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The complementary notion is called heteroscedasticity, also known as heterogeneity of variance. The spellings homoskedasticity and heteroskedasticity are also frequently used. “Skedasticity” comes from the Ancient Greek word “skedánnymi”, meaning “to scatter”.
Heteroscedasticity often occurs when there is a large difference among the sizes of the observations. A classic example of heteroscedasticity is that of income versus expenditure on meals. A wealthy person may eat inexpensive food sometimes and expensive food at other times. A poor person will almost always eat inexpensive food.
For any non-linear model (for instance logit and probit models), however, heteroskedasticity has more severe consequences: the maximum likelihood estimates of the parameters will be biased (in an unknown direction), as well as inconsistent (unless the likelihood function is modified to correctly take into account the precise form of ...
If, then, in a regression of on the natural logarithm of one or more of the regressors , we arrive at statistical significance for non-zero values on one or more of the ^, we reveal a connection between the residuals and the regressors. We reject the null hypothesis of homoscedasticity and conclude that heteroscedasticity is present.
Under the classical assumptions, ordinary least squares is the best linear unbiased estimator (BLUE), i.e., it is unbiased and efficient. It remains unbiased under heteroskedasticity, but efficiency is lost. Before deciding upon an estimation method, one may conduct the Breusch–Pagan test to examine the presence of heteroskedasticity.
Weighted least squares (WLS), also known as weighted linear regression, [1] [2] is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (heteroscedasticity) is incorporated into the regression.
Notice the relation between the variance and the mean, which implies, for example, heteroscedasticity in a linear model. Therefore, the goal is to find a function g {\displaystyle g} such that Y = g ( X ) {\displaystyle Y=g(X)} has a variance independent (at least approximately) of its expectation.
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.