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Chapter 14 Simple Linear Regression

Chapter 14 Simple Linear Regression. Chapter Outline. Simple Linear Regression Model Least Squares Method Coefficient of Determination Model Assumptions Testing for Significance. Simple Linear Regression.

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Chapter 14 Simple Linear Regression

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  1. Chapter 14Simple Linear Regression

  2. Chapter Outline • Simple Linear Regression Model • Least Squares Method • Coefficient of Determination • Model Assumptions • Testing for Significance

  3. Simple Linear Regression • Managerial decisions often are based on the relationship between two or more variables. • Regression analysis can be used to develop an equation showing how the variables are related. • The variable being predicted is called the dependent variable and is denoted by y. • The variables being used to predict the value of the dependent variable are called the independent variables and are denoted by x.

  4. Simple Linear Regression • Simple linear regression involves one independent variable and one dependent variable. • The relationship between the two variables is approximated by a straight line (hence, the ‘linear’ regression). • Regression analysis involving two or more independent variable is called multiple regression (covered in the next chapter).

  5. Simple Linear Regression Model • The equation that describes how y is related to x and an error term is called the regression model. • The simple linear regression model is: y = b0 + b1x +e where: • b0 and b1 are called parameters of the model, • e is a random variable called the error term.

  6. Simple Linear Regression Equation • The simple linear regression equation is: E(y) = 0 + 1x • Graph of the regression equation is a straight line. • 0 is the y intercept of the regression line. • 1 is the slope of the regression line. • E(y) is the expected value of y for a given value of x. • Please note that both 0 and 1 are population parameters, depicting the true relationship between y and x.

  7. Simple Linear Regression • Example: Stock Market Risk The systematic risk (a common risk shared by all the stocks) of stock market has different impacts on different stocks. Stocks that are more sensitive to systematic risk are riskier. We can conduct a regression analysis to estimate the sensitivity of an individual stock to the systematic market risk. On the next slide are shown the data for a sample of 20 most recent quarterly returns of Netflix and the SPY (an index fund that keeps track of S&P 500).

  8. Simple Linear Regression • Example: Stock Market Risk (data)

  9. Simple Linear Regression • Example: Stock Market Risk (Scatter Diagram) Trend Line

  10. Simple Linear Regression • Example: Stock Market Risk From the scatter diagram, we observe the following: • The plots are scattered around, indicating the relationship between the returns of SPY and Netflix is not perfect. • The trend line has a positive slope, indicating that the relationship is positive, i.e. as the returns of SPY go up, the returns of Netflix tend to go up too. • The vertical distance between a plot to the trend line is the difference between the actual return of Netflix and its estimated value, given an actual return of SPY. The difference is simply the estimated error, similar to y – E(y).

  11. Simple Linear Regression Equation • Positive Linear Relationship E(y) Regression line Intercept b0 Slope b1 is positive x

  12. Simple Linear Regression Equation • Negative Linear Relationship E(y) Intercept b0 Regression line Slope b1 is negative x x

  13. Simple Linear Regression Equation • No Relationship E(y) Intercept b0 Regression line Slope b1 is 0 x

  14. Estimated Simple Linear Regression Equation • The estimated simple linear regression equation: • The graph is called the estimated regression line. • b0 is the y intercept of the estimated regression line. • b1 is the slope of the estimated regression line. • is the estimated value of y for a given value of x. • Please note that b0 and b1 are sample estimates of 0 and 1, respectively, depicting the estimated sample relationship between y and x.

  15. Sample Data: x y x1 y1 . . . . xnyn Estimated Regression Equation Sample Statistics b0, b1 Estimation Process Regression Model y = b0 + b1x +e Regression Equation E(y) = b0 + b1x Unknown Parameters b0, b1 b0 and b1 provide estimates of b0 and b1

  16. ^ yi = estimated value of the dependent variable for the ith observation Least Squares Method • Least Squares Criterion where: yi = observed value of the dependent variable for the ith observation

  17. Least Squares Method • Least Squares Criterion • is the estimated error for the ith observation; • Take the square of means that it is the magnitude of the error not the sign of it (positive or negative) that matters; • The purpose of Least Squares Criterion is to find the b0 and b1 that minimize the sum of the square of estimated error for all the observations in the sample, i.e. the best-fit (with the smallest overall error) straight line that approximates the relationship between y and x.

  18. _ _ x = average value of independent variable y = average value of dependent variable Least Squares Method • Slope for the Estimated Regression Equation where: xi = value of independent variable for ith observation yi = value of dependent variable for ith observation

  19. Least Squares Method • y-Intercept for the Estimated Regression Equation

  20. Simple Linear Regression Quarterly Returns of SPY (x) Quarterly Returns of Netflix (y) • Example: Stock Market Risk 0.0630 0.1366 0.0470 0.0191 0.2537 -0.0302 0.3190 0.2693 Sx = 0.9143 Sy = 4.0419

  21. Estimated Regression Equation • Slope for the Estimated Regression Equation • y-Intercept for the Estimated Regression Equation • Estimated Regression Equation

  22. Estimated Regression Line – Stock Market Risk Example

  23. Coefficient of Determination • Relationship Among SST, SSR, SSE SST = SSR + SSE where: SST = total sum of squares (i.e. total variability of y) SSR = sum of squares due to regression (i.e. the variability of y that is explained by regression) SSE = sum of squares due to error (i.e. the variability of y that cannot be explained by regression)

  24. Coefficient of Determination The coefficient of determination is: r2 = SSR/SST r2 represents the percentage of total variability of y that is explained by regression.

  25. Coefficient of Determination r2 = SSR/SST = 0.404/2.741 = 0.147 The regression relationship is actually weak. Only 14.7% of the variability in the returns of Netflix can be explained by the linear relationship between the market returns (SPY) and the returns of Netflix.

  26. where: b1 = the slope of the estimated regression equation Sample Correlation Coefficient

  27. Sample Correlation Coefficient The sign of b1 in the equation is “+”.

  28. Assumptions About the Error Term e y = b0 + b1x +e 1. The error  is a random variable with mean of zero. 2. The variance of  , denoted by  2, is the same for all values of the independent variable. 3. The values of  are independent. 4. The error  is a normally distributed random variable.

  29. Test for Significance b1 determines the relationship between y and x. y = b0 + b1x +e To test for a significant regression relationship, we must conduct a hypothesis test to determine whether the value of b1 (slop) is zero. Two tests are commonly used: t Test F Test and Both the t test and F test require an estimate of s2, the variance of e in the regression model.

  30. Test for Significance • An Estimate of s2 • The mean square error (MSE) provides the estimate • (the sample variance s2 ) of s2. s2 = MSE = SSE/(n - 2) where:

  31. Test for Significance • An Estimate of s • To estimate s we take the square root of s2. • The resulting s is called the standard error of • the estimate.

  32. Test for Significance: t Test Hypotheses Test Statistic where

  33. Test for Significance: t Test Rejection Rule Reject H0 if p-value <a or t< -tor t>t where: tis based on a t distribution with n - 2 degrees of freedom • n is the number of observations in the regression; • 2 is the number of parameters (0 & 1) in the regression.

  34. Test for Significance: t Test 1. Determine the hypotheses. a = .05 2. Specify the level of significance. 3. Calculate the test statistic.

  35. Test for Significance: t Test 4. Determine whether to reject H0. p-Value approach t = 1.76 provides an area of .0473 in the upper tail. Hence, the p-value is 2*0.0473 = 0.0946. Since p-value is larger than 0.05, we will not reject H0. Critical Value approach For =5%, the critical value is 2.1 (a two-tailed test). Since our test statistic t = 1.76, which is less than 2.1, we will not reject H0.

  36. Confidence Interval for 1 • We can use a 95% confidence interval for 1 to test • the hypotheses just used in the t test. • H0 is rejected if the hypothesized value of 1 is not • included in the confidence interval for 1.

  37. where is the t value providing an area of a/2 in the upper tail of a t distribution with n - 2 degrees of freedom Confidence Interval for 1 • The form of a confidence interval for 1 is: t/2sb1 is the margin of error b1 is the point estimator

  38. = 2.87 ± 2.1(1.63) = 2.87 ± 3.42 Confidence Interval for 1 • Rejection Rule Reject H0 if 0 is not included in the confidence interval for 1. • 95% Confidence Interval for 1 or -0.55 to 6.29 • Conclusion 0 is included in the confidence interval. Do Not Reject H0

  39. Test for Significance: F Test Hypotheses Test Statistic Please note that the hypotheses of the F test are the same as the ones of the t test, which is always the case for a Simple Linear Regression (where there is only one independent variable.) F = MSR/MSE

  40. Test for Significance: F Test Rejection Rule Reject H0 if p-value <a or F>F where: F is based on an F distribution with 1 degree of freedom in the numerator and n - 2 degrees of freedom in the denominator

  41. ANOVA Table for A Regression Analysis p- Value Source of Variation Sum of Squares Degrees of Freedom Mean Square F Regression k - 1 SSR Error SSE nT - k Total SST nT - 1 nt is the number of observations. k is the number of parameters in a regression.

  42. ANOVA Table for A Regression Analysis Stock Market Risk Example - p- Value Source of Variation Sum of Squares Degrees of Freedom Mean Square F 0.404 3.11 0.095 Regression 1 0.404 Error 2.337 18 0.13 Total 2.741 19

  43. Test for Significance: F Test 1. Determine the hypotheses. a = .05 2. Specify the level of significance. F = MSR/MSE 3. Calculate the test statistic. F = MSR/MSE = 0.404/0.13 = 3.11 The relationship between the F value and the t value is F = t2, which is only true for simple linear regressions.

  44. Test for Significance: F Test 4. Determine whether to reject H0. p-Value approach F = 3.11 provides an area of .0946 in the upper tail. Hence, the p-value is 0.0946. Since p-value is larger than 0.05, we will not reject H0. Critical Value approach For =5%, the critical value is 4.41. Since our test statistic F = 3.11, which is less than 4.41, we will not reject H0.

  45. Some Cautions about the Interpretation of Significance Tests • Rejecting H0: b1 = 0 and concluding that the • relationship between x and y is significant does not enable us to conclude that a cause-and-effect • relationship is present between x and y. • Just because we are able to reject H0: b1 = 0 and • demonstrate statistical significance does not enable • us to conclude that there is a linear relationship • between x and y.

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