notebooks/VAR_vs_VARMAX_evaluation.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## VAR vs VARMAX Model Validation\n",
    "\n",
    "\n",
    "In this notebook, we will validate which model performs better given computational constraints: a VAR model retrained after each prediction or a VARMAX retrained every 20 predictions. To do so, we will retrun the model and functions constructed in [Systems_Identification_Fitting](Systems_Identification_Fitting.ipynb)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>marketPriceEth</th>\n",
       "      <th>marketPriceUsd</th>\n",
       "      <th>block_number</th>\n",
       "      <th>debtAvailableToSettle</th>\n",
       "      <th>globalDebt</th>\n",
       "      <th>globalDebtCeiling</th>\n",
       "      <th>systemSurplus</th>\n",
       "      <th>totalActiveSafeCount</th>\n",
       "      <th>RedemptionRateAnnualizedRate</th>\n",
       "      <th>RedemptionRateHourlyRate</th>\n",
       "      <th>RedemptionRateEightHourlyRate</th>\n",
       "      <th>RedemptionPrice</th>\n",
       "      <th>RaiDrawnFromSAFEs</th>\n",
       "      <th>collateral</th>\n",
       "      <th>debt</th>\n",
       "      <th>ETH Price (OSM)</th>\n",
       "      <th>block_timestamp</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.002481</td>\n",
       "      <td>4.362594</td>\n",
       "      <td>11860755</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2788.522358</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>0.068533</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.14</td>\n",
       "      <td>2788.440433</td>\n",
       "      <td>14.89236</td>\n",
       "      <td>2788.385291</td>\n",
       "      <td>1803.656430</td>\n",
       "      <td>2021-02-15 10:16:16+00:00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.002481</td>\n",
       "      <td>4.419266</td>\n",
       "      <td>11861008</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2788.657935</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>0.108751</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.14</td>\n",
       "      <td>2788.440433</td>\n",
       "      <td>14.89236</td>\n",
       "      <td>2788.385291</td>\n",
       "      <td>1763.974936</td>\n",
       "      <td>2021-02-15 11:13:28+00:00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.002481</td>\n",
       "      <td>4.419266</td>\n",
       "      <td>11861237</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2788.657935</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>0.108751</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.14</td>\n",
       "      <td>2788.440433</td>\n",
       "      <td>14.89236</td>\n",
       "      <td>2788.385291</td>\n",
       "      <td>1763.974936</td>\n",
       "      <td>2021-02-15 12:07:36+00:00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.001769</td>\n",
       "      <td>3.215446</td>\n",
       "      <td>11861502</td>\n",
       "      <td>0.0</td>\n",
       "      <td>3048.645955</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>0.116162</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.14</td>\n",
       "      <td>3048.440433</td>\n",
       "      <td>16.89236</td>\n",
       "      <td>3048.358490</td>\n",
       "      <td>1775.977923</td>\n",
       "      <td>2021-02-15 13:07:59+00:00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.001769</td>\n",
       "      <td>3.215446</td>\n",
       "      <td>11861791</td>\n",
       "      <td>0.0</td>\n",
       "      <td>3048.645955</td>\n",
       "      <td>115792089237316195423570985008687.9</td>\n",
       "      <td>0.116162</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.14</td>\n",
       "      <td>3048.440433</td>\n",
       "      <td>16.89236</td>\n",
       "      <td>3048.358490</td>\n",
       "      <td>1805.792735</td>\n",
       "      <td>2021-02-15 14:07:59+00:00</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   marketPriceEth  marketPriceUsd  block_number  debtAvailableToSettle  \\\n",
       "0        0.002481        4.362594      11860755                    0.0   \n",
       "1        0.002481        4.419266      11861008                    0.0   \n",
       "2        0.002481        4.419266      11861237                    0.0   \n",
       "3        0.001769        3.215446      11861502                    0.0   \n",
       "4        0.001769        3.215446      11861791                    0.0   \n",
       "\n",
       "    globalDebt                    globalDebtCeiling  systemSurplus  \\\n",
       "0  2788.522358  115792089237316195423570985008687.9       0.068533   \n",
       "1  2788.657935  115792089237316195423570985008687.9       0.108751   \n",
       "2  2788.657935  115792089237316195423570985008687.9       0.108751   \n",
       "3  3048.645955  115792089237316195423570985008687.9       0.116162   \n",
       "4  3048.645955  115792089237316195423570985008687.9       0.116162   \n",
       "\n",
       "   totalActiveSafeCount  RedemptionRateAnnualizedRate  \\\n",
       "0                     3                           1.0   \n",
       "1                     3                           1.0   \n",
       "2                     3                           1.0   \n",
       "3                     3                           1.0   \n",
       "4                     3                           1.0   \n",
       "\n",
       "   RedemptionRateHourlyRate  RedemptionRateEightHourlyRate  RedemptionPrice  \\\n",
       "0                       1.0                            1.0             3.14   \n",
       "1                       1.0                            1.0             3.14   \n",
       "2                       1.0                            1.0             3.14   \n",
       "3                       1.0                            1.0             3.14   \n",
       "4                       1.0                            1.0             3.14   \n",
       "\n",
       "   RaiDrawnFromSAFEs  collateral         debt  ETH Price (OSM)  \\\n",
       "0        2788.440433    14.89236  2788.385291      1803.656430   \n",
       "1        2788.440433    14.89236  2788.385291      1763.974936   \n",
       "2        2788.440433    14.89236  2788.385291      1763.974936   \n",
       "3        3048.440433    16.89236  3048.358490      1775.977923   \n",
       "4        3048.440433    16.89236  3048.358490      1805.792735   \n",
       "\n",
       "             block_timestamp  \n",
       "0  2021-02-15 10:16:16+00:00  \n",
       "1  2021-02-15 11:13:28+00:00  \n",
       "2  2021-02-15 12:07:36+00:00  \n",
       "3  2021-02-15 13:07:59+00:00  \n",
       "4  2021-02-15 14:07:59+00:00  "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# import libraries\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "from scipy import stats\n",
    "import math as m\n",
    "from sklearn.metrics import mean_squared_error\n",
    "import statsmodels.api as sm\n",
    "from statsmodels.tsa.api import VAR, VARMAX\n",
    "from sklearn.preprocessing import PowerTransformer\n",
    "import matplotlib.pyplot as plt\n",
    "import warnings\n",
    "import os\n",
    "warnings.filterwarnings(\"ignore\")\n",
    "\n",
    "%matplotlib inline \n",
    "\n",
    "os.chdir('..')\n",
    "\n",
    "states = pd.read_csv('data/states.csv')\n",
    "del states['Unnamed: 0']\n",
    "states.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 616,
   "metadata": {},
   "outputs": [],
   "source": [
    "# add additional state variables\n",
    "states['RedemptionPriceinEth'] = states['RedemptionPrice'] / states['ETH Price (OSM)']\n",
    "states['RedemptionPriceError'] = states['RedemptionPrice'] - states['marketPriceUsd']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 617,
   "metadata": {},
   "outputs": [],
   "source": [
    "# define constants (will come from cadCAD model but added here for calculations)\n",
    "params = {\n",
    "    'liquidation_ratio': 1.45,\n",
    "    'debt_ceiling': 1e9,\n",
    "    'uniswap_fee': 0.003,\n",
    "    'arbitrageur_considers_liquidation_ratio': True,\n",
    "}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- ## Create Arbtrageur data vector $u^*$ -->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 620,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Q</th>\n",
       "      <th>D</th>\n",
       "      <th>Rrai</th>\n",
       "      <th>Reth</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1249.422600</td>\n",
       "      <td>2.601571e+07</td>\n",
       "      <td>1.389236e+07</td>\n",
       "      <td>11977.048864</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1249.422600</td>\n",
       "      <td>2.601571e+07</td>\n",
       "      <td>1.384577e+07</td>\n",
       "      <td>12017.260094</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1249.422600</td>\n",
       "      <td>2.599152e+07</td>\n",
       "      <td>1.383046e+07</td>\n",
       "      <td>12030.502525</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1249.422600</td>\n",
       "      <td>2.598687e+07</td>\n",
       "      <td>1.382752e+07</td>\n",
       "      <td>12033.049146</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1249.422600</td>\n",
       "      <td>2.598554e+07</td>\n",
       "      <td>1.378488e+07</td>\n",
       "      <td>12070.754352</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1942</th>\n",
       "      <td>18.086319</td>\n",
       "      <td>3.383805e+03</td>\n",
       "      <td>7.636681e+02</td>\n",
       "      <td>1.509233</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1943</th>\n",
       "      <td>17.131152</td>\n",
       "      <td>3.115513e+03</td>\n",
       "      <td>7.942806e+02</td>\n",
       "      <td>1.436701</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1944</th>\n",
       "      <td>16.940119</td>\n",
       "      <td>3.061855e+03</td>\n",
       "      <td>8.004031e+02</td>\n",
       "      <td>1.422195</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1945</th>\n",
       "      <td>16.901912</td>\n",
       "      <td>3.051123e+03</td>\n",
       "      <td>8.016276e+02</td>\n",
       "      <td>1.419294</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1946</th>\n",
       "      <td>16.894271</td>\n",
       "      <td>3.048977e+03</td>\n",
       "      <td>8.018725e+02</td>\n",
       "      <td>1.418714</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>1947 rows × 4 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                Q             D          Rrai          Reth\n",
       "0     1249.422600  2.601571e+07  1.389236e+07  11977.048864\n",
       "1     1249.422600  2.601571e+07  1.384577e+07  12017.260094\n",
       "2     1249.422600  2.599152e+07  1.383046e+07  12030.502525\n",
       "3     1249.422600  2.598687e+07  1.382752e+07  12033.049146\n",
       "4     1249.422600  2.598554e+07  1.378488e+07  12070.754352\n",
       "...           ...           ...           ...           ...\n",
       "1942    18.086319  3.383805e+03  7.636681e+02      1.509233\n",
       "1943    17.131152  3.115513e+03  7.942806e+02      1.436701\n",
       "1944    16.940119  3.061855e+03  8.004031e+02      1.422195\n",
       "1945    16.901912  3.051123e+03  8.016276e+02      1.419294\n",
       "1946    16.894271  3.048977e+03  8.018725e+02      1.418714\n",
       "\n",
       "[1947 rows x 4 columns]"
      ]
     },
     "execution_count": 620,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# subset state variables for arbitrageur vector\n",
    "state_subset = states[['collateral','RaiDrawnFromSAFEs','RaiInUniswap','EthInUniswap']]\n",
    "\n",
    "# map state data to vector fields \n",
    "state_subset.columns = ['Q','D','Rrai','Reth']\n",
    "\n",
    "# alpha is the smoothing factor\n",
    "local = state_subset.ewm(alpha=0.8).mean()\n",
    "local"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 621,
   "metadata": {},
   "outputs": [],
   "source": [
    "# function to create coordinate transformations\n",
    "def coordinate_transformations(params,df,Q,R_eth,R_rai,D,RedemptionPrice,EthPrice):\n",
    "    '''\n",
    "    Description:\n",
    "    Function that takes in pandas dataframe and the names of columns\n",
    "    \n",
    "    Parameters:\n",
    "    df: pandas dataframe containing states information\n",
    "    Q: dataframe column name\n",
    "    R_eth: dataframe column name\n",
    "    R_rai: dataframe column name\n",
    "    D: dataframe column name\n",
    "    RedemptionPrice: dataframe column name\n",
    "    EthPrice: dataframe column name\n",
    "\n",
    "    Returns: Pandas dataframe with alpha, beta, gamma, delta transformed values\n",
    "    \n",
    "    Example:\n",
    "    \n",
    "    coordinate_transformations(params,states,'collateral','EthInUniswap','RaiInUniswap',\n",
    "                           'RaiDrawnFromSAFEs','RedemptionPrice','ETH Price (OSM)')[['alpha','beta','gamma','delta']]\n",
    "    '''\n",
    "    \n",
    "    # Calculate alpha\n",
    "    d = df[D].diff()\n",
    "    d.fillna(0,inplace=True)\n",
    "    df['d'] = d\n",
    "    \n",
    "    df['alpha'] = df['d'] / params['debt_ceiling']\n",
    "\n",
    "    # calculate beta\n",
    "    df['C_o'] = (df[RedemptionPrice]/states[EthPrice]) * params['liquidation_ratio']\n",
    "\n",
    "    q = df[Q].diff()\n",
    "    q.fillna(0,inplace=True)\n",
    "    df['q'] = q\n",
    "\n",
    "\n",
    "    df['C_1'] = (df['C_o'] * df[D]) - df[Q]\n",
    "\n",
    "    df['beta'] = (df['q'] - (df['C_o']*df['d']))/ df['C_1']\n",
    "\n",
    "    # calculate gamma\n",
    "    r = df[R_rai].diff()\n",
    "    r.fillna(0,inplace=True)\n",
    "    df['r'] = r\n",
    "\n",
    "    df['gamma'] = df['r']/df[R_rai]\n",
    "\n",
    "    # calculate delta\n",
    "    z = df[R_eth].diff()\n",
    "    z.fillna(0,inplace=True)\n",
    "    df['z'] = z\n",
    "\n",
    "    df['delta'] = df['z']/df[R_eth]\n",
    "    \n",
    "    return df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 622,
   "metadata": {},
   "outputs": [
    {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>alpha</th>\n",
       "      <th>beta</th>\n",
       "      <th>gamma</th>\n",
       "      <th>delta</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
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       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>-0.004041</td>\n",
       "      <td>0.004013</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-3.000000e-05</td>\n",
       "      <td>0.001201</td>\n",
       "      <td>-0.000699</td>\n",
       "      <td>0.000696</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-5.000000e-07</td>\n",
       "      <td>0.000020</td>\n",
       "      <td>-0.003815</td>\n",
       "      <td>0.003848</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1942</th>\n",
       "      <td>-1.646075e-06</td>\n",
       "      <td>0.108212</td>\n",
       "      <td>0.249397</td>\n",
       "      <td>-0.333334</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1943</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>-0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1944</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>-0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1945</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>-0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1946</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>-0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>1947 rows × 4 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "             alpha      beta     gamma     delta\n",
       "0     0.000000e+00  0.000000  0.000000  0.000000\n",
       "1     0.000000e+00  0.000000 -0.004041  0.004013\n",
       "2    -3.000000e-05  0.001201 -0.000699  0.000696\n",
       "3     0.000000e+00  0.000000  0.000000  0.000000\n",
       "4    -5.000000e-07  0.000020 -0.003815  0.003848\n",
       "...            ...       ...       ...       ...\n",
       "1942 -1.646075e-06  0.108212  0.249397 -0.333334\n",
       "1943  0.000000e+00 -0.000000  0.000000  0.000000\n",
       "1944  0.000000e+00 -0.000000  0.000000  0.000000\n",
       "1945  0.000000e+00 -0.000000  0.000000  0.000000\n",
       "1946  0.000000e+00 -0.000000  0.000000  0.000000\n",
       "\n",
       "[1947 rows x 4 columns]"
      ]
     },
     "execution_count": 622,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "transformed = coordinate_transformations(params,states,'collateral','EthInUniswap','RaiInUniswap',\n",
    "                           'RaiDrawnFromSAFEs','RedemptionPrice','ETH Price (OSM)')[['alpha','beta','gamma','delta']]\n",
    "\n",
    "transformed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 623,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
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       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Q</th>\n",
       "      <th>D</th>\n",
       "      <th>Rrai</th>\n",
       "      <th>Reth</th>\n",
       "      <th>RedemptionPrice</th>\n",
       "      <th>ETH Price (OSM)</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1249.422600</td>\n",
       "      <td>2.601571e+07</td>\n",
       "      <td>1.389236e+07</td>\n",
       "      <td>11977.048864</td>\n",
       "      <td>3.005443</td>\n",
       "      <td>3521.749280</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1249.422600</td>\n",
       "      <td>2.601571e+07</td>\n",
       "      <td>1.384577e+07</td>\n",
       "      <td>12017.260094</td>\n",
       "      <td>3.005450</td>\n",
       "      <td>3484.967617</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1249.422600</td>\n",
       "      <td>2.599152e+07</td>\n",
       "      <td>1.383046e+07</td>\n",
       "      <td>12030.502525</td>\n",
       "      <td>3.005465</td>\n",
       "      <td>3509.940000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1249.422600</td>\n",
       "      <td>2.598687e+07</td>\n",
       "      <td>1.382752e+07</td>\n",
       "      <td>12033.049146</td>\n",
       "      <td>3.005465</td>\n",
       "      <td>3438.463957</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1249.422600</td>\n",
       "      <td>2.598554e+07</td>\n",
       "      <td>1.378488e+07</td>\n",
       "      <td>12070.754352</td>\n",
       "      <td>3.005472</td>\n",
       "      <td>3464.125574</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1942</th>\n",
       "      <td>18.086319</td>\n",
       "      <td>3.383805e+03</td>\n",
       "      <td>7.636681e+02</td>\n",
       "      <td>1.509233</td>\n",
       "      <td>3.140000</td>\n",
       "      <td>1825.887144</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1943</th>\n",
       "      <td>17.131152</td>\n",
       "      <td>3.115513e+03</td>\n",
       "      <td>7.942806e+02</td>\n",
       "      <td>1.436701</td>\n",
       "      <td>3.140000</td>\n",
       "      <td>1793.770282</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1944</th>\n",
       "      <td>16.940119</td>\n",
       "      <td>3.061855e+03</td>\n",
       "      <td>8.004031e+02</td>\n",
       "      <td>1.422195</td>\n",
       "      <td>3.140000</td>\n",
       "      <td>1803.891149</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1945</th>\n",
       "      <td>16.901912</td>\n",
       "      <td>3.051123e+03</td>\n",
       "      <td>8.016276e+02</td>\n",
       "      <td>1.419294</td>\n",
       "      <td>3.140000</td>\n",
       "      <td>1803.891149</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1946</th>\n",
       "      <td>16.894271</td>\n",
       "      <td>3.048977e+03</td>\n",
       "      <td>8.018725e+02</td>\n",
       "      <td>1.418714</td>\n",
       "      <td>3.140000</td>\n",
       "      <td>1805.792735</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>1947 rows × 6 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                Q             D          Rrai          Reth  RedemptionPrice  \\\n",
       "0     1249.422600  2.601571e+07  1.389236e+07  11977.048864         3.005443   \n",
       "1     1249.422600  2.601571e+07  1.384577e+07  12017.260094         3.005450   \n",
       "2     1249.422600  2.599152e+07  1.383046e+07  12030.502525         3.005465   \n",
       "3     1249.422600  2.598687e+07  1.382752e+07  12033.049146         3.005465   \n",
       "4     1249.422600  2.598554e+07  1.378488e+07  12070.754352         3.005472   \n",
       "...           ...           ...           ...           ...              ...   \n",
       "1942    18.086319  3.383805e+03  7.636681e+02      1.509233         3.140000   \n",
       "1943    17.131152  3.115513e+03  7.942806e+02      1.436701         3.140000   \n",
       "1944    16.940119  3.061855e+03  8.004031e+02      1.422195         3.140000   \n",
       "1945    16.901912  3.051123e+03  8.016276e+02      1.419294         3.140000   \n",
       "1946    16.894271  3.048977e+03  8.018725e+02      1.418714         3.140000   \n",
       "\n",
       "      ETH Price (OSM)  \n",
       "0         3521.749280  \n",
       "1         3484.967617  \n",
       "2         3509.940000  \n",
       "3         3438.463957  \n",
       "4         3464.125574  \n",
       "...               ...  \n",
       "1942      1825.887144  \n",
       "1943      1793.770282  \n",
       "1944      1803.891149  \n",
       "1945      1803.891149  \n",
       "1946      1805.792735  \n",
       "\n",
       "[1947 rows x 6 columns]"
      ]
     },
     "execution_count": 623,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# add additional signals to arbitrageur state\n",
    "local['RedemptionPrice'] = states['RedemptionPrice']\n",
    "local['ETH Price (OSM)'] = states['ETH Price (OSM)']\n",
    "\n",
    "local"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 624,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>alpha</th>\n",
       "      <th>beta</th>\n",
       "      <th>gamma</th>\n",
       "      <th>delta</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>-0.003365</td>\n",
       "      <td>0.003346</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-2.419355e-05</td>\n",
       "      <td>0.000968</td>\n",
       "      <td>-0.001107</td>\n",
       "      <td>0.001101</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-4.652605e-06</td>\n",
       "      <td>0.000186</td>\n",
       "      <td>-0.000213</td>\n",
       "      <td>0.000212</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-1.323500e-06</td>\n",
       "      <td>0.000053</td>\n",
       "      <td>-0.003093</td>\n",
       "      <td>0.003124</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1942</th>\n",
       "      <td>-1.341458e-06</td>\n",
       "      <td>0.148292</td>\n",
       "      <td>0.200431</td>\n",
       "      <td>-0.240292</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1943</th>\n",
       "      <td>-2.682915e-07</td>\n",
       "      <td>0.029727</td>\n",
       "      <td>0.038541</td>\n",
       "      <td>-0.050485</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1944</th>\n",
       "      <td>-5.365830e-08</td>\n",
       "      <td>0.006036</td>\n",
       "      <td>0.007649</td>\n",
       "      <td>-0.010200</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1945</th>\n",
       "      <td>-1.073166e-08</td>\n",
       "      <td>0.001209</td>\n",
       "      <td>0.001528</td>\n",
       "      <td>-0.002044</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1946</th>\n",
       "      <td>-2.146332e-09</td>\n",
       "      <td>0.000242</td>\n",
       "      <td>0.000305</td>\n",
       "      <td>-0.000409</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>1947 rows × 4 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "             alpha      beta     gamma     delta\n",
       "0     0.000000e+00  0.000000  0.000000  0.000000\n",
       "1     0.000000e+00  0.000000 -0.003365  0.003346\n",
       "2    -2.419355e-05  0.000968 -0.001107  0.001101\n",
       "3    -4.652605e-06  0.000186 -0.000213  0.000212\n",
       "4    -1.323500e-06  0.000053 -0.003093  0.003124\n",
       "...            ...       ...       ...       ...\n",
       "1942 -1.341458e-06  0.148292  0.200431 -0.240292\n",
       "1943 -2.682915e-07  0.029727  0.038541 -0.050485\n",
       "1944 -5.365830e-08  0.006036  0.007649 -0.010200\n",
       "1945 -1.073166e-08  0.001209  0.001528 -0.002044\n",
       "1946 -2.146332e-09  0.000242  0.000305 -0.000409\n",
       "\n",
       "[1947 rows x 4 columns]"
      ]
     },
     "execution_count": 624,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "transformed_arbitrageur = coordinate_transformations(params,local,'Q','Reth','Rrai',\n",
    "                           'D','RedemptionPrice','ETH Price (OSM)')[['alpha','beta','gamma','delta']]\n",
    "\n",
    "transformed_arbitrageur"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 625,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\n",
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>alpha</th>\n",
       "      <th>beta</th>\n",
       "      <th>gamma</th>\n",
       "      <th>delta</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.000000e+00</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>-0.000676</td>\n",
       "      <td>0.000667</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-5.806452e-06</td>\n",
       "      <td>0.000233</td>\n",
       "      <td>0.000408</td>\n",
       "      <td>-0.000404</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4.652605e-06</td>\n",
       "      <td>-0.000186</td>\n",
       "      <td>0.000213</td>\n",
       "      <td>-0.000212</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>8.235004e-07</td>\n",
       "      <td>-0.000033</td>\n",
       "      <td>-0.000722</td>\n",
       "      <td>0.000724</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          alpha      beta     gamma     delta\n",
       "0  0.000000e+00  0.000000  0.000000  0.000000\n",
       "1  0.000000e+00  0.000000 -0.000676  0.000667\n",
       "2 -5.806452e-06  0.000233  0.000408 -0.000404\n",
       "3  4.652605e-06 -0.000186  0.000213 -0.000212\n",
       "4  8.235004e-07 -0.000033 -0.000722  0.000724"
      ]
     },
     "execution_count": 625,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def create_transformed_errors(transformed_states,transformed_arbitrageur):\n",
    "    '''\n",
    "    Description:\n",
    "    Function for taking two pandas dataframes of transformed states and taking the difference\n",
    "    to produce an error dataframe. \n",
    "    \n",
    "    Parameters:\n",
    "    transformed_states: pandas dataframe with alpha, beta, gamma, and delta features\n",
    "    transformed_arbitrageur: pandas dataframe with alpha, beta, gamma, and delta features\n",
    "\n",
    "    Returns:\n",
    "    error pandas dataframe and transformation object\n",
    "    \n",
    "    '''\n",
    "    alpha_diff = transformed_states['alpha'] - transformed_arbitrageur['alpha']\n",
    "    beta_diff = transformed_states['beta'] - transformed_arbitrageur['beta']\n",
    "    gamma_diff = transformed_states['gamma'] - transformed_arbitrageur['gamma']\n",
    "    delta_diff = transformed_states['delta'] - transformed_arbitrageur['delta']\n",
    "\n",
    "\n",
    "    e_u = pd.DataFrame(alpha_diff)\n",
    "    e_u['beta'] = beta_diff\n",
    "    e_u['gamma'] = gamma_diff\n",
    "    e_u['delta'] = delta_diff\n",
    "\n",
    "    e_u = e_u.astype(float)\n",
    "    \n",
    "    return e_u\n",
    "\n",
    "e_u = create_transformed_errors(transformed,transformed_arbitrageur)\n",
    "e_u.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 626,
   "metadata": {},
   "outputs": [],
   "source": [
    "def power_transformation(e_u):\n",
    "    '''\n",
    "    Definition:\n",
    "    Function to perform a power transformation on the coordinate \n",
    "    transformed differenced data\n",
    "    \n",
    "    Parameters:\n",
    "    e_u: Dataframe of coordinated transformed differenced data\n",
    "    \n",
    "    Required:\n",
    "    import pandas as pd\n",
    "    from sklearn.preprocessing import PowerTransformer\n",
    "    \n",
    "    Returns:\n",
    "    Transformed dataframe and transformation object\n",
    "    \n",
    "    Example:\n",
    "    transformed_df, pt = power_transformation(e_u)\n",
    "    '''\n",
    "    pt = PowerTransformer()\n",
    "    yeo= pd.DataFrame(pt.fit_transform(e_u),columns=e_u.columns)\n",
    "\n",
    "    return yeo, pt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 627,
   "metadata": {},
   "outputs": [],
   "source": [
    "e_u_yeo, pt = power_transformation(e_u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- ### Autogressive lag selection -->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 628,
   "metadata": {},
   "outputs": [],
   "source": [
    "# split data between train and test (in production deployment, can remove)\n",
    "split_point = int(len(e_u) * .8)\n",
    "train = e_u_yeo.iloc[0:split_point]\n",
    "test = e_u_yeo.iloc[split_point:]\n",
    "\n",
    "\n",
    "states_train = states.iloc[0:split_point]\n",
    "states_test = states.iloc[split_point:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 629,
   "metadata": {},
   "outputs": [],
   "source": [
    "def VARMAX_prediction(e_u,RedemptionPriceError,newRedemptionPriceError,steps=1,lag=1):\n",
    "    '''\n",
    "    Description:\n",
    "    Function to train and forecast a VARMAX model one step into the future\n",
    "    \n",
    "    Parameters:\n",
    "    e_u: errors pandas dataframe\n",
    "    RedemptionPriceErrorPrevious: 1d Numpy array of RedemptionPriceError values\n",
    "    newRedemptionPriceError: exogenous latest redemption price error signal - float\n",
    "    steps: Number of forecast steps. Default is 1\n",
    "    lag: number of autoregressive lags. Default is 1\n",
    "    \n",
    "    Returns:\n",
    "    Numpy array of transformed state changes\n",
    "    \n",
    "    Example\n",
    "    Y_pred = VARMAX_prediction(train,states_train['RedemptionPriceError'],\n",
    "                  states_test['RedemptionPriceError'][0:5],steps=5,lag=1)\n",
    "    '''\n",
    "    # instantiate the VARMAX model object from statsmodels \n",
    "    model = VARMAX(endog=e_u.values,exog=RedemptionPriceError,\n",
    "                   initialization='approximate_diffuse',measurement_error=True)\n",
    "\n",
    "    # fit model with determined lag values\n",
    "    results = model.fit(order=(lag,0))\n",
    "    \n",
    "    Y_pred = results.forecast(steps = steps, exog=newRedemptionPriceError)\n",
    "    \n",
    "    return Y_pred.values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 630,
   "metadata": {},
   "outputs": [],
   "source": [
    "def VAR_prediction(e_u,lag=1):\n",
    "    '''\n",
    "    Description:\n",
    "    Function to train and forecast a VAR model one step into the future\n",
    "    \n",
    "    Parameters:\n",
    "    e_u: errors pandas dataframe\n",
    "    lag: number of autoregressive lags. Default is 1\n",
    "    \n",
    "    Returns:\n",
    "    Numpy array of transformed state changes\n",
    "    \n",
    "    Example\n",
    "    VAR_prediction(e_u,6)    \n",
    "    '''\n",
    "    # instantiate the VAR model object from statsmodels \n",
    "    model = VAR(e_u)\n",
    "\n",
    "    # fit model with determined lag values\n",
    "    results = model.fit(lag)\n",
    "    \n",
    "    lag_order = results.k_ar\n",
    "    \n",
    "    Y_pred = results.forecast(e_u.values[-lag_order:],1)\n",
    "\n",
    "    \n",
    "    return Y_pred[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 631,
   "metadata": {},
   "outputs": [],
   "source": [
    "varmax_predictions = VARMAX_prediction(train,states_train['RedemptionPriceError'],\n",
    "                  states_test['RedemptionPriceError'].values[0:20],steps=20,lag=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 632,
   "metadata": {},
   "outputs": [],
   "source": [
    "var_predictions = []\n",
    "train_var = train.copy()\n",
    "for i in range(0,20): \n",
    "    var_predictions.append(VAR_prediction(train_var,15))\n",
    "    train_var = train_var.append(test.iloc[i])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 633,
   "metadata": {},
   "outputs": [],
   "source": [
    "def invert_power_transformation(pt,prediction):\n",
    "    '''\n",
    "    Definition:\n",
    "    Function to invert power transformation\n",
    "    \n",
    "    Parameters:\n",
    "    pt: transformation object\n",
    "    prediction: Numpy array of model state coordinate transformed percentage changes\n",
    "    \n",
    "    Required:\n",
    "    import pandas as pd\n",
    "    from sklearn.preprocessing import PowerTransformer\n",
    "    \n",
    "    Returns:\n",
    "    inverted transformation numpy array\n",
    "    \n",
    "    Example:\n",
    "    inverted_array = invert_power_transformation(pt,prediction)\n",
    "    \n",
    "    '''\n",
    "    # transform back into coordinate system\n",
    "    inverted = pt.inverse_transform(prediction)\n",
    "    \n",
    "    return inverted"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 634,
   "metadata": {},
   "outputs": [],
   "source": [
    "# invert the power transformation\n",
    "var_predictions = invert_power_transformation(pt,var_predictions)\n",
    "varmax_predictions = invert_power_transformation(pt,varmax_predictions)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Model Evaluation Comparision (in coordinate transformed and power transformed differenced space)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 635,
   "metadata": {},
   "outputs": [],
   "source": [
    "var_predictions_df = pd.DataFrame(var_predictions,columns=['alpha','beta','gamma','delta'])\n",
    "varmax_predictions_df = pd.DataFrame(varmax_predictions,columns=['alpha','beta','gamma','delta'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Alpha - Root Mean Square Error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 636,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "VAR performs better by -2.3483390703571416e-05\n"
     ]
    }
   ],
   "source": [
    "var_alpha_rmse = m.sqrt(mean_squared_error(var_predictions_df['alpha'], test.head(20)['alpha']))\n",
    "varmax_alpha_rmse = m.sqrt(mean_squared_error(varmax_predictions_df['alpha'], test.head(20)['alpha']))\n",
    "if var_alpha_rmse >= varmax_alpha_rmse:\n",
    "    print('VARMAX performs better by {}'.format(varmax_alpha_rmse-var_alpha_rmse))\n",
    "else:\n",
    "    print('VAR performs better by {}'.format(var_alpha_rmse-varmax_alpha_rmse))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Beta - Root Mean Square Error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 637,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "VAR performs better by -0.0004483374446575324\n"
     ]
    }
   ],
   "source": [
    "var_beta_rmse = m.sqrt(mean_squared_error(var_predictions_df['beta'], test.head(20)['beta']))\n",
    "varmax_beta_rmse = m.sqrt(mean_squared_error(varmax_predictions_df['beta'], test.head(20)['beta']))\n",
    "if var_beta_rmse >= varmax_beta_rmse:\n",
    "    print('VARMAX performs better by {}'.format(varmax_beta_rmse-var_beta_rmse))\n",
    "else:\n",
    "    print('VAR performs better by {}'.format(var_beta_rmse-varmax_beta_rmse))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Gamma - Root Mean Square Error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 638,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "VAR performs better by -0.00028606775500304715\n"
     ]
    }
   ],
   "source": [
    "var_gamma_rmse = m.sqrt(mean_squared_error(var_predictions_df['gamma'], test.head(20)['gamma']))\n",
    "varmax_gamma_rmse = m.sqrt(mean_squared_error(varmax_predictions_df['gamma'], test.head(20)['gamma']))\n",
    "if var_gamma_rmse >= varmax_gamma_rmse:\n",
    "    print('VARMAX performs better by {}'.format(varmax_gamma_rmse-var_gamma_rmse))\n",
    "else:\n",
    "    print('VAR performs better by {}'.format(var_gamma_rmse-varmax_gamma_rmse))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Delta - Root Mean Square Error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 639,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "VAR performs better by -0.00046840677073029663\n"
     ]
    }
   ],
   "source": [
    "var_delta_rmse = m.sqrt(mean_squared_error(var_predictions_df['delta'], test.head(20)['delta']))\n",
    "varmax_delta_rmse = m.sqrt(mean_squared_error(varmax_predictions_df['delta'], test.head(20)['delta']))\n",
    "if var_delta_rmse >= varmax_delta_rmse:\n",
    "    print('VARMAX performs better by {}'.format(varmax_delta_rmse-var_delta_rmse))\n",
    "else:\n",
    "    print('VAR performs better by {}'.format(var_delta_rmse-varmax_delta_rmse))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Model Evaluation Comparision as States"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 640,
   "metadata": {},
   "outputs": [],
   "source": [
    "def inverse_transformation_and_state_update(Y_pred,previous_state,params):\n",
    "    '''\n",
    "    Description:\n",
    "    Function to take system identification model prediction and invert transfrom and create new state\n",
    "    \n",
    "    Parameters:\n",
    "    y_pred: numpy array of transformed state changes\n",
    "    previous_state: pandas dataframe of previous state or 'current' state\n",
    "    params: dictionary of system parameters\n",
    "    \n",
    "    Returns:\n",
    "    pandas dataframe of new states \n",
    "    \n",
    "    Example:\n",
    "    inverse_transformation_and_state_update(Y_pred,previous_state,params)\n",
    "    '''\n",
    "    \n",
    "    d_star = Y_pred[0] * params['debt_ceiling']\n",
    "    \n",
    "    q_star = previous_state['C_o'] * params['debt_ceiling'] * Y_pred[0] + previous_state['C_1'] * Y_pred[1]\n",
    "    \n",
    "    r_star = Y_pred[2] * previous_state['gamma'] * previous_state['RaiInUniswap']\n",
    "\n",
    "    z_star = Y_pred[3] * previous_state['delta'] * previous_state['EthInUniswap']\n",
    "\n",
    "    new_state = pd.DataFrame(previous_state[['collateral','EthInUniswap','RaiInUniswap','RaiDrawnFromSAFEs']].to_dict())\n",
    "    new_state['Q'] = new_state['collateral'] + q_star\n",
    "    new_state['D'] = new_state['RaiDrawnFromSAFEs'] + d_star\n",
    "    new_state['R_Rai'] = new_state['RaiInUniswap'] + r_star\n",
    "    new_state['R_Eth'] = new_state['EthInUniswap'] + z_star\n",
    "    \n",
    "    return new_state[['Q','D','R_Rai','R_Eth']]\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 641,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Q</th>\n",
       "      <th>D</th>\n",
       "      <th>R_Rai</th>\n",
       "      <th>R_Eth</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1556</th>\n",
       "      <td>6081.169224</td>\n",
       "      <td>3.371551e+07</td>\n",
       "      <td>2.971972e+07</td>\n",
       "      <td>61661.047608</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1557</th>\n",
       "      <td>6110.991138</td>\n",
       "      <td>3.369966e+07</td>\n",
       "      <td>2.961955e+07</td>\n",
       "      <td>61701.017149</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1558</th>\n",
       "      <td>6120.952802</td>\n",
       "      <td>3.413544e+07</td>\n",
       "      <td>3.018807e+07</td>\n",
       "      <td>60746.179731</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1559</th>\n",
       "      <td>6087.703343</td>\n",
       "      <td>3.404582e+07</td>\n",
       "      <td>2.997961e+07</td>\n",
       "      <td>60989.915859</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1560</th>\n",
       "      <td>6088.850114</td>\n",
       "      <td>3.411418e+07</td>\n",
       "      <td>2.998927e+07</td>\n",
       "      <td>60974.824543</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1561</th>\n",
       "      <td>6085.478244</td>\n",
       "      <td>3.409787e+07</td>\n",
       "      <td>3.002661e+07</td>\n",
       "      <td>60899.884484</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1562</th>\n",
       "      <td>6141.965543</td>\n",
       "      <td>3.397362e+07</td>\n",
       "      <td>2.977962e+07</td>\n",
       "      <td>60663.356884</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1563</th>\n",
       "      <td>6091.936632</td>\n",
       "      <td>3.379863e+07</td>\n",
       "      <td>2.953994e+07</td>\n",
       "      <td>59906.282132</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1564</th>\n",
       "      <td>6126.153590</td>\n",
       "      <td>3.353541e+07</td>\n",
       "      <td>2.943921e+07</td>\n",
       "      <td>60270.621867</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1565</th>\n",
       "      <td>6131.086605</td>\n",
       "      <td>3.339909e+07</td>\n",
       "      <td>2.922044e+07</td>\n",
       "      <td>60086.935213</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1566</th>\n",
       "      <td>6096.528881</td>\n",
       "      <td>3.338932e+07</td>\n",
       "      <td>2.919057e+07</td>\n",
       "      <td>60163.041470</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1567</th>\n",
       "      <td>6102.356558</td>\n",
       "      <td>3.329873e+07</td>\n",
       "      <td>2.913818e+07</td>\n",
       "      <td>60212.233856</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1568</th>\n",
       "      <td>6138.249445</td>\n",
       "      <td>3.331039e+07</td>\n",
       "      <td>2.902142e+07</td>\n",
       "      <td>60285.465969</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1569</th>\n",
       "      <td>6116.777496</td>\n",
       "      <td>3.334030e+07</td>\n",
       "      <td>2.902173e+07</td>\n",
       "      <td>60263.943935</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1570</th>\n",
       "      <td>6112.744506</td>\n",
       "      <td>3.321958e+07</td>\n",
       "      <td>2.887157e+07</td>\n",
       "      <td>60452.220918</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1571</th>\n",
       "      <td>6111.791472</td>\n",
       "      <td>3.314914e+07</td>\n",
       "      <td>2.869285e+07</td>\n",
       "      <td>60585.218185</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1572</th>\n",
       "      <td>5897.338278</td>\n",
       "      <td>3.307860e+07</td>\n",
       "      <td>2.856717e+07</td>\n",
       "      <td>60586.909337</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1573</th>\n",
       "      <td>5849.262681</td>\n",
       "      <td>3.313946e+07</td>\n",
       "      <td>2.855823e+07</td>\n",
       "      <td>60606.974473</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1574</th>\n",
       "      <td>5849.389398</td>\n",
       "      <td>3.310479e+07</td>\n",
       "      <td>2.854731e+07</td>\n",
       "      <td>60630.105363</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1575</th>\n",
       "      <td>5853.111529</td>\n",
       "      <td>3.305654e+07</td>\n",
       "      <td>2.851072e+07</td>\n",
       "      <td>60714.019613</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
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      ],
      "text/plain": [
       "                Q             D         R_Rai         R_Eth\n",
       "1556  6081.169224  3.371551e+07  2.971972e+07  61661.047608\n",
       "1557  6110.991138  3.369966e+07  2.961955e+07  61701.017149\n",
       "1558  6120.952802  3.413544e+07  3.018807e+07  60746.179731\n",
       "1559  6087.703343  3.404582e+07  2.997961e+07  60989.915859\n",
       "1560  6088.850114  3.411418e+07  2.998927e+07  60974.824543\n",
       "1561  6085.478244  3.409787e+07  3.002661e+07  60899.884484\n",
       "1562  6141.965543  3.397362e+07  2.977962e+07  60663.356884\n",
       "1563  6091.936632  3.379863e+07  2.953994e+07  59906.282132\n",
       "1564  6126.153590  3.353541e+07  2.943921e+07  60270.621867\n",
       "1565  6131.086605  3.339909e+07  2.922044e+07  60086.935213\n",
       "1566  6096.528881  3.338932e+07  2.919057e+07  60163.041470\n",
       "1567  6102.356558  3.329873e+07  2.913818e+07  60212.233856\n",
       "1568  6138.249445  3.331039e+07  2.902142e+07  60285.465969\n",
       "1569  6116.777496  3.334030e+07  2.902173e+07  60263.943935\n",
       "1570  6112.744506  3.321958e+07  2.887157e+07  60452.220918\n",
       "1571  6111.791472  3.314914e+07  2.869285e+07  60585.218185\n",
       "1572  5897.338278  3.307860e+07  2.856717e+07  60586.909337\n",
       "1573  5849.262681  3.313946e+07  2.855823e+07  60606.974473\n",
       "1574  5849.389398  3.310479e+07  2.854731e+07  60630.105363\n",
       "1575  5853.111529  3.305654e+07  2.851072e+07  60714.019613"
      ]
     },
     "execution_count": 641,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "VAR_new_states = []\n",
    "index = -21\n",
    "for i in range(0,20): \n",
    "    previous_state = states.iloc[train_var.index[index:index+1]]\n",
    "    VAR_new_states.append(inverse_transformation_and_state_update(var_predictions[i],previous_state,params))\n",
    "    index += 1\n",
    "    \n",
    "VAR_new_states = pd.concat(VAR_new_states)\n",
    "VAR_new_states"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 642,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
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       "      <th></th>\n",
       "      <th>Q</th>\n",
       "      <th>D</th>\n",
       "      <th>R_Rai</th>\n",
       "      <th>R_Eth</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1556</th>\n",
       "      <td>6113.017030</td>\n",
       "      <td>3.373426e+07</td>\n",
       "      <td>2.971978e+07</td>\n",
       "      <td>61661.196053</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1557</th>\n",
       "      <td>6113.767936</td>\n",
       "      <td>3.369428e+07</td>\n",
       "      <td>2.961953e+07</td>\n",
       "      <td>61701.011749</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1558</th>\n",
       "      <td>6111.834120</td>\n",
       "      <td>3.427227e+07</td>\n",
       "      <td>3.018998e+07</td>\n",
       "      <td>60746.861009</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1559</th>\n",
       "      <td>6111.676794</td>\n",
       "      <td>3.409202e+07</td>\n",
       "      <td>2.997949e+07</td>\n",
       "      <td>60990.377376</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1560</th>\n",
       "      <td>6111.737270</td>\n",
       "      <td>3.412698e+07</td>\n",
       "      <td>2.998928e+07</td>\n",
       "      <td>60974.811810</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1561</th>\n",
       "      <td>6111.183947</td>\n",
       "      <td>3.412650e+07</td>\n",
       "      <td>3.002667e+07</td>\n",
       "      <td>60899.716552</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1562</th>\n",
       "      <td>6111.416858</td>\n",
       "      <td>3.394189e+07</td>\n",
       "      <td>2.977983e+07</td>\n",
       "      <td>60663.153212</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1563</th>\n",
       "      <td>6113.684414</td>\n",
       "      <td>3.379694e+07</td>\n",
       "      <td>2.954039e+07</td>\n",
       "      <td>59907.871761</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1564</th>\n",
       "      <td>6111.202474</td>\n",
       "      <td>3.352808e+07</td>\n",
       "      <td>2.943918e+07</td>\n",
       "      <td>60270.855448</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1565</th>\n",
       "      <td>6112.309691</td>\n",
       "      <td>3.336299e+07</td>\n",
       "      <td>2.922069e+07</td>\n",
       "      <td>60087.006328</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1566</th>\n",
       "      <td>6112.711526</td>\n",
       "      <td>3.336272e+07</td>\n",
       "      <td>2.919058e+07</td>\n",
       "      <td>60163.061682</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1567</th>\n",
       "      <td>6112.516972</td>\n",
       "      <td>3.333513e+07</td>\n",
       "      <td>2.913813e+07</td>\n",
       "      <td>60212.264822</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1568</th>\n",
       "      <td>6112.598914</td>\n",
       "      <td>3.334894e+07</td>\n",
       "      <td>2.902137e+07</td>\n",
       "      <td>60285.561837</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1569</th>\n",
       "      <td>6111.154189</td>\n",
       "      <td>3.334848e+07</td>\n",
       "      <td>2.902173e+07</td>\n",
       "      <td>60263.937977</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1570</th>\n",
       "      <td>6111.795664</td>\n",
       "      <td>3.320777e+07</td>\n",
       "      <td>2.887158e+07</td>\n",
       "      <td>60452.390642</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1571</th>\n",
       "      <td>6111.617205</td>\n",
       "      <td>3.314979e+07</td>\n",
       "      <td>2.869297e+07</td>\n",
       "      <td>60585.322346</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1572</th>\n",
       "      <td>5845.884475</td>\n",
       "      <td>3.306300e+07</td>\n",
       "      <td>2.856728e+07</td>\n",
       "      <td>60586.908513</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1573</th>\n",
       "      <td>5845.843064</td>\n",
       "      <td>3.311563e+07</td>\n",
       "      <td>2.855824e+07</td>\n",
       "      <td>60606.965541</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1574</th>\n",
       "      <td>5850.375399</td>\n",
       "      <td>3.310543e+07</td>\n",
       "      <td>2.854731e+07</td>\n",
       "      <td>60630.097761</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1575</th>\n",
       "      <td>5850.342911</td>\n",
       "      <td>3.305758e+07</td>\n",
       "      <td>2.851071e+07</td>\n",
       "      <td>60714.029589</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                Q             D         R_Rai         R_Eth\n",
       "1556  6113.017030  3.373426e+07  2.971978e+07  61661.196053\n",
       "1557  6113.767936  3.369428e+07  2.961953e+07  61701.011749\n",
       "1558  6111.834120  3.427227e+07  3.018998e+07  60746.861009\n",
       "1559  6111.676794  3.409202e+07  2.997949e+07  60990.377376\n",
       "1560  6111.737270  3.412698e+07  2.998928e+07  60974.811810\n",
       "1561  6111.183947  3.412650e+07  3.002667e+07  60899.716552\n",
       "1562  6111.416858  3.394189e+07  2.977983e+07  60663.153212\n",
       "1563  6113.684414  3.379694e+07  2.954039e+07  59907.871761\n",
       "1564  6111.202474  3.352808e+07  2.943918e+07  60270.855448\n",
       "1565  6112.309691  3.336299e+07  2.922069e+07  60087.006328\n",
       "1566  6112.711526  3.336272e+07  2.919058e+07  60163.061682\n",
       "1567  6112.516972  3.333513e+07  2.913813e+07  60212.264822\n",
       "1568  6112.598914  3.334894e+07  2.902137e+07  60285.561837\n",
       "1569  6111.154189  3.334848e+07  2.902173e+07  60263.937977\n",
       "1570  6111.795664  3.320777e+07  2.887158e+07  60452.390642\n",
       "1571  6111.617205  3.314979e+07  2.869297e+07  60585.322346\n",
       "1572  5845.884475  3.306300e+07  2.856728e+07  60586.908513\n",
       "1573  5845.843064  3.311563e+07  2.855824e+07  60606.965541\n",
       "1574  5850.375399  3.310543e+07  2.854731e+07  60630.097761\n",
       "1575  5850.342911  3.305758e+07  2.851071e+07  60714.029589"
      ]
     },
     "execution_count": 642,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "VARMAX_new_states = []\n",
    "index = -21\n",
    "for i in range(0,20): \n",
    "    previous_state = states.iloc[train_var.index[index:index+1]]\n",
    "    VARMAX_new_states.append(inverse_transformation_and_state_update(varmax_predictions[i],previous_state,params))\n",
    "    index += 1\n",
    "    \n",
    "VARMAX_new_states = pd.concat(VARMAX_new_states)\n",
    "VARMAX_new_states"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 643,
   "metadata": {},
   "outputs": [],
   "source": [
    "test_data = states.iloc[test.index[0:20]][['collateral','EthInUniswap','RaiInUniswap','RaiDrawnFromSAFEs']]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 644,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>collateral</th>\n",
       "      <th>EthInUniswap</th>\n",
       "      <th>RaiInUniswap</th>\n",
       "      <th>RaiDrawnFromSAFEs</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1557</th>\n",
       "      <td>6112.1732</td>\n",
       "      <td>61701.008091</td>\n",
       "      <td>2.961953e+07</td>\n",
       "      <td>3.369223e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1558</th>\n",
       "      <td>6112.1732</td>\n",
       "      <td>60746.845297</td>\n",
       "      <td>3.019005e+07</td>\n",
       "      <td>3.427256e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1559</th>\n",
       "      <td>6112.1732</td>\n",
       "      <td>60990.371859</td>\n",
       "      <td>2.997948e+07</td>\n",
       "      <td>3.409167e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1560</th>\n",
       "      <td>6112.1732</td>\n",
       "      <td>60974.811993</td>\n",
       "      <td>2.998928e+07</td>\n",
       "      <td>3.412678e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1561</th>\n",
       "      <td>6112.1732</td>\n",
       "      <td>60899.716831</td>\n",
       "      <td>3.002668e+07</td>\n",
       "      <td>3.412678e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1562</th>\n",
       "      <td>6112.1732</td>\n",
       "      <td>60663.156241</td>\n",
       "      <td>2.977982e+07</td>\n",
       "      <td>3.394178e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1563</th>\n",
       "      <td>6112.1732</td>\n",
       "      <td>59907.912377</td>\n",
       "      <td>2.954037e+07</td>\n",
       "      <td>3.379492e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1564</th>\n",
       "      <td>6112.1732</td>\n",
       "      <td>60270.860599</td>\n",
       "      <td>2.943917e+07</td>\n",
       "      <td>3.352893e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1565</th>\n",
       "      <td>6112.1732</td>\n",
       "      <td>60087.012619</td>\n",
       "      <td>2.922068e+07</td>\n",
       "      <td>3.336193e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1566</th>\n",
       "      <td>6112.1732</td>\n",
       "      <td>60163.059766</td>\n",
       "      <td>2.919058e+07</td>\n",
       "      <td>3.336193e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1567</th>\n",
       "      <td>6112.1732</td>\n",
       "      <td>60212.263728</td>\n",
       "      <td>2.913813e+07</td>\n",
       "      <td>3.333443e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1568</th>\n",
       "      <td>6112.1732</td>\n",
       "      <td>60285.559976</td>\n",
       "      <td>2.902136e+07</td>\n",
       "      <td>3.334813e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1569</th>\n",
       "      <td>6112.1732</td>\n",
       "      <td>60263.937922</td>\n",
       "      <td>2.902173e+07</td>\n",
       "      <td>3.334893e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1570</th>\n",
       "      <td>6112.1732</td>\n",
       "      <td>60452.386579</td>\n",
       "      <td>2.887157e+07</td>\n",
       "      <td>3.320728e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1571</th>\n",
       "      <td>6112.1732</td>\n",
       "      <td>60585.321023</td>\n",
       "      <td>2.869296e+07</td>\n",
       "      <td>3.314974e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1572</th>\n",
       "      <td>5847.1732</td>\n",
       "      <td>60586.908513</td>\n",
       "      <td>2.856727e+07</td>\n",
       "      <td>3.306341e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1573</th>\n",
       "      <td>5847.1732</td>\n",
       "      <td>60606.965445</td>\n",
       "      <td>2.855824e+07</td>\n",
       "      <td>3.311591e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1574</th>\n",
       "      <td>5851.6732</td>\n",
       "      <td>60630.097656</td>\n",
       "      <td>2.854731e+07</td>\n",
       "      <td>3.310571e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1575</th>\n",
       "      <td>5851.6732</td>\n",
       "      <td>60714.029269</td>\n",
       "      <td>2.851071e+07</td>\n",
       "      <td>3.305789e+07</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1576</th>\n",
       "      <td>5851.6732</td>\n",
       "      <td>60767.560015</td>\n",
       "      <td>2.853850e+07</td>\n",
       "      <td>3.308720e+07</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      collateral  EthInUniswap  RaiInUniswap  RaiDrawnFromSAFEs\n",
       "1557   6112.1732  61701.008091  2.961953e+07       3.369223e+07\n",
       "1558   6112.1732  60746.845297  3.019005e+07       3.427256e+07\n",
       "1559   6112.1732  60990.371859  2.997948e+07       3.409167e+07\n",
       "1560   6112.1732  60974.811993  2.998928e+07       3.412678e+07\n",
       "1561   6112.1732  60899.716831  3.002668e+07       3.412678e+07\n",
       "1562   6112.1732  60663.156241  2.977982e+07       3.394178e+07\n",
       "1563   6112.1732  59907.912377  2.954037e+07       3.379492e+07\n",
       "1564   6112.1732  60270.860599  2.943917e+07       3.352893e+07\n",
       "1565   6112.1732  60087.012619  2.922068e+07       3.336193e+07\n",
       "1566   6112.1732  60163.059766  2.919058e+07       3.336193e+07\n",
       "1567   6112.1732  60212.263728  2.913813e+07       3.333443e+07\n",
       "1568   6112.1732  60285.559976  2.902136e+07       3.334813e+07\n",
       "1569   6112.1732  60263.937922  2.902173e+07       3.334893e+07\n",
       "1570   6112.1732  60452.386579  2.887157e+07       3.320728e+07\n",
       "1571   6112.1732  60585.321023  2.869296e+07       3.314974e+07\n",
       "1572   5847.1732  60586.908513  2.856727e+07       3.306341e+07\n",
       "1573   5847.1732  60606.965445  2.855824e+07       3.311591e+07\n",
       "1574   5851.6732  60630.097656  2.854731e+07       3.310571e+07\n",
       "1575   5851.6732  60714.029269  2.851071e+07       3.305789e+07\n",
       "1576   5851.6732  60767.560015  2.853850e+07       3.308720e+07"
      ]
     },
     "execution_count": 644,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "test_data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Q - Root Mean Square Error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 645,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "VARMAX performs better by -3.4337187137228042\n"
     ]
    }
   ],
   "source": [
    "var_q_rmse = m.sqrt(mean_squared_error(VAR_new_states['Q'], test_data['collateral']))\n",
    "varmax_q_rmse = m.sqrt(mean_squared_error(VARMAX_new_states['Q'], test_data['collateral']))\n",
    "if var_q_rmse >= varmax_q_rmse:\n",
    "    print('VARMAX performs better by {}'.format(varmax_q_rmse-var_q_rmse))\n",
    "else:\n",
    "    print('VAR performs better by {}'.format(var_q_rmse-varmax_q_rmse))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## D - Root Mean Square Error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 646,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "VAR performs better by -3387.590170986572\n"
     ]
    }
   ],
   "source": [
    "var_d_rmse = m.sqrt(mean_squared_error(VAR_new_states['D'], test_data['RaiDrawnFromSAFEs']))\n",
    "varmax_d_rmse = m.sqrt(mean_squared_error(VARMAX_new_states['D'], test_data['RaiDrawnFromSAFEs']))\n",
    "if var_d_rmse >= varmax_d_rmse:\n",
    "    print('VARMAX performs better by {}'.format(varmax_d_rmse-var_d_rmse))\n",
    "else:\n",
    "    print('VAR performs better by {}'.format(var_d_rmse-varmax_d_rmse))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## R_Rai - Root Mean Square Error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 647,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "VAR performs better by -152.92185181335662\n"
     ]
    }
   ],
   "source": [
    "var_rai_rmse = m.sqrt(mean_squared_error(VAR_new_states['R_Rai'], test_data['RaiInUniswap']))\n",
    "varmax_rai_rmse = m.sqrt(mean_squared_error(VARMAX_new_states['R_Rai'], test_data['RaiInUniswap']))\n",
    "if var_rai_rmse >= varmax_rai_rmse:\n",
    "    print('VARMAX performs better by {}'.format(varmax_rai_rmse-var_rai_rmse))\n",
    "else:\n",
    "    print('VAR performs better by {}'.format(var_rai_rmse-varmax_rai_rmse))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## R_Eth - Root Mean Square Error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 648,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "VARMAX performs better by -0.15262955560427827\n"
     ]
    }
   ],
   "source": [
    "var_eth_rmse = m.sqrt(mean_squared_error(VAR_new_states['R_Eth'], test_data['EthInUniswap']))\n",
    "varmax_eth_rmse = m.sqrt(mean_squared_error(VARMAX_new_states['R_Eth'], test_data['EthInUniswap']))\n",
    "if var_eth_rmse >= varmax_eth_rmse:\n",
    "    print('VARMAX performs better by {}'.format(varmax_eth_rmse-var_eth_rmse))\n",
    "else:\n",
    "    print('VAR performs better by {}'.format(var_eth_rmse-varmax_eth_rmse))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 649,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Aggregated VARMAX RMSE in absolute values:\n",
      "348061.5987702837\n",
      "Aggregated VAR RMSE in absolute values:\n",
      "344524.67309575307\n"
     ]
    }
   ],
   "source": [
    "print('Aggregated VARMAX RMSE in absolute values:')\n",
    "print(abs(varmax_eth_rmse) + abs(varmax_rai_rmse) + abs(varmax_d_rmse) + abs(varmax_q_rmse))\n",
    "\n",
    "print('Aggregated VAR RMSE in absolute values:')\n",
    "print(abs(var_eth_rmse) + abs(var_rai_rmse) + abs(var_d_rmse) + abs(var_q_rmse))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exponential weighted moving average test grid\n",
    "\n",
    "#### Aggregated RMSE values with VARMAX(1) and VAR(6)\n",
    "VAR lags changed between alphas but for simplicity we held constant\n",
    "\n",
    "|Alpha   |VARMAX RMSE   | VAR RMSE  | \n",
    "|---|---|---|\n",
    "|0.1   |346,906.48   |356,511.48   |\n",
    "|0.2  |355,980.48   |353,095.32   |\n",
    "|0.3  |352,226.20   |350,923.13   |\n",
    "|0.4  |347,489.97   |348,758.57  |\n",
    "|0.5  |348,488.02  |347,155.68   |\n",
    "|0.6  |348,345.53   |346,199.91   |\n",
    "|0.7  |347,916.76   |345,765.18   |\n",
    "|**0.8**  |348,061.60   |**345,692.82**   |\n",
    "|0.9  |349,104.38   |346,223   |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Conclusion\n",
    "\n",
    "\n",
    "In this notebook we evaluated the performance of a VARMAX(1) model trained every 20 timesteps an the exogenous signal of redemption price error, against a VAR(15) model retrained everytime step. We decided on using a VAR(15) for the Rai Digital Twin because it performs slightly better, being able to retrain at every prediction timestep. "
   ]
  }
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