Slice Distribution of GARCH(1,1) 

Generalized autoregressive conditionally heteroscedastic process GARCHProcess is used to describe time series that exhibit volatility clustering phenomenon. The distribution of the time slice of a GARCH process has much heavier tails than the normal distribution. These two properties make GARCH process a very attractive choice to model financial time series, which display both of these phenomena.

Define a weakly stationary GARCHProcess.

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X proc1 = GARCHProcess[1, {.7}, {.2}];

Define a GARCHProcess with fixed initial value.

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X proc2 = GARCHProcess[1, {.7}, {.2}, {0}];

Simulate random samples from each process sliced at time 3.

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X sample1 = RandomVariate[proc1[3], 10^5]; sample2 = RandomVariate[proc2[3], 10^5];

Visualize sharply peaked probability density functions from data in the log scale.

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X Histogram[#, Automatic, "LogPDF"] & /@ {sample1, sample2}

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Compare to a NormalDistribution.

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X QuantilePlot[{sample1, sample2}, PlotLegends -> {"weakly stationary GARCH", "GARCH with fixed initial conditions"}]

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