smm_simulation.ipynb

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    "# SMM for LCM Topics Class\n",
    "Consider the following DPG:\n",
    "$$ Y_i = X_i'\\beta + e_i$$\n",
    "where $\\beta\\in\\mathbb{R}^k$ and we have $E[e_i|X_i]=0$.\n",
    "\n",
    "From the mean independence assumption we get the following identification result:\n",
    "$$ E[e_iX_i] = 0 \\\\ E[(Y_i-X_i'\\beta)X_i] = 0 \\\\ E[Y_iX_i] = E[X_iX_i']\\beta$$\n",
    "and hence $\\beta$ is identified as long as $E[X_iX_i']$ is invertible:\n",
    "$$ \\beta = E[X_iX_i']^{-1}E[X_iY_i].$$\n",
    "\n",
    "Now assume we didn't know about this last result.\n",
    "\n",
    "Usualy GMM would use the moment condition $E[e_iX_i]=0$ and be the solution to the following sample analogue problem:\n",
    "$$ \\hat{\\beta}_{GMM} \\equiv \\argmin_{b\\in\\mathbb{R}^k} \\sum_{i=1}^nx_i(y_i - x_i'b)W_n \\sum_{i=1}^n(y_i - x_i'b)x_i'$$\n",
    "where $W_n$ is a $k\\times k$ weight matrix.\n",
    "\n",
    "Now these can be solved exactly and we have $\\hat{\\beta}_{GMM} = \\hat{\\beta}_{OLS}$.\n",
    "\n",
    "## SMM Approach\n",
    "Now instead of solving these moment conditions directly, we instead fix some value $\\beta_0$ and\n",
    "- draw the unobserved component of the model, namely $e$ and compute $y$\n",
    "  - we draw these once at the beginning to not confound optimization over $\\beta$ with the drawing of $e$\n",
    "  - I use a normal distribution for $e$ but my guess is it doesn't really matter because of consistency (Does this depend on `n` or `reps` though?)\n",
    "- estimate the implied simulated moments, namely $E[Y_iX_i]$ by $\\hat{xy} \\equiv n^{-1}\\sum_{i=1}^ny_ix_i$\n",
    "- repeat this $R$ times and average over these estimated moments\n",
    "- evaluate a loss function (e.g. quadratic loss): $||\\hat{\\overline{xy}} - \\hat{xy}_{observed}||_2$.\n",
    "\n",
    "This loss is a function of $\\beta_0$ which we used to generate the simulated data/simulated moment. \n",
    "Thus, the goal becomes to minimize this criterion as a function of $\\beta_0$ by repeatedly doing the above for different $\\beta_0$.\n",
    "For example, we could do this for a grid of values $\\Beta$ and then just take the point $\\beta\\in \\Beta$ with the lowest criterion.\n",
    "Clearly, this grid search is inefficient and suffers from a curse of dimensionality.\n",
    "\n",
    "In practice, we will rely on routines used by `estimagic`. This means we will pass some function `criterion` to `estimagic.minimize` where that function only depends on $\\beta$.\n",
    "Tthe minimizer then optimizes this function over $\\beta$. The returned value is our solution $\\hat{\\beta}_{SMM}$ and we can compare it to the analytically computed $\\hat{\\beta}_{OLS}$."
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    "# Simulated methods of moments for LCM topics class\n",
    "# import estimagic\n",
    "from numpy.random import default_rng\n",
    "import numpy as np\n",
    "import plotly.express as px\n",
    "import pandas as pd\n",
    "from pathlib import Path\n",
    "\n",
    "import estimagic as em\n",
    "\n",
    "from smm import smm\n",
    "\n",
    "pd.options.mode.copy_on_write = True\n",
    "pd.options.future.infer_string = True\n",
    "pd.options.plotting.backend = \"plotly\"\n"
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      "C:\\Users\\budde\\projects\\lcm\\lcm_smm\n"
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   "source": [
    "this_dir = Path(\".\")\n",
    "this_dir = this_dir.resolve()\n",
    "root = this_dir\n",
    "print(root)\n",
    "bld = root / \"bld\"\n",
    "src = root / \"src\"\n",
    "\n",
    "bld.mkdir(parents=True, exist_ok=True)\n"
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      "Minimize with 3 free parameters terminated successfully after 55 criterion evaluations, 55 derivative evaluations and 38 iterations.\n",
      "\n",
      "The value of criterion improved from 230152513.91493684 to 1.9541359076876184e-06.\n",
      "\n",
      "The scipy_lbfgsb algorithm reported: CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL\n",
      "\n",
      "Independent of the convergence criteria used by scipy_lbfgsb, the strength of convergence can be assessed by the following criteria:\n",
      "\n",
      "                             one_step     five_steps \n",
      "relative_criterion_change  3.769e-11***  3.769e-11***\n",
      "relative_params_change      3.44e-12***  3.441e-12***\n",
      "absolute_criterion_change  3.769e-12***  3.769e-12***\n",
      "absolute_params_change     9.133e-12***  9.133e-12***\n",
      "\n",
      "(***: change <= 1e-10, **: change <= 1e-8, *: change <= 1e-5. Change refers to a change between accepted steps. The first column only considers the last step. The second column considers the last five steps.)\n"
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     "text": [
      "c:\\Users\\budde\\miniforge3\\envs\\epp\\Lib\\site-packages\\estimagic\\optimization\\optimize_result.py:131: FutureWarning:\n",
      "\n",
      "DataFrame.applymap has been deprecated. Use DataFrame.map instead.\n",
      "\n"
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    "np.random.seed(123)\n",
    "reps = 1000\n",
    "initial_value = np.array([1, 1, 1])\n",
    "data = pd.read_feather(src / \"data.feather\")\n",
    "\n",
    "res = smm(reps, data, initial_value)\n",
    "print(res)\n"
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   "source": [
    "\n",
    "fig = em.params_plot(\n",
    "    res,\n",
    "    max_evaluations=300,\n",
    ")\n",
    "fig.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "Analytic OLS: [2.8927337573355048, 194.25920102316675, -1.803510381105907]\n",
      "SMM: [2.8928048726557325, 194.2594787083691, -1.8030959120274355]\n",
      "Diff: [-7.11153202277437e-05, -0.00027768520234872085, -0.00041446907847153547]\n"
     ]
    }
   ],
   "source": [
    "beta_smm = res.params\n",
    "\n",
    "def _multivariate_ols(y, x):\n",
    "    \"\"\"Multivariate OLS regression.\"\"\"\n",
    "    return np.linalg.inv(x.T @ x) @ x.T @ y\n",
    "\n",
    "beta_ols = _multivariate_ols(data[\"Y\"], data[[\"A\", \"B\", \"C\"]])\n",
    "\n",
    "print(f\"Analytic OLS: {list(beta_ols)}\")\n",
    "print(f\"SMM: {list(beta_smm)}\")\n",
    "print(f\"Diff: {list(beta_ols - beta_smm)}\")\n"
   ]
  }
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