Heteroskedasticity

One of the standard assumptions of the classical linear regression model

$$y_i={\mathrm{\beta }}_0+{\mathrm{\beta }}_1{x}_{i1}+{\mathrm{\beta }}_2{x}_{i2}+\cdots +{\mathrm{\beta }}_k{x}_{ik}+\mathrm{\epsilon };i=1\cdots N$$

is that the variance of the error term $({\mathrm{\epsilon }}_i)$ is the same for all observations, that is $\mathrm{Var}(|x_{1i},x_{2i},\cdots ,x_{ki})={\sigma }^2$. The assumption of a constant error variance is known as homoskedasticity and its failure is referred to as heteroskedasticity, or unequal variance. Heteroskedasticity is expressed as $\mathrm{Var}(|x_{1i},x_{2i},\cdots ,x_{ki})={\sigma }_i^2$, where an i subscript on σ2 indicates that the variance of the error is no longer constant but may vary from observation to observation. Note, the alternate spellings homoscedasticity and heteroscedasticity are also commonly used. This entry first reviews when heteroskedasticity typically arises. The consequences of heteroskedastic errors are then discussed, followed by sections describing the detection of and solutions for heteroskedasticity.

Heteroskedasticity is often encountered when using cross-section data, when observations are made at a given point in time. The data may include units of observation such as individuals, families, firms, industries, cities, or countries where the observations may be subject to wide variations such as low-, medium-, and high-income families. For example, if households are surveyed to determine how their income influences their consumption expenditures, we would expect less variation in spending patterns for low-income households compared to wealthy households. Low-income households spend their income on necessities such as food, clothing, and rent. Wealthy households have more choice in how they spend their income and can afford to spend money on luxury items with some choosing to do so and others not. Consequently, wealthy households have a larger dispersion around average consumption than low-income households. Heteroskedasticity may also arise when grouped data are used rather than individual data. An example is when data on industry averages is used because data on individual firms are not available. An additional case that results in heteroskedastic errors would occur when the dependent variable is a qualitative or binary variable. This type of model is known as the linear probability model.