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Normal versus Noncentral Chi-square Asymptotics of Misspecified Models

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  • Chun, So Yeon
  • Alexander, Shapiro

Abstract

The noncentral chi-square approximation of the distribution of the likelihood ratio (LR) test statistic is a critical part of the methodology in structural equations modeling (SEM). Recently, it was argued by some authors that in certain situations normal distributions may give a better approximation of the distribution of the LR test statistic. The main goal of this paper is to evaluate the validity of employing these distributions in practice. Monte Carlo simulation results indicate that the noncentral chi-square distribution describes behavior of the LR test statistic well under small, moderate and even severe misspecifications regardless of the sample size (as long as it is sufficiently large), while the normal distribution, with a bias correction, gives a slightly better approximation for extremely severe misspecifications. However, neither the noncentral chi-square distribution nor the theoretical normal distributions give a reasonable approximation of the LR test statistics under extremely severe misspecifications. Of course, extremely misspecified models are not of much practical interest.

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 17310.

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Date of creation: Sep 2009
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Handle: RePEc:pra:mprapa:17310

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Keywords: Model misspecification; covariance structure analysis; maximum likelihood; generalized least squares; discrepancy function; noncentral chi-square distribution; normal distribution; factor analysis;

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  1. R. Golden, 2003. "Discrepancy Risk Model Selection Test theory for comparing possibly misspecified or nonnested models," Psychometrika, Springer, vol. 68(2), pages 229-249, June.
  2. Yuan, Ke-Hai & Hayashi, Kentaro & Bentler, Peter M., 2007. "Normal theory likelihood ratio statistic for mean and covariance structure analysis under alternative hypotheses," Journal of Multivariate Analysis, Elsevier, vol. 98(6), pages 1262-1282, July.
  3. Vuong, Quang H, 1989. "Likelihood Ratio Tests for Model Selection and Non-nested Hypotheses," Econometrica, Econometric Society, vol. 57(2), pages 307-33, March.
  4. McManus, Douglas A., 1991. "Who Invented Local Power Analysis?," Econometric Theory, Cambridge University Press, vol. 7(02), pages 265-268, June.
  5. Shapiro, Alexander, 2009. "Asymptotic normality of test statistics under alternative hypotheses," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 936-945, May.
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