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Here, we have analysed a GARCH(1,1) model with the aim to fit higher ordermoments for different companies' stock prices. When we assume a gaussianconditional distribution, we fail to capture any empirical data when fittingthe first three even moments of financial time series. We show instead that adouble gaussian conditional probability distribution better captures the higherorder 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 withthe corresponding moments from empirical data for different sectors of theeconomy. We found that the ability of the GARCH model with a double gaussianconditional distribution to fit higher order moments is dictated by the timewindow our data spans. We can only fit data collected within specific timewindow lengths and only with certain parameters of the conditional doublegaussian distribution. In order to incorporate the non-stationarity offinancial series, we assume that the parameters of the GARCH model have timedependence.

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