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Specification Tests in Econometrics

Why this mattered

Hausman’s paper turned model specification from a largely diagnostic or assumption-driven exercise into a general testing principle. The key insight was that, under correct specification, two estimators aimed at the same parameter can differ in efficiency but not systematically in probability: an efficient estimator and a merely consistent estimator should have a difference with mean zero in large samples. When that difference is too large, it is evidence that the assumptions giving the efficient estimator its advantage are false. This reframed misspecification as an estimable contrast between procedures, making it possible to test assumptions such as exogeneity, random effects, and simultaneous-equation restrictions with a common asymptotic logic.

The paradigm shift was practical as much as theoretical. Before this paper, many econometric modeling choices depended heavily on maintained assumptions that were difficult to interrogate within the same framework used for estimation. The Hausman test made those assumptions empirically contestable. It gave researchers a way to ask whether a convenient or efficient estimator, such as random effects or ordinary least squares under exogeneity, was defensible relative to a more robust alternative, such as fixed effects or instrumental variables. The wage-equation application in the paper illustrated the broader point: unobserved individual heterogeneity could be statistically detected as correlated with observed regressors, directly challenging standard specifications rather than merely warning that they might be fragile.

Its influence is visible across later econometrics because the paper supplied a reusable template: compare estimators that coincide under the null but diverge under the alternative. That template became central to panel-data methods, instrumental-variables diagnostics, tests of endogeneity, and broader generalized method of moments reasoning, where overidentifying restrictions and estimator contrasts play a similar role in assessing whether moment assumptions are credible. In this sense, the paper did not just introduce a named test; it helped establish specification testing as a routine part of empirical identification, linking estimation choices to falsifiable implications and shaping the standards by which later applied econometric work defended its causal and structural claims.

Abstract

Using the result that under the null hypothesis of no misspecification an asymptotically efficient estimator must have zero asymptotic covariance with its difference from a consistent but asymptotically inefficient estimator, specification tests are devised for a number of model specifications in econometrics. Local power is calculated for small departures from the null hypothesis. An instrumental variable test as well as tests for a time series cross section model and the simultaneous equation model are presented. An empirical model provides evidence that unobserved individual factors are present which are not orthogonal to the included right-hand-side variable in a common econometric specification of an individual wage equation.

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