(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 74687, 2664]*) (*NotebookOutlinePosition[ 75805, 2700]*) (* CellTagsIndexPosition[ 75761, 2696]*) (*WindowFrame->Normal*) Notebook[{ Cell["Generating time series in Mathematica", "Title"], Cell[CellGroupData[{ Cell["First, load in the required packages", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "It's important because otherwise you can't generate normally distributed \ shocks"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ $Needs["\"]$], "Input", PageWidth->PaperWidth, AspectRatioFixed->True] }, Open ]], Cell["\<\ This one is also useful in general, and the function EGARCHList \ below depends on some of its contents.\ \>", "Text"], Cell[BoxData[ $Needs["\"]$], "Input"], Cell["\<\ Note that this package also loads in some others, as shown below. \ This is important if you want to convert this notebook into package \ form.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ $$Packages$], "Input"], Cell[BoxData[ ${"Statistics`Common`DistributionsCommon`", "Statistics`DescriptiveStatistics`", "Statistics`NormalDistribution`", "Statistics`ContinuousDistributions`", "Global`", "System`"}$], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Random Walk \[LongDash]with or without drift"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "This is a very simple function, using the functional programming construct \ ", StyleBox["NestList[] ", FontWeight->"Bold"], "with a simple pure function. Note that the length of the series generated \ is constrained to be an integer, while the variables ", StyleBox["drift", FontWeight->"Bold"], " and ", StyleBox["sd", FontWeight->"Bold"], " -- the drift and the standard deviation of the shocks -- have default \ values. I haven't constrained the starting value (sv) to be numerical -- \ sometimes you might want to explore the properties of something in algebraic \ terms." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ $randomwalk[n_Integer, sv_, sd_: 1, drift_: 0] := Rest[NestList[#1 + Random[NormalDistribution[0, sd]] + drift\ &, sv, n]] $], "Input", PageWidth->PaperWidth, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["ARMA: generate artificial time series of any order", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Anyone who's done econometrics or time series work would be familiar with \ this sort of process:\n\n\t", Cell[BoxData[ $TraditionalForm \`y\_t = \[Sum]\+\(i = 1$\%p$ \[Phi]\_i$ y\_$t - 1$\ + \[Sum]\+$j = 0$\%q$ \[Theta]\_j$ \[Epsilon]\_$t - j$\)]], "\n\nThe \[Epsilon]'s are usually normally distributed, or at least iid. \ (should that be \"normally normally\")." }], "Text"], Cell[BoxData[ $ARMAList[n_Integer, sv_, sd : \((_Real | _Integer)$, pcoefs : {_, __}, qcoefs_?VectorQ, const_: 0] /; \ VectorQ[pcoefs] := \ With[{p = Length[pcoefs], q = Length[qcoefs], macoefs = Flatten[Reverse[Join[{1}, qcoefs]]], rpcoefs = Reverse[pcoefs], rawshocks = RandomArray[NormalDistribution[0, sd], n + Length[qcoefs]]}, With[{mabit = Expand\ /@ $(Take[ \((macoefs. \((\(RotateLeft[rawshocks, #]&$\ /@\ Range[0, q])\)) \), n])\)}, \n\t\t\t Rest[Last\ /@\ $(FoldList[Join[Rest[#1], {rpcoefs.#1\ + #2 + const}]&, Table[sv, {p}], mabit])$]\ ]]\)], "Input"], Cell["\<\ Special Cases for AR(1) processes. The pattern-matching engine \ will pick this up automatically, so you can put the single AR coefficient in \ list brackets or not as you see fit.\ \>", "Text"], Cell[BoxData[ $ARMAList[n_Integer, sv_, sd : \((_Real | _Integer)$, pcoefs : {_}, qcoefs_?VectorQ, const_: 0] /; \ VectorQ[pcoefs] := \ \n ARMAList[n, sv, sd, pcoefs\[LeftDoubleBracket]1\[RightDoubleBracket], qcoefs, const]\)], "Input"], Cell[BoxData[ $ARMAList[n_Integer, sv_, sd : \((_Real | _Integer)$, pcoef : $(_Real | _Integer | \ _Symbol)$, qcoefs_?VectorQ, const_: 0] := \ With[{q = Length[qcoefs], macoefs = Flatten[Reverse[Join[{1}, qcoefs]]], rawshocks = RandomArray[NormalDistribution[0, sd], n + Length[qcoefs]]}, With[{mabit = Expand\ /@ $(Take[ \((macoefs. \((\(RotateLeft[rawshocks, #]&$\ /@\ Range[0, q])\)) \), n])\)}, \n\t\t\t FoldList[pcoef\ *\ #1\ + #2 + const\ &, sv, mabit]\ ]]\)], "Input"], Cell["Special Cases for shock vectors that are already supplied.", "Text"], Cell[BoxData[ $ARMAList[\[Epsilon]_?VectorQ, sv_, pcoefs : {_, __}, qcoefs_?VectorQ, const_: 0] /; \ VectorQ[pcoefs] := \ With[{p = Length[pcoefs], q = Length[qcoefs], macoefs = Flatten[Reverse[Join[{1}, qcoefs]]], rpcoefs = Reverse[pcoefs], T = Length[\[Epsilon]] - Length[qcoefs]}, With[{mabit = Expand\ /@ \((Take[ \((macoefs. \((\(RotateLeft[\[Epsilon], #]&$\ /@\ Range[0, q])\)) \), T])\)}, \n\t\t\t Rest[Last\ /@\ $(FoldList[Join[Rest[#1], {rpcoefs.#1\ + #2 + const}]&, Table[sv, {p}], mabit])$]\ ]]\)], "Input"], Cell[BoxData[ $ARMAList[\[Epsilon]_?VectorQ, sv_, pcoef : {_}, qcoefs_?VectorQ, const_: 0] /; \ VectorQ[pcoefs] := \ \n ARMAList[\[Epsilon], sv, pcoef\[LeftDoubleBracket]1\[RightDoubleBracket], qcoefs, const]$], "Input"], Cell[BoxData[ $ARMAList[\[Epsilon]_?VectorQ, sv_, pcoef : \((_Real | _Integer | \ _Symbol)$, qcoefs_?VectorQ, const_: 0] := \ With[{q = Length[qcoefs], macoefs = Flatten[Reverse[Join[{1}, qcoefs]]], T = Length[\[Epsilon]] - Length[qcoefs]}, With[{mabit = Expand\ /@ $(Take[ \((macoefs. 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Cell[TextData[{ "These functions are based on the time series generation procedures written \ by Pedro J F de Lima of the Johns Hopkins University in the GAUSS programming \ language, available at ", StyleBox["ftp://speedy.econ.jhu.edu/public/gauss/st_proc.prc", FontFamily->"Courier", FontColor->RGBColor[0, 0, 1]], ". Note that the function tests to ensure that the coefficients are \ standardised, ie that \[Omega] =1 \[Dash] \[Sum]\[Alpha] \[Dash] \ \[Sum]\[Beta]. If this is not true the function definition will not match \ and nothing will happen." }], "Text"], Cell["With both AR and MA components, we have GARCH:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ $ARCHList[n_Integer, \[Omega]_Real, \[Alpha] : {__Real}, \[Beta] : {__Real}] /; \ \((\[Omega] == 1 - \((Plus@@\[Alpha])$ - $(Plus@@\[Beta])$)\) := \n \tWith[{m = Max[Length[\[Alpha]], Length[\[Beta]]]}, \n\t With[{q = Length[\[Alpha]], p = Length[\[Beta]], arev = Reverse[\[Alpha]], brev = Reverse[\[Beta]], z = RandomArray[NormalDistribution[0, 1], m + n]}, \n\t\t\t Module[{h = Table[1, {n + m}], e = Join[z\[LeftDoubleBracket]Range[m]\[RightDoubleBracket], Table[0, {n}]]}, \n\t\t\t\t Do[h\[LeftDoubleBracket]i\[RightDoubleBracket] = \[Omega]\ + $(\((e\[LeftDoubleBracket]Range[i - q, i - 1] \[RightDoubleBracket]\ )$^2)\).arev + h\[LeftDoubleBracket]Range[i - p, i - 1] \[RightDoubleBracket]\ .brev; \n\t\t\t\t $e\[LeftDoubleBracket]i\[RightDoubleBracket] = z\[LeftDoubleBracket]i\[RightDoubleBracket] + Sqrt[h\[LeftDoubleBracket]i\[RightDoubleBracket]]; $, {i, m + 1, n + m}]; \n\t\t\t\t{Drop[h, m], Drop[e, m]\ }]]] \)], "Input"], Cell[BoxData[ $General::"spell1" \( : \ $ "Possible spelling error: new symbol name \"\!$brev$\" is similar to \ existing symbol \"\!$arev$\"."\)], "Message"] }, Open ]], Cell["Without either AR or MA components we have white noise.", "Text"], Cell[BoxData[ $ARCHList[n_Integer, \[Omega]_Real, {}, {}] := ARCHList[n, \[Omega]]$], "Input"], Cell[BoxData[ $ARCHList[n_Integer, \[Omega]_Real] := { RandomArray[NormalDistribution[0, 1], n], Table[1, {n}]}$], "Input"], Cell["With only AR components, it's ARCH but not GARCH.", "Text"], Cell[BoxData[ $ARCHList[n_Integer, \[Omega]_Real, \[Alpha] : {__Real}, {}] := ARCHList[n, \[Omega], \[Alpha]]$], "Input"], Cell[BoxData[ $ARCHList[n_Integer, \[Omega]_Real, \[Alpha] : {__Real}] /; \((\[Omega] == 1 - \((Plus@@\[Alpha])$)\) := \n\t\t With[{q = Length[\[Alpha]], arev = Reverse[\[Alpha]], z = RandomArray[NormalDistribution[0, 1], q + n]}, \n\t\t\t Module[{h = Table[1, {n + q}], e = z}, \n\t\t\t\t Do[h\[LeftDoubleBracket]i\[RightDoubleBracket] = \[Omega]\ + $(\((e\[LeftDoubleBracket]Range[i - q, i - 1] \[RightDoubleBracket]\ )$^2)\).arev; \n\t\t\t\t $e\[LeftDoubleBracket]i\[RightDoubleBracket] = z\[LeftDoubleBracket]i\[RightDoubleBracket] + Sqrt[h\[LeftDoubleBracket]i\[RightDoubleBracket]]; $, {i, q + 1, n + q}]; \n\t\t\t\t{Drop[h, q], Drop[e, q]\ }]]\)], "Input"], Cell["The following cells generates plot of some GARCH data.", "Text"], Cell[BoxData[ $ListPlot[ \(ARCHList[1000, 0.05, {0.3, 0.2, 0.15}, {0.2, 0.1}] $\[LeftDoubleBracket]1\[RightDoubleBracket], PlotJoined -> True, Frame -> True, PlotRange -> All, PlotStyle -> {AbsoluteThickness[2], GrayLevel[0.5]}]\)], "Input"], Cell[BoxData[ $ListPlot[ \(ARCHList[1000, 0.05, {0.9, 0., 0.04}, {0.0, 0.01}] $\[LeftDoubleBracket]1\[RightDoubleBracket], PlotJoined -> True, Frame -> True, PlotRange -> All, PlotStyle -> {AbsoluteThickness[2], GrayLevel[0.5]}]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["EGARCH: artificial time series", "Section"], Cell[TextData[{ "These functions are based on the time series generation procedures written \ by Pedro J F de Lima of the Johns Hopkins University in the GAUSS programming \ language, available at ", StyleBox["ftp://speedy.econ.jhu.edu/public/gauss/st_proc.prc", FontFamily->"Courier", FontColor->RGBColor[0, 0, 1]], ". Note that I have used some of the functions in the standard ", StyleBox["Mathematica", FontSlant->"Italic"], " package \"LinearAlgebra`MatrixManipulation`\"." }], "Text"], Cell[BoxData[ $Needs["\"]$], "Input"], Cell[CellGroupData[{ Cell[BoxData[ $$Packages$], "Input"], Cell[BoxData[ ${"LinearAlgebra`MatrixManipulation`", "Statistics`Common`DistributionsCommon`", "Statistics`DescriptiveStatistics`", "Statistics`NormalDistribution`", "Statistics`ContinuousDistributions`", "Global`", "System`"}$], "Output"] }, Open ]], Cell[BoxData[ $EGARCHList[z : {__Real}, uncmean_, arpar_?VectorQ, mapar_?VectorQ, {\[Theta]1_, \[Theta]2_}] := \n\t With[{p = Length[arpar], q = Length[mapar] + 1, const = uncmean + \((1 - Plus@@arpar)$, n = Length[z]}, \ \n\t\t Module[{zlag, gz, \[Sigma]2}, \n\t\t\t zlag = $Take[RotateLeft[z, #], n]&$\ /@\ Range[0, q - 1]; \n\t\t\t gz = AppendRows[ZeroMatrix[q], \[Theta]1*zlag + \[Theta]2\ *\ $(Abs[zlag] - Sqrt[2./Pi])$]; \n\t\t\t\[Sigma]2 = ARMAList[const + Reverse[Join[{1}, mapar]].gz, uncmean, arpar, {0}]; \n \t\t\t{z*Sqrt[Exp[Drop[\[Sigma]2, q - 1]\ ]], Drop[\[Sigma]2, q - 1]}\ ]]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Fractional Noise: three methods.", "Section"], Cell[TextData[{ "These functions are based on the time series generation procedures written \ by Pedro J F de Lima of the Johns Hopkins University in the GAUSS programming \ language, available at ", StyleBox["ftp://speedy.econ.jhu.edu/public/gauss/gen_fnoi.prc", FontFamily->"Courier", FontColor->RGBColor[0, 0, 1]], ". There are three methods, finite Fourier transforms of Davies and Harte \ (1987), Hosking's recursive method (1984) and a method based on the Cholesky \ Decomposition. ( Note that this requires the standard Cholesky package to be \ read in first.) The choice between the three methods is handled as an option \ to the main function, then three subsidiary functions are used. These \ implementations are preliminary, as I cannot give any guarantees that I have \ understood the GAUSS code properly. In particular, the Hoskings method gives \ results that look rather different to the other two.\n" }], "Text"], Cell["This reads in the package.", "Text"], Cell[BoxData[ $Needs["\"]$], "Input"], Cell[TextData[{ "To do the Cholesky decomposition method, you also need this subsidiary \ function. \nToeplitz matrices are n \[Times] n symmetric matrices with \ elements ", Cell[BoxData[ $TraditionalForm\`a\_\(i, j$ = a\_$i - 1, j - 1$\)]], " for i,j = 2\[Ellipsis]n." }], "Text"], Cell[BoxData[ $createToeplitzMatrix[vec_?VectorQ] := \ With[{n = Length[vec]}, Array[vec\[LeftDoubleBracket]Abs[#1 - #2] + 1\[RightDoubleBracket]&, { n, n}]]$], "Input"], Cell[CellGroupData[{ Cell[BoxData[ $createToeplitzMatrix[{\[Alpha], \[Beta], \[Gamma], \[Delta], \[Epsilon]}] // MatrixForm$], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {"\[Alpha]", "\[Beta]", "\[Gamma]", "\[Delta]", "\[Epsilon]"}, {"\[Beta]", "\[Alpha]", "\[Beta]", "\[Gamma]", "\[Delta]"}, {"\[Gamma]", "\[Beta]", "\[Alpha]", "\[Beta]", "\[Gamma]"}, {"\[Delta]", "\[Gamma]", "\[Beta]", "\[Alpha]", "\[Beta]"}, {"\[Epsilon]", "\[Delta]", "\[Gamma]", "\[Beta]", "\[Alpha]"} }], ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell["It isn't the fastest, but it works.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ $Table[ Timing[createToeplitzMatrix[Table[Random[], {i}]]; ], {i, 50, 300, 50}] $], "Input"], Cell[BoxData[ ${{0.149999999999998578`\ Second, Null}, {0.583333333333333925`\ Second, Null}, {1.28333333333333321`\ Second, Null}, { 2.33333333333333392`\ Second, Null}, {3.63333333333333285`\ Second, Null}, {5.216666666666665`\ Second, Null}}$], "Output"] }, Open ]], Cell["This sets up the FFT method as the default.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ $Options[generateFractionalNoise] = {Method \[Rule] FiniteFourier}$], "Input"], Cell[BoxData[ ${Method \[Rule] FiniteFourier}$], "Output"] }, Open ]], Cell[BoxData[ $generateFractionalNoise[n_Integer, d_Real, opts___?OptionQ] /; \((d > \(-0.5$\ && \ d < 0.5)\) := \n\t Module[{way}, way = $Method /. {opts}$ /. Options[generateFractionalNoise]; \n\t\t Switch[way, FiniteFourier, fourierFNoise[n, d], \ Hoskings, hoskingsFNoise[n, d], CholeskiDecomp, cholFNoise[n, d]]\ ]\)], "Input"], Cell[BoxData[ $fourierFNoise[n_Integer, d_Real] /; \((d > \(-0.5$\ && \ d < 0.5)\) := \n\t Module[{gk, u, v, z, x1, auto}, With[{\[ScriptN] = n + 1}, With[{rhopart = Thread[$(Range[\[ScriptN]] - 1 + d)$/ $(Range[\[ScriptN]] - d)$], auto0 = Gamma[1 - 2\ d]/$(Gamma[1 - d]^2)$}, With[{\[Rho] = FoldList[Plus, First[rhopart], Rest[rhopart]]}, \n \t\t\t\tauto = auto0*\[Rho]; \n\t\t\t\t gk = Re[Fourier[ Join[{auto0}, Drop[auto, $-1$], Reverse[Drop[auto, $-2$]]]]\ ]; \n\t\t\t\t u = Thread[ Join[{\@2.}, Table[1, {\[ScriptN] - 2}], {\@2.}]* RandomArray[NormalDistribution[0, 1], \[ScriptN]]]; \n \t\t\t\tv = Join[{0.}, RandomArray[NormalDistribution[0, 1], \[ScriptN] - 2], { 0.}]; \n\t\t\t\t z = Join[Thread[u + $(I*v)$], Reverse[ Thread[Drop[Rest[u], $-1$] + $(I*Drop[Rest[v], \(-1$])\)]]]; \n\t\t\t\t x1 = 0.5*Re[Fourier[Thread[\@gk*z]]]/\@$\[ScriptN] - 1$; \n \t\t\t\tTake[x1, \[ScriptN] - 1] - $(Plus@@Take[x1, \[ScriptN] - 1])$/$(\[ScriptN] - 1)$]]]] \)], "Input"], Cell[BoxData[ $hoskingsFNoise[n_Integer, d_Real] /; \((d > \(-0.5$\ && \ d < 0.5)\) := \n\t Module[{vlist, \[Phi]tt, \[Phi]lists}, \n\t With[{k = Range[n], v0 = Gamma[1 - 2\ d]/$(Gamma[1 - d]^2)$}, \n \t\t\t\[Phi]tt = d/$(k - d)$; \n\t\t\t vlist = Drop[ FoldList[$(1 - \((#2)$^2\ )\)*\ #1\ &\ , v0, \[Phi]tt], $-1$]; \n\t\t\t \[Phi]lists = FoldList[Flatten[Join[{#1} - #2*\ Reverse[{#1}], {#2}]]&\ , \ First[\[Phi]tt], Rest[\[Phi]tt]]; \n\t\t\ \ Rest[Flatten[ Last[\n\t\t\t\t\t\t FoldList[ Join[{#1}, Flatten[{ \@$(#2\[LeftDoubleBracket]1\[RightDoubleBracket]) $\ *Random[NormalDistribution[0, 1]] + $(Flatten[{#2\[LeftDoubleBracket]2 \[RightDoubleBracket]}].Reverse[ Flatten[{#1}]])$}]]\ &, \ \@v0*Random[NormalDistribution[0, 1]], Transpose[{vlist, \[Phi]lists}]]\ ]]]\ ]]\)], "Input"], Cell[BoxData[ $\(cholFNoise[n_Integer, d_Real] /; \((d > 0.\ && \ d < 0.5)$ := \n\t With[{\[Tau] = Range[n], \[Sigma]e = 1}, \n\t With[{\tauto1 = Exp[LogGamma[1] + LogGamma[1 - 2\ d] - LogGamma[d] - LogGamma[1 - d] + LogGamma[d + \[Tau]] - LogGamma[1 - d + \[Tau]]]}, \n\t\t Transpose[ CholeskyDecomposition[createToeplitzMatrix[auto1]]].RandomArray[ NormalDistribution[0, 1], n]\ ]]\n\t\t\)\)], "Input"], Cell["\<\ The Fourier method seems to be fastest. The Timings below are from \ a Power Macintosh G3 233 MHz with 64 Mb of RAM and virtual memory on.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ $ListPlot[generateFractionalNoise[500, 0.49999], PlotJoined \[Rule] True] // Timing$], "Input"], Cell["{0.333333 Second}", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ $ListPlot[generateFractionalNoise[100, 0.49, Method \[Rule] Hoskings], PlotJoined \[Rule] True] // Timing$], "Input"], Cell["{0.233333 Second}", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ $ListPlot[ generateFractionalNoise[100, 0.49, Method \[Rule] CholeskiDecomp], PlotJoined \[Rule] True] // Timing$], "Input"], Cell["{2.13333 Second}", "Text"] }, Open ]] }, Closed]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 1152}, {0, 850}}, AutoGeneratedPackage->None, PrintingStyleEnvironment->"Working", WindowToolbars->{}, CellGrouping->Manual, WindowSize->{887, 800}, WindowMargins->{{98, Automatic}, {7, Automatic}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, MacintoshSystemPageSetup->"\<\ 00<0001804P000000`d26_oQon@3:`8g0dL5N`?P0080001804P000000]P2:001 0000I00000400`<30?l00BL?00400@0000000000000006P801T1T00000000000 00000000004000000000000000000000\>" ] (*********************************************************************** Cached data follows. 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