Sargan–Hansen test

The Sargan–Hansen test or Sargan's $J$ 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 $(m - k)$ degrees of freedom (where $m$ is the number of instruments and $k$ 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. (1975). "Testing for misspecification after estimating using instrumental variables". Mimeo (London School of Economics).
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.

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. Econometric Theory and Practice: Frontiers of Analysis and Applied Research. New York: Cambridge University Press. pp. 59–124. ISBN 0-521-80723-9.