Fixing Boundary Violations

7.1 Proofs of Boundary Constraints

This following appendix contains three proofs of the boundary constraints for pure independent variables, interaction variables, and dummy variables. The purpose is to illustrate that all of the boundary constraints can be generalized by (5.1).

Notation

$x_{j}^{\dagger}$: A column vector of the covariate matrix after being fixed at the minimum.
$\beta_{0}$: Baseline predicted vote share when all covariates are fixed at the minimum, except the dummies which are all set to $0$.
$\beta_{1}$ to $\beta_{4}$: Beta coefficient estimates for the pure independent variables, Previous Votes, Re-election, Effective Number of Parties, and Income, respectively.
$\beta_{5}$ to $\beta_{9}$: Beta coefficient estimates associated with the interaction variables, Economy, Trade Openness, Economy$\times$Trade Openness, Presidential Election, and Economy$\times$Presidential Election, respectively.
$\beta_{10}$ to $\beta_{13}$: Beta coefficient estimates for the regional dummy variables, Africa, Asia, Central and Eastern Europe, and Latin America and the Caribbean, respectively.

Case I: Pure Independent Variables

Given the overall predicted vote share
\begin{align*}
{{\hat{y}}^{\max }}&={{\hat{\beta }}_{0}}+\sum\limits_{i=1}^{4}{v_{i}^{+}{{{\hat{\beta }}}_{i}}x_{i}^{\dagger \max }}+\sum\limits_{j=5}^{9}{v_{j}^{+}{{{\hat{\beta }}}_{j}}x_{j}^{\dagger \max }}+\max \left( {{{\hat{\beta }}}_{10}},{{{\hat{\beta }}}_{11}},{{{\hat{\beta }}}_{12}},{{{\hat{\beta }}}_{13}} \right) \\
{{\hat{y}}^{\min }}&={{\hat{\beta }}_{0}}+\sum\limits_{i=1}^{4}{v_{i}^{-}{{{\hat{\beta }}}_{i}}x_{i}^{\dagger \max }}+\sum\limits_{j=5}^{9}{v_{j}^{-}{{{\hat{\beta }}}_{j}}x_{j}^{\dagger \max }}+\min \left( {{{\hat{\beta }}}_{10}},{{{\hat{\beta }}}_{11}},{{{\hat{\beta }}}_{12}},{{{\hat{\beta }}}_{13}} \right),
\end{align*}
we know
\begin{align*}
\hat{y}_{\sim j}^{\max }&={{\hat{y}}^{\max }}-v_{j}^{+}{{\hat{\beta }}_{j}}x_{j}^{\dagger \max } \\
\hat{y}_{\sim j}^{\min }&={{\hat{y}}^{\min }}-v_{j}^{-}{{\hat{\beta }}_{j}}x_{j}^{\dagger \max },
\end{align*}
where $j=\left\{ 1,2,3,4 \right\}$.

Since $a\le {{\hat{y}}^{\min }}\le {{\hat{y}}^{\max }}\le b$,
\begin{align*}
y_{\sim j}^{\max }+v_{j}^{+}{{\beta }_{j}}x_{j}^{\dagger \max }\le b \\
y_{\sim j}^{\min }+v_{j}^{-}{{\beta }_{j}}x_{j}^{\dagger \max }\ge a
\end{align*}
Therefore,
\begin{equation*}
\frac{a-\hat{y}_{\sim j}^{\min }}{x_{j}^{\dagger \max }}\le {{\beta }_{j}}\le \frac{b-\hat{y}_{\sim j}^{\max }} {x_{j}^{\dagger \max }}.
\end{equation*}
$\square$

Case II: Interaction Variables

To simplify the proof, we only present the upper bound constraint. The same proof can easily be applied to the lower bound constraint. In the following proof, we first deal with $\beta_{5}$, beta coefficient of Economy, one of the composition variables for Economy$\times$Trade Openness ($\beta_{7}$) and Economy$\times$Presidential Election ($\beta_{9}$).

Given
\begin{equation*}
\hat{y}_{\sim 5}^{\max }={{\hat{y}}^{\max }}-v_{5}^{+}{{\hat{\beta }}_{5}}x_{5}^{\dagger \max }-v_{7}^{+} {{\hat{\beta }}_{7}}x_{7}^{\dagger \max }-v_{9}^{+}{{\hat{\beta }}_{9}}x_{9}^{\dagger \max },
\end{equation*}
we know
\begin{equation*}
{{\hat{y}}^{\max }}-\hat{y}_{\sim 5}^{\max }=v_{5}^{+}{{\beta }_{5}}x_{5}^{\dagger \max }+v_{7}^{+} {{\beta }_{7}}x_{7}^{\dagger \max }+v_{9}^{+}{{\beta }_{9}}x_{9}^{\dagger \max }\le b-\hat{y}_{\sim 5}^{\max }.
\end{equation*}
Therefore,
\begin{equation*}
{{\beta }_{5}}\le \frac{b-\hat{y}_{\sim 5}^{\max }-v_{7}^{+}{{\beta }_{7}}x_{7}^{\dagger \max }-v_{9}^{+} {{\beta }_{9}}x_{9}^{\dagger \max }}{x_{5}^{\dagger \max }},
\end{equation*}
and (5.1) can be generalized to describe this boundary constraint if we change the definition of $\hat{y}_{\sim 5}^{\max }$ as (5.2) states.

For the resultant variables, such as Economy$\times$Trade Openness,
\begin{equation*}
\hat{y}_{\sim 7}^{\max }=
\begin{cases}
\begin{aligned}
{{\hat{y}}^{\max }}&-v_{5}^{+}{{\hat{\beta }}_{5}}x_{5}^{\dagger \max }\\ &-v_{7}^{+} {{\hat{\beta }}_{7}}x_{7}^{\dagger \max }-v_{9}^{+}{{\hat{\beta }}_{9}}x_{9}^{\dagger \max }
\end{aligned} &\text{if} \quad v_{6}^{+}{{\beta }_{6}}x_{6}^{\dagger \max }\ge v_{5}^{+} {{\beta }_{5}}x_{5}^{\dagger \max }+v_{9}^{+}{{\beta }_{9}}x_{9}^{\dagger \max } \\
{{\hat{y}}^{\max }}-v_{6}^{+}{{\hat{\beta }}_{6}}x_{6}^{\dagger \max }-v_{7}^{+} {{\hat{\beta }}_{7}}x_{7}^{\dagger \max } &\text{otherwise}
\end{cases}
\end{equation*}
If $v_{6}^{+}{{\hat{\beta }}_{6}}x_{6}^{\dagger \max }\ge v_{5}^{+}{{\hat{\beta }}_{5}}x_{5}^{\dagger \max } +v_{9}^{+}{{\hat{\beta }}_{9}}x_{9}^{\dagger \max }$,
\begin{align*}
&b-\hat{y}_{\sim 7}^{\max }\ge v_{5}^{+}{{\beta }_{5}}x_{5}^{\dagger \max }+v_{7}^{+} {{\beta }_{7}}x_{7}^{\dagger \max }+v_{9}^{+}{{\beta }_{9}}x_{9}^{\dagger \max }\\
&{{\hat{\beta }}_{7}}\le \frac{b-\hat{y}_{\sim 7}^{\max }-v_{5}^{+}{{{\hat{\beta }}}_{5}}x_{5}^{\dagger \max } -v_{9}^{+}{{{\hat{\beta }}}_{9}}x_{9}^{\dagger \max }}{x_{7}^{\dagger \max }};
\end{align*}
otherwise,
\begin{align*}
&b-\hat{y}_{\sim 7}^{\max }\ge v_{6}^{+}{{\hat{\beta }}_{6}}x_{6}^{\dagger \max }+v_{7}^{+} {{\hat{\beta }}_{7}}x_{7}^{\dagger \max }\\
&{{\hat{\beta }}_{7}}\le \frac{b-\hat{y}_{\sim 7}^{\max }-v_{6}^{+}{{{\hat{\beta }}}_{6}}x_{6}^{\dagger \max }} {x_{7}^{\dagger \max }}.
\end{align*}
Both cases can be generalized by (5.1) and (5.2).
$\square$

Case III: Regional Dummy Variables

For a regional dummy variable $x_{j}^{\dagger}$, where $j=\left\{ 10,11,12,13 \right\}$,
\begin{equation*}
\hat{y}_{\sim 10}^{\max }={{\hat{y}}^{\max }}-\max \left( {{{\hat{\beta }}}_{10}},{{{\hat{\beta }}}_{11}}, {{{\hat{\beta }}}_{12}},{{{\hat{\beta }}}_{13}} \right).
\end{equation*}
Thus, we can derive
\begin{equation*}
{{\hat{\beta }}_{j}}\le b-\hat{y}_{\sim 10}^{\max },
\end{equation*}
and this relationship can be generalized by (5.1) and (5.2) since $x_{\sim j}^{\dagger \max }=1$.
$\square$