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.Ill T-Mobile Wi-Fi 7:40 PM A Problems-Options 1. What is the difference between time value and intrinsic value? 2. What is the intrinsic value and time value of the following 3-month stock options? Assume the stock price is $45. 40 strike call option with a price of 7 45 strike call option with a price of 4 47 strike call option with a price of 2 40 strike put option with a price of 1 45 strike put option with a price of 3 47 strike put option with a price of 4 3. You own a stock currently at $50. You buy a 6-month put option with a 45 strike for a price of 4. You sell a 6-month call option with a 60 strike for a price of 3. Fill in the net profit table for the following stock prices at the end of 6 months. Stock price in 6 months Net value 65 60 55 50 45 40 4. You can buy or sell calls and puts. Which option has unlimited upside? Which option has unlimited downside? Which option has a limited upside? Which option has a limited downside? 5. You need to buy oil in 3 months. How can you create an option position using puts and calls to limit your exposure to a change in oil prices? 6 You need to call your Gosale clock in f manthe How can A X CIn 2008, the policymakers of the economy of Eastlandia projected the debt-GDP ratio and the ratio of the budget deficit to GDP for different scenarios for growth of the deficit over the next 10 years. Real GDP is currently $1,000 Billion per year and is expected to grow by 3% per year; the public debt is $300 billion at the beginning of the year, and the deficit is $30 billion in 2008 Year Real GDP Debt Budget Deficit Debt Budget Deficit Billions of $) Billions of $) % of GDP) (X GDP) 2008 1,000 300 30 200 1,030 30 2010 1,061 30 2011 1,093 DE 2012 1,126 30 2013 1,159 30 2014 1,194 30 2015 1,230 30 2016 1,267 30 2017 1,305 30 2018 1,344 30 ) Complete the table above if the deficit remains constant at $30 for the entire period. Remember that the government's debt will grow by the previous year's deficit. b) Now redo the table given that the deficit now grows by 3% every year during the period. Year Real GDP Deb Budget Deficit Debt Budget Deficit Billions of $) (Billions of $) (% of GDP) %% GDP) 2008 1,000 300 30 1009 1,030 2010 1,061 2011 1,093 2012 1,120 201 1,159 2014 1,194 2015 1,230 2016 1,267 2017 1,30 2018 1,344 c) What happens to the Debt-GDP ratio and the deficit-GDP ratio for the economy over time under the different scenarios? d) Which scenario would be preferable for this country? Why?(3) You and a classmate are assigned a project on which you will receive one combined grade. You each want to receive a good grade, but you also want to avoid hard work. In particular, here is the situation: If you both work hard, you get an A, which gives you each 40 units of happiness .If only one of you works hard, you both get a B, which gives you each 30 units of happiness. .If neither of you works hard, you both get a D, which gives each of you 10 units of happiness. . Working hard costs 25 units of happiness. (a) Fill in the following payoff matrix: Your decision Work Shirk Classmate's Work decision Shirk (b) What is the likely outcome? Explain your answer. T (c) If you get this classmate as your partner on a series of projects throughout the year, rather than only once, how might that change the outcome you predicted in part (b)
