Fixing Boundary Violations

5.2 Model Adjustment

Two adjustments are made to the TRMCO model. First, instead of centering by the means, the author adopts a model specification that fixes all independent variables at the minimum level for handling dummy variables.¹⁵ Centering dummy variables makes no sense in interpretation, especially for those unbalanced dummies that could cause numerical problems in parameter estimation.¹⁶ Therefore, the constant estimate of the OLS model is different from what was originally reported. Second, given that the truncated normal distribution is a more plausible distributional assumption, the criteria for model comparison should be based on the log pseudolikelihood function of the TRMCO model, as well as admissibility of the parameter estimates.¹⁷

Due to the spatial dependence of the sample, robust standard error is applied for hypothesis testing. For each model, the author compares three methods (OLS, TRM, and TRMCO), with three criteria: first, which methods produces the greatest values of the log pseudolikelihood function; second, whether the predicted value of vote-share is admissible; third, whether the parameter estimate complies to the boundary constraints. Only the answers satisfied with the latter two criteria are eligible for log pseudolikelihood comparison.

Regarding the numerical estimation of the TRMCO model, we use TRM's solution as the initial value. The number of maximum iterations and the tolerance value remain the same as $100$ and $10^{-4}$. In addition, since all independent variables are fixed at minimum, the boundary constraint for ${{\hat{\beta }}_{m}}$ is modified correspondingly

\begin{equation*}
\frac{a-\hat{y}_{\sim m}^{\min }}{x_{m}^{\max }-x_{m}^{\min }}\le {{\hat{\beta }}_{m}}\le \frac{b-\hat{y}_{\sim m}^{\max }}{x_{m}^{\max }-x_{m}^{\min }},
\tag{5.1}
\end{equation*}

where
\begin{align*}
&\hat{y}_{\sim m}^{\max }=
\begin{cases}
{{\hat{\beta }}_{0}}+\sum\limits_{j\ne m}{v_{j}^{+}{{{\hat{\beta }}}_{j}}{{\left( {{x}_{j}}-x_{j}^{\min } \right)}^{\max }}}+Max\left( {{\beta }_{d1}},\cdots ,{{\beta }_{d2}} \right) &\text{(independent vars)} \\
{{\hat{\beta }}_{0}}+\sum\limits_{}{v_{j}^{+}{{{\hat{\beta }}}_{j}}{{\left( {{x}_{j}}-x_{j}^{\min } \right)}^{\max }}} &\text{(regional dummies)},
\end{cases} \notag \\
&\hat{y}_{\sim m}^{\min }=
\begin{cases}
{{\hat{\beta }}_{0}}+\sum\limits_{j\ne m}{v_{j}^{-}{{{\hat{\beta }}}_{j}}{{\left( {{x}_{j}}-x_{j}^{\min } \right)}^{\max }}}+Min\left( {{\beta }_{d1}},\cdots ,{{\beta }_{d2}} \right) &\text{(independent vars)} \\
{{\hat{\beta }}_{0}}+\sum\limits_{}{v_{j}^{-}{{{\hat{\beta }}}_{j}}{{\left( {{x}_{j}}-x_{j}^{\min } \right)}^{\max }}} &\text{(regional dummies)},
\end{cases}
\tag{5.2}
\end{align*}
in which ${{\beta}_{dj}}$ represents a dummy variable, and we drop the terms ${{\left( {{x}_{j}}-x_{j}^{\min } \right)}^{\min }}$ since they are all zero.¹⁸ As the appendix makes evident, we can easily prove that (5.1) applies to both pure independent and interaction variables.

____________________

Footnote

¹⁵ We do not fix the original interaction variables to the minimum. Instead, we fix all the non-interaction variables at the minimum value first, and then compute the crossproducts to generate two interaction terms.

¹⁶ Consider that the regional dummy variable Africa only has 5.6% of the cases in the overall sample. If it is centered to the mean, the centered dummy only has the value of either $-.056$ or $.944$, of which the former is very small as a denominator and would sometimes lead to numerical problems in the estimation process.

¹⁷ Since the data are clustered samples and violate the i.i.d. assumption, the product of all the likelihood function, regardless of cluster dependence, is the so-called " pseudo-likelihood" function. (Strauss and Ikeda, 1990)

¹⁸ For the interaction term, the indicator variables $v_{j}^{+}$ and $v_{j}^{-}$ are not independently decided by its own beta coefficient. Rather, they are decided by the signs of the composing variable's beta coefficients. For instance, if $x_{7}^{*}=x_{5}^{*}\times x_{6}^{*}$, then $v_{7}^{+}=1$ when $sign({{\beta }_{5}})\times sign({{\beta }_{6}})>0\,$, and $v_{7}^{-}=1$ when $sign({{\beta }_{5}})\times sign({{\beta }_{6}})<0\,$.