Sargan–Hansen test explained

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] 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).

See also

• Durbin–Wu–Hausman test

Further reading

• Book: Davidson, Russell . James G. . McKinnon . Estimation and Inference in Econometrics . New York . Oxford University Press . 1993 . 0-19-506011-3 . 616–620 .
• Book: Verbeek, Marno . Marno Verbeek. A Guide to Modern Econometrics . New York . John Wiley & Sons . 2nd . 2004 . 0-470-85773-0 . 142–158 .
• Book: Kitamura, Yuichi . Specification Tests with Instrumental Variable and Rank Deficiency . Econometric Theory and Practice: Frontiers of Analysis and Applied Research . New York . Cambridge University Press . Dean . Corbae . 2006 . 0-521-80723-9 . 59–124 . https://books.google.com/books?id=mXitlEaAy8AC&pg=PA59 . etal.

Notes and References

1. Sargan . J. D. . 1958 . The Estimation of Economic Relationships Using Instrumental Variables . . 26 . 3 . 393–415 . 1907619 . 10.2307/1907619.
2. Book: Sargan, J. D. . 1975 . 1988 . Testing for misspecification after estimating using instrumental variables . Contributions to Econometrics . New York . Cambridge University Press . 0-521-32570-6 .
3. Hansen . Lars Peter . 1982 . Large Sample Properties of Generalized Method of Moments Estimators . Econometrica . 50 . 4 . 1029–1054 . 1912775 . 10.2307/1912775.
4. Book: Sargan, J. D. . 1988 . Lectures on Advanced Econometric Theory . Oxford . Basil Blackwell . 0-631-14956-2 .