Question 3 (16 pts.) To assess the efficacy of modest financial incentives, measured by the indicator variable Di, on immunization rates (1) among children, use the following experimental data: ! 4 i D (observed) Y (observed) D (hypothetical) Y (hypothetical) 1 1 0.9 0 0.6 2 0 0.3 1 0.8 3 1 0.7 0 0.2 0 0.1 1 0.4 a) Causal effect: i. (1 pts.) Define the causal effect using appropriate mathematical notation. : ii. (2 pts.) Interpret each of the two terms in the causal effect by specifying the relevant group, the state of nature and the outcome of interest. iii. (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 then substitute the corresponding numerical values. b) Selection bias: i. (1 pt.) Define the selection bias using appropriate mathematical notation. ii. (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. iii. (1 pt.) Describe what the two terms of the selection bias are comparing. iv. (2 pts.) Compute the selection bias. 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. c) (2 pts.) Compute the observed effect. d) (2 pts.) Briefly explain why the causal effect and the observed effect are of the same (opposite) sign. Question 3 (16 pts.) To assess the efficacy of modest financial incentives, measured by the indicator variable Di, on immunization rates (1) among children, use the following experimental data: ! 4 i D (observed) Y (observed) D (hypothetical) Y (hypothetical) 1 1 0.9 0 0.6 2 0 0.3 1 0.8 3 1 0.7 0 0.2 0 0.1 1 0.4 a) Causal effect: i. (1 pts.) Define the causal effect using appropriate mathematical notation. : ii. (2 pts.) Interpret each of the two terms in the causal effect by specifying the relevant group, the state of nature and the outcome of interest. iii. (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 then substitute the corresponding numerical values. b) Selection bias: i. (1 pt.) Define the selection bias using appropriate mathematical notation. ii. (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. iii. (1 pt.) Describe what the two terms of the selection bias are comparing. iv. (2 pts.) Compute the selection bias. 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. c) (2 pts.) Compute the observed effect. d) (2 pts.) Briefly explain why the causal effect and the observed effect are of the same (opposite) sign
