# Instructions

• There are four sets available.
• If your student ID number ends with 0 or 2, answer Set A.
• If your student ID number ends with 1 or 7, answer Set B.
• If your student ID number ends with 3, 6, or 8, answer Set C.
• If your student ID number ends with 4, 5, or 9, answer Set D.
• Upload your solutions to SPOC by 16:00 May 15, 2022 (Beijing time) to the correct submission link at SPOC.
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• Your reasoning and arguments should be clear. It should be in English.
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• It is best to do the quiz yourself without looking at any notes. But if you are going to look at notes, try to answer the quiz independently.

# Common setting of Quiz 8

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 $\widehat{\beta}_1\overset{p}{\to}\beta_1^o+\frac{\mathsf{Cov}\left(I_t,\varepsilon_t\right)}{\mathsf{Var}\left(I_t\right)}.\quad (**)$

# Quiz 8 Set A (10-15 minutes)

• 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.

# Quiz 8 Set B (10-15 minutes)

• 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?

# Quiz 8 Set C (10-15 minutes)

• 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 $$\beta_1^o$$.
• 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.