To Analyze the Situation of Simultaneous Causality Bias, Consider the Following

Essay

To analyze the situation of simultaneous causality bias, consider the following system of
equations: $$\begin{array} { c } Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i } \\\\X _ { i } = \gamma _ { 0 } + \gamma _ { 1 } Y _ { i } + v _ { i }\end{array}$$ Demonstrate the negative correlation between $X _ { i }$ and $u _ { i }$ for $\gamma _ { 1 } < 0$ , either through mathematics or by presenting an argument which starts as follows: "Imagine that $u _ { i }$ is negative."