# White test

In statistics, the White test is a statistical test that establishes whether the residual variance of a variable in a regression model is constant: that is for homoscedasticity.

This test, and an estimator for heteroscedasticity-consistent standard errors, were proposed by Halbert White in 1980.[1] These methods have become extremely widely used, making this paper one of the most cited articles in economics.[2]

In cases where the White test statistics is statistically significant, heteroscedasticity may not necessarily be the cause, but specification errors. In other words, “The White test can be a test of heteroscedasticity or specification error or both." If no cross product terms are introduced in the White test procedure, then this is a pure test of pure heteroscedasticity. If cross product are introduced in model, then it is a test of both heteroscedasticity and specification bias.[3]

## Testing constant variance

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. One then inspects the R2. The Lagrange multiplier (LM) test statistic is the product of the R2 value and sample size:

$\ LM = n \cdot R^2 .$

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 homoscedastic (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 homoscedasticity.

An alternative to the White test is the Breusch–Pagan test.

If homoscedasticity is rejected one can use heteroscedasticity-consistent standard errors.