Fixing Boundary Violations

3.1 Model Specification

The maximum likelihood estimate of the truncated regression model is a nonlinear constrained minimization optimization:⁹
\begin{align*}
Minimize \quad \quad \quad &-\log L\left( \boldsymbol{\beta} ,\sigma |\boldsymbol{{x}^{*}},{{y}_{i}} \right) \notag \\
Subject \quad to \qquad
&{{g}_{1}}={{\beta }_{0}}+\hat{y}_{\sim 0}^{\max }-b\le 0 \tag{3.1} \\
&{{g}_{2}}=a-{{\beta }_{0}}-\hat{y}_{\sim 0}^{\min }\le 0 \notag\\
&{{g}_{3}}=\beta_{0}-b\le 0 \notag\\
&{{g}_{4}}=-\beta_{0}+a\le 0 \notag\\
& \qquad \qquad \qquad \vdots \notag \\
&{{g}_{2m+3}}={{\beta }_{m}}-\min \left( \frac{b-\hat{y}_{\sim m}^{\max }}{x_{m}^{*\max }}, -\frac{\hat{y}_{\sim m}^{\min }-a}{x_{m}^{*\min }} \right)\le 0 \notag\\
&{{g}_{2m+4}}=-{{\beta }_{m}}+\max \left( \frac{a-\hat{y}_{\sim m}^{\min }}{x_{m}^{*\max }}, -\frac{\hat{y}_{\sim m}^{\max }-b}{x_{m}^{*\min }} \right)\le 0 \notag\\
&{{g}_{2m+5}}=\sigma -b+a\le 0 \notag\\
&{{g}_{2m+6}}=-\sigma +\kappa\le 0. \notag
\end{align*}

We can specify this problem in matrix terms:
\begin{equation*}
\boldsymbol{\gamma}=
\left(
\begin{matrix}
\boldsymbol{\beta}\\
\sigma
\end{matrix}
\right)
,{c}_{I}\left(\boldsymbol{\gamma} \right)=
\left(
\begin{matrix}
{{g}_{1}} \\
\vdots\\
{{g}_{2m+6}}
\end{matrix}
\right)
,\left( {{P}_{I}} \right)
\begin{cases}
{\min }f\left( \boldsymbol{\gamma} \right) \\
{{c}_{I}}\left( \boldsymbol{\gamma} \right)\le 0 \\
\boldsymbol{\gamma} \in \Omega
\end{cases},
\end{equation*}
where $\boldsymbol{\gamma}$ refers to the parameter vector being estimated, ${c}_{I}\left(\boldsymbol{\gamma} \right)$ is the vector of inequality constraints, $\Omega$ is the feasible parameter space, and ${{P}_{I}}$ represents the general minimization problem with only inequality constraints. The subscript $I$ represents inequality constraints hereafter.

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Footnote

⁹In this section, we adopt the centered model for the truncated regression. The covariate matrix is noted with an asterisk as $\boldsymbol{{x}^{*}}$.