3, is crucial for determining the shape and heaviness of the tails in a GARCH model's error distribution. Lower values correspond to heavier tails and fatter distributions, while higher values yield lighter, more normal-like distributions. The plot below showcases the distribution of estimated degrees of freedom across various series.


Exploring Impossibly Long Financial Series with GARCH Estimation

A study analyzed the variability of GARCH estimates on financial time series spanning an astonishing 100,000 observations. This length is equivalent to approximately four centuries of daily data or just over a year for 1-minute returns. In such cases, accounting for significant seasonality in volatility throughout the trading day becomes essential, requiring more sophisticated models than simple GARCH(1,1) with t-distributed errors.

Estimates' Distributions and Visualizations

The study examined the distributions of alpha and beta estimates, which correspond to the parameters of a GARCH(1,1) model. Alpha represents the degree of persistence in volatility shocks, while beta reflects the speed of mean reversion. It was found that these two values exhibit a strong positive correlation, meaning higher alpha estimates are associated with larger beta values.

Furthermore, the half-life—representing the time it takes for a shock to volatility to decay by half—was investigated. The distribution of estimated half-lives was found to be heavily right-skewed, indicating that most series exhibit relatively short half-life values, suggesting slower decay rates and increased persistence of volatility shocks.

Lastly, the study explored the degrees of freedom parameter, which is essential for determining the shape and heaviness of the tails in a GARCH model's error distribution. Lower values correspond to heavier tails and fatter distributions, while higher values yield lighter, more normal-like distributions. The distribution of estimated degrees of freedom across various series was analyzed, revealing valuable insights into these models' behavior and performance when applied to impossibly long financial time series.