Non-stationary GARCH modelling for fitting higher order moments of financial series within moving time windows

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 when fitting the first three even moments of financial time series. We show instead that a double gaussian conditional probability distribution better captures the higher order moments of the data. To demonstrate this point, we construct regions (phase diagrams), in the fourth and sixth order standardised moment space, where a GARCH(1,1) model can be used to fit these moments and compare them with the corresponding moments from empirical data for different sectors of the economy. We found that the ability of the GARCH model with a double gaussian conditional distribution to fit higher order moments is dictated by the time window our data spans. We can only fit data collected within specific time window lengths and only with certain parameters of the conditional double gaussian distribution. In order to incorporate the non-stationarity of financial series, we assume that the parameters of the GARCH model have time dependence.