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NEYMAN’S C(α) TEST FOR UNOBSERVED HETEROGENEITY

Published online by Cambridge University Press:  01 September 2015

Jiaying Gu
Jiaying Gu*
Affiliation:
University of Illinois at Urbana-Champaign
*
*Address correspondence to Jiaying Gu, Department of Economics, University of Illinois at Urbana-Champaign, 214 David Kinley Hall, 1407 W. Gregory Dr., Urbana, Illinois 61801, MC-707, USA; e-mail: gu17@illinois.edu.
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Abstract

A unified framework is proposed for tests of unobserved heterogeneity in parametric statistic models based on Neyman’s C(α) approach. Such tests are irregular in the sense that the first order derivative of the log likelihood with respect to the heterogeneity parameter is identically zero, and consequently the conventional Fisher information about the parameter is zero. Nevertheless, local asymptotic optimality of the C(α) tests can be established via LeCam’s differentiability in quadratic mean and the limit experiment approach. This leads to local alternatives of order n−1/4 . The scalar case result is already familiar from existing literature and we extend it to the multidimensional case. The new framework reveals that certain regularity conditions commonly employed in earlier developments are unnecessary, i.e. the symmetry or third moment condition imposed on the heterogeneity distribution. Additionally, the limit experiment for the multidimensional case suggests modifications on existing tests for slope heterogeneity in cross sectional and panel data models that lead to power improvement. Since the C(α) framework is not restricted to the parametric model and the test statistics do not depend on the particular choice of the heterogeneity distribution, it is useful for a broad range of applications for testing parametric heterogeneity.

Type
ARTICLES
Information
Econometric Theory , Volume 32 , Issue 6 , December 2016 , pp. 1483 - 1522
Copyright
Copyright © Cambridge University Press 2015 

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References

REFERENCES

Abelson, R. & Tukey, J. (1963) Efficient utilization of non-numerical information in quatitative analysis: General theory and the case of simple order. Annals of Mathematical Statistics 34, 1347–1369.CrossRefGoogle Scholar
Akharif, A. & Hallin, M. (2003) Efficient detection of random coefficients in autoregressive models. Annals of Statistics 31(2), 675–704.Google Scholar
Andrews, D. (1994) Empirical process methods in econometrics. In Engle, R. & McFadden, D.L. (eds.), Handbook of Econometrics , vol. 4, 2247–2294. Elsevier.Google Scholar
Bennala, N., Hallin, M., & Paindaveine, D. (2012) Pseudo-gaussian and rank-based optimal tests for random individual effects in large n small t panels. Journal of Econometrics 170, 50–67.CrossRefGoogle Scholar
Breusch, T. & Pagan, A. (1979) A simple test for heteroscedasticity and random coefficient variation. Econometrica 47(5), 1287–1294.Google Scholar
Breusch, T. & Pagan, A. (1980) The Lagrange multiplier test and its applications to model specification in econometrics. Review of Economic Studies 47, 239–253.CrossRefGoogle Scholar
Bühler, W. & Puri, P. (1966) On optimal asymptotic tests of composite hypotheses with several constraints. Z. Wahrscheinlichkeittstheorie verw. Geb. 5, 71–88.CrossRefGoogle Scholar
Cameron, A. & Trivedi, P. (1998) Regression Analysis of Count Data. Cambridge University Press.Google Scholar
Chen, H., Chen, J., & Kalbfleisch, J. (2001) A modified likelihood ratio test for homogeneity in finite mixture models. Journal of the Royal Statistical Society,Series B 63(1), 19–29.Google Scholar
Chernoff, H. (1954) On the distribution of the likelihood ratio. Annals of Mathematical Statistics 25(3), 573–578.CrossRefGoogle Scholar
Chesher, A. (1984) Testing for neglected heterogeneity. Econometrica 52(4), 865–872.CrossRefGoogle Scholar
Chesher, A. & Spady, R. (1991) Asymptotic expansions of the information matrix test statistic. Econometrica 59(3), 787–815.CrossRefGoogle Scholar
Cox, D. (1983) Some remarks on overdispersion. Biometrika 70(1), 269–274.Google Scholar
Cox, D. & Hinkley, D. (1974) Theoretical Statistics. Chapman and Hall.Google Scholar
Cramér, H. (1946) Mathematical Methods of Statistics. Princeton University Press.Google Scholar
Davidson, R. & MacKinnon, J. (1998) Graphical methods for investigating the size and power of hypothesis tests. The Manchester School 66(1), 1–26.Google Scholar
Dean, C. (1992) Testing for overdispersion in poisson and binomial regression models. Journal of the American Statistical Association 87, 451–457.Google Scholar
Dean, C. & Lawless, J. (1989) Tests for detecting overdispersion in poisson regression models. Journal of the American Statistical Association 84, 467–472.Google Scholar
Fisher, R. (1950) The significance of deviations from expectation in a poisson series. Biometrika 6, 17–24.Google Scholar
Gu, J., Koenker, R., & Volgushev, S. (2013) Testing for homogeneity in mixture models. ArXiv: 1302.1805[stat.ME].Google Scholar
Hallin, M. & Ley, C. (2014) Skew-symmetric distributions and fisher information: The double sin of the skew-normal. Bernoulli 20, 1432–1453.CrossRefGoogle Scholar
Hillier, G. (1986) Joint tests for zero restrictions on nonnegative regression coefficients. Biometrika 73(3), 657–669.Google Scholar
Honda, Y. (1988) A size correction to the lagrange multiplier test for heteroscedasticity. Journal of Econometrics 38, 375–386.Google Scholar
Horowitz, J. (1994) Bootstrap-based critical values for the information matrix test. Journal of Econometrics 61, 395–411.Google Scholar
Kiefer, N. (1984) A simple test for heterogeneity in exponential models of duration. Journal of Labor Economics 2(4), 539–549.Google Scholar
King, M. & Wu, P. (1997) Locally optimal one-sided tests for multiparameter hypotheses. Econometric Reviews 16, 131–156.Google Scholar
Lancaster, T. (1985) Generalized residuals and heterogenous duration models with applications to the weibull model. Journal of Econometrics 28, 155–169.Google Scholar
LeCam, L. (1972) Limits of experiments. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. I, pp. 245–261. University of California Press.Google Scholar
Lee, L. (1986) Specification test for poisson regression models. International Economic Review 27(3), 689–706.Google Scholar
Lee, L. & Chesher, A. (1986) Specification testing when score test statistics are identically zero. Journal of Econometrics 31, 121–149.Google Scholar
Lindsay, B.G. (1995) Mixture Models: Theory, Geometry and Applications. Institute of Mathematical Statistics.Google Scholar
Moran, P. (1973) Asymptotic properties of homogeneity tests. Biometrika 60(1), 79–85.Google Scholar
Neyman, J. (1959) Optimal asymptotic tests of composite statistical hypotheses. In Grenander, U. (ed.), Probability and Statistics, the Harald Cramer Volume. Wiley.Google Scholar
Neyman, J. (1979) C(α) tests and their use. Sankhyā: The Indian Journal of Statistics 41, 1–21.Google Scholar
Neyman, J. & Scott, E. (1966) On the use of C(α) optimal tests of composite hypotheses. Bulletin de lInstitut International de Statistique 41(1), 477–497.Google Scholar
Perlman, M. (1969) One-sided testing problems in multivariate analysis. Annals of Mathematical Statistics 40, 549–567.Google Scholar
Pesaran, M. & Yamagata, Y. (2008) Testing slope homogeneity in large panels. Journal of Econometrics 142, 50–93.Google Scholar
Pollard, D. (1997) Another look at differentiability in quadratic mean. In Pollard, D., Torgersen, E., & Yang, G. (eds.), Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics. Springer.Google Scholar
Rao, C. (1948) Large-sample test of statistical hypotheses concerning several parameters with applications to problems of estimation. Proceedings of the Cambridge Philosophical Society 44, 50–57.Google Scholar
Rotnitzky, A., Cox, D., Bottai, M., & Robins, J. (2000) Likelihood-based inference with singular information matrix. Bernoulli 6(2), 243–284.CrossRefGoogle Scholar
Schaafsma, W. & Smid, L. (1966) Most stringent somewhere most powerful tests against alternatives restricted by a number of linear alternatives. Annals of Mathematical Statistics 37, 1161–1172.Google Scholar
Shapiro, A. (1985) Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrika 72, 133–144.Google Scholar
Silvapulle, M. & Silvapulle, P. (1995) A score test against one-sided alternatives. Journal of the American Statistical Association 90, 342–349.CrossRefGoogle Scholar
Swamy, P. (1970) Efficient inference in a random coefficient regression model. Econometrica 38, 311–323.CrossRefGoogle Scholar
Turlach, B. & Weignessel, A. (2013) Functions to solve quadratic programming problems. R package version 1.5-5, 2013-04-17.Google Scholar
van der Vaart, A. (1998) Asymptotic Statistics. Cambridge University Press.Google Scholar
van der Vaart, A.W. & Wellner, J.A. (1996) Weak Convergence and Empirical Processes. New York: Springer. Series in Statistics.CrossRefGoogle Scholar
White, H. (1982) Maximum likelihood estimation of misspecified models. Econometrica 50, 1–25.Google Scholar
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