White test

To test for constant variance one undertakes an auxiliary regression analysis: this regresses the squared residuals from the original regression model onto a set of regressors that contain the original regressors along with their squares and cross-products.[3] One then inspects the R². The Lagrange multiplier (LM) test statistic is the product of the R² value and sample size:

${\displaystyle {\text{LM}}=nR^{2}.}$

This follows a chi-squared distribution, with degrees of freedom equal to P − 1, where P is the number of estimated parameters (in the auxiliary regression).

The logic of the test is as follows. First, the squared residuals from the original model serve as a proxy for the variance of the error term at each observation. (The error term is assumed to have a mean of zero, and the variance of a zero-mean random variable is just the expectation of its square.) The independent variables in the auxiliary regression account for the possibility that the error variance depends on the values of the original regressors in some way (linear or quadratic). If the error term in the original model is in fact homoskedastic (has a constant variance) then the coefficients in the auxiliary regression (besides the constant) should be statistically indistinguishable from zero and the R² should be “small". Conversely, a “large" R² (scaled by the sample size so that it follows the chi-squared distribution) counts against the hypothesis of homoskedasticity.

An alternative to the White test is the Breusch–Pagan test, where the Breusch-Pagan test is designed to detect only linear forms of heteroskedasticity. Under certain conditions and a modification of one of the tests, they can be found to be algebraically equivalent.[4]

If homoskedasticity is rejected one can use heteroskedasticity-consistent standard errors.

• In R, White's Test can be implemented using the white function of the skedastic package.[5]

• In Python, White's Test can be implemented using the het_white function of the statsmodels.stats.diagnostic.het_white [6]
• In Stata, the test can be implemented using the estat imtest, white function.[7]

• Heteroskedasticity
• Breusch–Pagan test
• Glejser test
• Goldfeld–Quandt test
• Park test

• Gujarati, Damodar N.; Porter, Dawn C. (2009). Basic Econometrics (Fifth ed.). New York: McGraw-Hill Irwin. pp. 386–88. ISBN 978-0-07-337577-9.
• Kmenta, Jan (1986). Elements of Econometrics (Second ed.). New York: Macmillan. pp. 292–298. ISBN 978-0-02-365070-3.
• Wooldridge, Jeffrey M. (2013). Introductory Econometrics: A Modern Approach (Fifth ed.). South-Western. pp. 269–70. ISBN 978-1-111-53439-4.