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Simple Linear Regression

Simple Linear Regression. Correlation. Correlation ( r ) measures the strength of the linear relationship between two sets of data (X,Y). The value for r is always between -1 and +1. Correlation helps answer the question:

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Simple Linear Regression

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  1. Simple Linear Regression

  2. Correlation • Correlation (r) measures the strength of the linear relationship between two sets of data (X,Y). • The value for r is always between -1 and +1. • Correlation helps answer the question: • if X is above its average value does Y tend to be above or below its average value? • If X increases does Y tend to increase or decrease?

  3. Scatter Plot

  4. Positive Correlation • If the correlation between two variables is positive (greater than 0), • When X is above its average, Y tends to be above its average • When X increases, Y tends to increase. r = 0.94

  5. Negative Correlation • If the correlation between two variables is negative (less than 0), • When X is above its average, Y tends to be below its average • When X increases, Y tends to decrease. r = -0.87

  6. Perfect Correlation If know X, know Y r = -1 r = 1

  7. No Correlation Knowing X does not help predicting Y r = -0.059

  8. Returns and Assets Managed • Correlation between the return of an mutual fund and the amount of assets managed? • Annual returns for 385 US equity mutual funds, year ending July 1998. Data provided by Lipper Analytic.

  9. Mutual Fund Name Assets($Mill) (X) Annual Return (Y) EQUITRUST:VAL GRO R 111.5 -4.09 FPA PARAMOUNT 572.2 -1.21 FRANKLIN VAL:VALUE;I 133.9 5.20 IMS CAPITAL VALUE FUND 13.0 6.31 HERITAGE:VALUE EQTY;A 20.0 6.65 ADVANTUS CORNERSTONE;A 113.9 6.69 PUTNAM NEW VALUE;B R 461.2 8.46 YACKTMAN FUND 796.0 8.59 GREENSPRING FUND 182.9 9.22 PIONEER II;A 7239.0 9.54 FRANKLIN ALL:MODERT;I 21.1 10.03 . . . . . . . . . Average = 1413.16 = 23.20 Standard Deviation SX = 4908.93 SY = 6.73 Retunes and Assets Managed • Sample of the Data

  10. Retunes and Assets Managed • Correlation r = 0.0339 • Does the size of Mutual Fund tell You anything about Expected Returns?

  11. Charictoristics of Correlation • Positive/Negative: Increase/Decrease • Positive: X increases, then Y increases • Negative: X increases, then Y decreases • Prefect Correlation • If know X, then know Y • All observations are on a straight line • No Corrrelation • No relationship between X and Y

  12. Correlation Quiz • Imagine that the correlation between price of a product and weekly sales is –0.8. The average price for the product was $1 and the average of the weekly sales was $200 per week. If the price for the product is set at $1.5 which of the following average weekly sales would be reasonable? • -100 • 200 • 240 • 160

  13. Regression Questions • Three Questions • What is the best estimate of a and b ? • Which line fits best? • Are a and bdifferent than zero? • Is there anything going on? • How much of Y is explained by X? • How much of the total variation of Y is explained by X?

  14. Best Guess? • If Knew X, what would be guess for Y? X = 78 Y=Average value of Y

  15. Best Guess? • If Knew X, what would be guess for Y? X = 78 Draw a line that “describes” X in terms of Y.

  16. Equation of a Straight Line • Intercept: Value of Y when X = 0:

  17. Equation of a Straight Line • Intercept: Value of Y when X = 0: • Weight when height = 0

  18. Equation of a Straight Line • Intercept: Value of Y when X = 0: • Weight when height = 0 • Sales when price = 0

  19. Equation of a Straight Line • Intercept: Value of Y when X = 0: • Weight when height = 0 • Sales when price = 0 • Slope: Change Y/Change X:

  20. Equation of a Straight Line • Intercept: Value of Y when X = 0: • Weight when height = 0 • Sales when price = 0 • Slope: Change Y/Change X: • Expected change in weight when height increases by 1

  21. Equation of a Straight Line • Intercept: Value of Y when X = 0: • Weight when height = 0 • Sales when price = 0 • Slope: Change Y/Change X: • Expected change in weight when height increases by 1 • Expected change in sales when price increases by 1

  22. Statistical Notation (Language) • Y is known as the dependent (or response) variable • Typcically we want to have some control over Y • X is know as the independent (or predictor) variable • Often we have some control over X – e.g. Price

  23. Best Straight Line? Choose: Choose:

  24. Minimize Forecast Error Forecast: Observation: Error:

  25. Minimize Forecast Error • Choose a and b so that they minimize the total error • In particular minimize the total sum of squared errors observed value (height for person i) expected value (height for person i) based on estimates of and

  26. What is the best estimate of a and b ? • Choose a and b so that they minimize the total error • In particular minimize the total sum of squared errors

  27. Best Line for Height vs Weight • Choose a and b so that they minimize the total error • Use a Statistical Software (e.g. SPSS)

  28. Interpreting Coefficients Height = 0, Weight = -100 Caution: Forecasting outside of Range of observed data is dangerous!!!!! Increase height by 1 inch, weight increases by 3.986 pounds

  29. The Best Line

  30. Best Line for Sales vs Price • Choose a and b so that they minimize the total error • Use a Statistical Software (e.g. SPSS)

  31. Interpreting Coefficients Price = 0, Sales = $303.86 Caution: Does this make sense? Increase price by $1, sales decrease by 53.84 units

  32. The Best Line

  33. Are a and bdifferent than zero? • Hypothesis Tests • Null Hypothesis: • a=0 • b=0: If this is true, no relationship between X and Y!!! • Statistical Software Calculates • t-statistic (very large or very small reject Null Hypothesis) • Significance Level = P-Value (sig < 0.05, reject Null Hypothesis)

  34. Are a and bdifferent than zero? • Statistical Software Calculates • t-statistic (far from zero reject Null Hypothesis) • Significance Level = P-Value (sig < 0.05, reject Null Hypothesis)

  35. Isbdifferent than zero? • t-statistic: 19.915 • Significance Level = P-Value: less than 0.000 • Reject Null Hypothesis: Reject idea that b = 0!!!

  36. How much of Y is explained by X? • R - square =% of variation of Y explained by the X • correlation = R • For Simple Linear Regression Only!!!

  37. Managerial Insight • What are the expected average sales for a week if price is set at $1?

  38. Managerial Insight What price would you have to set in order to get an average sales of $300 per store? Nonsense?

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