Fixing Boundary Violations: Applying Constrained
Optimization to the Truncated Regression Model
3.4 Hypothesis Testing
Hypothesis testing can be carried out by computing the empirical variance-covariance matrix
\begin{equation*}
Var\left( \boldsymbol{\hat{\gamma }} \right)={{\left( -\frac{{{\partial }^{2}}\left( \ln L \right)}{\partial
\boldsymbol{\hat{\gamma}}\partial {{\boldsymbol{\hat{\gamma }}}^{T}}} \right)}^{-1}},
\end{equation*}
where $\boldsymbol{\hat{\gamma}}={{\left( {{{\hat{\beta }}}_{0}},{{{\hat{\beta }}}_{1}},{{{\hat{\beta }}}_{2}},
\hat{\sigma } \right)}^{T}}$. When the i.i.d. assumption is violated, such as with cluster samples, the
robust standard error can be generated from
\begin{multline*}
Var{{\left( \boldsymbol{\hat{\beta }} \right)}_{robust}}={{\left( -\frac{1}
{{{{\hat{\sigma }}}^{2}}}\sum\limits_{i=1}^{n}{\sum\limits_{t=1}^{{{T}_{i}}}
{\boldsymbol{x_{it}^{*}}
\boldsymbol{x{{_{it}^{*}}}^{T}}}} \right)}^{-1}}\left[ \sum\limits_{i=1}^{n}{\left( \sum\limits_{t=1}^{{{T}_{i}}}
{\frac{1}{{{{\hat{\sigma }}}^{2}}}\boldsymbol{x_{it}^{*}}{{e}_{it}}} \right)\left( \sum\limits_{t=1}^{{{T}_{i}}}{\frac{1}
{{{{\hat{\sigma }}}^{2}}}{{e}_{it}}\boldsymbol{x{{_{it}^{*}}}^{T}}} \right)} \right]\\{{\left( -\frac{1}
{{{{\hat{\sigma }}}^{2}}}\sum\limits_{i=1}^{n}{\sum\limits_{t=1}^{{{T}_{i}}}
{\boldsymbol{x_{it}^{*}}
\boldsymbol{x{{_{it}^{*}}}^{T}}}} \right)}^{-1}},
\end{multline*}
where $T$ refers to a temporal or spatial unit, $n$ is the overall sample size, and $T_{i}$ is the sample size for
the $i$th unit. (Greene, 2008, 515)
To evaluate the sensitivity of hypothesis testing when different models are applied, the author compares the mean absolute deviation of the four parameter estimates and their corresponding $t$ statistics, a measure that indicates the variability of the regression result and the significance level of parameter estimates per trial.
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