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 = F( 1, 26) = Model | e e+11 Prob > F = Residual | e R-squared = Adj R-squared = Total | e e+09 Root MSE = MANU | Coef. Std. Err. t P>|t| [95% Conf. Interval] GDP | _cons |

2 Since the variance of the disturbance term in observation i is unobservable, the squared residual for that observation is used as a proxy. WHITE TEST FOR HETEROSCEDASTICITY. reg MANU GDP Source | SS df MS Number of obs = F( 1, 26) = Model | e e+11 Prob > F = Residual | e R-squared = Adj R-squared = Total | e e+09 Root MSE = MANU | Coef. Std. Err. t P>|t| [95% Conf. Interval] GDP | _cons |

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. WHITE TEST FOR HETEROSCEDASTICITY. reg MANU GDP Source | SS df MS Number of obs = F( 1, 26) = Model | e e+11 Prob > F = Residual | e R-squared = Adj R-squared = Total | e e+09 Root MSE = MANU | Coef. Std. Err. t P>|t| [95% Conf. Interval] GDP | _cons | predict EMANU, resid. gen EMANUSQ = EMANU*EMANU

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.) WHITE TEST FOR HETEROSCEDASTICITY. gen GDPSQ = GDP*GDP. reg EMANUSQ GDP GDPSQ Source | SS df MS Number of obs = F( 2, 25) = 3.35 Model | e e+18 Prob > F = Residual | e e+18 R-squared = Adj R-squared = Total | e e+18 Root MSE = 1.4e EMANUSQ | Coef. Std. Err. t P>|t| [95% Conf. Interval] GDP | GDPSQ | _cons | -4.21e e e e Test regression: regress squared residuals on the explanatory variables in the model, their squares, and their cross-products, omitting any duplicative variables.

5 In the present case we regress EMANUSQ on GDP and its square (and a constant). WHITE TEST FOR HETEROSCEDASTICITY Test regression: regress squared residuals on the explanatory variables in the model, their squares, and their cross-products, omitting any duplicative variables.. gen GDPSQ = GDP*GDP. reg EMANUSQ GDP GDPSQ Source | SS df MS Number of obs = F( 2, 25) = 3.35 Model | e e+18 Prob > F = Residual | e e+18 R-squared = Adj R-squared = Total | e e+18 Root MSE = 1.4e EMANUSQ | Coef. Std. Err. t P>|t| [95% Conf. Interval] GDP | GDPSQ | _cons | -4.21e e e e

6 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. WHITE TEST FOR HETEROSCEDASTICITY 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.. gen GDPSQ = GDP*GDP. reg EMANUSQ GDP GDPSQ Source | SS df MS Number of obs = F( 2, 25) = 3.35 Model | e e+18 Prob > F = Residual | e e+18 R-squared = Adj R-squared = Total | e e+18 Root MSE = 1.4e EMANUSQ | Coef. Std. Err. t P>|t| [95% Conf. Interval] GDP | GDPSQ | _cons | -4.21e e e e

7 R 2 is and n is 28. The test statistic is therefore 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. WHITE TEST FOR HETEROSCEDASTICITY. gen GDPSQ = GDP*GDP. reg EMANUSQ GDP GDPSQ Source | SS df MS Number of obs = F( 2, 25) = 3.35 Model | e e+18 Prob > F = Residual | e e+18 R-squared = Adj R-squared = Total | e e+18 Root MSE = 1.4e EMANUSQ | Coef. Std. Err. t P>|t| [95% Conf. Interval] GDP | GDPSQ | _cons | -4.21e e e e

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. WHITE TEST FOR HETEROSCEDASTICITY. gen GDPSQ = GDP*GDP. reg EMANUSQ GDP GDPSQ Source | SS df MS Number of obs = F( 2, 25) = 3.35 Model | e e+18 Prob > F = Residual | e e+18 R-squared = Adj R-squared = Total | e e+18 Root MSE = 1.4e EMANUSQ | Coef. Std. Err. t P>|t| [95% Conf. Interval] GDP | GDPSQ | _cons | -4.21e e e e

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. WHITE TEST FOR HETEROSCEDASTICITY. gen GDPSQ = GDP*GDP. reg EMANUSQ GDP GDPSQ Source | SS df MS Number of obs = F( 2, 25) = 3.35 Model | e e+18 Prob > F = Residual | e e+18 R-squared = Adj R-squared = Total | e e+18 Root MSE = 1.4e EMANUSQ | Coef. Std. Err. t P>|t| [95% Conf. Interval] GDP | GDPSQ | _cons | -4.21e e e e

Copyright Christopher Dougherty 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 Individuals studying econometrics on their own 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 or the University of London International Programmes distance learning course EC2020 Elements of Econometrics