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We study the effect of cigarette smoking on the child's birth weight via the simple linear regression Yr =30 +3131} +ui1f01' i: 1,...,n where Y is the birthweight and X is the number of packs smoked by the mother per day. (i) In this application, the unobserved term is capturing some latent genetic factor of the mother which affects both the mother's smoking behavior and infant's birhweight. You know the following identity holds for OLS estimator E]: It: 3141 + # =(Xa ) If you suspect the correlation between a and X is negative. is 31 more likely to over-estimate or under- estimate the true effect ,61? {I point) (ii) Referring to the concern in the previous part, we pick the average price of cigarettes as our instru- mental variable 21,-. What two properties do Z need to satisfy in order to be qualied as the instrumental variable? Suppose after you run a regression of the number of packs on the average price of cigarettes and nd an almost zero R-square. Which property is Z likely to violate? (1 point) (iii) Let Z = 111 El; 2;. The instrumental variable estimator 31 takes the following form: E?=1[Zi_ _)Yi 3': new tax; Now assume SLR.5 holds so that Var (a|X) = 0'2 and you can treat both X5 and Z; as xed numbers. Show that 31 has a larger variance that the simple OLS estimator 31. Le.1 Var (31) 2 Var (31). (2 points) Question 1 (understanding mean and variance of linear combinations of random variables) Let T_15 be the percentage of adult males who used tobacco products in 2015 in a country and T_10 be this percentage in 2010 in the same country. Define the random variable Z in the following way: Z =T_15 -T_10. We do not observe T_15 and T_10 for all countries of the world. We can only hope to get data from a random sample of n countries, where n is much smaller than the number of countries in the world. We want to estimate the E (Z) for the distribution of countries in the world. Each group member should attempt one of the following questions. The group can consult and improve the answer and only submit the improved answer, but the original person who attempted each part must be named. 1. What does the hypothesis E (Z) = 0 mean? After explaining what this hypothesis means, describe whether or not E (Z) = 0 implies T_15 = T_10 in every country in the world. Then, describe whether or not E (Z) = 0 implies -Er_15; = -Er_10; 1= 1 1= 1 for the n countries in the sample [Note that "Yes it does" or "No it doesn't" are not sufficient, you are expected to justify your answer.] 2. Using the result that sample average is an unbiased estimator of the population mean, show that iz = MET_15; - MELT_10; is an unbiased estimator of E (Z) . 3. Using the result that the variance of the sample average of a random sample of n observations from a distribution with mean / and variance o' is , compute the variance of /z = > >_,T_15; - " Ein T_10;, for a random sample of n = 40 countries, when Var (T_15) = Var(T_10) = 100, and p the correlation coefficient between 7_15 and T_10 is 0.8. 4. Suppose that we have obtained data on T_15 and T_10 for a sample n countries and computed Z; =T_15; -T_10; for i = 1, ..., n. Using the matrix formula for the OLS estimator, show that if we regress this variable on a constant only, the OLS estimate of the constant will be ! )_, T_15;- = ELIT_10.Question 1 (understanding mean and variance of linear combinations of random variables) Let T_15 be the percentage of adult males who used tobacco products in 2015 in a country and T_10 be this percentage in 2010 in the same country. Define the random variable Z in the following way: Z =T_15 -T_10. We do not observe T_15 and T_10 for all countries of the world. We can only hope to get data from a random sample of n countries, where n is much smaller than the number of countries in the world. We want to estimate the E (Z) for the distribution of countries in the world. Each group member should attempt one of the following questions. The group can consult and improve the answer and only submit the improved answer, but the original person who attempted each part must be named. 1. What does the hypothesis E (Z) = 0 mean? After explaining what this hypothesis means, describe whether or not E (Z) = 0 implies T_15 = T_10 in every country in the world. Then, describe whether or not E (Z) = 0 implies -Er_15; = -Er_10; 1= 1 1= 1 for the n countries in the sample [Note that "Yes it does" or "No it doesn't" are not sufficient, you are expected to justify your answer.] 2. Using the result that sample average is an unbiased estimator of the population mean, show that iz = MET_15; - MELT_10; is an unbiased estimator of E (Z) . 3. Using the result that the variance of the sample average of a random sample of n observations from a distribution with mean / and variance o' is , compute the variance of /z = > >_,T_15; - " Ein T_10;, for a random sample of n = 40 countries, when Var (T_15) = Var(T_10) = 100, and p the correlation coefficient between 7_15 and T_10 is 0.8. 4. Suppose that we have obtained data on T_15 and T_10 for a sample n countries and computed Z; =T_15; -T_10; for i = 1, ..., n. Using the matrix formula for the OLS estimator, show that if we regress this variable on a constant only, the OLS estimate of the constant will be ! )_, T_15;- = ELIT_10.5. For a utility function of the form U(x, ,x,)=min(5x, ,10x,), the individual will have excess x, for commodity bundles where a. x2 2x1 b. x2>5%, C X, >0.5x, d. x2>10%, 6. Matt enjoys Fuji and Gala apples and finds 1 Fuji apple to be just as good as 4 Gala apples. a. Matt's preferences over Fuji (F) and Gala (G) apples can be represented by the utility function U(F,G)=0.25F+G b. Matt's preferences over Fuji (F) and Gala (G) apples can be represented by the utility function U(F,G)=F+4G C. Matt's preferences over Fuji (F) and Gala (G) apples can be represented by the utility function U(F,G)=4F+.25G d. Matt's preferences over Fuji (F) and Gala (G) apples can be represented by the utility function U(F,G)=4F+G e. Both A and B f. Both C and D
