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 $\boldsymbol{{x}^{*}}$ .