Moments of GARCH(1,1) 

The value of a generalized autoregressive conditionally heteroscedastic process GARCHProcess has a heavy-tailed distribution with only a few finite moments of low order.

Fourth moment of a GARCHProcess with orders (1,1).

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X Moment[GARCHProcess[k, {a}, {b}][t], 4]

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Define the function to extract moment finiteness conditions.

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X mc[r_] := Reduce[Exists[{k, t}, {t, k} \[Element] Reals, Moment[GARCHProcess[k, {a}, {b}][t], r] \[Element] Reals], {a, b}]

Visualize the parameter conditions for moments to exist.

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X orders = {2, 4, 6, 8}; RegionPlot[Evaluate@Table[mc[r], {r, orders}], {a, 0, 1}, {b, 0, 1}, PlotLegends -> (Row[{"r = ", #}] & /@ orders), FrameLabel -> Automatic]

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Values of the first few even moments for a weakly stationary GARCHProcess.

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X lg3 = Log10[3.]; ticks = {Automatic, {{lg3, 3}, {1, 10}, {lg3 + 1, 30}, {2, 100}, {lg3 + 2, 300}, {3, 1000}}};

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X DiscretePlot[ Log10[Moment[GARCHProcess[.3, {.2}, {.3}][t], r]], {r, 0, 10, 2}, ExtentSize -> 1/2, PlotTheme -> "Grid", Ticks -> ticks]

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Compare to the values of the even moments for a non-weakly stationary GARCH with process initial values set to zero.

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X Grid[Partition[ Table[DiscretePlot[ Log10[Moment[GARCHProcess[.3, {.2}, {.3}, {}][t], r]], {r, 0, 10, 2}, ExtentSize -> 1/2, PlotTheme -> "Grid", Ticks -> ticks, PlotRange -> {-1, 4}, PlotLabel -> Row[{"t = ", t}]], {t, 1, 4}], 2]]

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