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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.

Suggested Citation

  • Chun, So Yeon & Alexander, Shapiro, 2009. "Normal versus Noncentral Chi-square Asymptotics of Misspecified Models," MPRA Paper 17310, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:17310
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    References listed on IDEAS

    as
    1. Vuong, Quang H, 1989. "Likelihood Ratio Tests for Model Selection and Non-nested Hypotheses," Econometrica, Econometric Society, vol. 57(2), pages 307-333, March.
    2. R. Golden, 2003. "Discrepancy Risk Model Selection Test theory for comparing possibly misspecified or nonnested models," Psychometrika, Springer;The Psychometric Society, vol. 68(2), pages 229-249, June.
    3. 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.
    4. McManus, Douglas A., 1991. "Who Invented Local Power Analysis?," Econometric Theory, Cambridge University Press, vol. 7(2), 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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    2. Hao Wu & Michael Browne, 2015. "Random Model Discrepancy: Interpretations and Technicalities (A Rejoinder)," Psychometrika, Springer;The Psychometric Society, vol. 80(3), pages 619-624, September.
    3. Hao Wu & Michael Browne, 2015. "Quantifying Adventitious Error in a Covariance Structure as a Random Effect," Psychometrika, Springer;The Psychometric Society, vol. 80(3), pages 571-600, September.
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    5. So Yeon Chun & Michael W. Browne & Alexander Shapiro, 2018. "Modified Distribution-Free Goodness-of-Fit Test Statistic," Psychometrika, Springer;The Psychometric Society, vol. 83(1), pages 48-66, March.
    6. Stanislav Kolenikov & Kenneth A. Bollen, 2012. "Testing Negative Error Variances," Sociological Methods & Research, , vol. 41(1), pages 124-167, February.

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    More about this item

    Keywords

    Model misspecification; covariance structure analysis; maximum likelihood; generalized least squares; discrepancy function; noncentral chi-square distribution; normal distribution; factor analysis;
    All these keywords.

    JEL classification:

    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General

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