Heteroscedasticity. Jamie Monogan. Intermediate Political Methodology. University of Georgia. Jamie Monogan (UGA) Heteroscedasticity POLS / 11

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1 Heteroscedasticity Jamie Monogan University of Georgia Intermediate Political Methodology Jamie Monogan (UGA) Heteroscedasticity POLS / 11

2 Objectives By the end of this meeting, participants should be able to: Define heteroscedasticity and describe the problems it produces. Identify when heteroscedasticity is present in real data analysis Use robust standard errors or feasible GLS to correct for heteroscedasticity. Jamie Monogan (UGA) Heteroscedasticity POLS / 11

3 What is Heteroscedasticity? AKA Heteroskedasticity The weak set of Gauss-Markov assumptions assumes the variance of the disturbances is homoscedastic: Var(u i ) = σ 2. A violation of this assumption means that the variance is not constant for all disturbances. We call this heteroscedastic error variance: Var(u i ) = σ 2 i. Why might this occur? Perhaps at higher values of outcome Y there is greater unexplained variation. Perhaps we have less certainty about individuals having certain input values X. The consequence: OLS estimates ˆβ are still unbiased. They are no longer efficient, however. Jamie Monogan (UGA) Heteroscedasticity POLS / 11

4 The Problem of Heteroscedasticity: A Visualization Homoscedasticity Heteroscedasticity Source: Gujarati & Porter 2009, Figures 11.1 & 11.2 (p. 366) Jamie Monogan (UGA) Heteroscedasticity POLS / 11

5 Exchange Rates: Marks to Pounds by Day A Real Example of Heteroscedasticity Citations: Greene, William H Econometric Analysis. 5th ed. Upper Saddle River, NJ: Prentice Hall. (p.239) Bollerslev, T Generalized Autoregressive Conditional Heteroscedasticity. Journal of Econometrics 31: Jamie Monogan (UGA) Heteroscedasticity POLS / 11

6 Modeling Swiss Government Employment A Real Example of Heteroscedasticity FWLS OLS Input Estimate SE Estimate SE GDP PR Intercept Notes: N=26, Breusch-Pagan test on OLS residuals 5.68 (p =.02). Citations: Leeman, Lucas & Jeff Gill Weighted Least Squares. In International Encyclopedia of Political Science. Bertrand Badie, Dirk Berg-Schlosser & Leonardo Morlino, eds. Thousand Oaks: Sage. Vatter, A., M. Freitag, C. Müller & M. Bühlmann Political, Social, and Economic Data of the Swiss Cantons Bern: University of Bern. Jamie Monogan (UGA) Heteroscedasticity POLS / 11

7 Identifying Heteroscedasticity Visual Diagnosis Plotting residuals (û) against: fitted values (Ŷ ). various predictors (X ). Plotting squared residuals. Hypothesis Tests Park test Glejser test Spearman s rank correlation test Goldfeld-Quandt test Breusch-Pagan(-Godfrey) test Jamie Monogan (UGA) Heteroscedasticity POLS / 11

8 Breusch-Pagan-Godfrey Test H 0 : homoscedasticity, H A : heteroscedasticity. Test with a 5 step process. 1 Estimate your regression model with OLS. Save the residuals (û 1, û 2,..., û n ). 2 Calculate the ML estimate of the error variance of regression: σ 2 = û 2 i /n. 3 Create a new variable by dividing the squared residuals by the error variance: p i = û 2 i / σ2. 4 Regress p i as a function of all variables, Z, that may account for heteroscedasticity. (Note: It may be that {Z} = {X }.) p i = α 1 + α 2 Z 2i + + α m Z mi + ν i 5 Calculate the explained sum of squares from the step four regression. Then compute the test statistic for testing the hypothesis: Θ = 1 2 ESS χ2 m 1 Jamie Monogan (UGA) Heteroscedasticity POLS / 11

9 Responding to Heteroscedasticity Huber-White standard errors. (Note: Huber 1976 & White 1980.) OLS estimates of parameters are unbiased. OLS estimates of the variance-covariance matrix of coefficients are inconsistent under heteroscedasticity. The Huber-White sandwich estimator is, however, consistent. (AKA robust standard errors. ) Only the standard errors change in this approach. OLS estimates are used, which are unbiased but inefficient. Weighted Least Squares A special case of Generalized Least Squares with no autocorrelation. GLS estimator: β = (X Ω 1 X) 1 X Ω 1 y We don t know Ω, though, so we turn to feasible Generalized Least Squares (fgls) and substitute Ω. Jamie Monogan (UGA) Heteroscedasticity POLS / 11

10 Feasible Weighted Least Squares Model the squared residuals with whatever you think explains variance: û 2 = ZΓ + ν. Again, it may be that {Z} = {X }. Save the predicted values. Call them w i. Form Ω as the diagonal n n matrix with w i as the Ω ii element. Inverting this is easy. Just take the reciprocal of every element: 1 w Ω w = w n Jamie Monogan (UGA) Heteroscedasticity POLS / 11

11 For Next Time Read Gujarati & Porter Chapter 12. Study the 2010 election data from Monogan s Dataverse. Describe a population regression function in which Republican share of the two party vote (creptwo) is a function of multiple predictors. Be ready to defend your choices. Estimate the regression model implied by your population regression model. Report these results in a neatly-formatted table. Evaluate whether there is heteroscedasticity in the residuals. Use a visual diagnostic and a test statistic. If these are at odds, which side do you fall down on? Whatever your conclusion, show me that you can conduct a remedial measure for heteroscedasticity. Report the results with Huber-White robust standard errors, or the results from Weighted Least Squares. Additional tips on regression diagnostics: Political Analysis Using R. Papers: Have you obtained your data? Computed descriptive statistics? Estimated your model using OLS? Jamie Monogan (UGA) Heteroscedasticity POLS / 11