Heteroskedasticity And Homoskedasticity
Both hetereoskedasticity and homoskedasticity are statistical terms used in econometric regression models. They are complementary notions. Both are useful to derive statistical pattern recognition and machine learning algorithms. Neither cause bias, but can indicate when bias is present, and may indicate unknown variables not being account for, may need to be accounted for.
Hetereoskedasticity is the variance of the error term, given the explanatory (e.g dependent variable - variable to be explained in a multiple regression model) variables, is not constant. Hetereoskedasticity is an assumption that the variance of the unobservable, in an OLS (ordinary least squares) regression or other regression model, conditional on X, is not constant. Hetereoskedasticity is also known as the 'non-constant variance'.
Hetereoskedasticity fails whenever the variance of unobervables changes across different segments of the population, where the segments are determined by different values of explanatory variables. An example, in a savings equation, heteroskedasticity is present if the variance of the unobserved factors affecting savings increases with income.
Heteroskedasticity is a measure used by econometricians to infer whether an OLS regression is useful. If heteroskedasticity is present it indicates that bias (difference between the expected value of an estimator and the population value that the estimator is supposed to be estimating) could be influencing the regression. Heteroskedasticity does not create bias, but rather indicates it may be present.
Econometricians have learned to adjust standard erros, t, F and LM statistics so they are valid in the presence of heteroskedasticity of unknown form (hetereoskedasticity that may depend on the explanatory variables in an unknown, arbitary form). These methods are valid whether or not the errors have a constant variance, at least in large samples, and are known as heteroskedasticity-robust procedures.
Homoskedasticity is defined as the errors in a regression model that have constant variance conditional on the explanatory variables. This assumption states that the variance of the unobservable, u, conditional on x, is constant. Homoskedasticity is also known as the 'constant variance'. This is distinct from from the zero conditional mean assumption, as there is a difference between the expected value of u and the variance of u. Homoskedasticity also plays no role in determining biasness.