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   "source": [
    "# Likelihood estimation\n",
    "\n",
    "This notebook shows how to do a simple maximum likelihood (ml) estimation with estimagic. As an illustrating example, we implement a simple linear regression model. This is the same example model used as in the method of moments notebook.\n",
    "\n",
    "We proceed in 4 steps:\n",
    "\n",
    "\n",
    "1. Create a data generating process\n",
    "2. Set up a likelihood function\n",
    "3. Maximize the likelihood function\n",
    "4. Calculate standard errors, confidence intervals, and p-values\n",
    "\n",
    "The user only needs to do step 1 and 2. The rest is done by `estimate_ml`. \n",
    "\n",
    "To be very clear: Estimagic is not a package to estimate linear models or other models that are implemented in Stata, statsmodels or anywhere else. Its purpose is to estimate parameters with custom likelihood or method of simulated moments functions. We just use an ordered logit model as an example of a very simple likelihood function.\n",
    "\n",
    "\n",
    "### Model:\n",
    "\n",
    "$$ y = \\beta_0 + \\beta_1 x + \\epsilon, \\text{ where } \\epsilon \\sim N(0, \\sigma^2)$$\n",
    "\n",
    "We aim to estimate $\\beta_0, \\beta_1, \\sigma^2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import estimagic as em\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "from scipy.stats import norm\n",
    "\n",
    "rng = np.random.default_rng(seed=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Create a data generating process"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def simulate_data(params, n_draws):\n",
    "    x = rng.normal(0, 1, size=n_draws)\n",
    "    e = rng.normal(0, params.loc[\"sd\", \"value\"], size=n_draws)\n",
    "    y = params.loc[\"intercept\", \"value\"] + params.loc[\"slope\", \"value\"] * x + e\n",
    "    return pd.DataFrame({\"y\": y, \"x\": x})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "true_params = pd.DataFrame(\n",
    "    data=[[2, -np.inf], [-1, -np.inf], [1, 1e-10]],\n",
    "    columns=[\"value\", \"lower_bound\"],\n",
    "    index=[\"intercept\", \"slope\", \"sd\"],\n",
    ")\n",
    "true_params"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = simulate_data(true_params, n_draws=100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Define the `loglike` function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def normal_loglike(params, data):\n",
    "    norm_rv = norm(\n",
    "        loc=params.loc[\"intercept\", \"value\"] + params.loc[\"slope\", \"value\"] * data[\"x\"],\n",
    "        scale=params.loc[\"sd\", \"value\"],\n",
    "    )\n",
    "    contributions = norm_rv.logpdf(data[\"y\"])\n",
    "    return {\"contributions\": contributions, \"value\": contributions.sum()}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A few remarks before we move on:\n",
    "\n",
    "1. There are numerically better ways to calculate the likelihood; we chose this implementation for brevity and readability. \n",
    "2. The loglike function takes params and other arguments. You are completely flexible with respect to the number and names of the other arguments as long as the first argument is params. \n",
    "3. The loglike function returns a dictionary with the entries \"contributions\" and \"value\". The \"contributions\" are the log likelihood evaluations of each individual in the dataset. The \"value\" are their sum. The \"value\" entry could be omitted, the \"contributions\" entry, however, is mandatory. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Estimate the model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "start_params = true_params.assign(value=[100, 100, 100])\n",
    "\n",
    "res = em.estimate_ml(\n",
    "    loglike=normal_loglike,\n",
    "    params=start_params,\n",
    "    optimize_options={\"algorithm\": \"scipy_lbfgsb\"},\n",
    "    loglike_kwargs={\"data\": data},\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "res.summary().round(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. What's in the results?\n",
    "\n",
    "`LikelihoodResult` objects provide attributes and methods to calculate standard errors, confidence intervals, and p-values. For all three, several methods are available. You can even calculate cluster robust standard errors. \n",
    "\n",
    "A few examples are:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "res.params"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "res.cov(method=\"robust\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "res.se()"
   ]
  }
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