Heteroskedasticity

1. Heteroskedasticity Lecture 17

2. Today’s plan • How to test for it: graphs, Park and Glejser tests • What we can do if we find heteroskedasticity • How to estimate in the presence of heteroskedasticity

3. Palm Beach County revisited • How far is Palm Beach an outlier? • Can the outlier be explained by heteroskedasticity? • If so, what are the consequences? • Heteroskedasticity will affect the variance of the regression line • It will consequently affect the variance of the estimated coefficients and estimated 95 percent confidence interval for the prediction (see Lecture 10). • L17.xls provides an example of how to work through a problem like this using Excel

4. Palm Beach County revisited (2) • Palm Beach is a good example to use since there are scale effects in the data • The voting pattern shows that the voting behavior and number of registered voters are related to the population in each county • As the county gets larger, voting patterns may diverge from what would be assumed given the number of registered voters • Note from the graph: as we move away from the origin, the difference between registered Reform voters and Reform votes cast increases • We’ll hypothesize that this will have an affect on heteroskedasticity

5. Notation • Heteroskedasticity is observed as cross-section variability in the data • data across units at point in time • In our notation, heteroskedasticity is: E(ei2)  2 • We can also write: E(ei2) = i2 • This means that we expect variable variance: the variance changes with each unit of observation

6. Consequences When heteroskedasticity is present: 1) OLS estimator is still linear 2) OLS estimator is still unbiased 3) OLS estimator is not efficient - the minimum variance property no longer holds 4) Estimates of the variances are biased 5) is not an unbiased estimator of sYX2 6) We can’t trust the confidence intervals or hypothesis tests (t-tests & F-tests): we may draw the wrong conclusions

7. Consequences (2) • When BLUE holds and there is homoskedasticity, the first-order condition gives: • With heteroskedasticity, we have: • If we substitute the equation for ci to both equations, we find: where

8. Cases • With homoskedasticity: around each point, the variance around the regression line is constant • With heteroskedasticity: around each point, the variance around the regression line varies with each value of the independent variable (with each i)

9. Detecting heteroskedasticity • There are three ways of detecting heteroskedastiticy: 1) Graphically 2) Park Test 3) Glejser Test

10. Graphical detection • Graph the errors (or error squared) against the independent variable(s). Note: you can use either e or e2 on the y-axis. • With homoskedasticity we have E(ei, X) = 0 : • The errors are independent of the independent variables • With heteroskedasticity we can get a variety of patterns • The errors show a systematic relationship with the independent variables

11. Graphical detection (2) • Using the Palm Beach example (L17.xls), the estimated regression equation was: • The errors of this equation, can be graphed against the number of registered Reform party voters, (the independent variable) • Graph shows that the errors increasing with the number of registered reform voters • While the graphs may be convincing, we also want to use a test to confirm this. We have two:

12. Park Test • Procedure: 1) Run regression Yi = a + bXi + ei despite the heteroskedasticity problem (it can also be multivariate) 2) Obtain residuals (ei), square them (ei2), and take their logs (ln ei2) 3) Run a spurious regression: 4) Do a hypothesis test on with H0: g1 = 0 5) Look at the results of the hypothesis test: • reject the null: you have heteroskedasticity • fail to reject the null: homoskedasticity, or which is a constant

13. Glejser Test • When we use the Glejser, we’re looking for a scaling effect • The procedure: 1) Run the regression (it can also be multivariate) 2) Collect ei terms 3) Take the absolute value of the errors 4) Regress |ei| against independent variable(s) • you can run different kinds of regressions:

14. Glejser Test (2) 4) [continued] • If heteroskedasticity takes one of these forms, this will suggest an appropriate transformation of the model • The null hypothesis is still H0: g1 = 0 since we’re testing for a relationship between the errors and the independent variables • We reach the same conclusions as in the Park Test

15. A cautionary note • The errors in the Park Test (vi) and the Glejser Test (ui) might also be heteroskedastic. • If this is the case, we cannot trust the hypothesis test H0: g1 = 0 or the t-test • If we find heteroskedastic disturbances in the data, what can we do? • Estimate the model Yi = a + bXi + ei using weighted least squares • We’ll look at two examples of weighted least squares: one where we know the true variance, and one where we don’t

16. Correction with known i2 • Given that the true variance is known and our model is: Yi = a + bXi + ei • Consider the following transformation of the model: • In the transformed model, let • So the expected value of the error squared is:

17. Correction with known i2 (2) • Given that there is heteroskedasticity, E(ei2) = i2 • thus: • In this simplistic example, we re-weighted model by the constant i • What this example shows: when the variance is known, we must transform our model to obtain a homoskedastic error term.

18. Correction with unknown i2 • Given an unknown variance, we need to state the ad-hoc but plausible assumptions with our variance i2(how the errors vary with the independent variable) • For example: we can assert that E(ei2) = 2Xi • Remember: Glejser Test allows us to choose a relationship between the errors and the independent variable

19. Correction with unknown i2 (2) • In this example you would transform the estimating equation by dividing through by to get: • Letting: • The expected value of this error squared is:

20. Correction with unknown i2 (3) • Recalling an earlier assumption, we find: • When we don’t know the true variance we re-scale the estimating equation by the independent variable

21. Returning to Palm Beach • On L17.xls we have presidential election data by county in Florida • To get a correct estimating equation, we can run a regression without Palm Beach if we think it’s an outlier. • Then we can see if we can obtain a prediction for the number of reform votes cast in Palm Beach • We can perform a Glejser Test for the regression excluding Palm Beach • We run a regression of the absolute value of the errors (|ei|)against registered Reform voters (Xi)

22. Returning to Palm Beach (2) • The t-test rejects the null • this indicates the presence of heteroskedasticity • We can re-scale the model in different ways or introduce a new independent variable (such as the total number of registered voters by county) • Keep transforming the model and running the Glejser Test • When we fail to reject the null: there is no longer heteroskedasticity in the model

23. Robust estimation • Heteroskedastic tests not used any more. Most software reports robust standard errors. Note that this is also the approach of the text book. • Have looked at tests for heteroskedasticity to get you used to weighted least squares. Important for the topics to come. • Robust standard errors report approximations to the estimation of the variance for the coefficient when there is a non-constant variance. It only holds for large samples. • Know that for a homoskedastic error term Var(ui|Xi) = s2: Var(b) = s2/Sxi2

24. Robust estimation (2) • Using analogous arguments, we can state that for the heteroskedastic case: Var(ui|Xi) = si2: Var(b) = si2 Sxi2 /(Sxi2)2 • This can be approximated (in the bi-variate model case) by: Var(b) = Sxi2ui2 /(Sxi2)2 • See L17_robust.xls and hetero.pdf to compare the results from calculating the robust standard error on the spreadsheet using EXCEL and the results from STATA for robust estimation.

25. Summary • Even with re-weighted equations, we might still have heteroskedastic errors • so we have to rerun the Glejser Test until we cannot reject the null • If we cannot reject the null, we may have to rethink our model transformation • if we suspect a scale effect, we may want to introduce new scaling variables • Variables from the re-scaled equation are comparable with the coefficients from the original model