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Otilia Boldeaa and Jan R. Magnusb,c
aDepartment of Econometrics & OR, Tilburg University, Tilburg, The Netherlands
bDepartment of Econometrics & Data Science, Vrije Universiteit, Amsterdam, The Netherlands
cTinbergen Institute, Amsterdam, The Netherlands
CONTACT Otilia Boldea [email protected] Department of Econometrics & OR, Tilburg University, Warandelaan 2, 5037 AB, Tilburg, The Netherlands.
Dante Amengual, Gabriele Fiorentini and Enrique Sentana kindly alerted us to an error in Theorem 2 in Boldea and Magnus (Citation2009), where we erroneously stated that the degrees of freedom of our proposed information matrix test is . The degrees of freedom are different, and a generalized inverse should be used in the definition of the IM test. Below is the corrected statement of Theorem 2, which is now a proposition because the proof follows from Amengual, Fiorentini, and Sentana (Citation2024a, Citation2024b).
Proposition A
(Information Matrix Test). Define the variance matrix:
where denotes the tth increment to the score, and
Then, evaluated at , and under the null hypothesis of correct specification, with
equal to the Moore-Penrose generalized inverse of
,
asymptotically follows a
-distribution with
degrees of freedom.
Our 2009 Appendix mistakenly suggested that the proof of Theorem 2 “follows from […] the development in Lancaster (Citation1984)”. The results in Lancaster (Citation1984) are insufficient to arrive at the correct degrees of freedom.
Instead, the proof of Proposition A follows directly from Amengual, Fiorentini, and Sentana (Citation2024a, Citation2024b). Amengual, Fiorentini, and Sentana (Citation2024a) show in their Proposition 1 that the IM test for a multivariate normal distribution (so for g = 1) coincides with a test that the expectations of all distinct third- and fourth-order multivariate Hermite polynomials are zero. Since there are unique kth order polynomials, and the Hermite polynomials are orthogonal, the IM test for g = 1 converges to a
-distribution with
degrees of freedom. Based on these results, Amengual, Fiorentini, and Sentana (Citation2024b) show in their Proposition 4 that a reparameterized version of our test converges to a
-distribution with
degrees of freedom.
The error in Theorem 2 also implies that we used the wrong critical values in . We reran the simulations in with the correct degrees of freedom, and we report them here. The corrected code can be found at https://github.com/oboldea/MultivariateNormalMixtures. The rest of the article is not affected.
Table 5 Size of IM test, simulation results.
References
- Amengual, A., Fiorentini, G., and Sentana, E. (2024a), “Multivariate Hermite Polynomials and Information Matrix Tests,” Econometrics and Statistics, forthcoming. DOI: 10.1016/j.ecosta.2024.01.005.
- Amengual, A., Fiorentini, G., and Sentana, E. (2024b), “The Information Matrix Test for Gaussian Mixtures,” CEMFI working paper 2401. Available at https://www.cemfi.es/ftp/wp/2401.pdf.
- Boldea, O., and Magnus, J. R. (2009), “Maximum Likelihood Estimation of the Multivariate Normal Mixture Model,” Journal of the American Statistical Association, 104, 1539–1549. DOI: 10.1198/jasa.2009.tm08273.
- Lancaster, A. (1984), “The Covariance Matrix of the Information Matrix Test,” Econometrica, 52, 1051–1053. DOI: 10.2307/1911198.