Nonstationary Generalised Autoregressive Conditional Heteroskedasticity Modelling for Fitting Higher Order Moments of Financial Series within Moving Time Windows

Figure 4

In (a), the phase diagram for (, ) space. The enclosed area between the black lines shows the region in which the GARCH-normal model is able to fit the fourth- and sixth-order standardised moment, whilst the rest of the space is where the values of and cannot be fitted by a GARCH-normal (1,1) model. The two highlighted points and show the examples of values of and that can be fitted by a GARCH-normal model and those that cannot, respectively. The other data points in the space represent the empirical data for several companies, truncated to of its overall length, incrementing in percents up to its full length. In (b), we show the histogram for point density on the higher order moment phase space, we detail the specifics of this calculation in A. We consider all possible windows with duration longer than 30 days. We show the histogram alongside the “GARCHable” region. The empirical data is shown for Lloyds Bank, GSK, Barclays Bank, Gold ETFs, S&P 500, DowJones, Rio Tinto, Bank of America, Oil, Natural Gas, Vale, Pfizer, and Citi Bank. This histogram shows the scattering of empirical data in a much clearer way than any error bar estimation, in which some information about the point distribution is lost.

(a)

(b)