multiplicative_functionals.ipynb

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   "source": [
    "# Multiplicative Functionals\n",
    "\n",
    "\n",
    "<a id='index-0'></a>"
   ]
  },
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    "## Contents\n",
    "\n",
    "- [Multiplicative Functionals](#Multiplicative-Functionals)  \n",
    "  - [Overview](#Overview)  \n",
    "  - [A Log-Likelihood Process](#A-Log-Likelihood-Process)  \n",
    "  - [Benefits from Reduced Aggregate Fluctuations](#Benefits-from-Reduced-Aggregate-Fluctuations)  "
   ]
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   "source": [
    "**Co-authors: Chase Coleman and Balint Szoke**"
   ]
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    "## Overview\n",
    "\n",
    "This lecture is a sequel to the [lecture on additive functionals](https://lectures.quantecon.org/py/additive_functionals.html)\n",
    "\n",
    "That lecture\n",
    "\n",
    "1. defined a special class of **additive functionals** driven by a first-order vector VAR  \n",
    "1. by taking the exponential of that additive functional, created an associated **multiplicative functional**  \n",
    "\n",
    "\n",
    "This lecture uses this special class to create and analyze two examples\n",
    "\n",
    "- A  **log likelihood process**, an object at the foundation of both frequentist and Bayesian approaches to statistical inference  \n",
    "- A version of Robert E. Lucas’s [[Luc03]](https://lectures.quantecon.org/py/zreferences.html#lucas-2003) and Thomas Tallarini’s [[Tal00]](https://lectures.quantecon.org/py/zreferences.html#tall2000) approaches to measuring the benefits of moderating aggregate fluctuations  "
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    "## A Log-Likelihood Process\n",
    "\n",
    "Consider a vector of additive functionals $ \\{y_t\\}_{t=0}^\\infty $\n",
    "described by\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "    x_{t+1} & = A x_t + B z_{t+1}\n",
    "    \\\\\n",
    "    y_{t+1} - y_t & = D x_{t} + F z_{t+1},\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "where $ A $ is a stable matrix, $ \\{z_{t+1}\\}_{t=0}^\\infty $ is\n",
    "an i.i.d. sequence of $ {\\cal N}(0,I) $ random vectors, $ F $ is\n",
    "nonsingular, and $ x_0 $ and $ y_0 $ are vectors of known\n",
    "numbers\n",
    "\n",
    "Evidently,\n",
    "\n",
    "$$\n",
    "x_{t+1} = \\left(A - B F^{-1}D \\right)x_t +\n",
    "B F^{-1} \\left(y_{t+1} - y_t \\right),\n",
    "$$\n",
    "\n",
    "so that $ x_{t+1} $ can be constructed from observations on\n",
    "$ \\{y_{s}\\}_{s=0}^{t+1} $ and $ x_0 $\n",
    "\n",
    "The distribution of $ y_{t+1} - y_t $ conditional on $ x_t $ is normal with mean $ Dx_t $ and nonsingular covariance matrix $ FF' $\n",
    "\n",
    "Let $ \\theta $ denote the vector of free parameters of the model\n",
    "\n",
    "These parameters pin down the elements of $ A, B, D, F $\n",
    "\n",
    "The **log likelihood function** of $ \\{y_s\\}_{s=1}^t $ is\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "    \\log L_{t}(\\theta)  =\n",
    "    & - {\\frac 1 2} \\sum_{j=1}^{t} (y_{j} - y_{j-1} -\n",
    "         D x_{j-1})'(FF')^{-1}(y_{j} - y_{j-1} - D x_{j-1})\n",
    "    \\\\\n",
    "    & - {\\frac t 2} \\log \\det (FF') - {\\frac {k t} 2} \\log( 2 \\pi)\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "Let’s consider the case of a scalar process in which $ A, B, D, F $ are scalars and $ z_{t+1} $ is a scalar stochastic process\n",
    "\n",
    "We let $ \\theta_o $ denote the “true” values of $ \\theta $, meaning the values that generate the data\n",
    "\n",
    "For the purposes of this exercise,  set $ \\theta_o = (A, B, D, F) = (0.8, 1, 0.5, 0.2) $\n",
    "\n",
    "Set $ x_0 = y_0 = 0 $"
   ]
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    "### Simulating sample paths\n",
    "\n",
    "Let’s write a program to simulate sample paths of $ \\{ x_t, y_{t} \\}_{t=0}^{\\infty} $\n",
    "\n",
    "We’ll do this by formulating the additive functional as a linear state space model and putting the [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) class to work\n",
    "\n",
    "> "
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    "\"\"\" \n",
    "\n",
    "@authors: Chase Coleman, Balint Skoze, Tom Sargent\n",
    "\n",
    "\"\"\"\n",
    "\n",
    "\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "import scipy.linalg as la\n",
    "import quantecon as qe\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.stats import lognorm\n",
    "\n",
    "\n",
    "class AMF_LSS_VAR:\n",
    "    \"\"\"\n",
    "    This class is written to transform a scalar additive functional\n",
    "    into a linear state space system.\n",
    "    \"\"\"\n",
    "    def __init__(self, A, B, D, F=0.0, ν=0.0):\n",
    "        # Unpack required elements\n",
    "        self.A, self.B, self.D, self.F, self.ν = A, B, D, F, ν\n",
    " \n",
    "        # Create space for additive decomposition\n",
    "        self.add_decomp = None\n",
    "        self.mult_decomp = None\n",
    " \n",
    "        # Construct BIG state space representation\n",
    "        self.lss = self.construct_ss()\n",
    " \n",
    "    def construct_ss(self):\n",
    "        \"\"\"\n",
    "        This creates the state space representation that can be passed\n",
    "        into the quantecon LSS class.\n",
    "        \"\"\"\n",
    "        # Pull out useful info\n",
    "        A, B, D, F, ν = self.A, self.B, self.D, self.F, self.ν\n",
    "        nx, nk, nm = 1, 1, 1\n",
    "        if self.add_decomp:\n",
    "            ν, H, g = self.add_decomp\n",
    "        else:\n",
    "            ν, H, g = self.additive_decomp()\n",
    " \n",
    "        # Build A matrix for LSS\n",
    "        # Order of states is: [1, t, xt, yt, mt]\n",
    "        A1 = np.hstack([1, 0, 0, 0, 0])       # Transition for 1\n",
    "        A2 = np.hstack([1, 1, 0, 0, 0])       # Transition for t\n",
    "        A3 = np.hstack([0, 0, A, 0, 0])       # Transition for x_{t+1}\n",
    "        A4 = np.hstack([ν, 0, D, 1, 0])       # Transition for y_{t+1}\n",
    "        A5 = np.hstack([0, 0, 0, 0, 1])       # Transition for m_{t+1}\n",
    "        Abar = np.vstack([A1, A2, A3, A4, A5])\n",
    " \n",
    "        # Build B matrix for LSS\n",
    "        Bbar = np.vstack([0, 0, B, F, H])\n",
    " \n",
    "        # Build G matrix for LSS\n",
    "        # Order of observation is: [xt, yt, mt, st, tt]\n",
    "        G1 = np.hstack([0, 0, 1, 0, 0])               # Selector for x_{t}\n",
    "        G2 = np.hstack([0, 0, 0, 1, 0])               # Selector for y_{t}\n",
    "        G3 = np.hstack([0, 0, 0, 0, 1])               # Selector for martingale\n",
    "        G4 = np.hstack([0, 0, -g, 0, 0])              # Selector for stationary\n",
    "        G5 = np.hstack([0, ν, 0, 0, 0])               # Selector for trend\n",
    "        Gbar = np.vstack([G1, G2, G3, G4, G5])\n",
    " \n",
    "        # Build H matrix for LSS\n",
    "        Hbar = np.zeros((1, 1))\n",
    " \n",
    "        # Build LSS type\n",
    "        x0 = np.hstack([1, 0, 0, 0, 0])\n",
    "        S0 = np.zeros((5, 5))\n",
    "        lss = qe.lss.LinearStateSpace(Abar, Bbar, Gbar, Hbar, mu_0=x0, Sigma_0=S0)\n",
    " \n",
    "        return lss\n",
    " \n",
    "    def additive_decomp(self):\n",
    "        \"\"\"\n",
    "        Return values for the martingale decomposition (Proposition 4.3.3.)\n",
    "            - ν         : unconditional mean difference in Y\n",
    "            - H         : coefficient for the (linear) martingale component (kappa_a)\n",
    "            - g         : coefficient for the stationary component g(x)\n",
    "            - Y_0       : it should be the function of X_0 (for now set it to 0.0)\n",
    "        \"\"\"\n",
    "        A_res = 1 / (1 - self.A)\n",
    "        g = self.D * A_res\n",
    "        H = self.F + self.D * A_res * self.B\n",
    " \n",
    "        return self.ν, H, g\n",
    " \n",
    "    def multiplicative_decomp(self):\n",
    "        \"\"\"\n",
    "        Return values for the multiplicative decomposition (Example 5.4.4.)\n",
    "            - ν_tilde  : eigenvalue\n",
    "            - H        : vector for the Jensen term\n",
    "        \"\"\"\n",
    "        ν, H, g = self.additive_decomp()\n",
    "        ν_tilde = ν + (.5) * H**2\n",
    " \n",
    "        return ν_tilde, H, g\n",
    " \n",
    "    def loglikelihood_path(self, x, y):\n",
    "        A, B, D, F = self.A, self.B, self.D, self.F\n",
    "        T = y.T.size\n",
    "        FF = F**2\n",
    "        FFinv = 1 / FF\n",
    "        temp = y[1:] - y[:-1] - D * x[:-1]\n",
    "        obs = temp * FFinv * temp\n",
    "        obssum = np.cumsum(obs)\n",
    "        scalar = (np.log(FF) + np.log(2 * np.pi)) * np.arange(1, T)\n",
    " \n",
    "        return (-0.5) * (obssum + scalar)\n",
    " \n",
    "    def loglikelihood(self, x, y):\n",
    "        llh = self.loglikelihood_path(x, y)\n",
    " \n",
    "        return llh[-1]"
   ]
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    "\n",
    "The heavy lifting is done inside the AMF_LSS_VAR class\n",
    "\n",
    "The following code adds some simple functions that make it straightforward to generate sample paths from an instance of AMF_LSS_VAR"
   ]
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    "def simulate_xy(amf, T):\n",
    "    \"Simulate individual paths.\"\n",
    "    foo, bar = amf.lss.simulate(T)\n",
    "    x = bar[0, :]\n",
    "    y = bar[1, :]\n",
    "\n",
    "    return x, y\n",
    "\n",
    "def simulate_paths(amf, T=150, I=5000):\n",
    "    \"Simulate multiple independent paths.\"\n",
    "\n",
    "    # Allocate space\n",
    "    storeX = np.empty((I, T))\n",
    "    storeY = np.empty((I, T))\n",
    "\n",
    "    for i in range(I):\n",
    "        # Do specific simulation\n",
    "        x, y = simulate_xy(amf, T)\n",
    "\n",
    "        # Fill in our storage matrices\n",
    "        storeX[i, :] = x\n",
    "        storeY[i, :] = y\n",
    "\n",
    "    return storeX, storeY\n",
    "\n",
    "def population_means(amf, T=150):\n",
    "    # Allocate Space\n",
    "    xmean = np.empty(T)\n",
    "    ymean = np.empty(T)\n",
    "\n",
    "    # Pull out moment generator\n",
    "    moment_generator = amf.lss.moment_sequence()\n",
    "\n",
    "    for tt in range (T):\n",
    "        tmoms = next(moment_generator)\n",
    "        ymeans = tmoms[1]\n",
    "        xmean[tt] = ymeans[0]\n",
    "        ymean[tt] = ymeans[1]\n",
    "\n",
    "    return xmean, ymean"
   ]
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   "source": [
    "Now that we have these functions in our took kit, let’s apply them to run some\n",
    "simulations\n",
    "\n",
    "In particular, let’s use our program to generate $ I = 5000 $ sample paths of length $ T = 150 $, labeled $ \\{ x_{t}^i, y_{t}^i \\}_{t=0}^\\infty $ for $ i = 1, ..., I $\n",
    "\n",
    "Then we compute averages of $ \\frac{1}{I} \\sum_i x_t^i $ and $ \\frac{1}{I} \\sum_i y_t^i $ across the sample paths and compare them with the population means of $ x_t $ and $ y_t $\n",
    "\n",
    "Here goes"
   ]
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\n",
      "text/plain": [
       "<Figure size 1008x576 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "A, B, D, F = [0.8, 1.0, 0.5, 0.2]\n",
    "amf = AMF_LSS_VAR(A, B, D, F=F)\n",
    "\n",
    "T = 150\n",
    "I = 5000\n",
    "\n",
    "# Simulate and compute sample means\n",
    "Xit, Yit = simulate_paths(amf, T, I)\n",
    "Xmean_t = np.mean(Xit, 0)\n",
    "Ymean_t = np.mean(Yit, 0)\n",
    "\n",
    "# Compute population means\n",
    "Xmean_pop, Ymean_pop = population_means(amf, T)\n",
    "\n",
    "# Plot sample means vs population means\n",
    "fig, ax = plt.subplots(2, figsize=(14, 8))\n",
    "\n",
    "ax[0].plot(Xmean_t, label=r'$\\frac{1}{I}\\sum_i x_t^i$', color=\"b\")\n",
    "ax[0].plot(Xmean_pop, label='$\\mathbb{E} x_t$', color=\"k\")\n",
    "ax[0].set_title('$x_t$')\n",
    "ax[0].set_xlim((0, T))\n",
    "ax[0].legend(loc=0)\n",
    "\n",
    "ax[1].plot(Ymean_t, label=r'$\\frac{1}{I}\\sum_i y_t^i$', color=\"b\")\n",
    "ax[1].plot(Ymean_pop, label='$\\mathbb{E} y_t$', color=\"k\")\n",
    "ax[1].set_title('$y_t$')\n",
    "ax[1].set_xlim((0, T))\n",
    "ax[1].legend(loc=0)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simulating log-likelihoods\n",
    "\n",
    "Our next aim is to write a program to simulate $ \\{\\log L_t \\mid \\theta_o\\}_{t=1}^T $\n",
    "\n",
    "We want as inputs to this program the *same* sample paths $ \\{x_t^i, y_t^i\\}_{t=0}^T $ that we  have already computed\n",
    "\n",
    "We now want to simulate $ I = 5000 $ paths of $ \\{\\log L_t^i  \\mid \\theta_o\\}_{t=1}^T $\n",
    "\n",
    "- For each path, we compute $ \\log L_T^i / T $  \n",
    "- We also compute $ \\frac{1}{I} \\sum_{i=1}^I \\log L_T^i / T $  \n",
    "\n",
    "\n",
    "Then we to compare these objects\n",
    "\n",
    "Below we plot the histogram of $ \\log L_T^i / T $ for realizations $ i = 1, \\ldots, 5000 $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def simulate_likelihood(amf, Xit, Yit):\n",
    "    # Get size\n",
    "    I, T = Xit.shape\n",
    "\n",
    "    # Allocate space\n",
    "    LLit = np.empty((I, T-1))\n",
    "\n",
    "    for i in range(I):\n",
    "        LLit[i, :] = amf.loglikelihood_path(Xit[i, :], Yit[i, :])\n",
    "\n",
    "    return LLit\n",
    "\n",
    "# Get likelihood from each path x^{i}, Y^{i}\n",
    "LLit = simulate_likelihood(amf, Xit, Yit)\n",
    "\n",
    "LLT = 1/T * LLit[:, -1]\n",
    "LLmean_t = np.mean(LLT)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "\n",
    "ax.hist(LLT)\n",
    "ax.vlines(LLmean_t, ymin=0, ymax=I//3, color=\"k\", linestyle=\"--\", alpha=0.6)\n",
    "plt.title(r\"Distribution of $\\frac{1}{T} \\log L_{T}  \\mid \\theta_0$\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that the log likelihood is almost always nonnegative, implying that $ L_t $ is typically bigger than 1\n",
    "\n",
    "Recall that the likelihood function is a pdf (probability density function) and **not** a probability measure, so it can take values larger than 1\n",
    "\n",
    "In the current case, the conditional variance of $ \\Delta y_{t+1} $, which equals  $ FF^T=0.04 $, is so small that the maximum value of the pdf is 2 (see the figure below)\n",
    "\n",
    "This implies that approximately $ 75\\% $ of the time (a bit more than one sigma deviation),  we should expect the **increment** of the log likelihood to be nonnegative\n",
    "\n",
    "Let’s see this in a simulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The pdf at +/- 1.175 sigma takes the value:  1.0001868966924388\n",
      "Probability of dL being larger than 1 is approx: 0.7600052842019751\n",
      "Fraction of dlogL being nonnegative in the sample is: 0.7603378378378378\n"
     ]
    }
   ],
   "source": [
    "normdist = sp.stats.norm(0, F)\n",
    "mult = 1.175\n",
    "print(f'The pdf at +/- {mult} sigma takes the value:  {normdist.pdf(mult * F)}')\n",
    "print(f'Probability of dL being larger than 1 is approx: {normdist.cdf(mult * F) - normdist.cdf(-mult * F)}')\n",
    "\n",
    "# Compare this to the sample analogue:\n",
    "L_increment = LLit[:, 1:] - LLit[:, :-1]\n",
    "r, c = L_increment.shape\n",
    "frac_nonegative = np.sum(L_increment >= 0) / (c * r)\n",
    "print(f'Fraction of dlogL being nonnegative in the sample is: {frac_nonegative}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let’s also plot the conditional pdf of $ \\Delta y_{t+1} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The pdf at +/- one sigma takes the value: 1.2098536225957168\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xgrid = np.linspace(-1, 1, 100)\n",
    "plt.plot(xgrid, normdist.pdf(xgrid))\n",
    "plt.title('Conditional pdf $f(\\Delta y_{t+1} \\mid x_t)$')\n",
    "print(f'The pdf at +/- one sigma takes the value: {normdist.pdf(F)}')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### An alternative parameter vector\n",
    "\n",
    "Now consider alternative parameter vector $ \\theta_1 = [A, B, D, F] = [0.9, 1.0, 0.55, 0.25] $\n",
    "\n",
    "We want to compute $ \\{\\log L_t \\mid \\theta_1\\}_{t=1}^T $\n",
    "\n",
    "The $ x_t, y_t $ inputs to this program should be exactly the **same** sample paths $ \\{x_t^i, y_t^i\\}_{t=0}^T $ that we we computed above\n",
    "\n",
    "This is because we want to generate data under the $ \\theta_o $ probability model but evaluate the likelihood under the $ \\theta_1 $ model\n",
    "\n",
    "So our task is to use our program to simulate $ I = 5000 $ paths of $ \\{\\log L_t^i  \\mid \\theta_1\\}_{t=1}^T $\n",
    "\n",
    "- For each path, compute $ \\frac{1}{T} \\log L_T^i $  \n",
    "- Then compute $ \\frac{1}{I}\\sum_{i=1}^I \\frac{1}{T} \\log L_T^i $  \n",
    "\n",
    "\n",
    "We want to compare these objects with each other and with the analogous objects that we computed above\n",
    "\n",
    "Then we want to interpret outcomes\n",
    "\n",
    "A function that we constructed can  handle these tasks\n",
    "\n",
    "The only innovation is that we must create an alternative model to feed in\n",
    "\n",
    "We will creatively call the new model `amf2`\n",
    "\n",
    "We make three graphs\n",
    "\n",
    "- the first sets the stage by repeating an earlier graph  \n",
    "- the second contains two histograms of values of  log likelihoods of the two models  over the period $ T $  \n",
    "- the third compares likelihoods under the true and alternative models  \n",
    "\n",
    "\n",
    "Here’s the code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Create the second (wrong) alternative model\n",
    "A2, B2, D2, F2 = [0.9, 1.0, 0.55, 0.25]   #  parameters for θ_1 closer to Θ_0\n",
    "amf2 = AMF_LSS_VAR(A2, B2, D2, F=F2)\n",
    "\n",
    "# Get likelihood from each path x^{i}, y^{i}\n",
    "LLit2 = simulate_likelihood(amf2, Xit, Yit)\n",
    "\n",
    "LLT2 = 1/(T-1) * LLit2[:, -1]\n",
    "LLmean_t2 = np.mean(LLT2)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "\n",
    "ax.hist(LLT2)\n",
    "ax.vlines(LLmean_t2, ymin=0, ymax=1400, color=\"k\", linestyle=\"--\", alpha=0.6)\n",
    "\n",
    "plt.title(r\"Distribution of $\\frac{1}{T} \\log L_{T}  \\mid \\theta_1$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let’s see a histogram of the log-likelihoods under the true and the alternative model (same sample paths)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/quantecon/anaconda3/lib/python3.7/site-packages/matplotlib/axes/_axes.py:6571: UserWarning: The 'normed' kwarg is deprecated, and has been replaced by the 'density' kwarg.\n",
      "  warnings.warn(\"The 'normed' kwarg is deprecated, and has been \"\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8, 6))\n",
    "\n",
    "plt.hist(LLT, bins=50, alpha=0.5, label='True', normed=True)\n",
    "plt.hist(LLT2, bins=50, alpha=0.5, label='Alternative', normed=True)\n",
    "plt.vlines(np.mean(LLT), 0, 10, color='k', linestyle=\"--\", linewidth= 4)\n",
    "plt.vlines(np.mean(LLT2), 0, 10, color='k', linestyle=\"--\", linewidth= 4)\n",
    "plt.legend()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we’ll plot the histogram of the difference in log likelihood ratio"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "LLT_diff = LLT - LLT2\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(8, 6))\n",
    "\n",
    "ax.hist(LLT_diff, bins=50)\n",
    "plt.title(r\"$\\frac{1}{T}\\left[\\log (L_T^i  \\mid \\theta_0) - \\log (L_T^i \\mid \\theta_1)\\right]$\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Interpretation\n",
    "\n",
    "These histograms of  log likelihood ratios illustrate  important features of **likelihood ratio tests** as tools for discriminating between statistical models\n",
    "\n",
    "- The log likeklihood is higher on average under the true model – obviously a very useful property  \n",
    "- Nevertheless, for a positive fraction of realizations, the log likelihood is higher for the incorrect than for the true model  \n",
    "\n",
    "\n",
    "> - in these instances, a likelihood ratio test mistakenly selects the wrong model  \n",
    "\n",
    "\n",
    "\n",
    "- These mechanics underlie the statistical theory of **mistake probabilities** associated with model selection tests based on  likelihood ratio  \n",
    "\n",
    "\n",
    "(In a subsequent lecture, we’ll use some of the code prepared in this lecture to illustrate mistake probabilities)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Benefits from Reduced Aggregate Fluctuations\n",
    "\n",
    "Now let’s turn to a new example of multiplicative functionals\n",
    "\n",
    "This example illustrates  ideas in the literatures on\n",
    "\n",
    "- **long-run risk** in the consumption based asset pricing literature (e.g., [[BY04]](https://lectures.quantecon.org/py/zreferences.html#bansal-yaron-2004), [[HHL08]](https://lectures.quantecon.org/py/zreferences.html#hhl-2008), [[Han07]](https://lectures.quantecon.org/py/zreferences.html#hansen-2007))  \n",
    "- **benefits of eliminating aggregate fluctuations** in representative agent macro models (e.g., [[Tal00]](https://lectures.quantecon.org/py/zreferences.html#tall2000), [[Luc03]](https://lectures.quantecon.org/py/zreferences.html#lucas-2003))  \n",
    "\n",
    "\n",
    "Let $ c_t $ be consumption at date $ t \\geq 0 $\n",
    "\n",
    "Suppose that $ \\{\\log c_t \\}_{t=0}^\\infty $ is an additive functional described by\n",
    "\n",
    "$$\n",
    "\\log c_{t+1} - \\log c_t = \\nu + D \\cdot x_t + F \\cdot z_{t+1}\n",
    "$$\n",
    "\n",
    "where\n",
    "\n",
    "$$\n",
    "x_{t+1} = A x_t + B z_{t+1}\n",
    "$$\n",
    "\n",
    "Here $ \\{z_{t+1}\\}_{t=0}^\\infty $ is an i.i.d. sequence of $ {\\cal N}(0,I) $ random vectors\n",
    "\n",
    "A representative household ranks consumption processes $ \\{c_t\\}_{t=0}^\\infty $ with a utility functional $ \\{V_t\\}_{t=0}^\\infty $ that satisfies\n",
    "\n",
    "\n",
    "<a id='equation-old1mf'></a>\n",
    "$$\n",
    "\\log V_t - \\log c_t = U \\cdot x_t + {\\sf u} \\tag{1}\n",
    "$$\n",
    "\n",
    "where\n",
    "\n",
    "$$\n",
    "U = \\exp(-\\delta) \\left[ I - \\exp(-\\delta) A' \\right]^{-1} D\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\n",
    "{\\sf u}\n",
    "  = {\\frac {\\exp( -\\delta)}{ 1 - \\exp(-\\delta)}} {\\nu} + \\frac{(1 - \\gamma)}{2} {\\frac {\\exp(-\\delta)}{1 - \\exp(-\\delta)}}\n",
    "\\biggl| D' \\left[ I - \\exp(-\\delta) A \\right]^{-1}B + F \\biggl|^2,\n",
    "$$\n",
    "\n",
    "Here $ \\gamma \\geq 1 $ is a risk-aversion coefficient and $ \\delta > 0 $ is a rate of time preference"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Consumption as a multiplicative process\n",
    "\n",
    "We begin by showing that consumption is a **multiplicative functional** with representation\n",
    "\n",
    "\n",
    "<a id='equation-old2mf'></a>\n",
    "$$\n",
    "\\frac{c_t}{c_0}\n",
    "= \\exp(\\tilde{\\nu}t )\n",
    "\\left( \\frac{\\tilde{M}_t}{\\tilde{M}_0} \\right)\n",
    "\\left( \\frac{\\tilde{e}(x_0)}{\\tilde{e}(x_t)} \\right) \\tag{2}\n",
    "$$\n",
    "\n",
    "where $ \\left( \\frac{\\tilde{M}_t}{\\tilde{M}_0} \\right) $ is a likelihood ratio process and $ \\tilde M_0 = 1 $\n",
    "\n",
    "At this point, as an exercise, we ask the reader please to verify the follow formulas for $ \\tilde{\\nu} $ and $ \\tilde{e}(x_t) $ as functions of $ A, B, D, F $:\n",
    "\n",
    "$$\n",
    "\\tilde \\nu =  \\nu + \\frac{H \\cdot H}{2}\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\n",
    "\\tilde e(x) = \\exp[g(x)] = \\exp \\bigl[ D' (I - A)^{-1} x \\bigr]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simulating a likelihood ratio process again\n",
    "\n",
    "Next, we want a program to simulate the likelihood ratio process $ \\{ \\tilde{M}_t \\}_{t=0}^\\infty $\n",
    "\n",
    "In particular, we want to simulate 5000 sample paths of length $ T=1000 $ for the case in which $ x $ is a scalar and $ [A, B, D, F] = [0.8, 0.001, 1.0, 0.01] $ and $ \\nu = 0.005 $\n",
    "\n",
    "After accomplishing this, we want to display a histogram of $ \\tilde{M}_T^i $ for\n",
    "$ T=1000 $\n",
    "\n",
    "Here is code that accomplishes these tasks"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The (min, mean, max) of additive Martingale component in period T is\n",
      "\t (-1.9690382123611578, 0.006547705117094718, 2.212063019486455)\n",
      "The (min, mean, max) of multiplicative Martingale component in period T is\n",
      "\t (0.12475222456050566, 1.0082098521350018, 8.163520579412458)\n"
     ]
    }
   ],
   "source": [
    "def simulate_martingale_components(amf, T=1000, I=5000):\n",
    "    # Get the multiplicative decomposition\n",
    "    ν, H, g = amf.multiplicative_decomp()\n",
    "\n",
    "    # Allocate space\n",
    "    add_mart_comp = np.empty((I, T))\n",
    "\n",
    "    # Simulate and pull out additive martingale component\n",
    "    for i in range(I):\n",
    "        foo, bar = amf.lss.simulate(T)\n",
    "\n",
    "        # Martingale component is third component\n",
    "        add_mart_comp[i, :] = bar[2, :]\n",
    "\n",
    "    mul_mart_comp = np.exp(add_mart_comp - (np.arange(T) * H**2) / 2)\n",
    "\n",
    "    return add_mart_comp, mul_mart_comp\n",
    "\n",
    "\n",
    "# Build model\n",
    "amf_2 = AMF_LSS_VAR(0.8, 0.001, 1.0, 0.01,.005)\n",
    "\n",
    "amc, mmc = simulate_martingale_components(amf_2, 1000, 5000)\n",
    "\n",
    "amcT = amc[:, -1]\n",
    "mmcT = mmc[:, -1]\n",
    "\n",
    "print(\"The (min, mean, max) of additive Martingale component in period T is\")\n",
    "print(f\"\\t ({np.min(amcT)}, {np.mean(amcT)}, {np.max(amcT)})\")\n",
    "\n",
    "print(\"The (min, mean, max) of multiplicative Martingale component in period T is\")\n",
    "print(f\"\\t ({np.min(mmcT)}, {np.mean(mmcT)}, {np.max(mmcT)})\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Comments\n",
    "\n",
    "- The preceding min, mean, and max of the cross-section of the date\n",
    "  $ T $ realizations of the multiplicative martingale component of\n",
    "  $ c_t $ indicate that the sample mean is close to its population\n",
    "  mean of 1  \n",
    "  \n",
    "  - This outcome prevails for all values of the horizon $ T $  \n",
    "  \n",
    "- The cross-section distribution of the multiplicative martingale\n",
    "  component of $ c $ at date $ T $ approximates a log normal\n",
    "  distribution well  \n",
    "- The histogram of the additive martingale component of\n",
    "  $ \\log c_t $ at date $ T $ approximates a normal distribution\n",
    "  well  \n",
    "\n",
    "\n",
    "Here’s a histogram of the additive martingale component"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/quantecon/anaconda3/lib/python3.7/site-packages/matplotlib/axes/_axes.py:6571: UserWarning: The 'normed' kwarg is deprecated, and has been replaced by the 'density' kwarg.\n",
      "  warnings.warn(\"The 'normed' kwarg is deprecated, and has been \"\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8, 6))\n",
    "\n",
    "ax.hist(amcT, bins=25, normed=True)\n",
    "plt.title(\"Histogram of Additive Martingale Component\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here’s a histogram of the multiplicative martingale component"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/quantecon/anaconda3/lib/python3.7/site-packages/matplotlib/axes/_axes.py:6571: UserWarning: The 'normed' kwarg is deprecated, and has been replaced by the 'density' kwarg.\n",
      "  warnings.warn(\"The 'normed' kwarg is deprecated, and has been \"\n"
     ]
    },
    {
     "data": {
      "image/png": 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ZkqTGGM6SJDXGcJYkqTGGsyRJjTGcJUlqjOEsSVJjDGdJkhpjOEuS1BjDWZKkxhjOkiQ1xnCWJKkxhrMkSY0xnCVJaozhLElSYwxnSZIaYzhLktQYw1mSpMYYzpIkNcZwliSpMYazJEmNMZwlSWqM4SxJUmMMZ0mSGmM4S5LUGMNZkqTGGM6SJDXGcJYkqTGGsyRJjTGcJUlqjOEsSVJjDGdJkhpjOEuS1BjDWZKkxhjOkiQ1xnCWJKkxhrMkSY0xnCVJaozhLElSYwxnSZIaYzhLktQYw1mSpMYYzpIkNcZwliSpMYazJEmNMZwlSWqM4SxJUmMMZ0mSGmM4S5LUGMNZkqTGGM6SJDXGcJYkqTGGsyRJjTGcJUlqjOEsSVJjDGdJkhpjOEuS1BjDWZKkxhjOkiQ1ZqRwTnJQkouTrExy9CTjPDvJhUkuSPKp2S1TkqSFY9FUIyTZAjgOeAqwGjg7ybKqunBgnH2A1wJPqKprk9zv7ipYkqT5bpSW8/7Ayqq6tKpuBU4CDhsa56XAcVV1LUBVXTm7ZUqStHCMEs67AqsGXq/uhw16MPDgJN9LcmaSg2arQEmSFpopu7WBTDCsJpjOPsABwG7Ad5I8oqquW29CyZHAkQB77LHHJhcrSdJCMErLeTWw+8Dr3YA1E4zzf6vqtqr6GXAxXVivp6qOr6qlVbV08eLF061ZkqR5bZRwPhvYJ8leSbYCDgeWDY3zBeA/AyTZma6b+9LZLFSSpIViynCuqtuBo4DTgIuAk6vqgiTHJjm0H+004OokFwLfBP66qq6+u4qWJGk+G+WcM1W1HFg+NOyYgecFvLp/SJKkGfAOYZIkNcZwliSpMYazJEmNMZwlSWqM4SxJUmMMZ0mSGmM4S5LUGMNZkqTGGM6SJDVmpDuEafNYcvSpM/r8ZW8/ZJYqkSSNky1nSZIaYzhLktQYw1mSpMYYzpIkNcZwliSpMYazJEmNMZwlSWqM4SxJUmMMZ0mSGmM4S5LUGMNZkqTGGM6SJDXGcJYkqTGGsyRJjTGcJUlqjOEsSVJjDGdJkhpjOEuS1BjDWZKkxhjOkiQ1xnCWJKkxhrMkSY0xnCVJaozhLElSYwxnSZIaYzhLktQYw1mSpMYYzpIkNcZwliSpMYazJEmNMZwlSWqM4SxJUmMMZ0mSGmM4S5LUGMNZkqTGGM6SJDXGcJYkqTGGsyRJjTGcJUlqjOEsSVJjDGdJkhpjOEuS1BjDWZKkxhjOkiQ1xnCWJKkxhrMkSY0xnCVJaozhLElSYwxnSZIaYzhLktSYkcI5yUFJLk6yMsnRGxnvWUkqydLZK1GSpIVlynBOsgVwHHAwsC9wRJJ9JxjvXsBfAGfNdpGSJC0ko7Sc9wdWVtWlVXUrcBJw2ATjvQX4B+DmWaxPkqQFZ5Rw3hVYNfB6dT/sTkn2A3avqi/NYm2SJC1Io4RzJhhWd76Z3AN4F/CaKSeUHJlkRZIVV1111ehVSpK0gIwSzquB3Qde7wasGXh9L+ARwBlJLgMeCyyb6KKwqjq+qpZW1dLFixdPv2pJkuaxUcL5bGCfJHsl2Qo4HFi27s2qur6qdq6qJVW1BDgTOLSqVtwtFUuSNM9NGc5VdTtwFHAacBFwclVdkOTYJIfe3QVKkrTQLBplpKpaDiwfGnbMJOMeMPOyJElauLxDmCRJjTGcJUlqjOEsSVJjDGdJkhpjOEuS1BjDWZKkxhjOkiQ1xnCWJKkxhrMkSY0xnCVJaozhLElSYwxnSZIaYzhLktQYw1mSpMYYzpIkNcZwliSpMYazJEmNMZwlSWqM4SxJUmMMZ0mSGmM4S5LUGMNZkqTGGM6SJDXGcJYkqTGGsyRJjTGcJUlqjOEsSVJjDGdJkhpjOEuS1BjDWZKkxhjOkiQ1xnCWJKkxhrMkSY0xnCVJaozhLElSYwxnSZIaYzhLktQYw1mSpMYYzpIkNWbRuAvQ7Fly9KkznsZlbz9kFiqRJM2ELWdJkhpjOEuS1BjDWZKkxhjOkiQ1xnCWJKkxhrMkSY0xnCVJaozhLElSYwxnSZIaYzhLktQYw1mSpMYYzpIkNcZwliSpMfPmf6Wajf+RSZKkFthyliSpMYazJEmNMZwlSWqM4SxJUmMMZ0mSGmM4S5LUmJHCOclBSS5OsjLJ0RO8/+okFyY5P8nXk+w5+6VKkrQwTBnOSbYAjgMOBvYFjkiy79Bo5wFLq+r3gVOAf5jtQiVJWihGaTnvD6ysqkur6lbgJOCwwRGq6ptV9dv+5ZnAbrNbpiRJC8co4bwrsGrg9ep+2GReDHx5JkVJkrSQjXL7zkwwrCYcMXkesBR40iTvHwkcCbDHHnuMWKIkSQvLKC3n1cDuA693A9YMj5TkQOBvgUOr6paJJlRVx1fV0qpaunjx4unUK0nSvDdKOJ8N7JNkryRbAYcDywZHSLIf8AG6YL5y9suUJGnhmDKcq+p24CjgNOAi4OSquiDJsUkO7Uf7R2B74DNJfphk2SSTkyRJUxjpv4ysquXA8qFhxww8P3CW65IkacHyDmGSJDXGcJYkqTGGsyRJjTGcJUlqjOEsSVJjDGdJkhpjOEuS1BjDWZKkxhjOkiQ1xnCWJKkxhrMkSY0xnCVJaozhLElSYwxnSZIaYzhLktQYw1mSpMYYzpIkNWbRuAtQW5YcfeqMPn/Z2w+ZpUokaeGy5SxJUmMMZ0mSGmM4S5LUGMNZkqTGGM6SJDXGcJYkqTGGsyRJjTGcJUlqjOEsSVJjDGdJkhpjOEuS1BjDWZKkxhjOkiQ1xnCWJKkxhrMkSY0xnCVJaozhLElSYwxnSZIaYzhLktQYw1mSpMYYzpIkNcZwliSpMYazJEmNMZwlSWrMonEXoPllydGnzujzl739kFmqRJLmLlvOkiQ1xnCWJKkxhrMkSY0xnCVJaozhLElSYwxnSZIaYzhLktQYw1mSpMYYzpIkNcZwliSpMYazJEmN8d7aaor35pYkW86SJDXHcJYkqTGGsyRJjfGcs+YVz1lLmg9sOUuS1JiRwjnJQUkuTrIyydETvL91kk/375+VZMlsFypJ0kIxZbd2ki2A44CnAKuBs5Msq6oLB0Z7MXBtVe2d5HDgHcBz7o6CpbvTTLvFwa5xSTM3yjnn/YGVVXUpQJKTgMOAwXA+DHhT//wU4L1JUlU1i7VKc4LnvSXN1CjhvCuwauD1auAxk41TVbcnuR64L/Dr2ShSWkhmo/U+Ex4cSOM3SjhngmHDLeJRxiHJkcCR/cu1SS4eYf7Ddmb+hP58WZb5shwwf5Zl2suRd8xyJTMzX7YHzJ9lmS/LAZt/WfYcdcRRwnk1sPvA692ANZOMszrJImBH4JrhCVXV8cDxoxY3kSQrqmrpTKbRivmyLPNlOWD+LIvL0Z75sizzZTmg7WUZ5Wrts4F9kuyVZCvgcGDZ0DjLgD/pnz8L+IbnmyVJmp4pW879OeSjgNOALYAPV9UFSY4FVlTVMuBDwCeSrKRrMR9+dxYtSdJ8NtIdwqpqObB8aNgxA89vBv54dkub1Iy6xRszX5ZlviwHzJ9lcTnaM1+WZb4sBzS8LLH3WZKktnj7TkmSGjOnwnmq24jOBUk+nOTKJD8Zdy0zlWT3JN9MclGSC5L85bhrmo4k90zygyQ/6pfjzeOuaSaSbJHkvCRfGnctM5HksiQ/TvLDJCvGXc9MJNkpySlJftp/Xx437po2VZKH9Nti3eOGJK8cd13TkeRV/Xf9J0lOTHLPcdc0bM50a/e3Ef03Bm4jChwxdBvR5iV5IrAW+HhVPWLc9cxEkvsD96+qc5PcCzgHeOYc3CYBtquqtUm2BL4L/GVVnTnm0qYlyauBpcAOVfX0cdczXUkuA5ZW1Zz/m9okHwO+U1Uf7P/qZduqum7cdU1X/3v8S+AxVfXzcdezKZLsSvcd37eqbkpyMrC8qj463srWN5daznfeRrSqbgXW3UZ0TqmqbzPB34DPRVV1eVWd2z+/EbiI7m5xc0p11vYvt+wfc+OodUiS3YBDgA+OuxZ1kuwAPJHur1qoqlvncjD3ngxcMteCecAiYJv+vhzbsuG9O8ZuLoXzRLcRnXNBMF/1/xPZfsBZ461kevqu4B8CVwJfrao5uRzAu4G/Ae4YdyGzoIDTk5zT311wrnogcBXwkf50wweTbDfuombocODEcRcxHVX1S+CdwC+Ay4Hrq+r08Va1obkUziPdIlSbX5Ltgc8Cr6yqG8Zdz3RU1e+q6lF0d8DbP8mcO+WQ5OnAlVV1zrhrmSVPqKpHAwcDf96fEpqLFgGPBt5XVfsBvwHm5DUzAH23/KHAZ8Zdy3QkuTddr+tewAOA7ZI8b7xVbWguhfMotxHVZtafo/0s8Mmq+ty465mpvrvxDOCgMZcyHU8ADu3P1Z4E/FGSE8Zb0vRV1Zr+3yuBz9Od2pqLVgOrB3pjTqEL67nqYODcqrpi3IVM04HAz6rqqqq6Dfgc8Pgx17SBuRTOo9xGVJtRfyHVh4CLquqfxl3PdCVZnGSn/vk2dF/en463qk1XVa+tqt2qagnd9+MbVdVci2AUSbbrLzKk7wJ+KjAn/8Khqn4FrErykH7Qk1n/v9yda45gjnZp934BPDbJtv1v2JPprpdpykh3CGvBZLcRHXNZmyzJicABwM5JVgNvrKoPjbeqaXsC8Hzgx/35WoDX9XeUm0vuD3ysvwL1HsDJVTWn/wxpHtgF+Hz328ki4FNV9ZXxljQjrwA+2TcsLgVeNOZ6piXJtnR/MfOycdcyXVV1VpJTgHOB24HzaPBOYXPmT6kkSVoo5lK3tiRJC4LhLElSYwxnSZIaYzhLktQYw1mSpMYYzpIkNcZwliSpMYazJEmN+f90+CTfhHLKVQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8, 6))\n",
    "\n",
    "ax.hist(mmcT, bins=25, normed=True)\n",
    "plt.title(\"Histogram of Multiplicative Martingale Component\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Representing the likelihood ratio process\n",
    "\n",
    "The likelihood ratio process $ \\{\\widetilde M_t\\}_{t=0}^\\infty $ can be represented as\n",
    "\n",
    "$$\n",
    "\\widetilde M_t = \\exp \\biggl( \\sum_{j=1}^t \\biggl(H \\cdot z_j -\\frac{ H \\cdot H }{2} \\biggr) \\biggr),  \\quad \\widetilde M_0 =1 ,\n",
    "$$\n",
    "\n",
    "where $ H =  [F + B'(I-A')^{-1} D] $\n",
    "\n",
    "It follows that $ \\log {\\widetilde M}_t \\sim {\\mathcal N} ( -\\frac{t H \\cdot H}{2}, t H \\cdot H ) $ and that consequently $ {\\widetilde M}_t $ is log normal\n",
    "\n",
    "Let’s plot the probability density functions for $ \\log {\\widetilde M}_t $ for\n",
    "$ t=100, 500, 1000, 10000, 100000 $\n",
    "\n",
    "Then let’s use the plots to  investigate how these densities evolve through time\n",
    "\n",
    "We will plot the densities of $ \\log {\\widetilde M}_t $ for different values of $ t $\n",
    "\n",
    "Note: `scipy.stats.lognorm` expects you to pass the standard deviation\n",
    "first $ (tH \\cdot H) $ and then the exponent of the mean as a\n",
    "keyword argument `scale` (`scale=`$ \\exp(-tH \\cdot H/2) $)\n",
    "\n",
    "- See the documentation [here](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.lognorm.html#scipy.stats.lognorm)  \n",
    "\n",
    "\n",
    "This is peculiar, so make sure you are careful in working with the log normal distribution\n",
    "\n",
    "Here is some code that tackles these tasks"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x1008 with 6 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def Mtilde_t_density(amf, t, xmin=1e-8, xmax=5.0, npts=5000):\n",
    "\n",
    "    # Pull out the multiplicative decomposition\n",
    "    νtilde, H, g = amf.multiplicative_decomp()\n",
    "    H2 = H * H\n",
    "\n",
    "    # The distribution\n",
    "    mdist = lognorm(np.sqrt(t * H2), scale=np.exp(-t * H2 / 2))\n",
    "    x = np.linspace(xmin, xmax, npts)\n",
    "    pdf = mdist.pdf(x)\n",
    "\n",
    "    return x, pdf\n",
    "\n",
    "\n",
    "def logMtilde_t_density(amf, t, xmin=-15.0, xmax=15.0, npts=5000):\n",
    "\n",
    "    # Pull out the multiplicative decomposition\n",
    "    νtilde, H, g = amf.multiplicative_decomp()\n",
    "    H2 = H * H\n",
    "\n",
    "    # The distribution\n",
    "    lmdist = norm(-t * H2 / 2, np.sqrt(t * H2))\n",
    "    x = np.linspace(xmin, xmax, npts)\n",
    "    pdf = lmdist.pdf(x)\n",
    "\n",
    "    return x, pdf\n",
    "\n",
    "\n",
    "times_to_plot = [10, 100, 500, 1000, 2500, 5000]\n",
    "dens_to_plot = map(lambda t: Mtilde_t_density(amf_2, t, xmin=1e-8, xmax=6.0), times_to_plot)\n",
    "ldens_to_plot = map(lambda t: logMtilde_t_density(amf_2, t, xmin=-10.0, xmax=10.0), times_to_plot)\n",
    "\n",
    "fig, ax = plt.subplots(3, 2, figsize=(8, 14))\n",
    "ax = ax.flatten()\n",
    "\n",
    "fig.suptitle(r\"Densities of $\\tilde{M}_t$\", fontsize=18, y=1.02)\n",
    "for (it, dens_t) in enumerate(dens_to_plot):\n",
    "    x, pdf = dens_t\n",
    "    ax[it].set_title(f\"Density for time {times_to_plot[it]}\")\n",
    "    ax[it].fill_between(x, np.zeros_like(pdf), pdf)\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These probability density functions illustrate a **peculiar property** of log likelihood ratio processes:\n",
    "\n",
    "- With respect to the true model probabilities, they have mathematical expectations equal to $ 1 $ for all $ t \\geq 0 $  \n",
    "- They almost surely converge to zero  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Welfare benefits of reduced random aggregate fluctuations\n",
    "\n",
    "Suppose in the tradition of a strand of macroeconomics (for example Tallarini [[Tal00]](https://lectures.quantecon.org/py/zreferences.html#tall2000), [[Luc03]](https://lectures.quantecon.org/py/zreferences.html#lucas-2003)) we want to estimate the welfare benefits from removing random fluctuations around trend growth\n",
    "\n",
    "We shall  compute how much initial consumption $ c_0 $ a representative consumer who ranks consumption streams according to [(1)](#equation-old1mf) would be willing to sacrifice to enjoy the consumption stream\n",
    "\n",
    "$$\n",
    "\\frac{c_t}{c_0} = \\exp (\\tilde{\\nu} t)\n",
    "$$\n",
    "\n",
    "rather than the stream described by equation [(2)](#equation-old2mf)\n",
    "\n",
    "We want to compute the implied percentage reduction in $ c_0 $ that the representative consumer would accept\n",
    "\n",
    "To accomplish this, we write a function that computes the coefficients $ U $\n",
    "and $ u $ for the original values of $ A, B, D, F, \\nu $, but\n",
    "also for the case that  $ A, B, D, F = [0, 0, 0, 0] $ and\n",
    "$ \\nu = \\tilde{\\nu} $\n",
    "\n",
    "Here’s our code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "def Uu(amf, δ, γ):\n",
    "    A, B, D, F, ν = amf.A, amf.B, amf.D, amf.F, amf.ν\n",
    "    ν_tilde, H, g = amf.multiplicative_decomp()\n",
    "\n",
    "    resolv = 1 / (1 - np.exp(-δ) * A)\n",
    "    vect = F + D * resolv * B\n",
    "\n",
    "    U_risky = np.exp(-δ) * resolv * D\n",
    "    u_risky = (np.exp(-δ) / (1 - np.exp(-δ))) * (ν + (.5) * (1 - γ) * (vect**2))\n",
    "\n",
    "    U_det = 0\n",
    "    u_det = (np.exp(-δ) / (1 - np.exp(-δ))) * ν_tilde\n",
    "\n",
    "    return U_risky, u_risky, U_det, u_det\n",
    "\n",
    "# Set remaining paramaters\n",
    "δ = 0.02\n",
    "γ = 2.0\n",
    "\n",
    "# Get coeffs\n",
    "U_r, u_r, U_d, u_d = Uu(amf_2, δ, γ)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The values of the two processes are\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "    \\log V^r_0 &= \\log c^r_0 + U^r x_0 + u^r\n",
    "     \\\\\n",
    "    \\log V^d_0 &= \\log c^d_0 + U^d x_0 + u^d\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "We look for the ratio $ \\frac{c^r_0-c^d_0}{c^r_0} $ that makes\n",
    "$ \\log V^r_0 - \\log V^d_0 = 0 $\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "    \\underbrace{ \\log V^r_0 - \\log V^d_0}_{=0} + \\log c^d_0 - \\log c^r_0\n",
    "      &= (U^r-U^d) x_0 + u^r - u^d\n",
    "    \\\\\n",
    " \\frac{c^d_0}{ c^r_0}\n",
    "     &= \\exp\\left((U^r-U^d) x_0 + u^r - u^d\\right)\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "Hence, the implied percentage reduction in $ c_0 $ that the\n",
    "representative consumer would accept is given by\n",
    "\n",
    "$$\n",
    "\\frac{c^r_0-c^d_0}{c^r_0} = 1 - \\exp\\left((U^r-U^d) x_0 + u^r - u^d\\right)\n",
    "$$\n",
    "\n",
    "Let’s compute this"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0809878812017448"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = 0.0  # initial conditions\n",
    "logVC_r = U_r * x0 + u_r\n",
    "logVC_d = U_d * x0 + u_d\n",
    "\n",
    "perc_reduct = 100 * (1 - np.exp(logVC_r - logVC_d))\n",
    "perc_reduct"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We find that the consumer would be willing to take a percentage reduction of initial consumption equal to around 1.081"
   ]
  }
 ],
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