Questions tagged [white-test]

White (1980) proposed a test for heteroskedasticity and model specification. The null hypothesis is that errors are (1) homoskedastic, (2) independent of regressors and (3) the model is well specified (in the sense that there are no omitted variables in the form of squares or cross-products of the original regressors). The alternative hypothesis is that either (1), (2) or (3) is violated.

The test uses an auxiliary regression where squared residuals from the original regression model are regressed onto a set of regressors that contain the original regressors along with their squares and cross-products. One then inspects the $R^2$. The Lagrange multiplier (LM) test statistic is the product of the $R^2$ value and sample size: $$ \text{LM}=nR^2. $$ Under the null hypothesis the test statistic follows a $\chi^2(p−1)$ distribution, 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 $R^2$ should be “small". Conversely, a “large" $R^2$ (scaled by the sample size so that it follows the $\chi^2$ distribution) counts against the hypothesis of homoskedasticity.

An alternative to the White test is the Breusch–Pagan test. Under certain conditions and a modification of one of the tests, they can be found to be algebraically equivalent.

The text above is largely based on Wikipedia's article "White test"

Reference: White, Halbert. "A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity." Econometrica: Journal of the Econometric Society (1980): 817-838.