This article presents three methods under different scenarios with least squares or maximum likelihood assumptions. All three methods can conditionally or fully solve the boundary violations problem. The analysis indicates that the results of current panel regression are not only subject to some methodological problems, but also sensitive to different centering methods. By applying the technique of constrained optimization, those methodological problems can be targeted and the best possible solution can successfully be reached. I suggest a mandatory use in one of the three methods when an empirical boundary violation occurs. The robustness of the result can be also checked by comparing the results from the current method and the maximum likelihood method that corrects demeaning bias.
While the application of constrained optimization can eliminate boundary violations, it costs more time and computing capacity to execute the revised methods, especially with the maximum likelihood assumption. Given the scope of this article, a systematic assessment is yet to be done with regard to different methods's performance under various conditions. We expect more future work in simulations, as well as empirical studies to illuminate to what extent the current panel regression suffers from boundary violations, and to what extent the revised models can successfully eliminate these violations. By possessing this knowledge, scholars in this field can provide more definite criteria to prevent political science research from reporting an illogical out-of-bound finding.