Explain the following.
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.will have to sell the property at a loss. Prots (in thousands of dollars) are shown in the following payoff table: State of Nature Rezoning Approved Rezoning Not Approved Decision Alternative .31 3'2 Purchase, ch 600 - 200 Do not purchase, d2 0 0 a. If the probability that the rezoning will be approved is 0.5, what decision is recom- mended? What is the expected prot? b. The investor can purchase an option to buy the land. Under the Option, the investor maintains the rights to purchase the land anytime during the next three months while learning more about possible resistance to the rezoning prOposal from area residents. Probabilities are as follows: Let H = High resistance to rezoning L = Low resistance to rezoning P(H) = 0.55 P(sl l H) = 0.18 P(sZIH) = 0.82 P(L) = 0.45 Pm l L) = 0.39 13(52 | r.) = 0.11 What is the optimal decision strategy if the investor uses the option period to learn more about the resistance from area residents before making the purchase decision? c. If the option will cost the investor an additional $10,000, should the investor purchase the option? Why or why not? What is the maximum that the investor should he will- ing to pay for the option
