Hostname: page-component-7479d7b7d-jwnkl Total loading time: 0 Render date: 2024-07-12T10:54:15.805Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Results for Generalized Wald Tests

Published online by Cambridge University Press:  11 February 2009

Donald W. K. Andrews
Donald W. K. Andrews
Affiliation:
Cowles Foundation for Research in Economics, Yale University
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents conditions under which a quadratic form based on a g-inverted weighting matrix converges to a chi-square distribution as the sample size goes to infinity. Subject to fairly weak underlying conditions, a necessary and sufficient condition is given for this result. The result is of interest because it is needed to establish asymptotic significance levels and local power properties of generalized Wald tests (i.e., Wald tests with singular limiting covariance matrices). Included in this class of tests are Hausman specification tests and various goodness-of-fit tests, among others. The necessary and sufficient condition is relevant to procedures currently in the econometrics literature because it illustrates that some results stated in the literature only hold under more restrictive assumptions than those given.

Type
Research Article
Information
Econometric Theory , Volume 3 , Issue 3 , June 1987 , pp. 348 - 358
Copyright
Copyright © Cambridge University Press 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Andrews, D.W.K.Chi-square diagnostic tests for econometric models: Introduction and applications. Journal of Econometrics (1988): forthcoming.Google Scholar
2.Andrews, D.W.K.Chi-square diagnostic tests for econometric models: Theory. Econometrica 56 (1988): forthcoming.CrossRefGoogle Scholar
3.Duncan, G.M. A generalized Durbin-Hausman test. Unpublished manuscript, Department of Economics, Washington State University, Pullman, Washington, 1983.Google Scholar
4.Hausman, J.A. & Taylor, W.E.. A generalized specification test. Economic Letters 8 (1981): 239–245.CrossRefGoogle Scholar
5.Heckman, J.J.The x2 goodness-of-fit statistic for models with parameters estimated from microdata. Econometrica 52 (1984): 1543–1547.CrossRefGoogle Scholar
6.Holly, A.A remark on Hausman's specification test. Econometrica 50 (1982): 749–759.CrossRefGoogle Scholar
7.Mitra, S.K.Generalized inverse of matrices and their application to linear models. Hand book of Statistics 1 (1980): 471–512.CrossRefGoogle Scholar
8.Mizon, G.E. & Richard, J.-F.. The encompassing principle and its application to testing non nested hypotheses. Econometrica 54 (1986): 657–678.CrossRefGoogle Scholar
9.Moore, D.S.Generalized inverses, Wald's method, and the construction of chi-squared tests of fit. Journal of the American Statistical Association 72 (1977): 131–137.CrossRefGoogle Scholar
10.Newey, W.K.Generalized method of moments specification testing. Journal of Econometrics 29 (1985): 229–256.CrossRefGoogle Scholar
11.Newey, W.K. & West, K.D.. A simple, positive definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55 (1987): 703–708.CrossRefGoogle Scholar
12.Rao, C.R.Linear statistical inference and its applications (2nd ed.). New York: John Wiley, 1973.CrossRefGoogle Scholar
13.Rao, C.R. & Mitra, S.K.. Generalized inverse of matrices and its applications. New York: John Wiley, 1971.Google Scholar
14.Seber, G.A.F.Linear regression analysis. New York: John Wiley, 1977.Google Scholar
15.Stewart, G.W.On the continuity of the generalized inverse. S1AM Journal of Applied Math ematics 17 (1969): 33–45.CrossRefGoogle Scholar
16.Tyler, D.E.Asymptotic inference for eigenvectors. Annals of Statistics 9 (1981): 725–736.CrossRefGoogle Scholar
17.Vuong, Q.H. Generalized inverses and asymptotic properties of Wald tests. Social Science Working Paper No. 607, California Institute of Technology, Pasadena, California, 1986.Google Scholar