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statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the White test is a
statistical test A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. Hypothesis testing allows us to make probabilistic statements about population parameters. ...
that establishes whether the
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
of the errors in a
regression model In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...
is constant: that is for homoskedasticity. This test, and an estimator for
heteroscedasticity-consistent standard errors The topic of heteroskedasticity-consistent (HC) standard errors arises in statistics and econometrics in the context of linear regression and time series analysis. These are also known as heteroskedasticity-robust standard errors (or simply robust ...
, were proposed by
Halbert White Halbert Lynn White Jr. (November 19, 1950 – March 31, 2012) was the Chancellor’s Associates Distinguished Professor of Economics at the University of California, San Diego, and a Fellow of the Econometric Society and the American Academy of ...
in 1980. These methods have become extremely widely used, making this paper one of the most cited articles in economics. In cases where the White test statistic is statistically significant, heteroskedasticity may not necessarily be the cause; instead the problem could be a specification error. In other words, the White test can be a test of heteroskedasticity or specification error or both. If no cross product terms are introduced in the White test procedure, then this is a test of pure heteroskedasticity. If cross products are introduced in the model, then it is a test of both heteroskedasticity and specification bias.


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
regressor Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...
s 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 = n R^2 . This follows a
chi-squared distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squa ...
, 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 In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
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-squared distribution) counts against the hypothesis of homoskedasticity. An alternative to the White test is the
Breusch–Pagan test In statistics, the Breusch–Pagan test, developed in 1979 by Trevor Breusch and Adrian Pagan, is used to test for heteroskedasticity in a linear regression model. It was independently suggested with some extension by R. Dennis Cook and Sanf ...
, 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. If homoskedasticity is rejected one can use
heteroskedasticity-consistent standard errors The topic of heteroskedasticity-consistent (HC) standard errors arises in statistics and econometrics in the context of linear regression and time series analysis. These are also known as heteroskedasticity-robust standard errors (or simply robus ...
.


Software implementations

* In R, White's Test can be implemented using the white_lm function of the skedastic package. * In
Python Python may refer to: Snakes * Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia ** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia * Python (mythology), a mythical serpent Computing * Python (pro ...
, White's Test can be implemented using the het_white function of the statsmodels.stats.diagnostic.het_white *In Stata, the test can be implemented using the estat imtest, white function.


See also

*
Heteroskedasticity In statistics, a sequence (or a vector) of random variables is homoscedastic () if all its random variables have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity. The ...
*
Breusch–Pagan test In statistics, the Breusch–Pagan test, developed in 1979 by Trevor Breusch and Adrian Pagan, is used to test for heteroskedasticity in a linear regression model. It was independently suggested with some extension by R. Dennis Cook and Sanf ...


References


Further reading

* * * {{cite book , last=Wooldridge , first=Jeffrey M. , author-link=Jeffrey Wooldridge , title=Introductory Econometrics: A Modern Approach , publisher=South-Western , edition=Fifth , year=2013 , isbn=978-1-111-53439-4 , pages=269–70 Statistical tests Regression diagnostics