Heteroskedasticity

Heteroskedasticity is a condition in regression analysis in which the variance of the error term is not constant across observations.

In a regression model:

$Y_i={\beta }_0+{\beta }_1{X}_{1,i}+\cdots +{\beta }_k{X}_{k,i}+{ϵ}_i$

homoskedasticity requires the conditional variance of the error term to be constant:

$\mathrm{Var}({ϵ}_i\mid 𝐗)={\sigma }^2$

Under heteroskedasticity, the conditional variance varies across observations:

$\mathrm{Var}({ϵ}_i\mid 𝐗)={\sigma }_i^2$

with:

${\sigma }_i^2\ne {\sigma }^2$

Example

Suppose a regression examines the relationship between income and household expenditure.

The variability of expenditure may be relatively small for households with lower incomes but substantially larger for households with higher incomes.

A simplified representation is:

$\mathrm{Var}({ϵ}_i\mid X_i)={\sigma }^2(X_i)$

where the variance of the error term depends on the value of the explanatory variable.

This creates a pattern in which residuals may become increasingly dispersed as the fitted values or explanatory variables increase.

Heteroskedasticity and ordinary least squares

Heteroskedasticity does not necessarily make ordinary least squares (OLS) coefficient estimates biased.

Under the appropriate exogeneity condition:

$E[{ϵ}_i\mid 𝐗]=0$

OLS coefficient estimates can remain unbiased even when the error variance is not constant.

However, heteroskedasticity affects the efficiency and estimated uncertainty of OLS.

In particular, conventional OLS standard errors that assume homoskedasticity may be incorrect.

Therefore, hypothesis tests and confidence intervals based on those standard errors may also be unreliable.

Robust standard errors

Heteroskedasticity-consistent standard errors can be used to obtain standard errors that are robust to certain forms of heteroskedasticity.

A simplified representation of the robust covariance estimator is:

$\widehat{\mathrm{Var}}(\widehat{𝜷})=({𝐗}^{𝖳}𝐗{)}^{-1}{𝐗}^{𝖳}\widehat{𝜴}𝐗({𝐗}^{𝖳}𝐗{)}^{-1}$

where $\widehat{𝜴}$ contains estimates of the observation-specific error variances.

Robust standard errors change the estimated uncertainty around the coefficients but generally do not change the OLS coefficient estimates themselves.

Detection

Heteroskedasticity can be investigated using residual analysis and statistical tests.

A residual-versus-fitted-value plot can reveal patterns in the dispersion of residuals.

For example, a widening residual spread may indicate non-constant error variance.

Common formal tests include:

• Breusch–Pagan test;
• White test;
• Goldfeld–Quandt test.

A simplified null hypothesis is:

$H_0:\mathrm{Var}({ϵ}_i\mid 𝐗)={\sigma }^2$

against an alternative in which the conditional variance is not constant.

Consequences

Depending on the regression setting, heteroskedasticity can:

• make conventional standard errors unreliable;
• invalidate conventional t-tests and F-tests based on those standard errors;
• make confidence intervals misleading;
• reduce the efficiency of OLS relative to other estimators under the relevant assumptions;
• indicate possible model misspecification or omitted variables.

Heteroskedasticity does not by itself establish that the regression coefficients are biased.

Heteroskedasticity in financial data

Heteroskedasticity is particularly relevant in financial and time-series data.

Asset returns often exhibit periods of relatively low volatility followed by periods of relatively high volatility.

A simplified representation is:

$\mathrm{Var}({ϵ}_t\mid {ℱ}_{t-1})={\sigma }_t^2$

where:

• ${ϵ}_t$ = error or innovation at time $t$;
• ${ℱ}_{t-1}$ = information available before time $t$;
• ${\sigma }_t^2$ = conditional variance at time $t$.

When volatility changes over time, the assumption of constant error variance may be inappropriate.

Models such as ARCH and GARCH explicitly model time-varying conditional variance.

Heteroskedasticity and GARCH models

A simple GARCH model can represent conditional variance as:

${\sigma }_t^2=\omega +\alpha {ϵ}_{t-1}^2+\beta {\sigma }_{t-1}^2$

where:

• ${\sigma }_t^2$ = conditional variance;
• $\omega$ = constant term;
• $\alpha$ = response of current volatility to the previous squared error;
• $\beta$ = persistence of conditional variance.

This allows volatility to vary over time rather than assuming:

${\sigma }_t^2={\sigma }^2$

for every period.

Homoskedasticity versus heteroskedasticity

The distinction can be summarized as:

$\text{Homoskedasticity}\to \mathrm{Var}({ϵ}_i\mid 𝐗)={\sigma }^2$

$\text{Heteroskedasticity}\to \mathrm{Var}({ϵ}_i\mid 𝐗)={\sigma }_i^2$

Thus:

$\text{Homoskedasticity}\to \text{Constant Error Variance}$

$\text{Heteroskedasticity}\to \text{Non-constant Error Variance}$

Limitations of robust standard errors

Robust standard errors address certain problems caused by non-constant error variance, but they do not solve every regression problem.

They do not, by themselves, correct:

• omitted-variable bias;
• endogeneity;
• incorrect functional form;
• measurement error;
• serial correlation unless an appropriate covariance estimator is used;
• non-stationarity;
• other forms of model misspecification.

Consequently, diagnosing heteroskedasticity should be part of broader regression-model assessment.

See also

• Regression analysis
• Linear regression
• Multiple linear regression
• Ordinary least squares
• Homoskedasticity
• Heteroskedasticity-consistent standard errors
• Breusch–Pagan test
• White test
• Autoregressive conditional heteroskedasticity
• Generalized autoregressive conditional heteroskedasticity
• Time series
• Econometrics
• Quantitative finance