1. 12.1 Heteroskedasticity: Remedies Normality Assumption

2. 12.2 Aims and Learning Objectives By the end of this session students should be able to: Use the weighted least squares procedure to deal with heteroskedasticity How reformulating the model may help remove apparent heteroskedasticity problems Understand what the normality assumption means and why it is important

3. 12.3 Y i =  +  2 X 2i +  3 X 3i + U i Regression Model Var(U i ) =   2 Homoskedasticity: Heteroskedasticity: Var(U i ) =  i  2

4. 12.4 Heteroskedasticity: Direct versus Indirect If we suspect the nature of heteroskedasticity to be indirect then we would re-specify the model. The correct remedy for indirect heteroskedasticity (e.g. caused by an omitted variable having a heteroskedastic component) is to make sure the model is correctly specified How can we deal with direct heteroskedasticity?

5. 12.5 Remedies for (Direct) Heteroskedasticity Weighted Least Squares Redefine the variables White’s Robust Standard Errors

6. 12.6 Weighted Least Squares Intuitively, we want to explicitly take account of the variability of consumption for different income groups We demonstrated in lecture 11, that OLS in the presence of heteroskedasticity is LUE but not BLUE OLS assigns equal weights (importance) to each observation We would like to use an estimation method such that observations with greater variability are given less weight than those with smaller variability

7. 12.7 Weighted Least Squares Var(U i ) =  i  2 Case 1: Disturbance term is known Divide by  i to obtain:

8. 12.8 The variance of the disturbance term in the transformed regression is Therefore The variance of the transformed disturbance term is now homoskedastic (and therefore BLUE)

9. 12.9 Weighted Least Squares wher e  i  2 =   2 d i The variance is assumed to be proportional to the value of d i Var(U i ) =  i  2 Case 2: Disturbance term is unknown

10. 12.10  i  2 =   2 d i std.dev. proportional to d i variance: standard deviation:  i  =   d i To correct for heteroskedasticity divide the model by d i var(U i ) =  i  2

11. 12.11 Y i =  1 * +   X i2 * +  3 X i3 * + V i * U i is heteroskedastic, but V i is homoskedastic.

12. 12.12 1. Decide which variable is proportional to the heteroskedasticity (d i could be one of the X variables or predicted Y). 2. Divide all terms in the original model by the square root of that variable (divide by d i ). 3. Run least squares on the transformed model which has new Y i, X i2 and X i3 variables but no intercept (in this example anyway). Weighted Least Squares Recap:

13. 12.13 Redefine the Variables Regress per capita consumption against per capita income Logarithmic transformation (e.g. log-linear model) Compresses the scales Easy interpretation

14. 12.14 White’s Standard Errors White’s estimator of the least squares variance: In large samples White’s standard error (square root of estimated variance) is a correct / accurate / consistent measure.

15. 12.15 Normality Tests Normality is necessary for the purposes of statistical inference (i.e. confidence intervals and hypothesis testing) Therefore if the assumption of normality does not hold then the variances and standard errors of our OLS estimators are invalid Important caveat: this applies only to “small” samples. In “large” samples if the distribution is non-normal we make use of the central limit theorem

16. 12.16 Preliminary Analysis (informal Tests) Histogram of the regression residuals Normal Probability Plot Normal probability paper (e.g. Minitab) Shape of the PDF of the random variable: Frequency against the residuals

17. 12.17 Formal Test for Normality Bera-Jarque (or Jarque-Bera) test (“large” sample test) where Normally distributed if S = 0 and K = 3

18. 12.18 Summary In this lecture we have: 1. Examined practical ways of removing the heteroskedasticity problem 2. Outlined the nature of the normality assumption and applied a suitable test