Explain the following questions and draw diagrams where necessary.
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) Simple linear regression is a statistical method that allows us to summarize and study relationships between two continuous (quantitative) variables: One variable, denoted X, is regarded as the predictor, explanatory, or independent variable. The other variable, denoted Y, is regarded as the response, outcome, or dependent variable. Suppose that we are given n-i.i.d observations { (x;, y;)}"_, from the assumed simple linear regression model Y = BIX + Bo + . Answer the following questions on simple linear regression. 5-a. Denote 1 and Bo as the point estimators of B, and Bo, respectively, that are obtained through the least squares method. Show, step by step, that the two point estimators are unbiased. Derive the least squares estimator of of and determine whether it is unbiased or not. Show your work step by step. 5-b. Calculate _'_1(yi - Bix; - Bo) (Bix, + Bo). Determine whether the point (X, Y) is on the line Y = 1X + Bo. Explain your reasoning mathematically. 5-c. Using the maximum likelihood estimation (MLE) technique, derive a point estimator for the coefficient B1 and the intercept Bo, respectively. Determine whether the point estimators that you obtained via MLE are unbiased or not. Justify your conclusion mathematically. 5-d. Calculate the variance of the four estimators from Questions 5-a and 5-c, respectively. Show your work step by step. 5-e. Suppose that we are using the simple linear regression model Y = B1 X + Bo + 1 while the true model is Y = 1X1 + B2X2 + Bo + 82 where Bo, B1, and B2 are constants. We assume that the distributions of &, and e2 are both N(0,02), i.e., normal distribution with variance o?. We further assume that the two noise variables are uncorrelated. Find the least squares estimator of B, in this case and determine whether the point estimator that you obtain is biased or not. If it is biased, calculate the bias.Econometrics questions 1. Labor economists often study the returns on investment in education (see, e.g., Card 1999). Suppose we have data on salaries of a set of people. some of whom went to college and some who did not. A simple model linking education to salary is Salary, = Bo + By College graduate, + er where the value of Salary, is the salary of person i and the value of College graduate; is 1 if person i graduated from college and is O' if person i did not. (a) What does Bo mean? What does , mean? (b) What is in the error term? (c) What are the conditions for the independent variable X to be endogenous? (d) Is the independent variable likely to be endogenous? Why or why not? (e) Explain how endogeneity could lead to incorrect inferences.What Is Fiscal Policy? Government budget constraint Bt - Bt-1 = rBt-1 + Gt - It deficits in year t interest payments primary deficit Bt = (1+r) Bt-1+ Gt - Tt Some remarks deficit (flow, Bt - Bt-1) v.s. debt (stock, Bt) evolution of debt-to-GDP ratio Bt Bt-1 Gt - Tt = (r -g) Bt-1 + Yt-1 Yt-1 Y+ change in debt ratio initial debt ratio primary deficit ratio Fiscal policy (changes in (Gt, Tt) to achieve macro objectives) v.s. automatic stabilizer
