{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "5FM1TC4FG4"
    },
    {
      "field": "uses",
      "target": "9MPM6E51WV"
    },
    {
      "field": "uses",
      "target": "V192HFRX5A"
    },
    {
      "field": "defines",
      "target": "F8T3MJX3VJ"
    }
  ],
  "body": "\n## Summary\n\nAn autoregressive–moving-average model, or ARMA($p$, $q$) model, combines an [[autoregressive model|2JC96FMGA1]] of order $p$ with a [[moving-average model|PEB9DN0AHA]] of order $q$.\nEach value is a linear combination of $p$ past values plus the current shock and $q$ past shocks.\nARMA models describe a wide range of stationary time series with few parameters, and they are the core of the Box–Jenkins approach to forecasting.\n\n## Explanations\n\n### L1 human\n\nAn autoregressive part lets the series remember its own past values, and a moving-average part lets recent shocks echo for a few steps.\nUsing both parts often describes the data with fewer numbers than using either part alone.\nFor example, $X_t = 0.5 X_{t-1} + \\varepsilon_t + 0.3 \\varepsilon_{t-1}$ is an ARMA($1$, $1$) model.\nForecasts from it decay towards the mean at the rate $0.5$ per step, after a first step that also uses the last shock.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "62M10",
      "91B84"
    ],
    "name": "Autoregressive–moving-average model",
    "same_as": {
      "wikidata": "Q290467",
      "wikipedia": "Autoregressive moving-average model"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Autoregressive_moving-average_model"
      }
    ],
    "tag": "B1HNX0YT0Z"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/B1HNX0YT0Z/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/B1HNX0YT0Z.md",
  "msc": [
    "62M10",
    "91B84"
  ],
  "name": "Autoregressive–moving-average model",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/B1H/B1HNX0YT0Z.md",
  "status": null,
  "tag": "B1HNX0YT0Z",
  "uri": "m2://B1HNX0YT0Z",
  "url": "https://tamnd.github.io/math2/wiki/autoregressive-moving-average-model/",
  "writing": {
    "tag": "B1HNX0YT0Z",
    "kind": "concept",
    "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
      }
    ]
  }
}