EC 532 Advanced Econometrics Lecture 1 : Heteroscedasticity Prof. Burak Saltoglu

2 Outline What is Heteroscedasticty Graphical Illustration of Heteroscedasticity Reasons for Heteroscedastic errors Consequneces of Heteroscedasticity Generalized Least Squares –GLS in Matrix Notation Testing Heteroscedasticity Remedies

3 Consequneces of Heteroscedasticity As we know under heteroscedastic error terms,

4 What is Heteroscedasticty Or more specifically for time series; means that the variance of disturbances do not change over time.

5 What is Heteroscedasticty The violation of this assumption is called as heteroscedasticity. In the case that the variances of all disturbances are not same, we say that the heteroscedasticity exists. Then

6 Graphical Illustration of Heteroscedasticity

7 X Y Density

8 Some Reasons to Heteroscedasticity The Error-Learning Models Improvement of data collecting (As data collecting techniques improve variances tend to reduce) Presence of outliers Misspecification of model Volatility clustering and news effect

9 The Consequneces of Heteroscedasticity for OLS At the presence of heteroscedasticity; –OLS estimators are still linear and unbiased estimators, but they are no longer the best. (BLUE) –The standard errors computed for the OLS are incorrect, then inference might be misleading.

10 Heteroscedasticity effect of income to household is it safe to assume that variability of consumption is stable for all income levels? suppose, variability of consumption increases with income in a relation such that

11 Consequences of Heteroscedasticity Properties of OLS Estimators: Assume an regression (1)Unbiasedness still holds since

12 Consequences of Heteroscedasticity OLS standard errors, which would be derived from σ 2 (X’X) -1 are incorrect since

13 Generalized Least Squares As we discussed the variance of observations might be different. But the OLS does not take into account the possibility of different variances. The method of GLS is OLS on the transformed variables that satisfies the assumptions. In GLS we assume the variance of each observation is known, and we divide all observations by their variances.

14 Generalized Least Squares Then; where is equal to 1 for each i.

15 Generalized Least Squares So the variance is; Now the residual is homoscedastic

16 An Example Let us assume that we have a model as; and we know there is a relation for error terms as;

17 An Example Now, for this case we can define the transformed form as;

18 An Example The variance;

19 An Example Let assume we have the following model; and, Then,

20 An Example

21 GLS in Matrix Notation If  is a symmetric and positive semi-definite; then there exists a non-singular matrix P such that ; If we set;

22 Properties of GLS

23 GLS in Matrix Notation

24 GLS in Matrix Notation When we faced with heteroscedasticity if we can find an nxn nonsingular transformation matrix T such that; then we multiply everything by T,

25 Detecting Heteroscedasticity 1.Goldfeld-Quandt Test This method applicable where one assumes the heteroscedastic variance is positively related one of variables. As in our previous example;

26 Detecting Heteroscedasticity Goldfeld-Quandt test proceed following steps; –Step1: Order the observations (lowest to highest) –Step2: Omit c central observations –Step3: Fit separate OLS regressions –Step4: Compute; where, k is the number of estimated parameters including the intercept

27 Detecting Heteroscedasticity follows an F-distribution and the null hypothesis of the test is that the residual is homoscedastic. Therefore if the is greater than the critical F value at the chosen significance level, we can reject the null and say the residual is heteroscedastic

28 Detecting Heteroscedasticity 2. Breusch-Pagan-Godfrey(BPG) BPG assumes that the error variance described as; where Z’s are some functions of non- stochastic variables. BPG is highly sensitive to normality assumption of residual term.

29 Detecting Heteroscedasticity The BPG proceeds as follows; –Step1:Run OLS and obtain residuals –Step2:Obtain –Step3: Generate series of p’s as; –Step5: Regress –Step4: Obtain ESS from previous step and calculate;

30 Detecting Heteroscedasticity The null hypothesis that the residual is homoscedastic. Therefore if our test statistic exceeds the critical value at the chosen significance level, we can reject the null hypothesis and we have sufficient evidence to say there is heteroscedasticity

31 Detecting Heteroscedasticity 3. White Test White test has no assumptions and easy to apply. Therefore it is commonly used test for the heteroscedasticity.

32 Detecting Heteroscedasticity White test proceed following steps; –Step1:Obtain residuals –Step2:Run auxiliary regression and obtain R- squared –Step3:Test the following; In the large samples

33 Detecting Heteroscedasticity The null hypothesis again claims that there is no heteroscedasticity Therefore if our test statistic exceeds the critical value at the chosen significance level, we can reject the null and have sufficient evidence to say there is heteroscedasticity

34 How to Deal with Heteroscedasticity WLS (Weighted Least Squares) White’s Heteroscedasticity consistent variances and standard errors Plausible assumption about heteroscedasticity pattern –Error variance is proportional to –Log transformation

35 END End of lecture