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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