Estimate AR persistence jointly

Estimate AR persistence jointly.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "from scipy import signal, stats\n",
    "import statsmodels.api as sm\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulate\n",
    "\n",
    "Simulate two autoregressive series with a shared persistence but different volatilities."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111dd7b90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Simulate data\n",
    "np.random.seed(5555)\n",
    "T = 1000\n",
    "\n",
    "true_phi_1 = 0.5\n",
    "true_phi_2 = 0.5\n",
    "\n",
    "sigma_1 = 1\n",
    "sigma_2 = 10\n",
    "\n",
    "eps_1 = np.random.normal(scale=sigma_1, size=T)\n",
    "eps_2 = np.random.normal(scale=sigma_2, size=T)\n",
    "\n",
    "y_1 = signal.lfilter([1], [1, -true_phi_1], eps_1)\n",
    "y_2 = signal.lfilter([1], [1, -true_phi_2], eps_2)\n",
    "y = np.c_[y_1, y_2]\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(10,3))\n",
    "ax.plot(y_1, label=r'$y_1$')\n",
    "ax.plot(y_2, label=r'$y_2$')\n",
    "ax.legend(loc='lower right');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Estimate separately"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                           Statespace Model Results                           \n",
      "==============================================================================\n",
      "Dep. Variable:                      y   No. Observations:                 1000\n",
      "Model:               SARIMAX(1, 0, 0)   Log Likelihood               -1428.399\n",
      "Date:                Fri, 29 May 2015   AIC                           2860.797\n",
      "Time:                        13:15:26   BIC                           2870.613\n",
      "Sample:                             0   HQIC                          2864.528\n",
      "                               - 1000                                         \n",
      "==============================================================================\n",
      "                             OPG                                              \n",
      "                 coef    std err          z      P>|z|      [95.0% Conf. Int.]\n",
      "------------------------------------------------------------------------------\n",
      "ar.L1          0.4629      0.029     16.137      0.000         0.407     0.519\n",
      "sigma2         1.0188      0.044     22.975      0.000         0.932     1.106\n",
      "==============================================================================\n"
     ]
    }
   ],
   "source": [
    "mod1 = sm.tsa.SARIMAX(y_1, order=(1,0,0), trend='n')\n",
    "res1 = mod1.fit()\n",
    "print res1.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                           Statespace Model Results                           \n",
      "==============================================================================\n",
      "Dep. Variable:                      y   No. Observations:                 1000\n",
      "Model:               SARIMAX(1, 0, 0)   Log Likelihood               -3729.215\n",
      "Date:                Fri, 29 May 2015   AIC                           7462.430\n",
      "Time:                        13:15:28   BIC                           7472.245\n",
      "Sample:                             0   HQIC                          7466.160\n",
      "                               - 1000                                         \n",
      "==============================================================================\n",
      "                             OPG                                              \n",
      "                 coef    std err          z      P>|z|      [95.0% Conf. Int.]\n",
      "------------------------------------------------------------------------------\n",
      "ar.L1          0.5017      0.026     19.449      0.000         0.451     0.552\n",
      "sigma2       101.5612      4.789     21.206      0.000        92.174   110.948\n",
      "==============================================================================\n"
     ]
    }
   ],
   "source": [
    "mod2 = sm.tsa.SARIMAX(y_2, order=(1,0,0), trend='n')\n",
    "res2 = mod2.fit()\n",
    "print res2.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Estimate together"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                           Statespace Model Results                           \n",
      "==============================================================================\n",
      "Dep. Variable:           ['y1', 'y2']   No. Observations:                 1000\n",
      "Model:                  JoinedExample   Log Likelihood               -5158.104\n",
      "Date:                Fri, 29 May 2015   AIC                          10322.208\n",
      "Time:                        13:18:06   BIC                          10336.931\n",
      "Sample:                             0   HQIC                         10327.804\n",
      "                               - 1000                                         \n",
      "==============================================================================\n",
      "                             OPG                                              \n",
      "                 coef    std err          z      P>|z|      [95.0% Conf. Int.]\n",
      "------------------------------------------------------------------------------\n",
      "phi            0.4827      0.019     24.982      0.000         0.445     0.521\n",
      "sigma2.y1      1.0193      0.044     23.036      0.000         0.933     1.106\n",
      "sigma2.y2    101.5670      4.801     21.153      0.000        92.156   110.978\n",
      "==============================================================================\n"
     ]
    }
   ],
   "source": [
    "from statsmodels.tsa.base import tsa_model\n",
    "from statsmodels.tsa.statespace.tools import (\n",
    "    constrain_stationary_univariate as csu,\n",
    "    unconstrain_stationary_univariate as usu\n",
    ")\n",
    "class JoinedExample(sm.tsa.statespace.MLEModel):\n",
    "    \n",
    "    def __init__(self, endog):\n",
    "        self.k_endog = endog.shape[1]\n",
    "        self.ssm = [sm.tsa.SARIMAX(endog[:,i], order=(1,0,0), trend='n') for i in range(self.k_endog)]\n",
    "        for ssm in self.ssm:\n",
    "            ssm.initialize_stationary()\n",
    "        \n",
    "        super(JoinedExample, self).__init__(endog, k_states=1)\n",
    "        self.initialize_approximate_diffuse()\n",
    "        self._initialize_representation()\n",
    "        \n",
    "    @property\n",
    "    def param_names(self):\n",
    "        return ['phi'] + ['sigma2.%s' % name for name in self.endog_names]\n",
    "        \n",
    "    @property\n",
    "    def start_params(self):\n",
    "        ssm_start_params = np.c_[[ssm.start_params for ssm in self.ssm]]\n",
    "        start_params = np.r_[\n",
    "            np.mean(ssm_start_params, axis=0)[0], # average starting AR\n",
    "            [ssm[1] for ssm in ssm_start_params]\n",
    "        ]\n",
    "        return start_params\n",
    "    \n",
    "    def filter(self, **kwargs):\n",
    "        return JoinedExampleResults(self, [ssm.filter() for ssm in self.ssm])\n",
    "    \n",
    "    def transform_params(self, unconstrained):\n",
    "        constrained = np.zeros(unconstrained.shape, unconstrained.dtype)\n",
    "        constrained[0] = csu(unconstrained[0:1])\n",
    "        constrained[1:] = unconstrained[1:]**2\n",
    "        return constrained\n",
    "    \n",
    "    def untransform_params(self, constrained):\n",
    "        unconstrained = np.zeros(constrained.shape, constrained.dtype)\n",
    "        unconstrained[0] = usu(constrained[0:1])\n",
    "        unconstrained[1:] = constrained[1:]**0.5\n",
    "        return unconstrained\n",
    "    \n",
    "    def loglike(self, params=None, average_loglike=False, transformed=True,\n",
    "                set_params=True, **kwargs):\n",
    "        loglike = 0\n",
    "        for i in range(self.k_endog):\n",
    "            ssm_params = np.r_[params[0], params[1+i]]\n",
    "            loglike += self.ssm[i].loglike(\n",
    "                ssm_params, average_loglike, transformed, set_params, **kwargs\n",
    "            )\n",
    "        return loglike\n",
    "\n",
    "    def loglikeobs(self, params=None, transformed=True, set_params=True, **kwargs):\n",
    "        loglikeobs = np.zeros(self.nobs, dtype=complex)\n",
    "        for i in range(self.k_endog):\n",
    "            if params is None:\n",
    "                ssm_params = None\n",
    "            else:\n",
    "                ssm_params = np.r_[params[0], params[1+i]]\n",
    "            loglikeobs += self.ssm[i].loglikeobs(\n",
    "                ssm_params, transformed, set_params, **kwargs\n",
    "            )\n",
    "        return loglikeobs\n",
    "    \n",
    "class JoinedExampleResults(sm.tsa.statespace.MLEResults):\n",
    "    def __init__(self, model, ssm_res):\n",
    "        super(JoinedExampleResults, self).__init__(model)\n",
    "        self.ssm_res = ssm_res\n",
    "        \n",
    "        self.llf_obs = model.loglikeobs()\n",
    "    \n",
    "mod = JoinedExample(y)\n",
    "res = mod.fit()\n",
    "print res.summary()"
   ]
  }
 ],
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