This is a follow-up study of my two previous works, "Indefinite Integration of the Gamma Integral" and "Fixing Boundary Violations". It relates to the former project because I apply the integration rule of the Gaussian distribution in maximum likelihood estimation, specifically in the deduction of gradient vectors and hessian matrices. However, I still present it as a "$\Phi $" function instead of "$h$" just in case the reader has not understood the "$h$" function. The relationship to the latter work is more straightforward, for both studies share the same target: regression with truncated normal variables. The difference is that the two studies deal with different data formats. "Fixing Boundary Violations" targets cross-sectional data, while this study focuses on time-series-cross-section (TSCS) data. As can be seen in this paper, solving boundary violations in TSCS data analysis is much more complicated than in cross-sectional data analysis.
Given the fact that I make no effort to earn academic credit, you can simply ignore my work if you believe I am plainly wrong. On the other hand, I am very happy to discuss the detail of this work to whoever is interested and also welcome any feedback. My only purpose is to improve this work and spread out the idea if my claim is tenable.