Explain the following questions.
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.1. Towards the end of the 20th century, the U.S. government wanted to save money by closing a small portion of its domestic military installations. While many people agreed that saving money was a desirable goal, people in areas potentially affected by a closing soon reacted negatively. Congress finally selected a panel whose task was to develop a list of installations to close, with the legislation specifying that Congress could not alter the list. Since the goal was to save money, why was this problem so hard to solve? 2. Your car gets 29 miles per gallon (mpg) at 60 miles per hour (mph) and 25 mpg a 70 mph. At what speed should you make a 525-mile trip: a. If gas costs $3 per gallon and your time is worth $18 per hour b. If gas costs $4 per gallon and your time is worth $12 per hour C. If gas costs $5 per gallon and your time is worth $9 per hour 3. A firm is planning to manufacture a new product. As the selling price is increased, the quantity that can be sold decreases. Numerically the sales department estimates: P = $475-0.250 Where P = selling price per unit and Q = quantity sold On the other hand, management estimates that the average unit cost of manufacturing and selling the product will decrease as the quantity sold increases. They estimate C = $480 + $22,500 Where C = cost to produce and sell Q per year The firm's management wishes to maximize profit. What quantity should be sold? How much profit will be made?QUESTION 48 Consider three items that might be included in GDP: (1) the estimated rental value of owner-occupied housing and (2) the purchase of newly constructed houses and (3) the rental of an apartment. How are these three items accounted for when GDP is calculated? All three items are included in the consumption component of GDP. Item (1) and item (3) are included in the consumption component of GDP, while item (2) is included in the investment component of GDP. G Items (1) and (2) are included in the investment component of GDP, while item (3) is included in the consumption component of GDP. Item (2) Is included in the investment component of GDP and item (3) is included in the consumption component of GOP, Item (1) is not included in GOP at all because it is considered a used good. QUESTION 49 How is the real rate of interest determined? The real interest rate is set by the central bank of a nation, such as our Federal Reserve. The real interest rate is determined by the equilibrium of the supply and demand for loanable funds. The real interest rate is determined by the nominal interest rate plus the inflation rate. The real interest rate is determined by the demand for private savings and the supply of public savings. QUESTION 50 Which of the following occurs when the Ford Motor Company removes a 2020 Ford Taurus from inventory and sells it to an American household for $22.000? U.5. consumption increases by $22,000, U.5. investment decreases by $22,000, and U.S. GDP does not change. U.5. consumption increases by $22,000, U.S. investment does not change, and U.S. GDP Increases by $22.000. U.S. consumption does not change, U.S. investment decreases by $22,000, and U.S. GDP decreases by $22,000. U.5. consumption increases by $22.000, U.5. investment increases by $22,000, and U.S. GDP increases by $44,000
