DurbinWuHausmanTest

The Durbin-Wu-Hausman test is a statistical procedure used in econometrics to assess whether one or more explanatory variables in a regression model are endogenous. Endogeneity occurs when an regressors correlate with the error term, leading to biased and inconsistent ordinary least squares (OLS) estimates. The test compares two sets of estimates: a consistent estimator based on instrumental variables (IV) or two-stage least squares (2SLS), and a potentially more efficient estimator such as OLS that would be inconsistent under endogeneity. If the exogeneity assumption holds, the two sets of estimates should be similar; a significant difference suggests endogeneity.

The test has historical roots in Durbin’s and Wu’s work, with Hausman providing a general framework for testing model specification by comparing two estimators. In practice, the most common implementation computes the difference between the OLS and IV coefficient estimates for the endogenous variable(s). The test statistic is formed from this difference and its estimated covariance matrix; under the null hypothesis of exogeneity, the statistic follows a chi-square distribution with degrees of freedom equal to the number of endogenous regressors being tested. There are robust variants that use heteroskedasticity-consistent covariance estimators.

Interpretation centers on whether to trust OLS or IV results. A small p-value leads to rejection of the null, indicating endogeneity and suggesting reliance on IV/2SLS estimates. A non-significant result implies that exogeneity cannot be rejected, making OLS a reasonable choice under the model assumptions. The test is widely used when instruments are available but its validity hinges on instrument relevance and exogeneity, as weak or invalid instruments can distort conclusions.