Nonlinear Hausman Specification Test
Also known as: Hausman specification test (nonlinear), nonlinear endogeneity test, Wu-Hausman test (nonlinear), NL-Hausman test
The Nonlinear Hausman test extends Hausman's (1978) endogeneity specification test to nonlinear models such as probit, logit, Tobit, and count-data regressions. It tests whether suspected regressors are endogenous — i.e., correlated with the error term — in a model where the outcome or the relationship is inherently nonlinear, ensuring that IV-corrected estimates are necessary.
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When to use it
Use the Nonlinear Hausman test when your structural model is inherently nonlinear — binary or ordered outcomes (probit/logit), censored outcomes (Tobit), count data (Poisson) — and you suspect that one or more regressors may be endogenous (e.g., due to omitted variables, simultaneity, or measurement error). Valid instruments that are correlated with the endogenous regressors but not with the structural error are required. Do not use this test when you lack valid instruments, when the outcome is continuous and linear (use the standard Hausman test instead), or when the model is just-identified and standard errors from the control function approach need correction via bootstrapping.
Strengths & limitations
- Extends the widely trusted Hausman framework to the full range of nonlinear econometric models.
- The control function implementation is computationally straightforward: it can be implemented in two stages using standard software.
- Provides a direct test of the endogeneity assumption rather than relying on informal arguments.
- Allows flexible detection of partial endogeneity — testing only specific regressors without assuming all are endogenous.
- If exogeneity is not rejected, the more efficient restricted nonlinear estimator can be justified formally.
- Requires valid instrumental variables; weak or invalid instruments lead to misleading test results.
- Standard errors from the two-step control function procedure require correction (bootstrapping or analytic adjustment) because the first-stage estimation introduces generated regressors.
- The test loses power in small samples where the difference between estimators is imprecisely estimated.
- In highly nonlinear settings, the Wald form of the test may have poor finite-sample size properties.
- Identification relies on exclusion restrictions that must be justified theoretically, not statistically.
Frequently asked
How is the Nonlinear Hausman test different from the standard (linear) Hausman test?
The standard Hausman test compares OLS and IV estimates in a linear model. In nonlinear models the OLS benchmark is no longer valid, so the test is typically implemented via the control function approach: first-stage residuals are added to the nonlinear model and their significance is tested. The logic is the same — comparing a consistent-under-H1 augmented model to the restricted model — but the mechanics are adapted to the nonlinear likelihood.
What is the control function approach and how does it implement the test?
The control function approach adds predicted residuals from a first-stage OLS regression of the endogenous variable on instruments into the structural nonlinear model. If these residuals are jointly significant (Wald test), the null of exogeneity is rejected. This is equivalent to testing for endogeneity while controlling for the source of endogeneity directly.
Why must I correct the standard errors in the second stage?
The first-stage residuals are estimated, not observed. Including estimated regressors in the second stage introduces additional uncertainty not captured by the nominal second-stage standard errors. Ignoring this leads to underestimated standard errors and over-rejection of the null. Bootstrapping the full two-step procedure is the most common remedy.
What happens if my instruments are weak?
Weak instruments — those with low first-stage partial F-statistic (below 10) — lead to imprecise first-stage residuals, which in turn make the Hausman test statistic unreliable. The test may fail to detect real endogeneity (low power) or reject spuriously. Strengthen identification by finding better instruments or using limited-information maximum likelihood.
Can I use the Nonlinear Hausman test with panel data?
Yes. In panel nonlinear models (e.g., random-effects probit), the Hausman principle compares random-effects and fixed-effects estimators to test for correlation between individual effects and regressors. Wooldridge (2010) provides extensions that combine panel structure with control function corrections for regressor endogeneity.
Sources
- Hausman, J. A. (1978). Specification tests in econometrics. Econometrica, 46(6), 1251–1271. DOI: 10.2307/1913827 ↗
- Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data (2nd ed.). MIT Press. ISBN: 978-0262232586
How to cite this page
ScholarGate. (2026, June 3). Nonlinear Hausman Specification Test. ScholarGate. https://scholargate.app/en/econometrics/nonlinear-hausman-test
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