29:null 2a:["$","title",null,{"children":"Identification of the mean in an AR(1) process | Mattias Villani | Observable"}] 2f:[] 2d:{"id":"38525a45d36d7a54","slug":"ident-mean-ar-1-process","title":"Identification of the mean in an AR(1) process","thumbnail":"3f8e82e871b4d5324b2e7658c03d28f6ee744892c668ac93920e14211a7e2572","access_level":"public","create_time":"$D2024-07-27T19:46:13.069Z","update_time":"$D2024-07-27T19:46:13.069Z","view_time":null,"first_public_time":"$D2024-07-27T19:46:13.069Z","has_unpublished_edits":null,"fork":{"id":"2c363584b53a6804","slug":"identification-ar-1-process","title":"Identification of the mean in an AR(1) process","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":0,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"education_2026"},"version":327,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null}],"members":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":1,"collection":{"id":"b4f68ca0e73a1ae6","slug":"time-series","title":"Time Series","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Identification of the mean in an AR(1) process","cells":[{"id":0,"value":"# Identification of the mean in an AR(1) process","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":311,"value":"> This notebook explores the likelihood for the mean/intercept and the AR parameter in the AR(1) process, with a view to identification issues.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":147,"value":"The AR(1) model in **standard form** is\n\n${tex`\nx_t = c + \\phi x_{t-1} + \\varepsilon_t, \\hspace{1cm} \\varepsilon_t \\sim N(0,\\sigma^2)\n`}\n\nwhere ${tex`c`} is the *intercept*.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":159,"value":"The AR(1) model in **steady-state form** is\n\n${tex`\nx_t = \\mu + \\phi(x_{t-1} - \\mu) + \\varepsilon_t, \\hspace{1cm} \\varepsilon_t \\sim N(0,\\sigma^2)\n`}\n\nwhere ${tex`\\mu = \\mathbb{E}(x_t)`} is the *unconditional mean* of the process.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":167,"value":"The steady-state form is attractive since the mean is more interpretable than an intercept, but, as shown by the widget below, the mean ${tex`\\mu`} becomes more and more weakly identified in the likelihood as ${tex`\\phi \\rightarrow 1`}. In the limit, the mean doesn't even exist.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":179,"value":"#### Data generating process for the simulated data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":18,"value":"viewof param = Inputs.form({\n mu: Inputs.range([-1, 1], {value: 0, label: tex`\\mu`, step: 0.1}),\n phi: Inputs.range([-0.999, 0.999], {value: 0, label: tex`\\phi`, step: 0.01}),\n sigma: Inputs.range([0.1, 2], {value: 0.5, label: tex`\\sigma`, step: 0.1}),\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":20,"value":"viewof T = Inputs.range([1, 1000], {label: \"time series length\", step: 1})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":210,"value":"The implied intercept in the data generating process is ${tex`c = `} ${intercept}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":169,"value":"bothplots = html`\n \n \n \n

\n ${pltmean}\n

\n ${pltintercept}\n

\n \n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"
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\n\n---\n#### 🙈 Appendix 🙈\n\nHere is just some code and details that you probably don't want to see.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":174,"value":"#### Detached plots","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":125,"value":"pltintercept = Plot.plot({\n title: \"Log likelihood in ϕ-c space\",\n width: 500, // Set the width\n height: 500, // Set the height equal to the width for a square aspect\n x: {label: \"c\", domain: [-1,1]},\n y: {label: \"ϕ\", domain: [-1.5,1.5]},\n color: {\n legend: true,\n scheme: \"viridis\",\n type: \"log\",\n label: \"loglik\",\n symmetric: false,\n //domain: [-1000,-700]\n //transform: d => Math.exp(d - maxloglikIntercept)\n },\n marks: [\n Plot.contour(loglikdataintercept, {x: \"c\", y: \"phi\", fill: \"loglik\", stroke: \"black\", blur: 4, thresholds: 100}),\n Plot.dot([{x: intercept, y: param.phi}], {x: \"x\", y: \"y\", fill: \"orange\", r: 4}),\n Plot.tip(loglikdataintercept, Plot.pointerX({x: \"c\", y: \"phi\", stroke: \"loglik\"}))\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":63,"value":"pltmean = Plot.plot({\n title: \"Log likelihood in ϕ-μ space\",\n width: 500, // Set the width\n height: 500, // Set the height equal to the width for a square aspect\n x: {label: \"μ\", domain: [-1,1]},\n y: {label: \"ϕ\", domain: [-1.5,1.5]},\n color: {\n legend: true,\n scheme: \"viridis\",\n type: \"log\",\n label: \"loglik\", \n symmetric: false, \n domain: [minloglik, maxloglik],\n //range: [\"red\",\"blue\"]\n //transform: d => d - maxloglik\n //transform: (d) => Math.exp(d)\n },\n marks: [\n Plot.contour(loglikdata, {x: \"mu\", y: \"phi\", fill: \"loglik\", stroke: \"black\", blur: 4, thresholds: 100}),\n Plot.dot([{x: param.mu, y: param.phi}], {x: \"x\", y: \"y\", fill: \"orange\", r: 4}),\n Plot.tip(loglikdata, Plot.pointerX({x: \"mu\", y: \"phi\", stroke: \"loglik\"}))\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":9,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":11,"value":"import {simulateAR, logLikeCondAR, logLikeCondARIntercept} from \"@mattiasvillani/statistics-tools\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":16,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":22,"value":"x = simulateAR(1000, param.mu, [param.phi], param.sigma)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":217,"value":"intercept = param.mu*(1-param.phi)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":14,"value":"loglikdatawithmax = {\n const timeseries = x.filter(d => d.time <=T).map(d => d.x);\n const mugrid = d3.range(-1, 1, 11/100);\n const phigrid = d3.range(-1.5, 1.5, 0.01);\n const logLik = []\n let maxloglik = -Infinity\n let minloglik = Infinity\n for (const mu of mugrid){\n for (const phi of phigrid){\n const logLikVal = logLikeCondAR(timeseries, mu, phi, param.sigma)\n logLik.push({mu: mu, phi: phi, loglik: logLikVal})\n maxloglik = d3.max([logLikVal, maxloglik])\n minloglik = d3.min([logLikVal, minloglik])\n }\n }\n return [logLik, minloglik, maxloglik]\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":42,"value":"loglikdata = loglikdatawithmax[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":117,"value":"loglikdatainterceptwithmax = {\n const timeseries = x.filter(d => d.time <=T).map(d => d.x);\n const cgrid = d3.range(-1, 1, 11/100);\n const phigrid = d3.range(-1.5, 1.5, 0.01);\n const logLik = []\n let maxloglik = -Infinity\n for (const c of cgrid){\n for (const phi of phigrid){\n const logLikVal = logLikeCondARIntercept(timeseries, c, phi, param.sigma)\n logLik.push({c: c, phi: phi, loglik: logLikVal})\n maxloglik = d3.max([logLikVal, maxloglik])\n }\n }\n return [logLik, maxloglik]\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":108,"value":"maxloglik = loglikdatawithmax[2]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":280,"value":"minloglik = loglikdatawithmax[1]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":120,"value":"maxloglikIntercept = loglikdatainterceptwithmax[1]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":122,"value":"loglikdataintercept = loglikdatainterceptwithmax[0]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W2f","stdlib":"1","titling":"auto"}} 2e:"$2d:body"