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2.3 Model Specification

Proper model specification is critical for the successful application of constrained optimization to the panel regression. While the demeaning operation is easy to apply, it generates some questions regarding interpretability, as well as logical soundness. For the covariates that can vary independently at the individual level, the demeaning operation experiences no problem, since the time mean represents the aggregate-level measure, which reflects the characteristic of the spatial units across time. However, when the covariates cannot vary independently, such as when they function as a time dummy or have involvement with an interaction term, the demeaning operation is problematic and unnecessary.

In Hansford and Gomez's study, there are 13 time dummies, and their values after demeaning operation range from -0.26 to 1. This scenario contradicts our intuition because a time dummy is either 0 or 1. Mathematically, after being demeaned, a time dummy should remain as 0 or 1 since
\begin{align*}
x_{it}^{d}-\bar{x}_{i}^{d}+\bar{\bar{x}}&=x_{it}^{d}-\frac{1}{T}-\frac{n}{n\cdot T} \\
&=x_{it}^{d},
\end{align*}
where the superscript $d$ denotes a time dummy. And here, we assume the data consists of balanced panels. Therefore, if time dummies have values other than 0 or 1, it indicates a missing value problem. For instance, the time dummy Yr2000 has a lowest value of -0.26 in the '48 observation #1471, since there are only three cases available in this county, including the case in 2000 (county id: #39075). On the other hand, the time dummy Yr2000 has a positive value of 0.070 in the '00 observation #25095, since only the '00 case is missing in this county (county id: #41049). Given that the demeaning operation is unnecessary and the result is difficult to interpret, we use the original data for the time dummies.

Another common issue in political studies is associated with interaction terms. However, we often neglect the nonlinear nature of the interaction model, especially where different centering methods are involved.12 If we center non-interaction variables before forming interaction terms, the regression result will be different from centering all variables after interaction terms were formed. Suppose ${{x}_{3}}={{x}_{1}}\times {{x}_{2}}$ and ${{x}_{1}}$ and ${{x}_{2}}$ are normally distributed, then the above case can be illustrated by the following equations:
\begin{align}
y &=\beta _{0}^{\left( 1 \right)}+\left( {{x}_{1}}-{{{\bar{x}}}_{1}} \right)\beta _{1}^{\left( 1 \right)} +\left( {{x}_{2}}-{{{\bar{x}}}_{2}} \right)\beta _{2}^{\left( 1 \right)}+\left( {{x}_{1}}-{{{\bar{x}}}_{1}} \right)\left( {{x}_{2}}-{{{\bar{x}}}_{2}} \right)\beta _{3}^{\left( 1 \right)} \tag{2.3} \\
& = \beta _{0}^{\left( 2 \right)}+\left( {{x}_{1}}-{{{\bar{x}}}_{1}} \right)\beta _{1}^{\left( 2 \right)}+\left( {{x}_{2}}-{{{\bar{x}}}_{2}} \right)\beta _{2}^{\left( 2 \right)}+\left( {{x}_{1}}{{x}_{2}}-\overline{{x_{1}}{x_{2}}} \right)\beta _{3}^{\left( 2 \right)}. \tag{2.4}
\end{align}
Apparently, $\beta _{i}^{\left( 1 \right)}\ne \beta _{i}^{\left( 2 \right)}$ for $i=1,2,3$, regardless of whether $y$ is distributed as untruncated or truncated normal.

When applying constrained optimization to panel regression, we suggest applying the fixed-at-minimum model if the regression includes dummy variables or interaction terms. The reason is twofold: first, the value of time dummies in many observations will approach zero as the temporal units increase, which will occasionally cause numerical problems; second, if we formulate the interaction term after fixing the composing variables at the minimum, given that the range of all variables is in the positive domain, we can simplify the specification of boundary constraints by eliminating the cases in which two negative composing variables result in a positive interaction term. Centering interaction variables, on the other hand, would complicate the specification of the admissible parameter space for $\beta$, and thus cause difficulty in parameter estimation.

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Footnote

12 We refer the centering methods to any model specification that involves linear transformation of the regression model.

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