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 , \alpha_0, has a time dependence.
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 ones obtained 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 length of time window our data spans. It is seen for this model that we can only fit data collected within specific time window lengths and only with certain parameters of the conditional double gaussian distribution . 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 moments. In order to overcome the non-stationarity of our modelling, we assume that the parameters of the GARCH model have time dependence.
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when fitting the first three even moments of financial time series
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with a double gaussian conditional distribution
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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
It is seen for this model that we can only fit data collected within specific time window lengths and only with certain parameters of the conditional double gaussian distribution
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