  Model specification searches and modifications are commonly employed in
covariance structure analysis (CSA) or structural equation modeling (SEM) to
improve the goodness-of-fit. However, these practices can be susceptible to
capitalizing on chance, as a model that fits one sample may not generalize to
another sample from the same population. This paper introduces the improved
Lagrange Multipliers (LM) test, which provides a reliable method for conducting
a thorough model specification search and effectively identifying missing
parameters. By leveraging the stepwise bootstrap method in the standard LM and
Wald tests, our data-driven approach enhances the accuracy of parameter
identification. The results from Monte Carlo simulations and two empirical
applications in political science demonstrate the effectiveness of the improved
LM test, particularly when dealing with small sample sizes and models with
large degrees of freedom. This approach contributes to better statistical fit
and addresses the issue of capitalization on chance in model specification.
