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Christopher Dougherty EC220 - Introduction to econometrics (chapter 7) Slideshow: White test for heteroscedasticity Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 7). [Teaching Resource] © 2012 The Author This version available at: http://learningresources.lse.ac.uk/133/ Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms. http://creativecommons.org/licenses/by-sa/3.0/ http://learningresources.lse.ac.uk/ WHITE TEST FOR HETEROSCEDASTICITY . reg MANU GDP Source | SS df MS -------------+-----------------------------Model | 1.1600e+11 1 1.1600e+11 Residual | 1.4312e+10 26 550462775 -------------+-----------------------------Total | 1.3031e+11 27 4.8264e+09 Number of obs F( 1, 26) Prob > F R-squared Adj R-squared Root MSE = = = = = = 28 210.73 0.0000 0.8902 0.8859 23462 -----------------------------------------------------------------------------MANU | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------GDP | .193693 .0133428 14.52 0.000 .1662665 .2211195 _cons | 603.9453 5699.677 0.11 0.916 -11111.91 12319.8 ------------------------------------------------------------------------------ The White test for heteroscedasticity looks for evidence of an association between the variance of the disturbance term and the regressors without assuming any specific relationship. 1 WHITE TEST FOR HETEROSCEDASTICITY . reg MANU GDP Source | SS df MS -------------+-----------------------------Model | 1.1600e+11 1 1.1600e+11 Residual | 1.4312e+10 26 550462775 -------------+-----------------------------Total | 1.3031e+11 27 4.8264e+09 Number of obs F( 1, 26) Prob > F R-squared Adj R-squared Root MSE = = = = = = 28 210.73 0.0000 0.8902 0.8859 23462 -----------------------------------------------------------------------------MANU | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------GDP | .193693 .0133428 14.52 0.000 .1662665 .2211195 _cons | 603.9453 5699.677 0.11 0.916 -11111.91 12319.8 ------------------------------------------------------------------------------ Since the variance of the disturbance term in observation i is unobservable, the squared residual for that observation is used as a proxy. 2 WHITE TEST FOR HETEROSCEDASTICITY . reg MANU GDP Source | SS df MS -------------+-----------------------------Model | 1.1600e+11 1 1.1600e+11 Residual | 1.4312e+10 26 550462775 -------------+-----------------------------Total | 1.3031e+11 27 4.8264e+09 Number of obs F( 1, 26) Prob > F R-squared Adj R-squared Root MSE = = = = = = 28 210.73 0.0000 0.8902 0.8859 23462 -----------------------------------------------------------------------------MANU | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------GDP | .193693 .0133428 14.52 0.000 .1662665 .2211195 _cons | 603.9453 5699.677 0.11 0.916 -11111.91 12319.8 -----------------------------------------------------------------------------. predict EMANU, resid . gen EMANUSQ = EMANU*EMANU We will perform the test using the manufacturing and GDP data used to illustrate the Goldfeld–Quandt test. We have regressed MANU on GDP and have saved the residuals as EMANU. We define EMANUSQ to be the squared residual. 3 WHITE TEST FOR HETEROSCEDASTICITY . gen GDPSQ = GDP*GDP . reg EMANUSQ GDP GDPSQ Source | SS df MS -------------+-----------------------------Model | 1.3183e+19 2 6.5913e+18 Residual | 4.9179e+19 25 1.9671e+18 -------------+-----------------------------Total | 6.2361e+19 27 2.3097e+18 Number of obs F( 2, 25) Prob > F R-squared Adj R-squared Root MSE = 28 = 3.35 = 0.0514 = 0.2114 = 0.1483 = 1.4e+09 -----------------------------------------------------------------------------EMANUSQ | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------GDP | 6271.896 2758.253 2.27 0.032 591.1687 11952.62 GDPSQ | -.0041155 .0022626 -1.82 0.081 -.0087754 .0005444 _cons | -4.21e+08 4.51e+08 -0.93 0.359 -1.35e+09 5.08e+08 ------------------------------------------------------------------------------ Test regression: regress squared residuals on the explanatory variables in the model, their squares, and their cross-products, omitting any duplicative variables. The test consists of regressing the squared residuals on the explanatory variables in the model, their squares, and their cross-products, omitting any duplicative variables. (For example, the square of a dummy variable would be duplicative.) 4 WHITE TEST FOR HETEROSCEDASTICITY . gen GDPSQ = GDP*GDP . reg EMANUSQ GDP GDPSQ Source | SS df MS -------------+-----------------------------Model | 1.3183e+19 2 6.5913e+18 Residual | 4.9179e+19 25 1.9671e+18 -------------+-----------------------------Total | 6.2361e+19 27 2.3097e+18 Number of obs F( 2, 25) Prob > F R-squared Adj R-squared Root MSE = 28 = 3.35 = 0.0514 = 0.2114 = 0.1483 = 1.4e+09 -----------------------------------------------------------------------------EMANUSQ | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------GDP | 6271.896 2758.253 2.27 0.032 591.1687 11952.62 GDPSQ | -.0041155 .0022626 -1.82 0.081 -.0087754 .0005444 _cons | -4.21e+08 4.51e+08 -0.93 0.359 -1.35e+09 5.08e+08 ------------------------------------------------------------------------------ Test regression: regress squared residuals on the explanatory variables in the model, their squares, and their cross-products, omitting any duplicative variables. In the present case we regress EMANUSQ on GDP and its square (and a constant). 5 WHITE TEST FOR HETEROSCEDASTICITY . gen GDPSQ = GDP*GDP . reg EMANUSQ GDP GDPSQ Source | SS df MS -------------+-----------------------------Model | 1.3183e+19 2 6.5913e+18 Residual | 4.9179e+19 25 1.9671e+18 -------------+-----------------------------Total | 6.2361e+19 27 2.3097e+18 Number of obs F( 2, 25) Prob > F R-squared Adj R-squared Root MSE = 28 = 3.35 = 0.0514 = 0.2114 = 0.1483 = 1.4e+09 -----------------------------------------------------------------------------EMANUSQ | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------GDP | 6271.896 2758.253 2.27 0.032 591.1687 11952.62 GDPSQ | -.0041155 .0022626 -1.82 0.081 -.0087754 .0005444 _cons | -4.21e+08 4.51e+08 -0.93 0.359 -1.35e+09 5.08e+08 ------------------------------------------------------------------------------ Test statistic: nR2, using R2 from this regression. Under H0, chi-squared statistic with degrees of freedom equal to the number of regressors, including the constant, minus one, in large samples. The test statistic is nR2, using R2 from this regression. Under the null hypothesis of no association, it is distributed as a chi-squared statistic with degrees of freedom equal to the number of regressors, including the constant, minus one, in large samples. 6 WHITE TEST FOR HETEROSCEDASTICITY . gen GDPSQ = GDP*GDP . reg EMANUSQ GDP GDPSQ Source | SS df MS -------------+-----------------------------Model | 1.3183e+19 2 6.5913e+18 Residual | 4.9179e+19 25 1.9671e+18 -------------+-----------------------------Total | 6.2361e+19 27 2.3097e+18 Number of obs F( 2, 25) Prob > F R-squared Adj R-squared Root MSE = 28 = 3.35 = 0.0514 = 0.2114 = 0.1483 = 1.4e+09 -----------------------------------------------------------------------------EMANUSQ | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------GDP | 6271.896 2758.253 2.27 0.032 591.1687 11952.62 GDPSQ | -.0041155 .0022626 -1.82 0.081 -.0087754 .0005444 _cons | -4.21e+08 4.51e+08 -0.93 0.359 -1.35e+09 5.08e+08 ------------------------------------------------------------------------------ nR 2 28 0.2114 5.92 52% ( 2) 5.99 R2 is 0.2114 and n is 28. The test statistic is therefore 5.92. The critical value of chi-squared with two degrees of freedom is 5.99 at the 5 percent level and so the null hypothesis of homoscedasticity is not rejected. 7 WHITE TEST FOR HETEROSCEDASTICITY . gen GDPSQ = GDP*GDP . reg EMANUSQ GDP GDPSQ Source | SS df MS -------------+-----------------------------Model | 1.3183e+19 2 6.5913e+18 Residual | 4.9179e+19 25 1.9671e+18 -------------+-----------------------------Total | 6.2361e+19 27 2.3097e+18 Number of obs F( 2, 25) Prob > F R-squared Adj R-squared Root MSE = 28 = 3.35 = 0.0514 = 0.2114 = 0.1483 = 1.4e+09 -----------------------------------------------------------------------------EMANUSQ | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------GDP | 6271.896 2758.253 2.27 0.032 591.1687 11952.62 GDPSQ | -.0041155 .0022626 -1.82 0.081 -.0087754 .0005444 _cons | -4.21e+08 4.51e+08 -0.93 0.359 -1.35e+09 5.08e+08 ------------------------------------------------------------------------------ nR 2 28 0.2114 5.92 52% ( 2) 5.99 Why has the White test failed to detect heteroscedasticity when the Goldfeld–Quandt test concluded that it was present at a high level of significance? One reason is that it is a large-sample test, and the sample is actually quite small. 8 WHITE TEST FOR HETEROSCEDASTICITY . gen GDPSQ = GDP*GDP . reg EMANUSQ GDP GDPSQ Source | SS df MS -------------+-----------------------------Model | 1.3183e+19 2 6.5913e+18 Residual | 4.9179e+19 25 1.9671e+18 -------------+-----------------------------Total | 6.2361e+19 27 2.3097e+18 Number of obs F( 2, 25) Prob > F R-squared Adj R-squared Root MSE = 28 = 3.35 = 0.0514 = 0.2114 = 0.1483 = 1.4e+09 -----------------------------------------------------------------------------EMANUSQ | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------GDP | 6271.896 2758.253 2.27 0.032 591.1687 11952.62 GDPSQ | -.0041155 .0022626 -1.82 0.081 -.0087754 .0005444 _cons | -4.21e+08 4.51e+08 -0.93 0.359 -1.35e+09 5.08e+08 ------------------------------------------------------------------------------ nR 2 28 0.2114 5.92 52% ( 2) 5.99 A second is that the White test tends to have low power — a price that one has to pay for its generality. These problems can be exacerbated by a loss of degrees of freedom if there are many explanatory variables in the original model. 9 Copyright Christopher Dougherty 2011. These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 7.2 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oup.com/uk/orc/bin/9780199567089/. Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course 20 Elements of Econometrics www.londoninternational.ac.uk/lse. 11.07.25