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.You may need to use the appropriate appendix table or technology to answer this question. Suppose In 2018, RAND Corporation researchers found that 77% of all individuals ages 66 to 65 are adequately prepared financially for retirement, Many financial planners have expressed concern that a smaller percentage of those in this age group who did not complete high school are adequately prepared financially for retirement. (a) Develop appropriate hypotheses such that rejection of N, will support the conclusion that the proportion of those who are adequately prepared financially for retirement is smaller for people in the 66-69 age group who did not complete high school than it is for the population of the 66-69 year old. (Enter In for a as needed.} (b) In a random sample of 300 people from the 65-69 age group who did not complete high school, 159 were not prepared financially for retirement. What is the p- value for your hypothesis test? Find the value of the test statistic. (Round your answer to two decimal places.) Find the p-value. (Round your answer to four decimal places.) p-value = (c] At a = 0.01, what is your conclusion? Do not reject He. We conclude that the percentage of 66- to 69-year-old individuals who are adequately prepared financially for retirement is smaller for those who did not complete high school. Reject Ho. We conclude that the percentage of 66- to 69-year-old individuals who are adequately prepared financially for retirement is smaller for those who did not complete high school. Reject Ho. We cannot conclude that the percentage of 66- to 69-year-old individuals who are adequately prepared financially for retirement is smaller for those who did not complete high school. Do not reject H,. We cannot conclude that the percentage of 66- to 69-vear-old individuals who are adequately pranarad financially far vatican2. Economic Growth. Suppose that Real GDP is given by the following ag- gregate production function: Y, = A, KO3 NOT a) Suppose that, in a given year, output grows by 3%, capital grows by 2.67% and labour grows by 1%. What is productivity growth? Bonus points to students who can derive the growth accounting formula. Hint: You need to make use of the following approxi- mation for a growth rate in variable X: gx = In X - In X-1 b) Suppose that the savings rate, s, is 0.5, the depreciation rate, d, is 0.09 and population growth, n, is 0.01. Productivity, A, is equal to 2. i. Derive the per capita aggregate production function and plot it on a diagram with k, = ~ on the horizontal axis and y, = # on the vertical axis. ii. Plot savings per capita, M, on the same graph. iii. Plot the amount of investment required to keep capital per capita constant. c) Solve for a steady state equilibrium. In particular, find capital, output, consumption, savings and investment in per capita terms. Show this equilibrium on your graph from part a). d) The golden rule savings rate is the one where consumption per capita is maximized in the steady state. Find the savings rate which is associated with the golden rule consumption level. Is it higher or lower than the savings rate used in part c)? Show the golden rule steady state on your graph. Hint: Consumption per capita in the steady state is given by: c = (1 -$)A You will need to use calculus to find the savings rate that maximizes consumption. Take the derivative with respect to the savings rate and set the first order condition equal to zero. Make use of the product and chain rules for the derivative with respect to s. You do not need to check the second order condition