Fitting higher order moments of empirical financial series with GARCH models Luke De Clerk Department of Physics, Loughborough University, Leicestershire, LE11 3TU, United Kingdom Sergey Savel’ev Department of Physics, Loughborough University, Leicestershire, LE11 3TU, United Kingdom Abstract Here we have analysed a GARCH(1,1) model with the aim to fit higher order moments for different companies’ stock prices. When we assume a gaussian conditional distribution, we fail to capture any empirical data. We show instead that a double gaussian conditional probability better captures the higher order moments of the data. To demonstrate this point, we construct regions (phase diagrams) in higher order moment space, where a GARCH(1,1) model can be used to fit the higher order moments and compare this with empirical data from different sectors of the economy. We found that, the ability of the GARCH model to fit higher order moments is dictated by the time window our data spans. Primarily, if we have a time series, using a GARCH(1,1) model with a double gaussian conditional probability (a GARCH-double-normal model) we cannot necessarily fit the statistical moments of the time series. Highlighting, that the GARCH-double-normal model only allows fitting of specific lengths of time series. This is indicated by the migration of the companies’ data out of the region of the phase diagram where GARCH is able to fit these higher order moments. In order to overcome the non-stationarity of our modelling, we assume that one of the parameters of the GARCH model, α 0 , has a time dependence. Keywords: GARCH, Phase Diagrams, Double Gaussian, Empirical Data JEL Classification: C10, C40, C50, C80 1. Introduction Modelling of financial time series is a very extensive area of research. In general, there are large over simplifications of a financial time series, so there has been research into the modelling of time series with a plethora of different mathematical tools. The main issues are; the large tails present in most financial time series, the heteroskedasticity of volatility and the conditional second order moments of price change, volatility; demonstrating the presence of a further stochastic dynamic in addition to those of price change. These stochastic processes evolve in the same system but have different time scales; a fast changing stochastic process (the price) and a slow changing stochastic process (the volatility). This motivates the creation of the Autoregressive Conditional Heteroskedasticity models (ARCH) by Engle [1] and later generalised (GARCH) by Bollersev [2]. Preprint submitted to Elsevier October 25, 2021 arXiv:2102.11627v1 [econ.EM] 23 Feb 2021