Heteroskedasticity

In statistics, a sequence or a vector of random variables is heteroskedastic if the random variables in the sequence or vector may have different variances. The complement is called homoskedasticity. (In America, it is usually spelled homoscedastic. It is an exception to the rule that American spellings are usually more faithful to the etymologies than British spellings.)

When using a variety of techniques in statistics, 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 will be true if the observations of the error term are assumed to be drawn from identical distributions.

Heteroskedasticity (aka skewedness, opposite: homoskedasticity) is a violation of this assumption. 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. Heteroskedasticity is often studied as part of econometrics, which frequently deals with data exhibiting it. It comes in two forms, pure and impure. Because there are so many types of each, most textbooks limit themselves to dealing with heteroskedasticity in general, or one or two examples.

Consequences

The consequences are similar to serial correlation.

1. When OLS to is applied heteroskedastic models it is no longer a minimum variance estimator. The variances and standard errors are understated.
2. The variance of the sample betas increases.

Examples

Heteroskedasticity often occurs when there is a large difference between the size of observations.

1. [1] cites a cross sectional example: Comparing states with widely differing populations, such as Rhode Island and California.
2. Imagine you are watching a rocket take off nearby and measuring the distance it has travelled once each second. In the first couple of seconds your measurements may be accurate to the nearest centimeter, say. However, 5 minutes later as the rocket recedes into space, the accuracy of your measurements may only be good to 100m, because of the increased distance, atmospheric distortion and a variety of other factors. The data you collect would exhibit heteroskedasticity.

References

There are a great many references. Most statistics text books will include at least some material on heteroskedasticity.

1. Studenmund, A.H. Using Econometrics 2nd Ed. ISBN 0-673-52125-7. Devotes a chapter to heteroskedasticity.