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In statistics, a collection of random variables is heteroscedastic (or 'heteroskedastic';〔The spellings ''homoskedasticity'' and ''heteroskedasticity'' are also frequently used. J. Huston McCulloch argued that there should be a ‘k’ in the middle of the word and not a ‘c’. His argument was that the word had been constructed in English directly from Greek roots rather than coming into the English language indirectly via the French. See 〕 from Ancient Greek “different” and ' “dispersion”) if there are sub-populations 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 existence of heteroscedasticity is a major concern in the application of regression analysis, including the analysis of variance, as it can invalidate statistical tests of significance that assume that the modelling errors are uncorrelated and uniform—hence that their variances do not vary with the effects being modeled. For instance, while the ordinary least squares estimator is still unbiased in the presence of heteroscedasticity, it is inefficient because the true variance and covariance are underestimated. Similarly, in testing for differences between sub-populations using a location test, some standard tests assume that variances within groups are equal. Because heteroscedasticity concerns expectations of the second moment of the errors, its presence is referred to as misspecification of the second order. ==Definition== Suppose there is a sequence of random variables ''t''=1''n'' and a sequence of vectors of random variables, ''t''=1n. In dealing with conditional expectations of ''Y''''t'' given ''X''''t'', the sequence ''t''=1''n'' is said to be heteroscedastic if the conditional variance of Yt 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 cross-sectional or time series measurements. Heteroscedasticity is often studied as part of econometrics, which frequently deals with data exhibiting it. While the influential 1980 paper by Halbert White used the term "heteroskedasticity" rather than "heteroscedasticity",〔 〕 the latter spelling has been employed more frequently in later works. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Heteroscedasticity」の詳細全文を読む スポンサード リンク
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