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.Question 1 There are 10,000 identical individuals in the market for commodity X, each with a demand function given by Qodx = 12 - 2Px, and 1000 identical producers of commodity X, each with a function given by Qsx = 20 Px. where Qox is an individual's quantity demanded, Qsx is a single producer's quantity supplied, and Px is the price of the commodity. (a) Find the market demand function (QDx) and the market supply function (QSx) for commodity X (b) Determine the market demand schedule and the market supply schedule of commodity X (for whole dollar prices) and from them find the equilibrium price and the equilibrium quantity. (c) Plot, on one set of axes, the market demand curve and the market supply curve for commodity X and show the equilibrium point (d) Obtain the equilibrium price and the equilibrium quantity mathematically. (e) Explain why the equilibrium condition is considered stable. (f) Determine the elasticity of demand for commodity X at the equilibrium point. Question 2 Suppose that from the condition of equilibrium in Question 1, there is an increase in consumers' incomes (ceteris paribus) so that a new market demand curve is given by QDx = 140,000 - 20,000Px. (a) Derive the new market demand schedule (b) Show the new market demand curve on the graph used in Question 1(c) (c) State the new equilibrium price and equilibrium quantity for commodity X (d) Determine the Income Elasticity at the original equilibrium price and at the new equilibrium price. (Assume that the increase in income is 106). (e) In the light of your answer to 2 (d), comment on the nature of commodity X.rnr r'- r" 1.The concept of GDP is used by policy makers for many purposes that include keeping track ofthe growth of our economy and forecasting a recession; but it is also abused by some. Discuss some of the applications of GDP which may be considered improper use or abuse of the concept. 2." Even though the Great Recession ofcially ended, the unemployment rate is still considered high.' Discuss. 3.Discuss how the distribution of income among various groups of income earners have changed in this country during the past 5D years. in your opinion, do we need to initiate any policy to address the distribution of income? Make sure you provide references to anyconclusions you arrive at. 4.Many people feet the Fed, which is largely independent of Congress and the President, is too powerful. In fact, the Chairman of the Fed Board of Governors, Ben Bernanke, has been called the second most powerful person in the country. Do you think the Fed is too powerful? Should Congress have more oversight of the Fed? Share your thoughts. 5."N either monetary policy nor fiscal policy alone can be effective in formulating sound economic policies for recession.' Do you agree or disagree? Share your thoughts. Effou now should realize how important it is for economists to try and predict future conditions of the LLB. economy. Economists are also interested in the performance of the economy at a more local level. Discuss how local and national forecasts play an important role in the State and Federal levels. If here have been significant discussions on the government's scal policy during the Great Recession. in particular, many did not like the government's actions in salvaging the auto industry in providing guaranteed loans to GM and Chrysler. What is your opinion? Provide economic reasons for your opinion. 2. 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.vi payments accounts, still PLEsuit IsOnly foreign currency traded with the Canadian dollar. 7. For each of the following situations, outline the effect on the price of the Canadian dollar in terms of US dollars and draw a demand and supply graph that illustrates the changes that occur in the foreign exchange market for the Canadian dollar. . A contractionary monetary policy initiated by the Bank of Canada raises Canadian interest rates. b. Canada's real output rises at a time when real output in the United States is falling. c. Americans (but not Canadians) find Canada a more attractive place to make financial investments. d. Given Canada's aging population, more Canadian "snowbirds" travel to the United States each winter. e. Due to a credit crisis that affects US financial institutions more than it does Canadian ones, Canada's attractiveness as a destination for direct and portfolio investment increases. f. The Bank of Canada initiates an expansionary monetary policy that reduces Canadian interest rates