Strict Exogeneity.ipynb

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   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import graphviz as gr\n",
    "import statsmodels.formula.api as smf"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b1e807e3",
   "metadata": {},
   "source": [
    "## Strict Exogeneity"
   ]
  },
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    "g = gr.Digraph()\n",
    "\n",
    "g.edge(\"ui\", \"y1\")\n",
    "g.edge(\"ui\", \"y2\")\n",
    "g.edge(\"ui\", \"y3\")\n",
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   "source": [
    "np.random.seed(123)\n",
    "n = 1000\n",
    "\n",
    "ui = np.random.normal(0, 1, n)\n",
    "unit = np.array(range(n))\n",
    "\n",
    "x1 = np.random.normal(ui, 1, n)\n",
    "y1 = np.random.normal(x1+ui, 1, n)\n",
    "\n",
    "x2 = np.random.normal(ui+x1, 1, n)\n",
    "y2 = np.random.normal(x2+ui, 1, n)\n",
    "\n",
    "x3 = np.random.normal(ui+x1+x2, 1, n)\n",
    "y3 = np.random.normal(x3+ui, 1, n)\n",
    "\n",
    "df = pd.concat([\n",
    "    pd.DataFrame(dict(y=y1, x=x1, t=1, i=unit)),\n",
    "    pd.DataFrame(dict(y=y2, x=x2, t=2, i=unit)),\n",
    "    pd.DataFrame(dict(y=y3, x=x3, t=3, i=unit)),\n",
    "])"
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   "source": [
    "m = smf.ols(\"y~x\", data=df).fit()\n",
    "m.params[\"x\"]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
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    "m = smf.ols(\"y~x+C(i)\", data=df).fit()\n",
    "m.params[\"x\"]"
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   "cell_type": "markdown",
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   "source": [
    "## Feedback\n",
    "\n",
    "$$\n",
    "Y_{it-1} \\rightarrow X_{it}\n",
    "$$"
   ]
  },
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       "<!-- x3&#45;&gt;y3 -->\n",
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       "<title>x3&#45;&gt;y3</title>\n",
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       "</g>\n",
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     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = gr.Digraph()\n",
    "\n",
    "g.edge(\"ui\", \"y1\")\n",
    "g.edge(\"ui\", \"y2\")\n",
    "g.edge(\"ui\", \"y3\")\n",
    "\n",
    "g.edge(\"ui\", \"x1\")\n",
    "g.edge(\"ui\", \"x2\")\n",
    "g.edge(\"ui\", \"x3\")\n",
    "\n",
    "g.edge(\"x1\", \"y1\")\n",
    "g.edge(\"x2\", \"y2\")\n",
    "g.edge(\"x3\", \"y3\")\n",
    "\n",
    "g.edge(\"x1\", \"x2\")\n",
    "g.edge(\"x1\", \"x3\")\n",
    "g.edge(\"x2\", \"x3\")\n",
    "\n",
    "\n",
    "g.edge(\"y1\", \"x2\")\n",
    "g.edge(\"y2\", \"x3\")\n",
    "\n",
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "647c6903",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-12-15T19:05:32.014768Z",
     "start_time": "2022-12-15T19:05:32.005704Z"
    }
   },
   "outputs": [],
   "source": [
    "np.random.seed(123)\n",
    "\n",
    "n = 1000\n",
    "\n",
    "ui = np.random.normal(0, 1, n)\n",
    "unit = np.array(range(n))\n",
    "\n",
    "x1 = np.random.normal(ui, 1, n)\n",
    "y1 = np.random.normal(x1+ui, 1, n)\n",
    "\n",
    "x2 = np.random.normal(ui+x1+y1, 1, n)\n",
    "y2 = np.random.normal(x2+ui, 1, n)\n",
    "\n",
    "x3 = np.random.normal(ui+x2+y1+y2, 1, n)\n",
    "y3 = np.random.normal(x3+ui, 1, n)\n",
    "\n",
    "df = pd.concat([\n",
    "    pd.DataFrame(dict(y=y1, x=x1, t=1, i=unit, ui=ui)),\n",
    "    pd.DataFrame(dict(y=y2, x=x2, t=2, i=unit, ui=ui)),\n",
    "    pd.DataFrame(dict(y=y3, x=x3, t=3, i=unit, ui=ui)),\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "c80f6f91",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-12-15T19:05:35.950808Z",
     "start_time": "2022-12-15T19:05:32.016256Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9862243767610324"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = smf.ols(\"y~x+C(i)\", data=df).fit()\n",
    "m.params[\"x\"]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7a8e4ee8",
   "metadata": {},
   "source": [
    "Expanding a residual $e_{it}$ on $Y_{it}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "00c11725",
   "metadata": {},
   "source": [
    "## Carryover\n",
    "\n",
    "$$\n",
    "X_{it-1} \\rightarrow Y_{it}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "a8f4f0c2",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-12-15T19:05:36.178445Z",
     "start_time": "2022-12-15T19:05:35.952585Z"
    }
   },
   "outputs": [
    {
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     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = gr.Digraph()\n",
    "\n",
    "g.edge(\"ui\", \"y1\")\n",
    "g.edge(\"ui\", \"y2\")\n",
    "g.edge(\"ui\", \"y3\")\n",
    "\n",
    "g.edge(\"ui\", \"x1\")\n",
    "g.edge(\"ui\", \"x2\")\n",
    "g.edge(\"ui\", \"x3\")\n",
    "\n",
    "g.edge(\"x1\", \"y1\")\n",
    "g.edge(\"x2\", \"y2\")\n",
    "g.edge(\"x3\", \"y3\")\n",
    "\n",
    "g.edge(\"x1\", \"x2\")\n",
    "g.edge(\"x1\", \"x3\")\n",
    "g.edge(\"x2\", \"x3\")\n",
    "\n",
    "\n",
    "g.edge(\"x1\", \"y2\")\n",
    "g.edge(\"x2\", \"y3\")\n",
    "\n",
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "2d2415dc",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-12-15T19:05:36.189028Z",
     "start_time": "2022-12-15T19:05:36.180156Z"
    }
   },
   "outputs": [],
   "source": [
    "np.random.seed(123)\n",
    "\n",
    "n = 1000\n",
    "\n",
    "ui = np.random.normal(0, 1, n)\n",
    "unit = np.array(range(n))\n",
    "\n",
    "x1 = np.random.normal(ui, 1, n)\n",
    "y1 = np.random.normal(x1+ui, 1, n)\n",
    "\n",
    "x2 = np.random.normal(ui+x1, 1, n)\n",
    "y2 = np.random.normal(x2+x1+ui, 1, n)\n",
    "\n",
    "x3 = np.random.normal(ui+x1+x2, 1, n)\n",
    "y3 = np.random.normal(x2+x3+ui, 1, n)\n",
    "\n",
    "df = pd.concat([\n",
    "    pd.DataFrame(dict(y=y1, x=x1, t=1, i=unit, ui=ui)),\n",
    "    pd.DataFrame(dict(y=y2, x=x2, t=2, i=unit, ui=ui)),\n",
    "    pd.DataFrame(dict(y=y3, x=x3, t=3, i=unit, ui=ui)),\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "cb7e5f04",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-12-15T19:05:39.847800Z",
     "start_time": "2022-12-15T19:05:36.190305Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.5592315346170307"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = smf.ols(\"y~x+C(i)\", data=df).fit()\n",
    "m.params[\"x\"]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "cf5827d1",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-12-15T19:05:43.050456Z",
     "start_time": "2022-12-15T19:05:39.849046Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0258800114132918"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = smf.ols(\"y~x+x_minus1+C(i)\",\n",
    "            data=df.assign(x_minus1 = df.groupby(\"i\")[\"x\"].shift(1))).fit()\n",
    "\n",
    "m.params[\"x\"]"
   ]
  }
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   "kernels_config": {
    "python": {
     "delete_cmd_postfix": "",
     "delete_cmd_prefix": "del ",
     "library": "var_list.py",
     "varRefreshCmd": "print(var_dic_list())"
    },
    "r": {
     "delete_cmd_postfix": ") ",
     "delete_cmd_prefix": "rm(",
     "library": "var_list.r",
     "varRefreshCmd": "cat(var_dic_list()) "
    }
   },
   "types_to_exclude": [
    "module",
    "function",
    "builtin_function_or_method",
    "instance",
    "_Feature"
   ],
   "window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}