2a:null 2b:["$","title",null,{"children":"Time-varying Beta model with dynamic shrinkage process prior | Mattias Villani | Observable"}] 31:T558,The **time-varying Beta model with dynamic shrinkage process** is of the form ${tex.block` \begin{aligned} x_t \vert \theta_t & \overset{\mathrm{ind}}{\sim} \mathrm{Beta} \big(\mu_t \psi, (1-\mu_t)\psi \big) \\ \mu_t &= \frac{\exp(\theta_t)}{1 + \exp(\theta_t)} \\ \theta_t &= \theta_{t-1} + \nu_t, \hspace{1.75cm} \nu_t \overset{ind}{\sim} N(0, \tau^2 \exp(h_t)) \\ h_t &= \phi \log h_{t-1} + \eta_t, \hspace{1cm} \eta_t \overset{iid}{\sim}Z(\alpha, \beta, 0, \sigma_\eta) \end{aligned} `} where ${tex`\psi`} is a static precision-type parameter since ${tex`\mathbb{V}(x_t)=\frac{\mu_t(1-\mu_t)}{1+\psi}`}, and ${tex`Z(\alpha, \beta, \mu, \sigma)`} is the heavy tailed [Z-distribution](https://observablehq.com/@mattiasvillani/z-distribution). The dynamic shrinkage process is a time series extension of the horseshoe prior from regression. The connection can be seen by setting the persistence of the local volatility ${tex`\phi`} to zero. It can be shown that ${tex.block` \eta_t \sim Z(1/2, 1/2, 0, 1) \Longleftrightarrow \lambda_t = \exp(h_t/2) = \exp(\eta_t/2) \sim C⁺(0,1), `} so that the **local standard deviation** ${tex`\lambda_t`} follows the half Cauchy in the original horseshoe prior. The global standard deviation ${tex`\tau`} can also be assigned the ${tex`C⁺(0,1)`} prior as in the original horseshoe, or fixed to some suitable value. 32:T860,timeplot = { let xt = simulatedData.filter(d => d.type == "time series, xₜ").map((d) => d.x); let mut = simulatedData.filter(d => d.type == "data mean, μₜ").map((d) => d.x); let taulambdat = simulatedData.filter(d => d.type == "total stdev, τ⋅λₜ").map((d) => d.x); const y2 = d3.scaleLinear(d3.extent(taulambdat), d3.extent(xt)); return Plot.plot({ //title: md`Local level model with error stdev ${tex`\sigma_\varepsilon = ${input.sigmaeps}`} and // global stdev ${tex`\tau = ${input.tau.toFixed(5)}`}`, y: {axis: "left", label: "xₜ, exp(θₜ)", domain: d3.extent(xt.concat(mut))}, color: { legend: true, domain: ["time series, xₜ", "data mean, μₜ", "total standard deviation, τ⋅λₜ"], range: [colors[0], colors[1], colors[2]] //scheme: d3.Paired }, marks: [ Plot.axisY(y2.ticks(), {filter: plotselector.includes("total stdev, τ⋅λₜ"), color: colors[2], anchor: "right", label: labelrightaxis, y: y2, tickFormat: y2.tickFormat()}), Plot.ruleX([0]), Plot.ruleX([input.n], {filter: plotselector.includes("total stdev, τ⋅λₜ")}), Plot.ruleY([d3.min(simulatedData.map(d => d.x))]), Plot.lineY(simulatedData, {filter: d => d.type == "time series, xₜ" && plotselector.includes("time series, xₜ"), x: "time", y: "x", stroke: "type", strokeWidth: 2}), Plot.dot(simulatedData, {filter: d => d.type == "time series, xₜ" && plotselector.includes("time series, xₜ"), x: "time", y: "x", stroke: "type", strokeWidth: 0.5, fill: "type"}), Plot.lineY(simulatedData, {filter: d => d.type == "data mean, μₜ" && plotselector.includes("data mean, μₜ"), x: "time", y: "x", stroke: "type", strokeWidth: 2}), Plot.lineY(simulatedData, Plot.mapY((D) => D.map(y2), { filter: d => d.type == "total stdev, τ⋅λₜ" && plotselector.includes("total stdev, τ⋅λₜ"), x: "time", y: "x", stroke: colors[2]})) ] }) }33:[] 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{value: 0.01, step: 0.001, label: tex`\\text{global std, }\\tau`}),\n phi: Inputs.range([-0.99, 0.99], {value: 0.7, step: 0.01, label: tex`\\text{persistence, }\\phi`}),\n alpha: Inputs.range([Number.EPSILON, 3], {value: 0.5, step: 0.1, label: tex`\\alpha`}),\n beta: Inputs.range([Number.EPSILON, 3], {value: 0.5, step: 0.1, label: tex`\\beta`}),\n sigma: Inputs.range([Number.EPSILON, 3], {value: 1, step: 0.1, label: tex`\\sigma_\\eta`})\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1282,"value":"viewof plotselector = Inputs.checkbox([\"time series, xₜ\", \"data mean, μₜ\", \"total stdev, τ⋅λₜ\"], {label: \"plot:\", value: [\"time series, xₜ\", \"data mean, μₜ\"]})","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":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":13,"value":"
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\n\n---\n## 🙈 Appendix 🙈\nHere follow code and details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":15,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":11,"value":"math = require(\"mathjs\");","pinned":false,"hidden":false,"mode":"ojs"},{"id":1185,"value":"import {colors, simBetaDSP} from \"@mattiasvillani/statistics-tools\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":816,"value":"jStat = require(\"jstat\");","pinned":false,"hidden":false,"mode":"ojs"},{"id":96,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":1399,"value":"showdist = plotselector","pinned":false,"hidden":false,"mode":"ojs"},{"id":1013,"value":"n = input.n","pinned":false,"hidden":false,"mode":"ojs"},{"id":1394,"value":"labelrightaxis = {\n if (plotselector.includes(\"total stdev, τ⋅λₜ\")){ return \"τ⋅λₜ\"}\n else{\n return \"\"\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":94,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md"},{"id":519,"value":"simulatedData = {\n simulatebutton\n return simBetaDSP(n, input.psi, input.tau, input.phi, input.alpha, input.beta, input.sigma, input.initval)\n}","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}} 2f:"$2e:body"