Fixing Boundary Violations: Applying Constrained
Optimization to the Truncated Regression Model
2.1 The Basics of Truncated Regression Model
The TRM model can be described with the following specifications. Suppose we have n
i.i.d. observations of y_{i}, which follows the truncated normal distribution TN\left( \mu_{i}
,{{\sigma }^{2}};a,b \right), where \mu_{i}, \sigma, a, and b are location parameter, scale parameter,
lower limit, and upper limit, respectively. We add m covariates ({{x}_{1},\cdots,{x}_{m}}) in the model to
explain \mu_{i} for each observation i by assuming {{\mu }_{i}}=\boldsymbol{{{x}_{i}}\beta} . Therefore,
the likelihood function is
\begin{equation*}
L\equiv\prod\limits_{i=1}^{n}{\left\{\frac{\exp\left(\frac{-\left( {{y}_{i}}-\boldsymbol{{{x}_{i}}\beta} \right)}
{2{{\sigma }^{2}}} \right)}{\int_{a}^{b}{\exp \left( \frac{-\left( y-\boldsymbol{{{x}_{i}}\beta} \right)}
{2{{\sigma }^{2}}} \right)dy}} \right\}}.
\end{equation*}
Using \Phi \left( \cdot \right) to replace the cdf function of the normal distribution. We can derive the loglikelihood
\begin{equation*} \log L=-\sum\limits_{i=1}^{n}{\ln {{D}_{i}}-\frac{1}{2{{\sigma }^{2}}}}\sum\limits_{i=1}^{n}{{{\left( {{y}_{i}} -\boldsymbol{x_{i}\beta} \right)}^{2}}}, \end{equation*}
where {{D}_{i}}=\sqrt{2\pi}\sigma\left[\Phi\left(\frac{b-\boldsymbol{x_{i}\beta}}{\sigma}\right)-\Phi \left(\frac{a-\boldsymbol{x_{i}\beta}}{\sigma } \right) \right].
With a few manipulations, we can deduce the gradient vector and the Hessian matrix, and apply the generalized Gauss-Newton algorithm to derive maximum likelihood estimates of \boldsymbol{\hat{\beta}} and \hat{\sigma}. (Hausman and Wise, 1977, 936)
Notice that the above model does not specify any constraints on the dependent variable y_{i}, regression coefficient \boldsymbol{\beta}, and scale parameter \sigma. However, those parameters do have certain theoretical constraints that need to be specified. Those constraints can be categorized into three types: (i) boundary limits of the dependent variable, (ii) admissible parameter space of the independent variables, and (iii) constraints of the scale parameter.
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