What is heteroscedasticity?

Heteroscedasticity in statistics refers to the nonconstant standard deviations of a predicted variable over distinct, independent variables or periods. Residual errors tend to widen over time with heteroskedasticity, as shown below.

Conditional and unconditional heteroskedasticity occur. Conditional heteroskedasticity finds nonconstant volatility linked to earlier periods, such as daily volatility. Unconditional heteroskedasticity is structural volatility variation unrelated to past-period volatility. Use unconditional heteroskedasticity when future high and low volatility are known.

Heteroskedasticity does not skew coefficient estimates but makes them less exact, which raises the risk that they are farther from the actual population value.

Fundamentals of Heteroskedasticity

Financial markets exhibit conditional heteroskedasticity in stock and bond prices. Predicting the volatility of these equities over time is impossible. An example of unconditional heteroskedasticity is energy use, which has seasonal fluctuations.

Heteroskedasticity in statistics refers to the impact of scattering on the error variance within a sample with at least one independent variable. These fluctuations may be used to assess the margin of error between data sets, such as predicted and actual outcomes, by measuring the data point departure from the mean.

According to Chebyshev’s inequality, most data points in a dataset must be within a certain number of standard deviations from the mean to be meaningful. This gives random variable mean deviation probability rules.

A random variable’s likelihood of being inside those points depends on the number of standard deviations. A meaningful range of two standard deviations may need at least 75% of the data points. Data quality concerns can generate deviations over the minimum.

The opposite of heteroskedasticity is homoskedasticity. Homoskedasticity is when the residual term variance is constant or close to it. Homoskedasticity is a linear regression assumption. This ensures accurate estimates, appropriate prediction bounds for the dependent variable, and correct confidence intervals and p-values for the parameters.

Types Heteroscedasticity

Unconditional

Predictable unconditional heteroskedasticity can relate to cyclical variables. This can include more holiday retail sales or air conditioner repair calls in warmer months.

If variance shifts are not seasonal, they can be linked to events or prognostic indicators. A new smartphone model may raise sales as the activity is cyclical and dependent on the event but not the season.

Heteroskedasticity can also occur as data approaches a boundary, which reduces the data range and requires a lower variance.

Conditional

Natural conditional heteroskedasticity is unpredictable. No hint suggests analysts should expect data to grow more or less dispersed. Financial goods often exhibit conditional heteroskedasticity since not all changes are due to events or seasons.

In stock markets, conditional heteroskedasticity was highly correlated with volatility yesterday. This model describes high- and low-volatility periods.

Special Considerations

Financial Models and Heteroskedasticity

Regression models explain security and investment portfolio performance using heteroskedasticity. The Capital Asset Pricing Model (CAPM) is a popular model that compares stock performance to market volatility. This approach now includes size, momentum, quality, and style (value versus growth).

Add these predictor factors to explain dependent variable variation. CAPM explains portfolio performance. The CAPM model’s architects knew their model couldn’t explain an exciting anomaly: high-quality equities, less volatile than low-quality stocks, performed better than projected. CAPM suggests higher-risk stocks beat lower-risk ones.

In other words, high-volatility equities should defeat low-volatility ones. However, less volatile, high-quality companies outperformed CAPM.

Later, other researchers added quality as a “factor” to the CAPM model, which included size, style, and momentum. Adding this element to the model explained low-volatility stock performance anomalies. Multi-factor models underpin factor investing and smart beta.

Conclusion

• Statistics defines heteroskedasticity as the nonconstant standard error of a variable across time.
• As shown above, residual errors with heteroskedasticity fan out over time.
• Heteroskedasticity can affect the validity of econometric research and financial models like CAPM by violating linear regression modeling assumptions.