Fixing Boundary Violations

4.1 Experiment Settings

To understand how the TRM model performs with or without boundary constraints, three simulation tests are carried out to evaluate the validity of parameter estimation. For each simulation, two independent variables (${{x}_{1}}$ and ${{x}_{2}}$) are included in the regression model. The dependent variable was randomly generated as truncated normal following ${{y}_{i}}\sim TN\left( \mu ,\sigma ;0,1 \right)$, where $\mu \sim U\left( 0.05,0.95 \right)$ and $\sigma \sim U\left( {\min \left( 1-\mu ,\mu \right)}/{5}\;, \max \left( 1-\mu ,\mu \right) \right)$. The independent variables are generated by varying explanatory power, degree of nonlinearity, and degree of multicollinearity. The sampling scheme can be specified as:
\begin{align*}
\text{Simulation I} \qquad {{x}_{1}}&=U,\quad {{x}_{2}}=U\cdot {{w}_{1}}+y\left( 1-{{w}_{1}} \right) \\
\text{Simulation II} \qquad {{x}_{1}}&=U, \quad {{x}_{2}}=U\cdot {{w}_{1}}+logit(y)\left( 1-{{w}_{1}} \right) \\
\text{Simulation III} \qquad {{x}_{1}}&=U\cdot {{w}_{2}}+{{x}_{2}}\left( 1-{{w}_{2}} \right), \quad {{x}_{2}} =U\cdot {{w}_{1}}+y\left( 1-{{w}_{1}} \right),
\end{align*}
where $U$ refers to a continuous uniform random variable following $U\left( 0,1 \right)$, $logit(y)=\ln \left( {y}/{1-y}\; \right)$, ${{w}_{i}}\in \left( 0,1 \right)$, and the sampling process of three simulations are independent. Except for $x_{2}$ in Simulation II, $x_{j}$ is bounded within $(0,1)$.¹²

For the first simulation, the linear relationship between ${{x}_{2}}$ and $y$ is assumed and the explanatory power is completely decided by $w_{1}$ given the independence of ${{x}_{1}}$ and $y$. When ${w_{1}}$ approaches $0$, the random part of ${{x}_{2}}$ is zero, and the deterministic part of ${{x}_{2}}$ makes r-squared approach $1$. When ${w_{1}}$ approaches $1$, ${{x}_{2}}$ is entirely composed of the random part and r-squared approaches $0$. For the second simulation, a nonlinear logistic relationship is set to the relationship of ${{x}_{2}}$ and $y$, while ${{x}_{1}}$ is independent from $y$. When ${w_{1}}$ approaches $0$, $x_{2}$ only contains the deterministic part, and it causes strong ceiling and floor effects, which seriously violate boundary restrictions when a linear regression is applied. On the other hand, when ${w_{1}}$ approaches $1$, the floor and ceiling effects vanish, and ${{x}_{2}}$ and $y$ becomes independent. For the third simulation, $x_{2}$ is first drawn with varying degree of explanatory power (decided by $w_{1}$), and then $x_{1}$ is drawn with a certain ratio (decided by $w_{2}$) of the deterministic part $x_{2}$ and the random part $U$. When $w_{2}$ approaches $0$, $x_{1}$ is perfectly collinear with $x_{2}$. When $w_{2}$ approaches $1$, $x_{1}$ and $x_{2}$ are completely independent of each other.

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Footnote

¹² The logit transformation in the deterministic part of $x_{2}$ enlarges its range from -8.66 to 13.79 in Simulation II.