Changing Variance

Heteroskedasticity is an important issue encountered in statistics and econometric models, particularly in regression analyses. One of the fundamental assumptions of the classical linear regression model is that the error terms have constant variance (homoskedasticity). If the variance of the error terms is not constant—that is, if variance varies across observations—this condition is called heteroskedasticity.

Theoretical Background

Under the assumption of constant variance, the variance of the error terms is equal for all observations:

When this assumption is violated, although the classical Ordinary Least Squares (OLS) estimators remain unbiased and consistent, they are no longer efficient (minimum variance). This reduces the reliability of the estimates and renders standard errors and statistical tests (t and F tests) invalid.

Causes of Heteroskedasticity

There are several possible causes of heteroskedasticity:

1. Omitted important variables,
2. Structural heterogeneity commonly observed in cross-sectional data (e.g., differences in spending across income levels),
3. Seasonality in time series data,
4. Mispecified dependent variables,
5. Samples composed of subgroups with different structures (heterogeneous population)

Detection Methods

Both graphical and statistical tests are used to detect heteroskedasticity. The most commonly used methods include:

• White Test,
• Breusch-Pagan-Godfrey Test,
• Park Test,
• Goldfeld-Quandt Test,
• Glejser Test,
• Graphical Residual Analysis (residuals versus fitted values)

For example, in the White test, the squared residuals are regressed on the original independent variables, their squares, and their cross-products. If the resulting nR² statistic exceeds a critical chi-square value, the hypothesis of constant variance is rejected.

Solution Methods

Weighted Least Squares (WLS)

The most well-known solution to heteroskedasticity is to reapply OLS by assigning weights inversely proportional to the variances of the error terms. The WLS method restores the efficiency condition by stabilizing the variance:

Here, W is the weight matrix constructed based on the inverse variances of the error terms.

Transformations

Data transformations are applied to stabilize variance. The most commonly used methods include:

• Logarithmic transformation (Y* = lnY),
• Square root transformation (Y* = √Y),
• Inverse transformation (Y* = 1/Y),
• Arcsin transformation (particularly for proportion data)

Conditional Variance Models (ARCH/GARCH)

In time series analysis, specialized models developed to address heteroskedasticity include ARCH (Autoregressive Conditional Heteroskedasticity) and GARCH (Generalized ARCH). The ARCH model, developed by Engle (1982), links variance to the squares of past error terms, while the GARCH model, developed by Bollerslev (1986), links variance to both past error terms and past variances.