Fixing Boundary Violations: Applying Constrained
Optimization to the Truncated Regression Model
2.4 Type III Boundary Constraints
Type III boundary constraints are about the scale parameter $\sigma$. While often the
constraints are not effective, we can consider adopting the full truncation range as the upper limit and an
arbitrary small positive number ($\kappa$) as the lower limit8
\begin{equation*}
\kappa \le \hat{\sigma }\le b-a.
\end{equation*}
If $\sigma$ approaches infinity, $y_{i}$ will approach the uniform distribution. When the optimization result gives
an upper boundary value of $\hat{\sigma}$, it signifies a violation of the distribution assumption and means that
$y_{i}$ does not fit the truncated normal assumption well. For the lower limit constraint, if $\sigma$ approaches
zero or becomes negative, this indicates a negative variance resulting from the non-positive definite Hessian. Many
possible explanations can account for this problem, but its occurrence is usually associated with an ill-specified
model, and thus regarded as a failed estimate.
In this article, we separate the OLS out-of-bounds violation from the type I violation. The former happens when the OLS estimate generates an inadmissible predicted value to an empirical observation; the latter is identified when any possible predicted value falls outside the boundary. Apparently, a type I violation is defined with a more rigid standard, and it encompasses the OLS out-of-bounds violation.
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Footnote
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