{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $q \\ge 1$ be an integer.\nA real [[time series|EA3MF4VTEH]] $(\\varepsilon_t)_{t \\in \\Z}$ is an *ARCH($q$) process* if it satisfies the following conditions.\n\n1. $\\varepsilon_t = \\sigma_t z_t$ for all $t$, where $(z_t)_{t \\in \\Z}$ are [[independent and identically distributed|00MRN0FARV]] random variables with mean $0$ and variance $1$.\n2. $\\sigma_t^2 = \\alpha_0 + \\sum_{i=1}^{q} \\alpha_i \\varepsilon_{t-i}^2$ for all $t$, where $\\alpha_0 \u003e 0$ and $\\alpha_1, \\dots, \\alpha_q \\ge 0$ are constants and $\\sigma_t \\ge 0$.\n3. $z_t$ is independent of $(\\varepsilon_s)_{s \u003c t}$ for all $t$.\n\nThen $\\sigma_t^2$ is the [[conditional variance|0VGPX25NXQ]] of $\\varepsilon_t$ given the past $(\\varepsilon_s)_{s \u003c t}$.\n\n## Remarks\n\nThe GARCH($p$, $q$) model replaces clause 2 by $\\sigma_t^2 = \\alpha_0 + \\sum_{i=1}^{q} \\alpha_i \\varepsilon_{t-i}^2 + \\sum_{j=1}^{p} \\beta_j \\sigma_{t-j}^2$ with $\\beta_j \\ge 0$.\n\nExample: take $q = 1$, $\\alpha_0 = 1$ and $\\alpha_1 = 1/2$.\nIf the process is weakly stationary with variance $v$, then taking expectations in clause 2 gives $v = 1 + v/2$, so $v = 2$.\nIn general a weakly stationary ARCH($q$) process exists if and only if $\\alpha_1 + \\cdots + \\alpha_q \u003c 1$, and then its variance is $\\alpha_0 / (1 - \\alpha_1 - \\cdots - \\alpha_q)$.\n\nNon-example: the rule $\\sigma_t^2 = 1 - \\frac12 \\varepsilon_{t-1}^2$ fails clause 2 because $\\alpha_1 \u003c 0$.\nIt gives a negative \"variance\" as soon as $\\abs{\\varepsilon_{t-1}} \u003e \\sqrt{2}$.\n\nAn ARCH process is uncorrelated in time, but it is not independent: the squares $\\varepsilon_t^2$ follow an autoregressive pattern.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "MPN10SB26R",
    "kind": "definition",
    "msc": [
      "62M10",
      "91G70",
      "62P20"
    ],
    "name": "Autoregressive conditional heteroskedasticity",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity"
      }
    ],
    "tag": "P3YM2DDGX9",
    "uses": [
      "EA3MF4VTEH",
      "0VGPX25NXQ",
      "D8VX6XQ3TT",
      "00MRN0FARV"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/P3YM2DDGX9/",
  "links": [
    {
      "field": "defines",
      "target": "MPN10SB26R"
    },
    {
      "field": "uses",
      "target": "EA3MF4VTEH"
    },
    {
      "field": "uses",
      "target": "0VGPX25NXQ"
    },
    {
      "field": "uses",
      "target": "D8VX6XQ3TT"
    },
    {
      "field": "uses",
      "target": "00MRN0FARV"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/P3YM2DDGX9.md",
  "msc": [
    "62M10",
    "91G70",
    "62P20"
  ],
  "name": "Autoregressive conditional heteroskedasticity",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/P3Y/P3YM2DDGX9.md",
  "status": null,
  "tag": "P3YM2DDGX9",
  "uri": "m2://P3YM2DDGX9",
  "url": "https://tamnd.github.io/math2/wiki/autoregressive-conditional-heteroskedasticity-definition/",
  "writing": {
    "tag": "P3YM2DDGX9",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}