational study faces endogeneity. A good empirical paper does not say "we do not need to worry about endogeneity"—it says "we have honestly analyzed the threats of endogeneity, and under this premise, the best available evidence points to X."

One-sentence conclusion:

"The Durbin-Wu-Hausman test is a screwdriver in your toolbox—it can tell you whether OLS and IV differ statistically. But the endogeneity argument requires not a screwdriver—but a complete set of architectural blueprints: where does endogeneity come from (omitted variables? reverse causality? measurement error?), in which direction does the bias point (is the effect of X overestimated or underestimated?), what strategies have you used to address it (panel FE? IV? DID? natural experiments?), and whether the conclusions still hold under different assumptions. Between 'I tested for endogeneity, p < 0.05' and 'I analyzed the sources, direction, and magnitude of endogeneity, and stress-tested it under multiple strategies'—reviewers and readers care about the latter. The former is merely a supplementary note you add to the paper after completing the latter."

VIII. Presentation Suggestions for Bilibili/WeChat Official Account

• Bilibili video: It is recommended to use the core metaphor of "detective vs. referee." Opening: two characters—one is the "referee" (the DWH test), holding up a sign with a p-value, saying after the match ends, "There is no significant difference between OLS and IV—the match is valid!" The other is the "detective" (the researcher), who keeps asking questions before and during the match: "Who was omitted?" "Who is influencing our X in the opposite direction?" "How was X measured?" The visuals contrast the "referee" casually waving the sign with the "detective" carefully inspecting the scene. Voiceover: "The DWH test is a referee—it announces the result after the match ends. But the endogeneity argument requires a detective—who keeps questioning before and during the match: why exactly are X and ε correlated? How large is this correlation? In which direction does the bias point? The referee sees only one statistic—the detective sees the entire case." Act 1 "Why testing is not enough": three scenarios where DWH fails. Scenario 1—the IV is weak (first-stage F = 3.2), the variance of the IV estimate is enormous, DWH is insignificant. Annotation: "It is not that OLS is exogenous—it is that the IV is too weak, and the test lacks power to detect the difference." Scenario 2—the IV is also endogenous, and its bias direction happens to be opposite to that of OLS → OLS and IV are close, DWH is insignificant. Annotation: "It is not that both are correct—it is that both are wrong, but the errors happen to make them look consistent." Scenario 3—N is extremely large, and a tiny OLS-IV difference → DWH is significant. Annotation: "Significance does not equal endogeneity—it may just be that large N amplifies sampling noise." Act 2 "The correct endogeneity analysis": a three-step process—(1) draw the three sources of endogeneity (omitted variables → red line connecting both X and Y; reverse causality → arrow from Y to X pointing in the wrong direction; measurement error → X* is shaky/blurry), (2) judge the direction of bias (arrow pointing to upper or lower bound), (3) convergence of multiple strategies—three independent identification strategies each give coefficients in the same direction with similar magnitudes → they converge at this conclusion. Voiceover: "You are not 'testing' endogeneity—you are 'analyzing' endogeneity. Analysis is harder than testing—but it is also more honest."

• WeChat Official Account: Create a symptom-treatment chart for the three sources of endogeneity (omitted variables, reverse causality, measurement error) with corresponding response strategies. Create warning cards for the three failure scenarios of the DWH test. Create a side-by-side comparison chart for the paradigm shift from "testing" to "analysis" (old paradigm: DWH → choose model vs. new paradigm: analyze sources → identification strategy → multiple checks → sensitivity). Create a quick judgment card for the three rules for judging bias direction (omitted variables, reverse causality, measurement error). Create a method card with usage templates for Oster's δ and the Altonji-Elder-Taber ratio. Create a scenario comparison card for argumentation strategies under different data scenarios (OLS only, with panel, with IV). Create a correction card for the four common misconceptions.

• Recommended titles:

  • Main title: 《Should Endogeneity Be "Tested" or "Analyzed"?—The Position of the DWH Test at the Empirical Frontier》
  • Alternative title: 《The Textbook Tells You to Run the Durbin-Wu-Hausman Test—But Your Paper May Not Need It》
  • New media title: 《DWH Test p < 0.05—Is This Evidence of Endogeneity, or a Statistical Coincidence?》

• Key quotes:

  "The question the DWH test answers is 'whether OLS and IV differ'—not 'whether X is endogenous.' And what you need to answer is the latter—a question about an unobservable error term. For what is unobservable—no test can make the judgment for you. What you need is analysis, not a p-value."

  "Endogeneity is not a binary state of 'whether it exists in your data'—it is a continuous question of 'how much, from where, and in which direction.' The DWH test is a binary switch—it can only tell you 'on or off.' But it cannot tell you whether behind the switch lies omitted variables, reverse causality, or measurement error—let alone the direction and approximate magnitude of the bias."

  "Judging the direction of bias is the most underappreciated tool in endogeneity analysis. You may not be able to say how large the bias is—but you can usually say which way it points. If OLS overestimates the true effect—you have an upper bound. If OLS underestimates—you have a lower bound. An unknown effect with bounds is closer to knowledge than an unknown effect without bounds."

  "Multiple identification strategies are the gold standard of the modern endogeneity argument. The instrumental variable identifies on my turf, DID identifies on your turf, panel FE identifies on his turf—three independent pieces of evidence converge on the same conclusion. This is not the persuasiveness of a single p-value—it is the persuasiveness of three independent lines of evidence. What convinces reviewers and readers is never a single test—but a complete argumentative structure that attacks the same problem from multiple angles."