Fixing Boundary Violations

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

⁸We set $\kappa=0.001$ in this paper.