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Hypothesis tests with a repeatedly singular information matrix

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  • Sentana, Enrique
  • Amengual, Dante
  • Bei, Xinyue

Abstract

We study score-type tests in likelihood contexts in which the nullity of the information matrix under the null is larger than one, thereby generalizing earlier results in the literature. Examples include multivariate skew normal distributions, Hermite expansions of Gaussian copulas, purely non-linear predictive regressions, multiplicative seasonal time series models and multivariate regression models with selectivity. Our proposal, which involves higher order derivatives, is asymptotically equivalent to the likelihood ratio but only requires estimation under the null. We conduct extensive Monte Carlo exercises that study the finite sample size and power properties of our proposal and compare it to alternative approaches.

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  • Sentana, Enrique & Amengual, Dante & Bei, Xinyue, 2020. "Hypothesis tests with a repeatedly singular information matrix," CEPR Discussion Papers 14415, C.E.P.R. Discussion Papers.
    • Handle: RePEc:cpr:ceprdp:14415
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    1. James J. Heckman, 1976. "The Common Structure of Statistical Models of Truncation, Sample Selection and Limited Dependent Variables and a Simple Estimator for Such Models," NBER Chapters, in: Annals of Economic and Social Measurement, Volume 5, number 4, pages 475-492, National Bureau of Economic Research, Inc.
    2. Davidson, James, 1994. "Stochastic Limit Theory: An Introduction for Econometricians," OUP Catalogue, Oxford University Press, number 9780198774037.
    3. Dante Amengual & Enrique Sentana & Zhanyuan Tian, 2022. "Gaussian Rank Correlation and Regression," Advances in Econometrics, in: Essays in Honor of M. Hashem Pesaran: Panel Modeling, Micro Applications, and Econometric Methodology, volume 43, pages 269-306, Emerald Group Publishing Limited.
    4. Lee, Lung-Fei & Chesher, Andrew, 1986. "Specification testing when score test statistics are identically zero," Journal of Econometrics, Elsevier, vol. 31(2), pages 121-149, March.
    5. T. S. Breusch & A. R. Pagan, 1980. "The Lagrange Multiplier Test and its Applications to Model Specification in Econometrics," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 47(1), pages 239-253.
    6. Amsler, Christine & Prokhorov, Artem & Schmidt, Peter, 2016. "Endogeneity in stochastic frontier models," Journal of Econometrics, Elsevier, vol. 190(2), pages 280-288.
    7. Davidson, Russell & MacKinnon, James G., 1994. "Graphical Methods for Investigating the Size and Power of Hypothesis Tests," Queen's Economics Department Working Papers 273307, Queen's University - Department of Economics.
    8. Prosper Dovonon & Eric Renault, 2013. "Testing for Common Conditionally Heteroskedastic Factors," Econometrica, Econometric Society, vol. 81(6), pages 2561-2586, November.
    9. Sargan, J D, 1983. "Identification and Lack of Identification," Econometrica, Econometric Society, vol. 51(6), pages 1605-1633, November.
    10. Amemiya, Takeshi, 1984. "Tobit models: A survey," Journal of Econometrics, Elsevier, vol. 24(1-2), pages 3-61.
    11. Dante Amengual & Enrique Sentana, 2020. "Is a Normal Copula the Right Copula?," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 38(2), pages 350-366, April.
    12. Davidson, Russell & MacKinnon, James G, 1998. "Graphical Methods for Investigating the Size and Power of Hypothesis Tests," The Manchester School of Economic & Social Studies, University of Manchester, vol. 66(1), pages 1-26, January.
    13. Horowitz, Joel L. & Savin, N. E., 2000. "Empirically relevant critical values for hypothesis tests: A bootstrap approach," Journal of Econometrics, Elsevier, vol. 95(2), pages 375-389, April.
    14. Arellano-Valle, Reinaldo B. & Azzalini, Adelchi, 2008. "The centred parametrization for the multivariate skew-normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 99(7), pages 1362-1382, August.
    15. Yanqin Fan & Andrew J. Patton, 2014. "Copulas in Econometrics," Annual Review of Economics, Annual Reviews, vol. 6(1), pages 179-200, August.
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    1. Dante Amengual & Xinyue Bei & Enrique Sentana, 2022. "Normal but skewed?," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 37(7), pages 1295-1313, November.
    2. Amengual, Dante & Bei, Xinyue & Carrasco, Marine & Sentana, Enrique, 2025. "Score-type tests for normal mixtures," Journal of Econometrics, Elsevier, vol. 248(C).

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    Keywords

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    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C34 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Truncated and Censored Models; Switching Regression Models

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