You would like to evaluate impact of COVID restrictions (e.g., shelter-in-place) on the log-consumption in restaurants. You have an IID sample of counties in the US where for each county i you observe the variables D; and G;, where D; denotes whether the county has OVID restrictions (D; = 1) or not (D; = 0) and G; denotes the log-consumption in restaurants. You think that the relationship between Gi and D; is linear, so that Gi = Bo + B1 Di +U;. (2) with U; = -W+6 where To >0 and TW; reflects the "fear to get sick; W; is the number of positive COVID cases in the county; and is an IID mean 0 shock. 1. (7 points) Do you think there is reason to suspect that the error V; may be correlated with the regressor Di in this setting? Why or why not? 2. (7 points) Let Zi be the number of ICU beds on county i. A friend, who has taken 141 in the past, suggests that a consistent estimator can be obtained by means of using Zas an instrument for Di (a) Can you think of an explanation for why it might be reasonable to assume that Cou(Ui, Zi) = O and Cou(Di, 2) = Osp #0? (b) Con you think of an explanation for why it might be reasonable to assume that Cou(U;, Z;) # O? 3. (8 points) Suppose you know that E[W|D, 2] = Z. Based on this, construct an estimator that is consistent for B1. Hint: Think about whether W; - E[WDi, Zi] is correlated with D. Based on this, try to modify the regression model in 2. You would like to evaluate impact of COVID restrictions (e.g., shelter-in-place) on the log-consumption in restaurants. You have an IID sample of counties in the US where for each county i you observe the variables D; and G;, where D; denotes whether the county has OVID restrictions (D; = 1) or not (D; = 0) and G; denotes the log-consumption in restaurants. You think that the relationship between Gi and D; is linear, so that Gi = Bo + B1 Di +U;. (2) with U; = -W+6 where To >0 and TW; reflects the "fear to get sick; W; is the number of positive COVID cases in the county; and is an IID mean 0 shock. 1. (7 points) Do you think there is reason to suspect that the error V; may be correlated with the regressor Di in this setting? Why or why not? 2. (7 points) Let Zi be the number of ICU beds on county i. A friend, who has taken 141 in the past, suggests that a consistent estimator can be obtained by means of using Zas an instrument for Di (a) Can you think of an explanation for why it might be reasonable to assume that Cou(Ui, Zi) = O and Cou(Di, 2) = Osp #0? (b) Con you think of an explanation for why it might be reasonable to assume that Cou(U;, Z;) # O? 3. (8 points) Suppose you know that E[W|D, 2] = Z. Based on this, construct an estimator that is consistent for B1. Hint: Think about whether W; - E[WDi, Zi] is correlated with D. Based on this, try to modify the regression model in 2
