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Welcome to BUAD 310

Welcome to BUAD 310. Instructor: Kam Hamidieh Lecture 18, Monday March 31, 2014. Agenda & Announcement. Today: Finish off slides from last time Begin Simple Linear Regression, Chapters 19-22 Reading: 19: All of it 20: Only 20.1, 20.2 (Skim the rest)

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Welcome to BUAD 310

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  1. Welcome to BUAD 310 Instructor: Kam Hamidieh Lecture 18, Monday March 31, 2014

  2. Agenda & Announcement • Today: • Finish off slides from last time • Begin Simple Linear Regression, Chapters 19-22 • Reading: • 19: All of it • 20: Only 20.1, 20.2 (Skim the rest) • 21: All of it (We’ll cover the basics of 21.4) • HW 5 is due on Wednesday April 2, 5 PM. Please do not ask for extensions. BUAD 310 - Kam Hamidieh

  3. About HWs • Start early. • Watch my videos. • Read the notes and the book BEFORE attempting the homeworks. • Some homework problems may be harder than the ones in class ones but not always. The homework ones may come across harder if you dive in cold! • Come to office hours if you are struggling. BUAD 310 - Kam Hamidieh

  4. Exam 2 • It will be in class on Wednesday April 16th, 2014. • Coverage: • Lecture 12, March 3 to the end of lecture 21, April 9. • HW 4, 5, & 6 • You’ll have 33(?) multiple choice questions & use Scantron. I’ll pass out the Scantron sheets today. • Cheat Sheet: TWO 8.5 by 11 inches, both sides, hand written. You can put anything you wish on it. • Our rules and procedure will be very similar to the first exam. • NOTE: NO CELL PHONES OR COMMUNICATION DEVICES WILL BE ALLOWED! BUAD 310 - Kam Hamidieh

  5. Last Time • Chi-Squared tests: • Test of goodness-of-fit: Do we think the distribution of a categorical variable is (statistically) equal to some preconceived distribution? • Test of independence: Are two categorical variables dependent? • Test stat: • The degrees of freedom depended on the test. • Use chi-square distribution to get the p-value. BUAD 310 - Kam Hamidieh

  6. Simple Linear Regression • Start with two quantitative variables: • Y = dependent, response variable • X = independent, factor, covariate, explanatory, or predictor variable • Goal: Find a straight line that describes how Y changes as X changes Y X BUAD 310 - Kam Hamidieh

  7. Simple Linear Regression • Simple: One Y vs. one X • Linear: We use a straight line. • “Regression”: This guy coined it.Galton observed that tall parentstended to have shorter offspring: heights “regressed” to the average. BUAD 310 - Kam Hamidieh

  8. Why do we care? (Goals?) • We may want to see if there is a relationship between X and Y. • If there is a relationship: • We can see how Y changes as X changes. • We can predict Y for a given X. BUAD 310 - Kam Hamidieh

  9. Apple & SP500 & Oil Data obtained from: finance.yahoo.com & www.eia.gov, daily % changes, 5/31/2013 to 10/21/2013 BUAD 310 - Kam Hamidieh

  10. Statistical Inference BUAD 310 - Kam Hamidieh

  11. Statistical Inference BUAD 310 - Kam Hamidieh

  12. Simple Linear Regression Model (SRM) Observed values of Y are linearly related to values of X by: Individual: Mean: ε‘sis called the error terms. B0 is the (population) intercept. B1is the (population) slope. The unknown population parameters are: B0 , B1 , and σε BUAD 310 - Kam Hamidieh

  13. Estimating The Population Line • The line E[Y|X=xi] = B0+ B1xi is called the population line. It is not observed. • Our technical goal will be to estimate the line by estimating the population coefficients B0and B1. BUAD 310 - Kam Hamidieh

  14. SRM Error Part Line Part Y yi B1 εi B0 + B1xi B0 X xi BUAD 310 - Kam Hamidieh

  15. SRM Says that Y’s are normally distributed. Says that mean of Y’s at a fixed X follows a line. Y X xi BUAD 310 - Kam Hamidieh

  16. Believability • Simple Linear Regression model is a statistical model: • When statisticians (AKA data scientists?!?!) model the relationship between X and Y, they do not imply that the model is the actual reality. • Why model then?“All models are wrong but some are useful.” GEP Box ? X Y + randomness BUAD 310 - Kam Hamidieh

  17. Estimating The Population Line • Once we have estimated the line, we re-expressed it as: • The y-hat is called the fitted or predicted or estimated y value. • b0, b1 , estimate B0, B1, and E[Y|X=xi] respectively. BUAD 310 - Kam Hamidieh

  18. Residuals A residual is the vertical distance of an observed point from the estimated line: The residuals ei’s estimate the unobserved errors ε. BUAD 310 - Kam Hamidieh

  19. Back to Apple and S&P 500 (0.005, 0.032) 0.00363 The residual at x = 0.005: Predicted value ≈ 0.00363 BUAD 310 - Kam Hamidieh

  20. Interpretation of Coefficients • The slope b1tells us on average how much of an increase (or decrease) there is for when the x variable increases by one unit. WHY? • The intercept b0tells us the value of when the x = 0. WHY? • For Apple vs. S&P 500: • Slope: On average, when S&P 500 gains 1% in a day, Apply goes up by 0.465% in a day. • Intercept: On average, when S&P 500 is unchanged in a day, Apply goes up by 0.13% in a day. BUAD 310 - Kam Hamidieh

  21. In Class Exercise 1 We have estimated the slope to be 1 and the intercept to be -1/3. • Draw the line. • What are your b0, b1? • What do b0, b1 estimate? • Write down the estimated regression line. • What are your observed y values and their corresponding x values? • What are your fitted values? • What are your residuals? BUAD 310 - Kam Hamidieh

  22. Method of Least Squares • The method of least squares estimates B0, and B1 by minimizing the distance between observed y values and values that come from the estimated regression line: • Least: minimize • Squares: you are minimize a bunch of squared terms. • WHY? Pick b0, b1 so this is as small as possible. BUAD 310 - Kam Hamidieh

  23. Intuition Which line is the “best”? BUAD 310 - Kam Hamidieh

  24. More Intuition • Why squared? Why not look at say absolute values? • Why look at the vertical distance as opposed to “direct” (dashed) distance? Y X BUAD 310 - Kam Hamidieh

  25. Least Squares Results By calculus or other methods: r = sample correlation between y and x values (see slide 8 of lecture 10) SX = sample standard deviation of x values SY = Sample standard deviation of y values = mean of x values = mean of y values BUAD 310 - Kam Hamidieh

  26. apple-sp500-returns.txt BUAD 310 - Kam Hamidieh

  27. In Class Exercise 2 What’s wrong? • The slope describes the change in x for a change in y. (Hint: see slide 20) • The population regression line is expressed as + . (Hint: see slide 13) • A couple of the parameters of the SRM are b0, b1. (Hint: see slide 12) • Large b1 values imply large correlation r. (Hint: see slide 25) • b1 and r will have opposite signs. (Hint: see slide 25) • Errors estimate the residuals. (Hint: see slide 18) • Fitted values always equal observed values. (Hint: see slide 18) • We obtain estimates of b0, b1 by minimizing the residuals. (Hint: see slide 22) BUAD 310 - Kam Hamidieh

  28. Next Time • Continue with simple linear regression BUAD 310 - Kam Hamidieh

  29. For Fun… This is a question one of our students – Thanks to Jake S. - got during his summer internship interview: “A battleship has 2 rockets which of each can either hit or miss their target. If both are fired simultaneously at an enemy, what is the chance that the enemy is hit?” Many people would say 50% since the rockets can either hit or miss. What is the correct answer? BUAD 310 - Kam Hamidieh

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