Heteroscedasticity - Definition

Definition

Suppose there is a sequence of random variables {Y_t}_t=1n and a sequence of vectors of random variables, {X_t}_t=1n. In dealing with conditional expectations of Y_t given X_t, the sequence {Y_t}_t=1n is said to be heteroscedastic if the conditional variance of Y_t given X_t, changes with t. Some authors refer to this as conditional heteroscedasticity to emphasize the fact that it is the sequence of conditional variances that changes and not the unconditional variance. In fact it is possible to observe conditional heteroscedasticity even when dealing with a sequence of unconditional homoscedastic random variables, however, the opposite does not hold. If the variance changes only because of changes in value of X and not because of a dependence on the index t, the changing variance might be described using a scedastic function. (This last sentence is not clear: if "variance" means "variance of Y_t conditional on X_t", it says that the variance is a function of X_t, and not of t; but are then the variables Y_t heteroscedastic?)

When using some statistical techniques, such as ordinary least squares (OLS), a number of assumptions are typically made. One of these is that the error term has a constant variance. This might not be true even if the error term is assumed to be drawn from identical distributions.

For example, the error term could vary or increase with each observation, something that is often the case with cross-sectional or time series measurements. Heteroscedasticity is often studied as part of econometrics, which frequently deals with data exhibiting it. White's influential paper used "heteroskedasticity" instead of "heteroscedasticity" whereas the latter has been used in later works.