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In statistics, a collection of random variables is heteroscedastic (often spelled heteroskedastic,^{[1]} and commonly pronounced with a hard k regardless of spelling) if there are subpopulations that have different variabilities from others. Here "variability" could be quantified by the variance or any other measure of statistical dispersion. Thus heteroscedasticity is the absence of homoscedasticity.
The possible existence of heteroscedasticity is a major concern in the application of regression analysis, including the analysis of variance, because the presence of heteroscedasticity can invalidate statistical tests of significance that assume that the modelling errors are uncorrelated and normally distributed and that their variances do not vary with the effects being modelled. Similarly, in testing for differences between subpopulations using a location test, some standard tests assume that variances within groups are equal.
Tests for the possible presence of heteroscedasticity are outlined below.
The term means "differing variance" and comes from the Greek "hetero" ('different') and "skedasis" ('dispersion').
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Suppose there is a sequence of random variables {Y_{t}}_{t=1}^{n} and a sequence of vectors of random variables, {X_{t}}_{t=1}^{n}. In dealing with conditional expectations of Y_{t} given X_{t}, the sequence {Y_{t}}_{t=1}^{n} is said to be heteroskedastic 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.
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 crosssectional or time series measurements. Heteroscedasticity is often studied as part of econometrics, which frequently deals with data exhibiting it. White's influential paper^{[2]} used "heteroskedasticity" instead of "heteroscedasticity" whereas the latter has been used in later works.^{[3]}
Heteroscedasticity does not cause ordinary least squares coefficient estimates to be biased, although it can cause ordinary least squares estimates of the variance (and, thus, standard errors) of the coefficients to be biased, possibly above or below the true or population variance. Thus, regression analysis using heteroscedastic data will still provide an unbiased estimate for the relationship between the predictor variable and the outcome, but standard errors and therefore inferences obtained from data analysis are suspect. Biased standard errors lead to biased inference, so results of hypothesis tests are possibly wrong. As an example of the consequence of biased standard error estimation which OLS will produce if heteroscedasticity is present, a researcher might find compelling results against the rejection of a null hypothesis at a given significance level as statistically significant, when that null hypothesis was actually uncharacteristic of the actual population (i.e., make a type II error).
Under certain assumptions, the OLS estimator has a normal asymptotic distribution when properly normalized and centered (even when the data does not come from a normal distribution). This result is used to justify using a normal distribution, or a chi square distribution (depending on how the test statistic is calculated), when conducting a hypothesis test. This holds even under heteroscedasticity. More precisely, the OLS estimator in the presence of heteroscedasticity is asymptotically normal, when properly normalized and centered, with a variancecovariance matrix that differs from the case of homoscedasticity. In 1980, White^{[2]} proposed a consistent estimator for the variancecovariance matrix of the asymptotic distribution of the OLS estimator. This validates the use of hypothesis testing using OLS estimators and White's variancecovariance estimator under heteroscedasticity.
Heteroscedasticity is also a major practical issue encountered in ANOVA problems.^{[4]} The F test can still be used in some circumstances.^{[5]}
However, it has been said that students in econometrics should not overreact to heteroskedasticity.^{[3]} One author wrote, "unequal error variance is worth correcting only when the problem is severe."^{[6]} And another word of caution was in the form, "heteroscedasticity has never been a reason to throw out an otherwise good model."^{[7]}^{[3]}
With the advent of heteroscedasticityconsistent standard errors allowing for inference without specifying the conditional second moment of error term, testing conditional homoscedasticity is not as important as in the past.^{[citation needed]}
The econometrician Robert Engle won the 2003 Nobel Memorial Prize for Economics for his studies on regression analysis in the presence of heteroscedasticity, which led to his formulation of the autoregressive conditional heteroscedasticity (ARCH) modeling technique.^{[citation needed]}
There are several methods to test for the presence of heteroscedasticity. Although tests for heteroscedasticity between groups can formally be considered as a special case of testing within regression models, some tests have structures specific to this case.
These tests consist of a test statistic (a mathematical expression yielding a numerical value as a function of the data), a hypothesis that is going to be tested (the null hypothesis), an alternative hypothesis, and a statement about the distribution of statistic under the null hypothesis.
Many introductory statistics and econometrics books, for pedagogical reasons, present these tests under the assumption that the data set in hand comes from a normal distribution. A great misconception is the thought that this assumption is necessary. Most of the methods of detecting heteroscedasticity outlined above modified for use even when the data do not come from a normal distribution. In many cases, this assumption can be relaxed, yielding a test procedure based on the same or similar test statistics but with the distribution under the null hypothesis evaluated by alternative routes: for example, by using asymptotic distributions which can be obtained from asymptotic theory,^{[citation needed]} or by using resampling.
There are three common corrections for heteroscedasticity:
Heteroscedasticity often occurs when there is a large difference among the sizes of the observations.
The immediate generalisation of the above, which considers only scalar random variables, is to multivariate heteroscedasticity. One version of this is to use covariance matrices as the multivariate measure of dispersion. Several authors have considered tests in this context, for both regression and groupeddata situations.^{[12]}^{[13]}
Most statistics textbooks will include at least some material on heteroscedasticity. Some examples are:

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