Munich Personal RePEc Archive Normal versus Noncentral Chi - square Asymptotics of Misspecified Models

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 misspeci cations regardless of the sample size (as long as it is su ciently large), while the normal distribution, with a bias correction, gives a slightly better approximation for extremely severe misspeci cations. However, neither the noncentral chi-square distribution nor the theoretical normal distributions give a reasonable approximation of the LR test statistics under extremely severe misspeci cations. Of course, extremely misspeci ed models are not of much practical interest.