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Recall our consumption-income example: \[\begin{eqnarray}C_t &=& \beta_0^o+\beta_1^o I_t+\varepsilon_t \\ I_t &=&C_t+D_t,\end{eqnarray}\] where \(C_t\) is consumption, \(I_t\) is income, and \(D_t\) is non-consumption.

Assume, for convenience, that \[\left(\begin{array}{c} D\\ \varepsilon \end{array}\right)\sim N\left(\left(\begin{array}{c} \mu_{D}\\ 0 \end{array}\right),\left(\begin{array}{cc} \sigma_{D}^{2} & 0\\ 0 & \sigma_{\varepsilon}^{2} \end{array}\right)\right) \quad (*).\] Assume we have IID draws from this distribution.

Let \(\widehat{\beta}_1\) be the least squares estimator of \(\beta_1^o\). We have shown in class that \[\widehat{\beta}_1\overset{p}{\to}\beta_1^o+\frac{\mathsf{Cov}\left(I_t,\varepsilon_t\right)}{\mathsf{Var}\left(I_t\right)}.\quad (**)\]

Answer the following questions:

- Which specific part of \((*)\) is required for \((**)\) to hold?
- For this setting and with the proper choice of \(Z_t\), a vector of instrumental variables, derive and simplify \(Q_{ZX}\) for the setting you have.
- What is the rank of \(Q_{ZX}\)? Do we need conditions to ensure that it has the appropriate rank? Discuss.

Answer the following questions:

- Which specific part of \((*)\) is required for \((**)\) to hold?
- Obtain down the reduced form for \(C_t\) and \(I_t\).
- Explain why the reduced forms are linear regressions.
- Discuss how you can use the reduced forms to uniquely identify \(\beta_1^o\). Are there restrictions for your strategy to work?

Answer the following questions:

- Which specific part of \((*)\) is required for \((**)\) to hold?
- Let \(e_t\) be the OLS residual from a least squares regression of \(C_t\) on \(I_t\). A classmate of yours suggests that we can define a new estimator \[\widetilde{\beta}_1=\widehat{\beta}_1-\frac{\displaystyle\frac{1}{n}\sum_{t=1}^n (I_t-\overline{I})(e_t-\overline{e})}{\displaystyle\frac{1}{n}\sum_{t=1}^n (I_t-\overline{I})^2}\] in order to remove the problem in (**). Explain why this procedure cannot allow you to consistently estimate \(\beta_1^o\).

Answer the following questions:

- Which specific part of \((*)\) is required for \((**)\) to hold?
- A classmate of yours suggests that \(D_t^2\) could be part of the vector \(Z_t\) discussed in class. Do you agree?
- If you agree, explain how to consistently estimate \(\beta^o\) in this situation.
- If you do not agree, explain why your classmate is wrong.