1. Statistics for Business and Economics 7 th Edition Chapter 11 Simple Regression Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-1

2. Chapter Goals After completing this chapter, you should be able to: Explain the simple linear regression model Obtain and interpret the simple linear regression equation for a set of data Describe R 2 as a measure of explanatory power of the regression model Understand the assumptions behind regression analysis Explain measures of variation and determine whether the independent variable is significant Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-2

3. Chapter Goals After completing this chapter, you should be able to: Calculate and interpret confidence intervals for the regression coefficients Use a regression equation for prediction Form forecast intervals around an estimated Y value for a given X Use graphical analysis to recognize potential problems in regression analysis Explain the correlation coefficient and perform a hypothesis test for zero population correlation (continued) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-3

4. Overview of Linear Models In many economic and business problems a specific functional relationship is needed to obtain numerical results: If 250 workers are employed in a factory, how many units would be produced during an average day? If a developing country increases its fertilizer production by 1,000,000 tons, how much increase in grain production should be expected? If a firm sets the price of its product at $10 per unit, what would be the mean level of sales? (continued) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-4

5. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Overview of Linear Models We can approximate the desired functional relationships by a linear equation, Y = β 0 + β 1 X Where Y is the dependent variable and X is the independent variable β 0 is the Y-intercept β 1 is the slope 11.1 Ch. 11-5

6. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Least Squares Regression Estimates for coefficients β 0 and β 1 are found using a Least Squares Regression technique The least-squares regression line, based on sample data, is Where b 1 is the slope of the line and b 0 is the y- intercept: Ch. 11-6

7. Introduction to Regression Analysis Regression analysis is used to: Predict the value of a dependent variable based on the value of at least one independent variable Explain the impact of changes in an independent variable on the dependent variable Dependent variable: the variable we wish to explain (also called the endogenous variable) Independent variable: the variable used to explain the dependent variable (also called the exogenous variable) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-7

8. Linear Regression Model The relationship between X and Y is described by a linear function Changes in Y are assumed to be caused by changes in X Linear regression population equation model Where  0 and  1 are the population model coefficients and  is a random error term. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-8 11.2

9. Simple Linear Regression Model Linear component The population regression model: Population Y intercept Population Slope Coefficient Random Error term Dependent Variable Independent Variable Random Error component Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-9

10. Simple Linear Regression Model (continued) Random Error for this X i value Y X Observed Value of Y for X i Predicted Value of Y for X i XiXi Slope = β 1 Intercept = β 0 εiεi Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-10

11. Simple Linear Regression Equation The simple linear regression equation provides an estimate of the population regression line Estimate of the regression intercept Estimate of the regression slope Estimated (or predicted) y value for observation i Value of x for observation i The individual random error terms e i have a mean of zero Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-11

12. Least Squares Estimators b 0 and b 1 are obtained by finding the values of b 0 and b 1 that minimize the sum of the squared differences between y and : Differential calculus is used to obtain the coefficient estimators b 0 and b 1 that minimize SSE Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-12 11.3

13. Least Squares Estimators The slope coefficient estimator is And the constant or y-intercept is The regression line always goes through the mean x, y (continued) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-13

14. Finding the Least Squares Equation The coefficients b 0 and b 1, and other regression results in this chapter, will be found using a computer Hand calculations are tedious Statistical routines are built into Excel Other statistical analysis software can be used Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-14

15. Linear Regression Model Assumptions The true relationship form is linear (Y is a linear function of X, plus random error) The error terms, ε i are independent of the x values The error terms are random variables with mean 0 and constant variance, σ 2 (the constant variance property is called homoscedasticity) The random error terms, ε i, are not correlated with one another, so that Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-15

16. Interpretation of the Slope and the Intercept b 0 is the estimated average value of y when the value of x is zero (if x = 0 is in the range of observed x values) b 1 is the estimated change in the average value of y as a result of a one-unit change in x Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-16

17. Simple Linear Regression Example A real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet) A random sample of 10 houses is selected Dependent variable (Y) = house price in $1000s Independent variable (X) = square feet Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-17

18. Sample Data for House Price Model House Price in $1000s (Y) Square Feet (X) 2451400 3121600 2791700 3081875 1991100 2191550 4052350 3242450 3191425 2551700 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-18

19. Graphical Presentation House price model: scatter plot Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-19

20. Regression Using Excel Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-20 Excel will be used to generate the coefficients and measures of goodness of fit for regression Data / Data Analysis / Regression

21. Regression Using Excel Data / Data Analysis / Regression Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-21 (continued) Provide desired input:

22. Excel Output Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-22

23. Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 The regression equation is: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall (continued) Ch. 11-23

24. Graphical Presentation House price model: scatter plot and regression line Slope = 0.10977 Intercept = 98.248 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-24

25. Interpretation of the Intercept, b 0 b 0 is the estimated average value of Y when the value of X is zero (if X = 0 is in the range of observed X values) Here, no houses had 0 square feet, so b 0 = 98.24833 just indicates that, for houses within the range of sizes observed, $98,248.33 is the portion of the house price not explained by square feet Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-25

26. Interpretation of the Slope Coefficient, b 1 b 1 measures the estimated change in the average value of Y as a result of a one- unit change in X Here, b 1 =.10977 tells us that the average value of a house increases by.10977($1000) = $109.77, on average, for each additional one square foot of size Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-26

27. Measures of Variation Total variation is made up of two parts: Total Sum of Squares Regression Sum of Squares Error Sum of Squares where: = Average value of the dependent variable y i = Observed values of the dependent variable i = Predicted value of y for the given x i value Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-27 11.4

28. Measures of Variation SST = total sum of squares Measures the variation of the y i values around their mean, y SSR = regression sum of squares Explained variation attributable to the linear relationship between x and y SSE = error sum of squares Variation attributable to factors other than the linear relationship between x and y (continued) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-28

29. Measures of Variation (continued) xixi y X yiyi SST =  (y i - y) 2 SSE =  (y i - y i ) 2  SSR =  (y i - y) 2  _ _ _ y  Y y _ y  Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-29

30. Coefficient of Determination, R 2 The coefficient of determination is the portion of the total variation in the dependent variable that is explained by variation in the independent variable The coefficient of determination is also called R-squared and is denoted as R 2 note: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-30

31. Examples of Approximate r 2 Values r 2 = 1 Y X Y X Perfect linear relationship between X and Y: 100% of the variation in Y is explained by variation in X Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-31

32. Examples of Approximate r 2 Values Y X Y X 0 < r 2 < 1 Weaker linear relationships between X and Y: Some but not all of the variation in Y is explained by variation in X Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-32

33. Examples of Approximate r 2 Values r 2 = 0 No linear relationship between X and Y: The value of Y does not depend on X. (None of the variation in Y is explained by variation in X) Y X r 2 = 0 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-33

34. Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 58.08% of the variation in house prices is explained by variation in square feet Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-34

35. Correlation and R 2 The coefficient of determination, R 2, for a simple regression is equal to the simple correlation squared Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-35

36. Estimation of Model Error Variance An estimator for the variance of the population model error is Division by n – 2 instead of n – 1 is because the simple regression model uses two estimated parameters, b 0 and b 1, instead of one is called the standard error of the estimate Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-36

37. Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-37

38. Comparing Standard Errors YY X X s e is a measure of the variation of observed y values from the regression line The magnitude of s e should always be judged relative to the size of the y values in the sample data i.e., s e = $41.33K is moderately small relative to house prices in the $200 - $300K range Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-38

39. Inferences About the Regression Model Inferences from regression analysis will help us understand the process being modeled and make decisions about the process. We assume that the random model error, ε i, are normally distributed with variance σ 2. y i is also normally distributed with the same variance. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-39 11.5

40. Inferences About the Regression Model The variance of the regression slope coefficient (b 1 ) is estimated by where: = Estimate of the standard error of the least squares slope Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-40 11.5

41. Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-41

42. Comparing Standard Errors of the Slope Y X Y X is a measure of the variation in the slope of regression lines from different possible samples Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-42

43. Varance of the Slope Coefficient Higher the distance of the points from the regression line measured by s e, the greater the variance for b 1 The greater the spread in the X values, the smaller the variance for b 1 Smaller variance slope coefficient estimators imply a better regression model. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-43

44. Inference about the Slope: t Test t test for a population slope Is there a linear relationship between X and Y? Null and alternative hypotheses H 0 : β 1 = 0(no linear relationship) H 1 : β 1  0(linear relationship does exist) Test statistic where: b 1 = regression slope coefficient β 1 = hypothesized slope s b1 = standard error of the slope Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-44

45. Inference about the Slope: t Test House Price in $1000s (y) Square Feet (x) 2451400 3121600 2791700 3081875 1991100 2191550 4052350 3242450 3191425 2551700 Estimated Regression Equation: The slope of this model is 0.1098 Does square footage of the house affect its sales price? (continued) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-45

46. Inferences about the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1  0 From Excel output: CoefficientsStandard Errort StatP-value Intercept98.2483358.033481.692960.12892 Square Feet0.109770.032973.329380.01039 t b1b1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-46

47. Inferences about the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1  0 Test Statistic: t = 3.329 There is sufficient evidence that square footage affects house price From Excel output: Reject H 0 CoefficientsStandard Errort StatP-value Intercept98.2483358.033481.692960.12892 Square Feet0.109770.032973.329380.01039 tb1b1 Decision: Conclusion: Reject H 0  /2=.025 -t n-2,α/2 Do not reject H 0 0  /2=.025 -2.30602.3060 3.329 d.f. = 10-2 = 8 t 8,.025 = 2.3060 (continued) t n-2,α/2 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-47

48. Inferences about the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1  0 P-value = 0.01039 There is sufficient evidence that square footage affects house price From Excel output: Reject H 0 CoefficientsStandard Errort StatP-value Intercept98.2483358.033481.692960.12892 Square Feet0.109770.032973.329380.01039 P-value Decision: P-value < α so Conclusion: (continued) This is a two-tail test, so the p-value is P(t > 3.329)+P(t < -3.329) = 0.01039 (for 8 d.f.) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-48

49. Confidence Interval Estimate for the Slope Confidence Interval Estimate of the Slope: Excel Printout for House Prices: At 95% level of confidence, the confidence interval for the slope is (0.0337, 0.1858) CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 d.f. = n - 2 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-49

50. Confidence Interval Estimate for the Slope Since the units of the house price variable is $1000s, we are 95% confident that the average impact on sales price is between $33.70 and $185.80 per square foot of house size CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 This 95% confidence interval does not include 0. Conclusion: There is a significant relationship between house price and square feet at the.05 level of significance (continued) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-50

51. F-Test for Significance F Test statistic: where where F follows an F distribution with k numerator and (n – k - 1) denominator degrees of freedom (k = the number of independent variables in the regression model) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-51

52. Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 With 1 and 8 degrees of freedom P-value for the F-Test Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-52

53. F-Test for Significance H 0 : β 1 = 0 H 1 : β 1 ≠ 0  =.05 df 1 = 1 df 2 = 8 Test Statistic: Decision: Conclusion: Reject H 0 at  = 0.05 There is sufficient evidence that house size affects selling price 0  =.05 F.05 = 5.32 Reject H 0 Do not reject H 0 Critical Value: F  = 5.32 (continued) F Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-53

54. Prediction The regression equation can be used to predict a value for y, given a particular x For a specified value, x n+1, the predicted value is Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-54 11.6

55. Predictions Using Regression Analysis Predict the price for a house with 2000 square feet: The predicted price for a house with 2000 square feet is 317.85($1,000s) = $317,850 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-55

56. Relevant Data Range When using a regression model for prediction, only predict within the relevant range of data Relevant data range Risky to try to extrapolate far beyond the range of observed X’s Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-56

57. Estimating Mean Values and Predicting Individual Values Y X x i y = b 0 +b 1 x i  Confidence Interval for the expected value of y, given x i Prediction Interval for an single observed y, given x i Goal: Form intervals around y to express uncertainty about the value of y for a given x i y  Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-57

58. Confidence Interval for the Average Y, Given X Confidence interval estimate for the expected value of y given a particular x i Notice that the formula involves the term so the size of interval varies according to the distance x n+1 is from the mean, x Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-58

59. Prediction Interval for an Individual Y, Given X Confidence interval estimate for an actual observed value of y given a particular x i This extra term adds to the interval width to reflect the added uncertainty for an individual case Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-59

60. Estimation of Mean Values: Example Find the 95% confidence interval for the mean price of 2,000 square-foot houses Predicted Price y i = 317.85 ($1,000s)  Confidence Interval Estimate for E(Y n+1 |X n+1 ) The confidence interval endpoints are 280.66 and 354.90, or from $280,660 to $354,900 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-60

61. Estimation of Individual Values: Example Find the 95% confidence interval for an individual house with 2,000 square feet Predicted Price y i = 317.85 ($1,000s)  Confidence Interval Estimate for y n+1 The confidence interval endpoints are 215.50 and 420.07, or from $215,500 to $420,070  Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-61

62. Correlation Analysis Correlation analysis is used to measure strength of the association (linear relationship) between two variables Correlation is only concerned with strength of the relationship No causal effect is implied with correlation Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-62 11.7

63. Correlation Analysis The population correlation coefficient is denoted ρ (the Greek letter rho) The sample correlation coefficient is where Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-63

64. Hypothesis Test for Correlation To test the null hypothesis of no linear association, the test statistic follows the Student’s t distribution with (n – 2 ) degrees of freedom: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-64

65. Decision Rules Lower-tail test: H 0 : ρ  0 H 1 : ρ < 0 Upper-tail test: H 0 : ρ ≤ 0 H 1 : ρ > 0 Two-tail test: H 0 : ρ = 0 H 1 : ρ ≠ 0 Hypothesis Test for Correlation  /2  -t  -t  /2 tt t  /2 Reject H 0 if t < -t n-2,  Reject H 0 if t > t n-2,  Reject H 0 if t < -t n-2,    or t > t n-2,  Where has n - 2 d.f. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-65

66. Graphical Analysis The linear regression model is based on minimizing the sum of squared errors If outliers exist, their potentially large squared errors may have a strong influence on the fitted regression line Be sure to examine your data graphically for outliers and extreme points Decide, based on your model and logic, whether the extreme points should remain or be removed Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-66 11.9

67. Chapter Summary Introduced the linear regression model Reviewed correlation and the assumptions of linear regression Discussed estimating the simple linear regression coefficients Described measures of variation Described inference about the slope Addressed estimation of mean values and prediction of individual values Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 11-67