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Chapter 5

Chapter 5. Heteroskedasticity. What is in this Chapter?. How do we detect this problem What are the consequences of this problem? What are the solutions?. What is in this Chapter?.

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Chapter 5

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  1. Chapter 5 Heteroskedasticity

  2. What is in this Chapter? • How do we detect this problem • What are the consequences of this problem? • What are the solutions?

  3. What is in this Chapter? • First, We discuss tests based on OLS residuals, likelihood ratio test, G-Q test and the B-P test. The last one is an LM test. • Regarding consequences, we show that the OLS estimators are unbiased but inefficient and the standard errors are also biased, thus invalidating tests of significance

  4. What is in this Chapter? • Regarding solutions, we discuss solutions depending on particular assumptions about the error variance and general solutions. • We also discuss transformation of variables to logs and the problems associated with deflators, both of which are commonly used as solutions to the heteroskedasticity problem.

  5. 5.1 Introduction • The homoskedasticity=variance of the error terms is constant • The heteroskedasticity=variance of the error terms is non-constant • Illustrative Example  • Table 5.1 presents consumption expenditures (y) and income (x) for 20 families. Suppose that we estimate the equation by ordinary least squares. We get (figures in parentheses are standard errors)

  6. 5.1 Introduction

  7. 5.1 Introduction

  8. 5.1 Introduction

  9. 5.1 Introduction

  10. 5.1 Introduction

  11. 5.1 Introduction • The residuals from this equation are presented in Table 5.3 • In this situation there is no perceptible increase in the magnitudes of the residuals as the value of x increases • Thus there does not appear to be a heteroskedasticity problem.

  12. 5.2 Detection of Heteroskedasticity

  13. 5.2 Detection of Heteroskedasticity

  14. 5.2 Detection of Heteroskedasticity

  15. 5.2 Detection of Heteroskedasticity

  16. 5.2 Detection of Heteroskedasticity

  17. 5.2 Detection of Heteroskedasticity

  18. 5.2 Detection of Heteroskedasticity • Some Other Tests • Likelihood Ratio Test • Goldfeld and Quandt Test • Breusch-Pagan Test

  19. 5.2 Detection of Heteroskedasticity • Likelihood Ratio Test

  20. 5.2 Detection of Heteroskedasticity • Goldfeld and Quandt Test • If we do not have large samples, we can use the Goldfeld and Quandt test. • In this test we split the observations into two groups — one corresponding to large values of x and the other corresponding to small values of x — • Fit separate regressions for each and then apply an F-test to test the equality of error variances. • Goldfeld and Quandt suggest omitting some observations in the middle to increase our ability to discriminate between the two error variances.

  21. 5.2 Detection of Heteroskedasticity • Breusch-Pagan Test

  22. 5.2 Detection of Heteroskedasticity

  23. 5.2 Detection of Heteroskedasticity • Illustrative Example

  24. 5.2 Detection of Heteroskedasticity

  25. 5.2 Detection of Heteroskedasticity

  26. 5.2 Detection of Heteroskedasticity

  27. 5.2 Detection of Heteroskedasticity

  28. 5.2 Detection of Heteroskedasticity

  29. 5.3 Consequences of Heteroskedasticity

  30. 5.3 Consequences of Heteroskedasticity

  31. 5.3 Consequences of Heteroskedasticity

  32. 5.3 Consequences of Heteroskedasticity

  33. 5.3 Consequences of Heteroskedasticity

  34. 5.3 Consequences of Heteroskedasticity

  35. 5.3 Consequences of Heteroskedasticity

  36. 5.3 Consequences of Heteroskedasticity

  37. 5.3 Consequences of Heteroskedasticity

  38. 5.4 Solutions to the Heteroskedasticity Problem • There are two types of solutions that have been suggested in the literature for the problem of heteroskedasticity:  • Solutions dependent on particular assumptions about σi. • General solutions. • We first discuss category 1. Here we have two methods of estimation: weighted least squares (WLS) and maximum likelihood (ML).

  39. 5.4 Solutions to the Heteroskedasticity Problem • WLS

  40. 5.4 Solutions to the Heteroskedasticity Problem Thus the constant term in this equation is the slope coefficient in the original equation.

  41. 5.4 Solutions to the Heteroskedasticity Problem

  42. 5.4 Solutions to the Heteroskedasticity Problem

  43. 5.4 Solutions to the Heteroskedasticity Problem • If we make some specific assumptions about the errors, say that they are normal • We can use the maximum likelihood method, which is more efficient than the WLS if errors are normal

  44. 5.4 Solutions to the Heteroskedasticity Problem

  45. 5.4 Solutions to the Heteroskedasticity Problem

  46. 5.4 Solutions to the Heteroskedasticity Problem • Illustrative Example

  47. 5.4 Solutions to the Heteroskedasticity Problem

  48. 5.4 Solutions to the Heteroskedasticity Problem

  49. 5.5 Heteroskedasticity and the Use of Deflators • There are two remedies often suggested and used for solving the heteroskedasticity problem:  • Transforming the data to logs • Deflating the variables by some measure of "size."

  50. 5.5 Heteroskedasticity and the Use of Deflators

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