What is the null hypothesis?

The specified regressors can be treated as exogenous under the maintained model.

Does rejection prove the instruments are valid?

No. Their validity is part of the conditions supporting the test's interpretation.

Must Durbin and Wu-Hausman versions give identical p-values?

No; their variance estimates and finite-sample reference calculations can differ.

AI Glossary / Statistics and probability / Advanced

Durbin - Wu - Hausman test

Definition

The Durbin-Wu-Hausman test is a family of econometric tests used to assess whether specified regressors can be treated as exogenous in a model that also has an instrumental-variable formulation. The central idea is to compare estimators that should agree asymptotically under exogeneity but can differ when a regressor is correlated with the structural error. In the familiar setting, ordinary least squares is compared with instrumental variables. Valid interpretation depends on the instruments and model assumptions; the test does not independently certify them.

What it means

A regression coefficient can be distorted when an explanatory variable responds to unobserved influences on the outcome. Instrumental variables may address that problem if suitable instruments exist. Under the null hypothesis of exogeneity and the relevant model conditions, both ordinary least squares and instrumental-variable estimates target the same parameter. A difference too large to attribute to their joint sampling variability provides evidence against that null. This is a conditional specification check, not a direct observation of an unmeasured confounder or a universal test of causal validity.

How it works

Specify the potentially endogenous regressors, included exogenous variables and excluded instruments. Estimate the relevant models using the same observations and compatible specifications. One formulation builds a quadratic statistic from the coefficient difference and its covariance. Under the classical efficient-estimator assumptions, that covariance can be expressed as the difference between the estimators' covariance matrices. Another formulation adds first-stage residuals to an augmented outcome regression and tests their coefficients. Durbin score and Wu-Hausman F versions need not give identical finite-sample results. Heteroskedasticity or clustering requires a suitable robust procedure rather than automatic use of the classical formulas.

Three practical examples

Demand research: examine whether a price regressor can be treated as exogenous when a defensible instrument has been specified.
Operational analysis: compare an ordinary regression with an instrumental-variable model for a potentially jointly determined input.
Model review: distinguish a test of regressor exogeneity from tests of instrument relevance or overidentifying restrictions.

Think of it this way

Think of comparing two measuring methods that should converge on the same target under a shared set of assumptions. A substantial disagreement raises questions about those assumptions. However, the comparison is only informative if the supposedly dependable reference method has a credible foundation of its own.

Worked example

Suppose a hypothetical scalar coefficient is estimated as 1.0 by ordinary least squares and 1.5 by instrumental variables. Assume the classical Hausman covariance relationship is valid, with estimated variances 0.04 and 0.09 respectively. The coefficient difference is 0.5 and its variance is 0.09 - 0.04 = 0.05. The statistic is 0.5 squared/0.05 = five. Against an asymptotic chi-square reference with one degree of freedom, the tail probability is approximately 0.02535. Under the stipulated assumptions, this rejects exogeneity at 5%. The example does not justify simply subtracting any two reported variances: the covariance identity requires specific conditions. It also does not show that the instruments are valid, prove a causal effect of 1.5 or provide an exact small-sample probability.

Illustrative case study

Fictional case: Rowan Analytics studies the relationship between staffing and output, recognizing that managers may increase staffing when unobserved demand rises. The team proposes an instrument, documents its substantive justification and checks its relevance. A properly specified exogeneity test becomes one part of the assessment, alongside sensitivity analysis and model diagnostics. Analysts report the test's conditional interpretation rather than saying it proves the instrumental-variable estimate is correct. This invented case illustrates a disciplined workflow and does not endorse a particular staffing instrument or causal conclusion.

Common misconceptions and mistakes

Do not treat failure to reject as proof of exogeneity, especially with weak instruments or limited information. Invalid instruments can undermine the comparison. Do not confuse this test with Durbin-Watson autocorrelation testing, the Durbin incomplete-block rank test or an overidentification test. A negative or ill-conditioned estimated covariance difference is a warning about the calculation and assumptions, not a meaningful negative chi-square result to interpret casually. Different samples, controls or functional forms can create coefficient differences unrelated to the intended test.

Business use

Businesses using observational data should make the model's identification assumptions explicit before allowing a test result to drive decisions. Record why each instrument should influence the regressor and why it should not directly enter the outcome error. Report which exogeneity test and covariance estimator were used, together with the number of tested regressors. If the evidence is weak or conflicting, preserve that uncertainty rather than choosing the model with the preferred business implication. Statistical diagnostics supplement substantive knowledge; they do not manufacture a credible instrument.

Check your understanding

Why is the worked example's difference variance 0.05 rather than 0.13?

Answer: It uses the stipulated efficient-estimator covariance identity. Adding the variances would incorrectly assume the two estimates are independent despite being calculated from the same data.

Related terms

Instrumental variables
Ordinary Least Squares
endogeneity
exogeneity
two-stage least squares
Hausman specification test
weak instruments

Learn next

distinguish instrument relevance, exclusion assumptions and regressor exogeneity, then compare classical and robust versions of the augmented-regression test in a consistent model specification.

Sources

Sources: Stata, ivregress postestimation, exogeneity tests and robust alternatives: https://www.stata.com/manuals/rivregresspostestimation.pdf ; Stata, historical augmented-regression explanation: https://www.stata.com/support/faqs/statistics/durbin-wu-hausman-test/ ; Dufour and Tchatoka, Exogeneity tests, incomplete models, weak identification and non-Gaussian distributions: https://arxiv.org/abs/1701.07050 .

Previous: Durbin-Watson statistic Next: Dutch eligibility trace