Figure 1.

Relations Modeled in Simulations

Note: $\text{expit}(\text{x})=\frac{exp(\text{x})}{1+exp(\text{x})}$ Note: ε₁, …, ε_T−1, ε are independent Normally distributed random variables each with mean 0 and variance 1, and ${\epsilon }_Y=\varphi \left({\displaystyle \prod _{t=0}^{T-2}}{\delta }_t\right)X+{\displaystyle \sum _{s=1}^{T-2}}\varphi \left({\displaystyle \prod _{t=s}^{T-2}}{\delta }_t\right){\epsilon }_s+\varphi {\epsilon }_{T-1}+\epsilon$ is a Normally distributed random variable with mean zero and variance: $\mathit{Var}({\epsilon }_Y)={\varphi }^2{\left({\displaystyle \prod _{t=0}^{T-2}}{\delta }_t\right)}^2+{\displaystyle \sum _{s=1}^{T-2}}{\varphi }^2{\left({\displaystyle \prod _{t=s}^{T-2}}{\delta }_t\right)}^2+{\varphi }^2+1$