Consider the regression model (y_{i}=beta_{1}+beta_{2} x_{i}+beta_{3} z_{i}+beta_{4} q_{i}+e_{i}), where (Eleft(e_{i} mid mathbf{X} ight)=0), with (mathbf{X}) representing all

Consider the regression model \(y_{i}=\beta_{1}+\beta_{2} x_{i}+\beta_{3} z_{i}+\beta_{4} q_{i}+e_{i}\), where \(E\left(e_{i} \mid \mathbf{X}\right)=0\), with \(\mathbf{X}\) representing all observations on \(x, z\), and \(q\). Suppose \(z_{i}\) is unobservable and omitted from the equation, but conditional mean independence \(E\left(z_{i} \mid x_{i}, q_{i}\right)=E\left(z_{i} \mid q_{i}\right)\) holds, with \(E\left(z_{i} \mid q_{i}\right)=\delta_{1}+\delta_{2} q_{i}\).

a. Show that \(E\left(y_{i} \mid x_{i}, q_{i}\right)=\left(\beta_{1}+\beta_{3} \delta_{1}\right)+\beta_{2} x_{i}+\left(\beta_{3} \delta_{2}+\beta_{4}\right) q_{i}\).

b. i. Is it possible to get a consistent estimate of the causal effect of \(x_{i}\) on \(y_{i}\) ?

ii. Is it possible to get a consistent estimate of the causal effect of \(z_{i}\) on \(y_{i}\) ?

iii. Is it possible to get a consistent estimate of the causal effect of \(q_{i}\) on \(y_{i}\) ?