In statistics, a sequence of random variables is homoscedastic () if all its random variables have the same finite variance; this is also known as homogeneity of variance. The complementary notion is called heteroscedasticity, also known as heterogeneity of variance. The term originates from the Ancient Greek σκεδάννυμι skedánnymi, 'to scatter'. Assuming a variable is homoscedastic when in reality it is heteroscedastic () results in unbiased but inefficient point estimates and in biased estimates of standard errors, and may result in overestimating the goodness of fit as measured by the Pearson coefficient. The existence of heteroscedasticity is a major concern in regression analysis and the analysis of variance, as it invalidates statistical tests of significance which assume that the modelling errors all have the same variance. While the ordinary least squares (OLS) estimator is still unbiased in the presence of heteroscedasticity, it is inefficient and inference based on the assumption of homoskedasticity is misleading. In that case, generalized least squares (GLS) was frequently used in the past. Nowadays, standard practice in econometrics is to include heteroskedasticity-consistent standard errors instead of using GLS, as GLS can exhibit strong bias in small samples if the actual skedastic function is unknown. Because heteroscedasticity concerns expectations of the second moment of the errors, its presence is referred to as misspecification of the second order. The econometrician Robert Engle was awarded 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.
Definition Consider the linear regression equation y i = x i β i + ε i , i = 1 , … , N , {\displaystyle y_{i}=x_{i}\beta _{i}+\varepsilon _{i},\ i=1,\ldots ,N,} where the dependent random variable y i {\displaystyle y_{i}} equals the deterministic variable x i {\displaystyle x_{i}} times coefficient β i {\displaystyle \beta _{i}} plus a random disturbance term ε i {\displaystyle \varepsilon _{i}} that has mean zero. The disturbances are homoscedastic if the variance of ε i {\displaystyle \varepsilon _{i}} is a constant σ 2 {\displaystyle \sigma ^{2}} ; otherwise, they are heteroscedastic. In particular, the disturbances are heteroscedastic if the variance of ε i {\displaystyle \varepsilon _{i}} depends on i {\displaystyle i} or on the value of x i {\displaystyle x_{i}} . One way they might be heteroscedastic is if σ i 2 = x i σ 2 {\displaystyle \sigma _{i}^{2}=x_{i}\sigma ^{2}} (an example of a scedastic function), so the variance is proportional to the value of x {\displaystyle x} . More generally, if the variance-covariance matrix of disturbance ε i {\displaystyle \varepsilon _{i}} across i {\displaystyle i} has a nonconstant diagonal, the disturbance is heteroscedastic. The matrices below are covariances when there are just three observations across time. The disturbance in matrix A is homoscedastic; this is the simple case where OLS is the best linear unbiased estimator. The disturbances in matrices B and C are heteroscedastic. In matrix B, the variance is time-varying, increasing steadily across time; in matrix C, the variance depends on the value of x {\displaystyle x} . The disturbance in matrix D is homoscedastic because the diagonal variances are constant, even though the off-diagonal covariances are non-zero and ordinary least squares is inefficient for a different reason: serial correlation.
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