31:[] 2f:{"title":"Stochastic volatility time series model","cells":[{"id":0,"value":"# Stochastic volatility time series model","pinned":false,"hidden":false,"mode":"md"},{"id":194,"value":"Many time series, particularly in the finance, has a variance that is changing over time. Furthermore, it is common to find volatility clustering in the data, meaning that once the the variance is high (turbulent stock market) it tends to remain high for a while and vice versa. The basic stochastic volatility (SV) model tries to capture this:\n\n${tex`\n\\begin{aligned}\ny_t &= \\mu_y + \\varepsilon_t, \\hspace{2.5cm} \\varepsilon_t \\overset{\\mathrm{indep}}{\\sim}N(0,\\exp(z_t)), \\\\ \nz_t &= \\mu_z + \\phi(z_{t-1} -\\mu_z) + \\eta_t, \\quad \\eta_t \\overset{\\mathrm{iid}}{\\sim}N(0, \\sigma^2_\\eta)\n\\end{aligned}\n`}\nNote that ${tex`\\sigma^2_t = \\exp(z_t)`} is guaranteed to be positive for all *t*, as required.\n","pinned":false,"hidden":false,"mode":"md"},{"id":463,"value":"#### Settings","pinned":false,"hidden":false,"mode":"md"},{"id":162,"value":"viewof input = Inputs.form({\n n: Inputs.range([10, 1000], {value: 100, step: 1, label: \"time series length\"}),\n muy: Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\\text{process mean }\\mu_y`}),\n muz: Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\\text{mean log variance }\\mu_z`}),\n phi: Inputs.range([-0.99, 0.99], {value: 0.8, step: 0.01, label: tex`\\text{AR(1) parameter }\\phi`}),\n sigmaeta: Inputs.range([0.01, 2], {value: 1, step: 0.1, label: tex`\\text{std innovation, }\\sigma_\\eta`})\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1118,"value":"viewof plotstd = Inputs.toggle({label: \"plot standard deviation exp(zₜ/2)\", value: true})","pinned":false,"hidden":false,"mode":"ojs"},{"id":781,"value":"viewof simulatebutton = Inputs.button(\"Simulate new data\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1041,"value":"timeplot = Plot.plot({\n color: {\n legend: true,\n domain: [\"time series, yₜ\", \"log standard deviation, σₜ\"],\n //scheme: d3.Paired\n },\n marks: [\n Plot.ruleX([0]),\n Plot.ruleY([d3.min(simulatedData.map(d => d.y))]),\n Plot.lineY(simulatedData, {filter: plotstd || (d => d.type == \"time series, yₜ\"), x: \"time\", y: \"y\", stroke: \"type\", \n strokeWidth: 2, tip: true})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":13,"value":"
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\n\n---\n## 🙈 Appendix 🙈\nHere follow code and details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":15,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":11,"value":"math = require(\"mathjs\");","pinned":false,"hidden":false,"mode":"ojs"},{"id":816,"value":"jStat = require(\"jstat\");","pinned":false,"hidden":false,"mode":"ojs"},{"id":96,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":1013,"value":"n = input.n","pinned":false,"hidden":false,"mode":"ojs"},{"id":94,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md"},{"id":1004,"value":"function simStochVol(n, muy, muz, phi, sigmaeta) {\n\n simulatebutton\n\n const data = [];\n let z = muz;\n for (let t = 0; t < n; t++) {\n z = muz + phi*(z - muz) + sigmaeta*d3.randomNormal()();\n data.push({time: t + 1, y: Math.exp(z/2), type: \"log standard deviation, σₜ\"});\n data.push({time: t + 1, y: muy + Math.exp(z/2)*d3.randomNormal()(), type: \"time series, yₜ\"});\n }\n\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":519,"value":"simulatedData = simStochVol(n, input.muy, input.muz, input.phi, input.sigmaeta)","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W31","stdlib":"1","titling":"auto"}