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N M Fn question 3 (33 pts.) Michael is a health economics student interested in estimating the effect of health insurance D on the number of doctor visits by estimating the population regression function (PRF) Yi = A Di + ui by using the sample reported in Table 2. Enrollment in a health insura coverage nce plan is voluntary. Table 2 1 0 1 Michael has heard that observational studies produce biased results, while randomized experiments deliver the causal effect if an exhaustive set of samples are produced. He conducts a set of experiments, which are accurately reported in Table 3. However, Michaels grasp of the analysis of randomized experiments is somewhat shaky and he needs assistance in answering a set of questions. Table 3 Treatment group (Ti-1) | Control group (Ti = 0) 0) Sample 1 Sample 2Table 3 Treatment group(T1) Control group (T0) Sample 1 Sample 2 1 Sample 3 2 Sample 4 5 2 1 3 Sample 5 3 Sample 6 (2 pts.) Michael would like to use his experimental results in Table 3 to decompose the observed effect in Table 2 into the causal effect and the selection bias. List the individuals and the samples in Table 3 that must be used for the computation of the causal effect in Table 2. a)

b) Causal effect: (2 pts.) Define the causal effect using appropriate mathematical notatio (2 pts.) the state of nature and the outcome of interest. Interpret each of the two terms in the causal effect by specifying the relevant group, 1 pt.) Describe what the two terms of the causal effect are comparing iv. (2 pts.) Compute the causal effect. In your answer, first expand each expected value by using the appropriate letters and subscripts for each observation and the corresponding numerical values. n substitute the

using the appropriate letters and subscripts tor eal corresponding numerical values V (2 pts.) Briefly explain why the causal effect could be found from the exhaustive set of samples but not necessarily from an individual sample c) Observed effect: (2 pts.) Define the observed effect using appropriate mathematical notation.

Fn Alt Alt pts.) Interpret each of the two terms in the observed effect by specifying the re oupthe state of nature and the outcome of interest. he relevant ii. (1 pt.) Describe what the two terms of the observed effect are comparing. iv. (2 pts.) Compute the observed effect. In your answer, first expand each expected value by using the appropriate letters and subscripts for each observation and then substitute the corresponding numerical values.

pcthe observed effect. In your answer, first expand each expected vaide Dy using the appropriate letters and subscripts for each observation and then substitute the corresponding numerical values d) Selection bias: i. (2 pts.) Define the selection bias using appropriate mathematical notation. (2 pts.) Interpret each of the two terms in the selection bias by specifying the relevant group, the state of nature and the outcome of interest. ii. (1 pt.) Describe what the two terms of the selection bias are comparing.

Alt a pts.) Compute the selection bias. In your answer, first expand each expected value py using the appropriate letters and subscripts for each observation and then subst corresponding numerical values e) (2 pts.) sign. Briefly explain why the causal effect and the observed effect are of the same (opposite) 1) (2 pts) Demonstrate that the indicator variable Tt could be re written as the population regression function (PRF): α + ρΤί + ui.

8) (2 pts.) Demonstrate that the observed effect equals the causal effect in the PRF inf) if (2 pts.) Briefly explain why randomizing enrollment into a health insurance plan ensures that 10

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