Sargan–Hansen test

The Sargan–Hansen test or Sargan's ${\displaystyle J}$ test is a statistical test used for testing over-identifying restrictions in a statistical model. It was proposed by John Denis Sargan in 1958,[1] and several variants were derived by him in 1975.[2] Lars Peter Hansen re-worked through the derivations and showed that it can be extended to general non-linear GMM in a time series context.[3]

The Sargan test is based on the assumption that model parameters are identified via a priori restrictions on the coefficients, and tests the validity of over-identifying restrictions. The test statistic can be computed from residuals from instrumental variables regression by constructing a quadratic form based on the cross-product of the residuals and exogenous variables.[4]^:132–33 Under the null hypothesis that the over-identifying restrictions are valid, the statistic is asymptotically distributed as a chi-square variable with ${\displaystyle (m-k)}$ degrees of freedom (where ${\displaystyle m}$ is the number of instruments and ${\displaystyle k}$ is the number of endogenous variables).

This version of the Sargan statistic was developed for models estimated using instrumental variables from ordinary time series or cross-sectional data. When longitudinal ("panel data") data are available, it is possible to extend such statistics for testing exogeneity hypotheses for subsets of explanatory variables.[5] Testing of over-identifying assumptions is less important in longitudinal applications because realizations of time varying explanatory variables in different time periods are potential instruments, i.e., over-identifying restrictions are automatically built into models estimated using longitudinal data.

See also

• Durbin–Wu–Hausman test

References

1. Sargan, J. D. (1958). "The Estimation of Economic Relationships Using Instrumental Variables". Econometrica. 26 (3): 393–415. doi:10.2307/1907619. JSTOR 1907619.
2. Sargan, J. D. (1988) [1975]. "Testing for misspecification after estimating using instrumental variables". Contributions to Econometrics. New York: Cambridge University Press. ISBN 0-521-32570-6.
3. Hansen, Lars Peter (1982). "Large Sample Properties of Generalized Method of Moments Estimators". Econometrica. 50 (4): 1029–1054. doi:10.2307/1912775. JSTOR 1912775.
4. Sargan, J. D. (1988). Lectures on Advanced Econometric Theory. Oxford: Basil Blackwell. ISBN 0-631-14956-2.
5. Bhargava, Alok (1991). "Identification and panel data models with endogenous regressors". Review of Economic Studies. 58 (1): 129–140. doi:10.2307/2298050. JSTOR 2298050.

Further reading

• Davidson, Russell; McKinnon, James G. (1993). Estimation and Inference in Econometrics. New York: Oxford University Press. pp. 616–620. ISBN 0-19-506011-3.
• Verbeek, Marno (2004). A Guide to Modern Econometrics (2nd ed.). New York: John Wiley & Sons. pp. 142–158. ISBN 0-470-85773-0.
• Kitamura, Yuichi (2006). "Specification Tests with Instrumental Variable and Rank Deficiency". In Corbae, Dean; et al. (eds.). Econometric Theory and Practice: Frontiers of Analysis and Applied Research. New York: Cambridge University Press. pp. 59–124. ISBN 0-521-80723-9.