In statistics, heteroskedasticity (or heteroscedasticity) occurs when the standard deviations of a predicted variable, observed across different values of an independent variable or over time, are non-constant. A clear sign of heteroskedasticity is when residual errors fan out over time.
Heteroskedasticity manifests in two primary forms: conditional and unconditional. Conditional heteroskedasticity reveals non-constant volatility linked to previous periods’ volatility. Unconditional heteroskedasticity, however, pertains to structural changes in volatility unrelated to past variability, identifiable in future periods of high and low volatility.
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Key Insights
- Heteroskedasticity (or heteroscedasticity) is evident when a variable’s standard errors, observed over time, are inconsistent.
- A visual clue of heteroskedasticity is the residual errors fanning out over time.
- This irregularity violates assumptions of linear regression modeling, impacting the validity of econometric and financial models like the CAPM. Although heteroskedasticity doesn’t introduce bias in coefficient estimates, it reduces their precision, raising the possibility of inaccurate population values.
Exploratory Overview of Heteroskedasticity
In finance, conditional heteroskedasticity is often seen in stock and bond prices whose volatility is unpredictable over periods. Unconditional heteroskedasticity suits discussions on seasonal variability, such as electricity usage patterns.
In statistics, heteroskedasticity refers to error variance or the degree of scattering within at least one independent variable in a sample. These variations help calculate the margin of error, reflecting the deviation of data points from the mean.
A dataset’s relevance is often devoid dependent on most data points falling within a specific range of standard deviations described by Chebyshev’s theorem (Chebyshev’s inequality). For instance, if a range of two standard deviations contains at least 75% of data points, it is deemed valid. Data quality issues often cause deviations beyond these requirements.
Heteroskedasticity’s counterpart, homoskedasticity, indicates consistent variance in residual terms and is a fundamental assumption in linear regression modeling to ensure reliable estimates and valid predictions.
Types of Heteroskedasticity
Unconditional Heteroskedasticity
This form is predictable and linked to cyclical variables. Examples include increased retail sales during holidays or higher air conditioner repairs in summer. It also covers events causing shifts not tied to traditional seasons, like new smartphone releases, which create cyclical data changes conditioned on these events.
Heteroskedasticity is evident even when approaching data boundaries where variance narrows due to boundary restrictions.
Conditional Heteroskedasticity
Conversely, conditional heteroskedasticity is inherently unpredictable. No evident markers suggest rising or falling data scatter trends. Financial products often exhibit conditional heteroskedasticity, where changes can’t solely be attributed to predictable events.
A typical application is stock market volatility. Current volatility is often related to prior periods, explaining high or low volatility streaks.
Financial Modeling Implications of Heteroskedasticity
Understanding heteroskedasticity is crucial in regression modeling, especially in investments. Regression models often elucidate the performance of securities and portfolios, with the Capital Asset Pricing Model (CAPM) being a prime example. CAPM divides a stock’s performance based on its market-relative volatility. Extensions include other predictability variables like size, momentum, quality, and style (value vs. growth).
These variables help explain variance in the dependent variable. For instance, though CAPM suggested high-risk stocks outperformed, high-quality, typically low-volatility stocks actually did better, contrary to CAPM. By adding ‘quality’ as another ‘factor,’ newer multi-factor models resolved this anomaly, leading to refined factor investing and smart beta strategies.
Related Terms: homoskedasticity, volatility, econometrics, CAPM, multi-factor models
References
- Fidelity. “Understanding Factor-based Investing”.
