Heteroskedasticity: Nature and Detection

1. Heteroskedasticity: Nature and Detection

2. Aims and Learning Objectives • By the end of this session students should be able to: • Explain the nature of heteroskedasticity • Understand the causes and consequences of • heteroskedasticity • Perform tests to determine whether a regression • model has heteroskedastic errors

3. Nature of Heteroskedasticity Heteroskedasticity is a systematic pattern in the errors where the variances of the errors are not constant. Ordinary least squares assumes that all observations are equally reliable.

4. Regression Model Yi = 1 + 2Xi + Ui Var(Ui) = 2 Homoskedasticity: Or E(Ui2) = 2 Var(Ui) = i2 Heteroskedasticity: Or E(Ui2) = i2

5. Homoskedastic pattern of errors Consumption Yi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xi Income

6. The Homoskedastic Case Yi f(Yi) Consumption . . . . X1 X2 X3 X4 Xi Income

7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heteroskedastic pattern of errors Consumption . Yi Xi Income

8. The Heteroskedastic Case Yi f(Yi) Consumption . . rich people . poor people X1 X2 X3 Xi Income

9. Causes of Heteroskedasticity Common Causes • Direct • Scale Effects • Structural Shift • Learning Effects • Indirect • Omitted Variables • Outliers • Parameter Variation

10. Consequences of Heteroskedasticity 1. Ordinary least squares estimators still linear and unbiased. 2. Ordinary least squares estimators not efficient. 3. Usual formulas give incorrect standard errors for least squares. 4. Confidence intervals and hypothesis tests based on usual standard errors are wrong.

11. ^ ^ Yi = 1 + 2Xi + ei Var(ei) = i2 heteroskedasticity: Formula for ordinary least squares variance (homoskedastic disturbances): Formula for ordinary least squares variance (heteroskedastic disturbances): Therefore when errors are heteroskedastic ordinary least squares estimators are inefficient (i.e. not “best”)

12. Detecting Heteroskedasticity ei2 : squared residuals provide proxies for Ui2 Preliminary Analysis • Data - Heteroskedasticity often occurs in cross • sectional data (exceptions: ARCH, panel data) • Graphical examination of residuals - plot ei or • ei2 against each explanatory variable or against • predicted Y

13. ei 0 . . Xi Residual Plots Plot residuals against one variable at a time after sorting the data by that variable to try to find a heteroskedastic pattern in the data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14. Formal Tests for Heteroskedasticity The Goldfeld-Quandt Test 1. Sort data according to the size of a potential proportionality factor d (largest to smallest) 2. Omit the middle r observations 3. Run separate regressions on first n1 observations and last n2 observations 4. If disturbances are homoskedastic then Var(Ui) should be the same for both samples.

15. Ho: 12 =22 H1: 12 >22 The Goldfeld-Quandt Test 5. Specify null and alternative hypothesis 6. Test statistic Compare test statistic value with critical value from F-distribution table

16. White’s Test 1. Estimate And obtain the residuals 2.Run the following auxiliary regression: 3. Calculate White test statistic from auxiliary regression 4. Obtain critical value from 2 distribution (df = no. of explanatory variables in auxiliary regression) 5. Decision rule: if test statistic > critical 2 value then reject null hypothesis of no heteroskedasticity

17. Summary In this lecture we have: 1. Analysed the theoretical causes and consequences of heteroskedasticity 2. Outlined a number of tests which can be used to detect the presence of heteroskedastic errors