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# Christopher Dougherty EC220 - Introduction to econometrics (chapter 7) Slideshow: White test for heteroscedasticity Original citation: Dougherty, C. (2012)

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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/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://creativecommons.org/licenses/by-sa/3.0/ http://learningresources.lse.ac.uk/

WHITE TEST FOR HETEROSCEDASTICITY 1 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.. reg MANU GDP Source | SS df MS Number of obs = 28 -------------+------------------------------ F( 1, 26) = 210.73 Model | 1.1600e+11 1 1.1600e+11 Prob > F = 0.0000 Residual | 1.4312e+10 26 550462775 R-squared = 0.8902 -------------+------------------------------ Adj R-squared = 0.8859 Total | 1.3031e+11 27 4.8264e+09 Root MSE = 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 ------------------------------------------------------------------------------

WHITE TEST FOR HETEROSCEDASTICITY 2 Since the variance of the disturbance term in observation i is unobservable, the squared residual for that observation is used as a proxy.. reg MANU GDP Source | SS df MS Number of obs = 28 -------------+------------------------------ F( 1, 26) = 210.73 Model | 1.1600e+11 1 1.1600e+11 Prob > F = 0.0000 Residual | 1.4312e+10 26 550462775 R-squared = 0.8902 -------------+------------------------------ Adj R-squared = 0.8859 Total | 1.3031e+11 27 4.8264e+09 Root MSE = 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 ------------------------------------------------------------------------------

WHITE TEST FOR HETEROSCEDASTICITY 3 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.. reg MANU GDP Source | SS df MS Number of obs = 28 -------------+------------------------------ F( 1, 26) = 210.73 Model | 1.1600e+11 1 1.1600e+11 Prob > F = 0.0000 Residual | 1.4312e+10 26 550462775 R-squared = 0.8902 -------------+------------------------------ Adj R-squared = 0.8859 Total | 1.3031e+11 27 4.8264e+09 Root MSE = 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

WHITE TEST FOR HETEROSCEDASTICITY 4 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.). gen GDPSQ = GDP*GDP. reg EMANUSQ GDP GDPSQ Source | SS df MS Number of obs = 28 -------------+------------------------------ F( 2, 25) = 3.35 Model | 1.3183e+19 2 6.5913e+18 Prob > F = 0.0514 Residual | 4.9179e+19 25 1.9671e+18 R-squared = 0.2114 -------------+------------------------------ Adj R-squared = 0.1483 Total | 6.2361e+19 27 2.3097e+18 Root MSE = 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.

WHITE TEST FOR HETEROSCEDASTICITY 5 In the present case we regress EMANUSQ on GDP and its square (and a constant).. gen GDPSQ = GDP*GDP. reg EMANUSQ GDP GDPSQ Source | SS df MS Number of obs = 28 -------------+------------------------------ F( 2, 25) = 3.35 Model | 1.3183e+19 2 6.5913e+18 Prob > F = 0.0514 Residual | 4.9179e+19 25 1.9671e+18 R-squared = 0.2114 -------------+------------------------------ Adj R-squared = 0.1483 Total | 6.2361e+19 27 2.3097e+18 Root MSE = 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.

WHITE TEST FOR HETEROSCEDASTICITY 6. gen GDPSQ = GDP*GDP. reg EMANUSQ GDP GDPSQ Source | SS df MS Number of obs = 28 -------------+------------------------------ F( 2, 25) = 3.35 Model | 1.3183e+19 2 6.5913e+18 Prob > F = 0.0514 Residual | 4.9179e+19 25 1.9671e+18 R-squared = 0.2114 -------------+------------------------------ Adj R-squared = 0.1483 Total | 6.2361e+19 27 2.3097e+18 Root MSE = 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 ------------------------------------------------------------------------------ The test statistic is nR 2, using R 2 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. Test statistic: nR 2, using R 2 from this regression. Under H 0, chi-squared statistic with degrees of freedom equal to the number of regressors, including the constant, minus one, in large samples.

WHITE TEST FOR HETEROSCEDASTICITY 7 R 2 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.. gen GDPSQ = GDP*GDP. reg EMANUSQ GDP GDPSQ Source | SS df MS Number of obs = 28 -------------+------------------------------ F( 2, 25) = 3.35 Model | 1.3183e+19 2 6.5913e+18 Prob > F = 0.0514 Residual | 4.9179e+19 25 1.9671e+18 R-squared = 0.2114 -------------+------------------------------ Adj R-squared = 0.1483 Total | 6.2361e+19 27 2.3097e+18 Root MSE = 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 ------------------------------------------------------------------------------

WHITE TEST FOR HETEROSCEDASTICITY 8 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.. gen GDPSQ = GDP*GDP. reg EMANUSQ GDP GDPSQ Source | SS df MS Number of obs = 28 -------------+------------------------------ F( 2, 25) = 3.35 Model | 1.3183e+19 2 6.5913e+18 Prob > F = 0.0514 Residual | 4.9179e+19 25 1.9671e+18 R-squared = 0.2114 -------------+------------------------------ Adj R-squared = 0.1483 Total | 6.2361e+19 27 2.3097e+18 Root MSE = 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 ------------------------------------------------------------------------------

WHITE TEST FOR HETEROSCEDASTICITY 9 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.. gen GDPSQ = GDP*GDP. reg EMANUSQ GDP GDPSQ Source | SS df MS Number of obs = 28 -------------+------------------------------ F( 2, 25) = 3.35 Model | 1.3183e+19 2 6.5913e+18 Prob > F = 0.0514 Residual | 4.9179e+19 25 1.9671e+18 R-squared = 0.2114 -------------+------------------------------ Adj R-squared = 0.1483 Total | 6.2361e+19 27 2.3097e+18 Root MSE = 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 ------------------------------------------------------------------------------

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/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 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/lsewww.londoninternational.ac.uk/lse. 11.07.25

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