Analysing the Symmetric GARCH model across different sample sizes

In the world of econometrics, the GARCH model has the desired ability to model the changing variance (heteroskedasticity) of a financial time series. Therefore, the primary goal of this paper is to investigate changes in volatility, parameter estimates and forecasting error across different window lengths as this may indicate an appropriate sample size to use when fitting an S-GARCH model. After examining the 6489 1-day log-returns on the FTSE/JSE-ALSI between 27 December 1995 and 15 December 2021, an average estimate for volatility of 0.193 670 was calculated. Furthermore, given that the rolling window methodology was applied across 20 different window lengths, a total of 90 000 S-GARCH models were fit to the data. Among others, key results include volatility estimates across most window lengths taking longer to settle after the Global Financial Crisis (GFC) than after the COVID-19 pandemic. In terms of parameter estimates, values for α and β under a window length of 100 trading days were often calculated infinitely close to zero and one respectively, indicating a strong possibility of the optimising algorithm arriving at local maxima of the likelihood function. With the exceptionally low p-values under the Jarque-Bera and Kolmogorov-Smirnov tests, substantial motivation was provided for the use of the Student's t-distribution when fitting GARCH models. Given the results obtained, it is clear that a window length between 100 and 500 can produce fitted models that are unstable and that a window length between 1000 and 2000 may be too large to account for market shocks.