doc/examples/py_double_ml_rdflex.ipynb

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    "# DoubleML for Flexible Covariate Adjustment in Regression Discontinuity Designs (RDD)\n",
    "\n",
    "This notebook demonstrates how to use RDD designs within ``DoubleML``. Our implementation, ``RDFlex``, is based on the paper _\"Flexible Covariate Adjustments in Regression Discontinuity Designs\"_ by [Noack, Olma and Rothe (2024)](https://arxiv.org/abs/2107.07942). \n",
    "\n",
    "In regression discontinuity designs (RDD) treatment assignment is determined by a continuous running variable $S$ (or \"score\") crossing a known threshold $c$ (or \"cutoff\"). We aim to estimate the local average treatment effect\n",
    "\n",
    "$$\\theta_{0} = \\mathbb{E}[Y_i(1)-Y_i(0)\\mid S = c]$$\n",
    "\n",
    "at the cutoff value. We therefore assume, that individuals are not able to manipulate their score in the neighborhood of the cutoff and that there is a discontinuity in the outcome that can be explained solely by the score."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "import plotly.express as px\n",
    "import plotly.graph_objs as go\n",
    "from statsmodels.nonparametric.kernel_regression import KernelReg\n",
    "\n",
    "from lightgbm import LGBMRegressor, LGBMClassifier\n",
    "\n",
    "from rdrobust import rdrobust\n",
    "\n",
    "import doubleml as dml\n",
    "from doubleml.rdd import RDFlex\n",
    "from doubleml.rdd.datasets import make_simple_rdd_data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sharp RDD\n",
    "\n",
    "In the sharp design, the treatment assignment is deterministic given the score. Namely, all the individuals with a score higher than the cutoff, receive the treatment $$D_i = \\mathbb{I}[S_i > c].$$\n",
    "\n",
    "### Generate Data\n",
    "\n",
    "The function ``make_simple_rdd_data()`` can be used to generate data of a rather standard RDD setting. If we set ``fuzzy = False``, the generated data follows a sharp RDD. We also generate covariates $X$ that can be used to adjust the estimation at a later stage.\n",
    "In this example, the cutoff is set to $c = 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>y</th>\n",
       "      <th>d</th>\n",
       "      <th>score</th>\n",
       "      <th>x0</th>\n",
       "      <th>x1</th>\n",
       "      <th>x2</th>\n",
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       "      <th>0</th>\n",
       "      <td>1.524407</td>\n",
       "      <td>0.0</td>\n",
       "      <td>-1.085631</td>\n",
       "      <td>0.484418</td>\n",
       "      <td>-0.639093</td>\n",
       "      <td>-0.556110</td>\n",
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       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>11.843350</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.997345</td>\n",
       "      <td>0.399618</td>\n",
       "      <td>0.589009</td>\n",
       "      <td>0.906710</td>\n",
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       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1.278325</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.282978</td>\n",
       "      <td>-0.382700</td>\n",
       "      <td>0.097267</td>\n",
       "      <td>-0.706374</td>\n",
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       "      <th>3</th>\n",
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       "      <td>0.0</td>\n",
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       "      <td>-0.126210</td>\n",
       "      <td>-0.477759</td>\n",
       "      <td>-0.302400</td>\n",
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       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5.924320</td>\n",
       "      <td>0.0</td>\n",
       "      <td>-0.578600</td>\n",
       "      <td>0.606731</td>\n",
       "      <td>0.606459</td>\n",
       "      <td>0.409011</td>\n",
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      "text/plain": [
       "           y    d     score        x0        x1        x2\n",
       "0   1.524407  0.0 -1.085631  0.484418 -0.639093 -0.556110\n",
       "1  11.843350  1.0  0.997345  0.399618  0.589009  0.906710\n",
       "2   1.278325  1.0  0.282978 -0.382700  0.097267 -0.706374\n",
       "3  -0.316781  0.0 -1.506295 -0.126210 -0.477759 -0.302400\n",
       "4   5.924320  0.0 -0.578600  0.606731  0.606459  0.409011"
      ]
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     "execution_count": 2,
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    }
   ],
   "source": [
    "np.random.seed(123)\n",
    "\n",
    "data_dict = make_simple_rdd_data(n_obs=1000, fuzzy=False)\n",
    "\n",
    "cov_names = ['x' + str(i) for i in range(data_dict['X'].shape[1])]\n",
    "df = pd.DataFrame(\n",
    "    np.column_stack((data_dict['Y'], data_dict['D'], data_dict['score'], data_dict['X'])),\n",
    "    columns=['y', 'd', 'score'] + cov_names,\n",
    ")\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Comparing the observed outcomes, we can clearly see a discontinuity at the cutoff value of $c = 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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    "fig = px.scatter(\n",
    "    x=df['score'],\n",
    "    y=df['y'],\n",
    "    color=df['d'].astype(bool),\n",
    "    labels={\n",
    "        \"x\": \"Score\",   \n",
    "        \"y\": \"Outcome\",\n",
    "        \"color\": \"Treatment\"\n",
    "    },\n",
    "    title=\"Scatter Plot of Outcome vs. Score by Treatment Status\"\n",
    ")\n",
    "\n",
    "fig.update_layout(\n",
    "    xaxis_title=\"Score\",\n",
    "    yaxis_title=\"Outcome\"\n",
    ")\n",
    "fig.show()"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Oracle Values and Comparisons\n",
    "\n",
    "As the data is generated we have access to the \"true\" potential outcomes. This enables us to observe the individual treatment effect (ITE)\n",
    "\n",
    "$$Y_i(1) - Y_i(0)$$\n",
    "\n",
    "for all individuals. As the effect at the cutoff is the average of these ITEs\n",
    "\n",
    "$$\\theta_{0} = \\mathbb{E}[Y_i(1)-Y_i(0)\\mid S = c],$$\n",
    "\n",
    "the ITEs can be used in a kernel regression to get an oracle estimator at the cutoff, as a local weighted average."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The oracle ATE is estimated as 0.977950766577146 at the cutoff c=0.\n"
     ]
    },
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           "x": 0.05
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        "title": {
         "text": "Locally Linear Kernel Regression of ITE on Score"
        },
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          "text": "Score"
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         "title": {
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   "source": [
    "ite = data_dict['oracle_values']['Y1'] - data_dict['oracle_values']['Y0']\n",
    "score = data_dict['score']\n",
    "\n",
    "oracle_model = KernelReg(endog=ite, exog=score, reg_type='ll', var_type='c', ckertype='gaussian')\n",
    "\n",
    "score_grid = np.linspace(-1, 1, 100)\n",
    "oracle_effects_sharp, _ = oracle_model.fit(score_grid)\n",
    "\n",
    "scatter = go.Scatter(\n",
    "    x=score,\n",
    "    y=ite,\n",
    "    mode='markers',\n",
    "    name='ITE',\n",
    "    marker=dict(color='blue')\n",
    ")\n",
    "line = go.Scatter(\n",
    "    x=score_grid,\n",
    "    y=oracle_effects_sharp,\n",
    "    mode='lines',\n",
    "    name='Average Effect Estimate',\n",
    "    line=dict(color='red')\n",
    ")\n",
    "\n",
    "\n",
    "fig = go.Figure(data=[scatter, line])\n",
    "fig.update_layout(\n",
    "    title='Locally Linear Kernel Regression of ITE on Score',\n",
    "    xaxis_title='Score',\n",
    "    yaxis_title='Effect',\n",
    "    legend=dict(x=0.8, y=0.2)\n",
    ")\n",
    "\n",
    "print(f\"The oracle ATE is estimated as {oracle_model.fit([0])[0][0]} at the cutoff c={0}.\")\n",
    "\n",
    "fig.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### RDD with Linear Adjustment\n",
    "\n",
    "The standard RDD estimator for the sharp design takes the form \n",
    "\n",
    "$$\\hat{\\theta}_{\\text{SRD}}(h) = \\sum_{i=1}^n w_i(h)(Y_i-X_i^T\\hat{\\gamma}_h)$$\n",
    "\n",
    "where $w_i(h)$ are local linear regression weights that depend on the data through the realizations of the running variable $S_i$ only and $h>0$ is a bandwidth. $\\hat{\\gamma}_h$ is a minimizer.\n",
    "\n",
    "The packages ``rdrobust`` implements this estimation with a linear adjustment."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Call: rdrobust\n",
      "Number of Observations:                  1000\n",
      "Polynomial Order Est. (p):                  1\n",
      "Polynomial Order Bias (q):                  2\n",
      "Kernel:                            Triangular\n",
      "Bandwidth Selection:                    mserd\n",
      "Var-Cov Estimator:                         NN\n",
      "\n",
      "                                Left      Right\n",
      "------------------------------------------------\n",
      "Number of Observations           520        480\n",
      "Number of Unique Obs.            520        480\n",
      "Number of Effective Obs.         272        241\n",
      "Bandwidth Estimation            0.69       0.69\n",
      "Bandwidth Bias                 1.096      1.096\n",
      "rho (h/b)                      0.629      0.629\n",
      "\n",
      "Method             Coef.     S.E.   t-stat    P>|t|       95% CI      \n",
      "-------------------------------------------------------------------------\n",
      "Conventional       1.247    0.485     2.57   1.017e-02     [0.296, 2.198]\n",
      "Robust                 -        -    2.293   2.184e-02       [0.19, 2.42]\n"
     ]
    },
    {
     "data": {
      "text/plain": []
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rdrobust_sharp = rdrobust(y=df['y'], x=df['score'], fuzzy=df['d'], covs=df[cov_names], c=0.0)\n",
    "rdrobust_sharp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### RDD with flexible adjustment\n",
    "\n",
    "[Noack, Olma and Rothe (2024)](https://arxiv.org/abs/2107.07942) propose an estimator that reduces the variance of the above esimator, using a flexible adjustment of the outcome by machine learning. For more details, see our User Guide. The estimator here takes the form \n",
    "\n",
    "$$\\hat{\\theta}_{\\text{RDFlex}}(h) = \\sum_{i=1}^n w_i(h)M_i(\\eta),\\quad M_i(\\eta) = Y_i - \\eta(X_i),$$\n",
    "\n",
    "with $\\eta(\\cdot)$ being potentially nonlinear adjustment functions.\n",
    "\n",
    "We initialize a `DoubleMLData` object using the usual package syntax:\n",
    "\n",
    " - `y_col` refers to the observed outcome, on which we want to estimate the effect at the cutoff\n",
    " - `s_col` refers to the score\n",
    " - `x_cols` refers to the covariates to be adjusted for\n",
    " - `d_cols` is an indicator whether an observation is treated or not. In the sharp design, this should be identical to an indicator whether an observation is left or right of the cutoff ($D_i = \\mathbb{I}[S_i > c]$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "dml_data = dml.DoubleMLData(df, y_col='y', d_cols='d', x_cols=cov_names, s_col='score')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The ``RDFlex`` object is intialized with only one learner, that adjusts the outcome based on the covariates."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Method             Coef.     S.E.     t-stat       P>|t|           95% CI\n",
      "-------------------------------------------------------------------------\n",
      "Conventional      1.091     0.120     9.072    1.169e-19  [0.855, 1.327]\n",
      "Robust                 -        -     6.206    5.429e-10  [0.639, 1.229]\n"
     ]
    }
   ],
   "source": [
    "ml_g = LGBMRegressor(n_estimators=500, learning_rate=0.01, verbose=-1)\n",
    "\n",
    "rdflex_sharp = RDFlex(dml_data,\n",
    "                      ml_g,\n",
    "                      fuzzy=False,\n",
    "                      n_folds=5,\n",
    "                      n_rep=1)\n",
    "rdflex_sharp.fit(n_iterations=2)\n",
    "\n",
    "print(rdflex_sharp)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is visible that the flexible adjustment decreases the standard error in the estimation and therefore provides tighter confidence intervals. For coverage simulations, see the [DoubleML Coverage Repository](https://docs.doubleml.org/doubleml-coverage/dev/rdd/rdd.html).\n",
    "\n",
    "`RDFlex` uses a iterative fitting approach to determine a preliminary bandwidth selections for the local adjustments. The default number of iterations is `n_iterations=2`, according to [Noack, Olma and Rothe (2024)](https://arxiv.org/abs/2107.07942)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Fuzzy RDD\n",
    "\n",
    "In the fuzzy RDDs, the treatment assignment is still deterministic given the score ($T_i = \\mathbb{I}[S_i > c]$).\n",
    "However, in the neighborhood of the cutoff, there is a probability of observations not picking up the treatment they were assigned. These observations cause the probability jump of treatment at the cutoff to be smaller than 1.\n",
    "\n",
    "### Generate Data\n",
    "\n",
    "The function ``make_simple_rdd_data()`` with ``fuzzy = True`` generates basic data for the fuzzy case. The cutoff is still set to $c = 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>y</th>\n",
       "      <th>d</th>\n",
       "      <th>score</th>\n",
       "      <th>x0</th>\n",
       "      <th>x1</th>\n",
       "      <th>x2</th>\n",
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       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>2.131533</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.496714</td>\n",
       "      <td>-0.665035</td>\n",
       "      <td>-0.790864</td>\n",
       "      <td>0.272860</td>\n",
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       "      <th>1</th>\n",
       "      <td>10.104291</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.138264</td>\n",
       "      <td>0.412951</td>\n",
       "      <td>-0.936828</td>\n",
       "      <td>0.872424</td>\n",
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       "      <th>2</th>\n",
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       "      <td>0.082593</td>\n",
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       "      <th>3</th>\n",
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       "      <td>1.0</td>\n",
       "      <td>1.523030</td>\n",
       "      <td>0.741938</td>\n",
       "      <td>0.428174</td>\n",
       "      <td>0.603456</td>\n",
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       "      <th>4</th>\n",
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       "      <td>1.0</td>\n",
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       "      <td>0.629650</td>\n",
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      "text/plain": [
       "           y    d     score        x0        x1        x2\n",
       "0   2.131533  1.0  0.496714 -0.665035 -0.790864  0.272860\n",
       "1  10.104291  1.0 -0.138264  0.412951 -0.936828  0.872424\n",
       "2   3.337677  1.0  0.647689 -0.896057  0.082593  0.418121\n",
       "3   7.767124  1.0  1.523030  0.741938  0.428174  0.603456\n",
       "4   4.209740  1.0 -0.234153 -0.321100  0.629650 -0.839770"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.seed(42)\n",
    "\n",
    "data_dict = make_simple_rdd_data(n_obs=1000, fuzzy=True)\n",
    "\n",
    "cov_names = ['x' + str(i) for i in range(data_dict['X'].shape[1])]\n",
    "df = pd.DataFrame(\n",
    "    np.column_stack((data_dict['Y'], data_dict['D'], data_dict['score'], data_dict['X'])),\n",
    "    columns=['y', 'd', 'score'] + cov_names,\n",
    ")\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Comparing the observed outcomes, the discontinuity is much less pronounced than in the sharp case."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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   "source": [
    "fig = px.scatter(\n",
    "    x=df['score'],\n",
    "    y=df['y'],\n",
    "    color=df['d'].astype(bool),\n",
    "    labels={\n",
    "        \"x\": \"Score\",   \n",
    "        \"y\": \"Outcome\",\n",
    "        \"color\": \"Treatment\"\n",
    "    },\n",
    "    title=\"Scatter Plot of Outcome vs. Score by Treatment Status\"\n",
    ")\n",
    "\n",
    "fig.update_layout(\n",
    "    xaxis_title=\"Score\",\n",
    "    yaxis_title=\"Outcome\"\n",
    ")\n",
    "fig.show()"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Oracle Values and Comparisons\n",
    "\n",
    "As in the sharp design, the ITEs can be used to obtain an oracle estimator at the cutoff.\n",
    "\n",
    "Remark that in the fuzzy design (ADD Assumptions), we can interpret the ratio as the treatment effect on compliers such that we have to restrict the oracle computation on the corresponding subpoipulation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The oracle ATE on compliers is estimated as 0.9087280080883967 at the cutoff c=0.\n"
     ]
    },
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        "title": {
         "text": "Locally Linear Kernel Regression of ITE on Score"
        },
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         "title": {
          "text": "Score"
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         "title": {
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     "metadata": {},
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   ],
   "source": [
    "complier_mask = ((data_dict[\"score\"] < 0) & (data_dict[\"D\"] == False)) | ((data_dict[\"score\"] > 0) & (data_dict[\"D\"] == True))\n",
    "ite = data_dict['oracle_values']['Y1'][complier_mask] - data_dict['oracle_values']['Y0'][complier_mask]\n",
    "score = data_dict['score'][complier_mask]\n",
    "\n",
    "oracle_model = KernelReg(endog=ite, exog=score, reg_type='ll', var_type='c', ckertype='gaussian')\n",
    "\n",
    "score_grid = np.linspace(-1, 1, 100)\n",
    "oracle_effects_fuzzy, _ = oracle_model.fit(score_grid)\n",
    "\n",
    "scatter = go.Scatter(\n",
    "    x=score,\n",
    "    y=ite,\n",
    "    mode='markers',\n",
    "    name='ITE',\n",
    "    marker=dict(color='blue')\n",
    ")\n",
    "line = go.Scatter(\n",
    "    x=score_grid,\n",
    "    y=oracle_effects_fuzzy,\n",
    "    mode='lines',\n",
    "    name='Average Effect Estimate',\n",
    "    line=dict(color='red')\n",
    ")\n",
    "\n",
    "\n",
    "fig = go.Figure(data=[scatter, line])\n",
    "fig.update_layout(\n",
    "    title='Locally Linear Kernel Regression of ITE on Score',\n",
    "    xaxis_title='Score',\n",
    "    yaxis_title='Effect',\n",
    "    legend=dict(x=0.8, y=0.2)\n",
    ")\n",
    "\n",
    "print(f\"The oracle ATE on compliers is estimated as {oracle_model.fit([0])[0][0]} at the cutoff c={0}.\")\n",
    "\n",
    "fig.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### RDD with Linear Adjustment\n",
    "\n",
    "The standard RDD estimator for the fuzzy design takes the form \n",
    "\n",
    "$$\\hat{\\theta}_{\\text{FRD}}(h) = \\frac{\\hat{\\theta}_{\\text{SRD}}(h)}{\\hat{\\theta}_{\\text{D}}(h)} = \\frac{\\sum_{i=1}^n w_i(h)(Y_i-X_i^T\\hat{\\gamma}_{Y, h})}{\\sum_{i=1}^n w_i(h)(D_i-X_i^T\\hat{\\gamma}_{D, h})}$$\n",
    "\n",
    "Under the assumption, that there are no *Defiers* (= individuals that would always pick the opposite treatment of their assigned one), this effect can be interpreted as *Average Treatment Effect on the Treated*. The package ``rdrobust`` implements this estimation with a linear adjustment."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Call: rdrobust\n",
      "Number of Observations:                  1000\n",
      "Polynomial Order Est. (p):                  1\n",
      "Polynomial Order Bias (q):                  2\n",
      "Kernel:                            Triangular\n",
      "Bandwidth Selection:                    mserd\n",
      "Var-Cov Estimator:                         NN\n",
      "\n",
      "                                Left      Right\n",
      "------------------------------------------------\n",
      "Number of Observations           490        510\n",
      "Number of Unique Obs.            490        510\n",
      "Number of Effective Obs.         244        263\n",
      "Bandwidth Estimation           0.653      0.653\n",
      "Bandwidth Bias                 1.023      1.023\n",
      "rho (h/b)                      0.638      0.638\n",
      "\n",
      "Method             Coef.     S.E.   t-stat    P>|t|       95% CI      \n",
      "-------------------------------------------------------------------------\n",
      "Conventional       2.798     3.98    0.703   4.821e-01   [-5.003, 10.598]\n",
      "Robust                 -        -    0.717   4.735e-01   [-5.806, 12.502]\n"
     ]
    },
    {
     "data": {
      "text/plain": []
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rdrobust_fuzzy = rdrobust(y=df['y'], x=df['score'], fuzzy=df['d'], covs=df[cov_names], c=0.0)\n",
    "rdrobust_fuzzy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The fuzzy design usually has much larger standard errors than the sharp design, as the jump in treatment probability adds further estimation uncertainty."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### RDD with flexible adjustment\n",
    "\n",
    "[Noack, Olma and Rothe (2024)](https://arxiv.org/abs/2107.07942) propose an estimator that reduces the variance of the above esimator, using a flexible adjustment of the outcome by ML. For more details, see our user guide. The estimator here takes the form \n",
    "\n",
    "$$\\hat{\\theta}_{\\text{RDFlex, FRD}}(h) = \\frac{\\sum_{i=1}^n w_i(h)(Y_i - \\hat{\\eta}_Y(X_i))}{\\sum_{i=1}^n w_i(h)(D_i - \\hat{\\eta}_D(X_i))},$$\n",
    "\n",
    "\n",
    "with $\\eta_Y(\\cdot), \\eta_D(\\cdot)$ being potentially nonlinear adjustment functions.\n",
    "\n",
    "We initialize a `DoubleMLData` object using the usual package syntax:\n",
    "\n",
    " - `y_col` refers to the observed outcome, on which we want to estimate the effect at the cutoff\n",
    " - `s_col` refers to the score\n",
    " - `x_cols` refers to the covariates to be adjusted for\n",
    " - `d_cols` is an indicator whether an observation is treated or not. In the fuzzy design, this should __not__ be identical to an indicator whether an observation is left or right of the cutoff ($D_i \\neq \\mathbb{I}[S_i > c]$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "dml_data = dml.DoubleMLData(df, y_col='y', d_cols='d', x_cols=cov_names, s_col='score')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This time, we also have to specify a classifier that adjusts the treatment assignment probabilities."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Method             Coef.     S.E.     t-stat       P>|t|           95% CI\n",
      "-------------------------------------------------------------------------\n",
      "Conventional      3.475     2.231     1.558    1.193e-01  [-0.897, 7.847]\n",
      "Robust                 -        -     2.021    4.330e-02  [0.185, 12.112]\n"
     ]
    }
   ],
   "source": [
    "ml_g = LGBMRegressor(n_estimators=500, learning_rate=0.01, verbose=-1)\n",
    "ml_m = LGBMClassifier(n_estimators=500, learning_rate=0.01, verbose=-1)\n",
    "\n",
    "rdflex_fuzzy = RDFlex(dml_data,\n",
    "                      ml_g,\n",
    "                      ml_m,\n",
    "                      fuzzy=True,\n",
    "                      n_folds=5,\n",
    "                      n_rep=1)\n",
    "rdflex_fuzzy.fit(n_iterations=2)\n",
    "\n",
    "print(rdflex_fuzzy)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Also for the fuzzy case, we observe a significant decrease in estimation standard error."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Advanced: Global and Local Learners, Stacked Ensembles\n",
    "\n",
    "By default, ``RDFlex`` fits ML methods using kernel weights, resulting in a \"local\" fit around the cutoff. If the adjustment should also include \"global\" information from the full support of $S$ available in the data, the use of a the ``GlobalLearner`` wrapper is recommended.\n",
    "\n",
    "The ``GlobalLearner`` allows to ignore the weights and fit the ML method on the full support of the data, even if weights are provided.\n",
    "\n",
    "The learners can also be stacked. All learners have to support the `sample_weight` in their `fit` method. By stacking and using local and global learners, it is possible to further tune the estimation and potentially reduce standard errors even further."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "from doubleml.utils import GlobalRegressor, GlobalClassifier\n",
    "\n",
    "from sklearn.linear_model import LinearRegression, LogisticRegression\n",
    "from sklearn.ensemble import RandomForestRegressor, RandomForestClassifier\n",
    "from sklearn.ensemble import StackingClassifier, StackingRegressor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "reg_estimators = [\n",
    "    ('lr local', LinearRegression()),\n",
    "    ('rf local', RandomForestRegressor()),\n",
    "    ('lr global', GlobalRegressor(base_estimator=LinearRegression())),\n",
    "    ('rf global', GlobalRegressor(base_estimator=RandomForestRegressor()))\n",
    "]\n",
    "\n",
    "class_estimators = [\n",
    "    ('lr local', LogisticRegression()),\n",
    "    ('rf local', RandomForestClassifier()),\n",
    "    ('lr global', GlobalClassifier(base_estimator=LogisticRegression())),\n",
    "    ('rf global', GlobalClassifier(base_estimator=RandomForestClassifier()))\n",
    "]\n",
    "\n",
    "ml_g = StackingRegressor(\n",
    "    estimators=reg_estimators,\n",
    "    final_estimator=LinearRegression(),\n",
    ")\n",
    "\n",
    "ml_m = StackingClassifier(\n",
    "    estimators=class_estimators,\n",
    "    final_estimator=LogisticRegression(),\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Method             Coef.     S.E.     t-stat       P>|t|           95% CI\n",
      "-------------------------------------------------------------------------\n",
      "Conventional      3.622     2.036     1.779    7.527e-02  [-0.369, 7.613]\n",
      "Robust                 -        -     1.725    8.453e-02  [-0.689, 10.806]\n"
     ]
    }
   ],
   "source": [
    "rdflex_fuzzy_stack = RDFlex(dml_data,\n",
    "                            ml_g,\n",
    "                            ml_m,\n",
    "                            fuzzy=True,\n",
    "                            n_folds=5,\n",
    "                            n_rep=1)\n",
    "rdflex_fuzzy_stack.fit(n_iterations=2)\n",
    "\n",
    "print(rdflex_fuzzy_stack)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For a conclusion, we look at a visualization of the estimated coefficient and the confidence intervals. We see that by using the flexible adjustment, it is possible to shrink confidence bands."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "df_sharp = pd.DataFrame({\"coef\": [rdrobust_sharp.coef.values[0][0], rdflex_sharp.coef[0]],\n",
    "                         \"CI lower\": [rdrobust_sharp.ci.values[0][0], rdflex_sharp.confint().values[0][0]],\n",
    "                         \"CI upper\": [rdrobust_sharp.ci.values[0][1], rdflex_sharp.confint().values[0][1]],\n",
    "                         \"method\": [\"Linear Adj.\", \"Flexible Adj.\"]})\n",
    "df_fuzzy = pd.DataFrame({\"coef\": [rdrobust_fuzzy.coef.values[0][0], rdflex_fuzzy.coef[0], rdflex_fuzzy_stack.coef[0]],\n",
    "                         \"CI lower\": [rdrobust_fuzzy.ci.values[0][0], rdflex_fuzzy.confint().values[0][0], rdflex_fuzzy_stack.confint().values[0][0]],\n",
    "                         \"CI upper\": [rdrobust_fuzzy.ci.values[0][1], rdflex_fuzzy.confint().values[0][1], rdflex_fuzzy_stack.confint().values[0][1]],\n",
    "                         \"method\": [\"Linear Adj.\", \"Flexible Adj.\", \"Flexible Adj. (Stacked)\"]})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 1000x400 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "fig, axes = plt.subplots(nrows=1, ncols=2, figsize=(10, 4))\n",
    "\n",
    "axes[0].errorbar(\n",
    "    df_sharp['method'],\n",
    "    df_sharp['coef'],\n",
    "    yerr=(df_sharp['coef'] - df_sharp['CI lower'], df_sharp['CI upper'] - df_sharp['coef']),\n",
    "    fmt='o',\n",
    "    capsize=5,\n",
    "    capthick=2\n",
    ")\n",
    "axes[0].set_title('Sharp Design')\n",
    "axes[0].set_ylabel('Coefficient')\n",
    "axes[0].set_xlabel('Method')\n",
    "axes[0].axhline(oracle_effects_sharp[49:51].mean(), linestyle=\"--\", color=\"r\")\n",
    "\n",
    "axes[1].errorbar(\n",
    "    df_fuzzy['method'],\n",
    "    df_fuzzy['coef'],\n",
    "    yerr=(df_fuzzy['coef'] - df_fuzzy['CI lower'], df_fuzzy['CI upper'] - df_fuzzy['coef']),\n",
    "    fmt='o',\n",
    "    capsize=5,\n",
    "    capthick=2\n",
    ")\n",
    "axes[1].set_title('Fuzzy Design')\n",
    "axes[1].set_ylabel('Coefficient')  \n",
    "axes[1].set_xlabel('Method')\n",
    "axes[1].axhline(oracle_effects_fuzzy[49:51].mean(), linestyle=\"--\", color=\"r\")\n",
    "\n",
    "plt.tight_layout()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "dml_rdd",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.12.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}