Question III Consider the causal model With the settings X, and X, from a certain distribution, we obtain Y = (X, X.). Define the average cause effect as 5 (@ ) = E[C(x,, X.)l where C (#;: 20) = 32; Assume that X. ~ N(0, 1) and X, = X, +U where Um N(0, 1) and U is independent of X.. (a) Suppose Find S (@ ,) in terms of a, G y and of. (b ) Now suppose Find S (x ,) in terms of a, 8 y and of. (c) Define BE [Y X.= =] Is S (x , ) = (2,) under the model of c (, ) in (3)? (d) 1: 3 (@) = 3 (2,) under the model of c(, ) in (b)? Question IV (Optional) In = 1994 research paper, two economists examined the impact of looks on earnings using interviewers' ratings of respondents' physical appearance. They used date in which respondents reported their wages, and the interviewers rated the respondent's appearance, using five categories: (1. homely; 2. quite plain, 3. average, 4. good looking and 6. strikingly handsome or beautiful). Download the file beauty.xla from the course web page. The file contains the following variables: hourly earning, looks, female (is equal to 1 if female, 0 otherwise) and years of education. Assume that a population consists of all individuals in the data set. (a) Estimate the following regression emhr = a+ 3 , looks ty , graeductu (1) wing the population data (ie., use all data). What is the value of (the estimated) ? Denote this value as 3- (b) (i) Draw a simple random sample of size 100. (mi) Estimate (1) using the sample you obtained. (mi) Test the null hypothesis Ho : 3 = 9, against H1 : 9 = 8, wing a = 10% s: the size of the test. Let reject be a dummy variable indicating whether No is rejected (reject=1 if He is rejected)