{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "<a id='multiplicative-functionals'></a>\n",
    "<div id=\"qe-notebook-header\" style=\"text-align:right;\">\n",
    "        <a href=\"https://quantecon.org/\" title=\"quantecon.org\">\n",
    "                <img style=\"width:250px;display:inline;\" src=\"https://assets.quantecon.org/img/qe-menubar-logo.svg\" alt=\"QuantEcon\">\n",
    "        </a>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Multiplicative Functionals\n",
    "\n",
    "\n",
    "<a id='index-0'></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 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)  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Co-authored with Chase Coleman and Balint Szoke"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Overview\n",
    "\n",
    "This lecture is a sequel to the [lecture on additive functionals](https://lectures.quantecon.org/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/zreferences.html#lucas-2003) and Thomas Tallarini’s [[Tal00]](https://lectures.quantecon.org/zreferences.html#tall2000) approaches to measuring the benefits of moderating aggregate fluctuations  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A Log-Likelihood Process\n",
    "\n",
    "Consider a vector of additive functionals $ \\{y_t\\}_{t=0}^\\infty $ 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 $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 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 [LSS](https://github.com/QuantEcon/QuantEcon.jl/blob/master/src/lss.jl) struct to work"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Setup"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "hide-output": true
   },
   "outputs": [],
   "source": [
    "using InstantiateFromURL\n",
    "github_project(\"QuantEcon/quantecon-notebooks-julia\", version = \"0.3.0\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "using LinearAlgebra, Statistics\n",
    "using Distributions, Parameters, Plots, QuantEcon\n",
    "import Distributions: loglikelihood\n",
    "gr(fmt = :png);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "loglikelihood (generic function with 4 methods)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AMF_LSS_VAR = @with_kw (A, B, D, F = 0.0, ν = 0.0, lss = construct_ss(A, B, D, F, ν))\n",
    "\n",
    "function construct_ss(A, B, D, F, ν)\n",
    "    H, g = additive_decomp(A, B, D, F)\n",
    "\n",
    "    # Build A matrix for LSS\n",
    "    # Order of states is: [1, t, xt, yt, mt]\n",
    "    A1 = [1 0 0 0 0]       # Transition for 1\n",
    "    A2 = [1 1 0 0 0]       # Transition for t\n",
    "    A3 = [0 0 A 0 0]       # Transition for x_{t+1}\n",
    "    A4 = [ν 0 D 1 0]       # Transition for y_{t+1}\n",
    "    A5 = [0 0 0 0 1]       # Transition for m_{t+1}\n",
    "    Abar = vcat(A1, A2, A3, A4, A5)\n",
    "\n",
    "    # Build B matrix for LSS\n",
    "    Bbar = [0, 0, B, F, H]\n",
    "\n",
    "    # Build G matrix for LSS\n",
    "    # Order of observation is: [xt, yt, mt, st, tt]\n",
    "    G1 = [0 0 1 0 0]               # Selector for x_{t}\n",
    "    G2 = [0 0 0 1 0]               # Selector for y_{t}\n",
    "    G3 = [0 0 0 0 1]               # Selector for martingale\n",
    "    G4 = [0 0 -g 0 0]              # Selector for stationary\n",
    "    G5 = [0 ν 0 0 0]               # Selector for trend\n",
    "    Gbar = vcat(G1, G2, G3, G4, G5)\n",
    "\n",
    "    # Build LSS struct\n",
    "    x0 = [0, 0, 0, 0, 0]\n",
    "    S0 = zeros(5, 5)\n",
    "    return LSS(Abar, Bbar, Gbar, mu_0 = x0, Sigma_0 = S0)\n",
    "end\n",
    "\n",
    "function additive_decomp(A, B, D, F)\n",
    "    A_res = 1 / (1 - A)\n",
    "    g = D * A_res\n",
    "    H = F + D * A_res * B\n",
    "\n",
    "    return H, g\n",
    "end\n",
    "\n",
    "function multiplicative_decomp(A, B, D, F, ν)\n",
    "    H, g = additive_decomp(A, B, D, F)\n",
    "    ν_tilde = ν + 0.5 * H^2\n",
    "\n",
    "    return ν_tilde, H, g\n",
    "end\n",
    "\n",
    "function loglikelihood_path(amf, x, y)\n",
    "    @unpack A, B, D, F = amf\n",
    "    T = length(y)\n",
    "    FF = F^2\n",
    "    FFinv = inv(FF)\n",
    "    temp = y[2:end] - y[1:end-1] - D*x[1:end-1]\n",
    "    obs =  temp .* FFinv .* temp\n",
    "    obssum = cumsum(obs)\n",
    "    scalar = (log(FF) + log(2pi)) * (1:T-1)\n",
    "    return -0.5 * (obssum + scalar)\n",
    "end\n",
    "\n",
    "function loglikelihood(amf, x, y)\n",
    "    llh = loglikelihood_path(amf, x, y)\n",
    "    return llh[end]\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The heavy lifting is done inside the AMF_LSS_VAR struct\n",
    "\n",
    "The following code adds some simple functions that make it straightforward to generate sample paths from an instance of AMF_LSS_VAR"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "population_means (generic function with 2 methods)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function simulate_xy(amf, T)\n",
    "    foo, bar = simulate(amf.lss, T)\n",
    "    x = bar[1, :]\n",
    "    y = bar[2, :]\n",
    "    return x, y\n",
    "end\n",
    "\n",
    "function simulate_paths(amf, T = 150, I = 5000)\n",
    "    # Allocate space\n",
    "    storeX = zeros(I, T)\n",
    "    storeY = zeros(I, T)\n",
    "\n",
    "    for i in 1: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",
    "    end\n",
    "\n",
    "    return storeX, storeY\n",
    "end\n",
    "\n",
    "function population_means(amf, T = 150)\n",
    "    # Allocate Space\n",
    "    xmean = zeros(T)\n",
    "    ymean = zeros(T)\n",
    "\n",
    "    # Pull out moment generator\n",
    "    moment_generator = moment_sequence(amf.lss)\n",
    "    for (tt, x) = enumerate(moment_generator)\n",
    "        ymeans = x[2]\n",
    "        xmean[tt] = ymeans[1]\n",
    "        ymean[tt] = ymeans[2]\n",
    "        if tt == T\n",
    "            break\n",
    "        end\n",
    "    end\n",
    "    return xmean, ymean\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
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"
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F = 0.2\n",
    "amf = AMF_LSS_VAR(A = 0.8, B = 1.0, D = 0.5, 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 = mean(Xit, dims = 1)\n",
    "Ymean_t = mean(Yit, dims = 1)\n",
    "\n",
    "# Compute population means\n",
    "Xmean_pop, Ymean_pop = population_means(amf, T)\n",
    "\n",
    "# Plot sample means vs population means\n",
    "plt_1 = plot(Xmean_t', color = :blue, label = \"1/I sum_i x_t^i\")\n",
    "plot!(plt_1, Xmean_pop, color = :black, label = \"E x_t\")\n",
    "plot!(plt_1, title = \"x_t\", xlim = (0, T), legend = :bottomleft)\n",
    "\n",
    "plt_2 = plot(Ymean_t', color = :blue, label = \"1/I sum_i x_t^i\")\n",
    "plot!(plt_2, Ymean_pop, color = :black, label = \"E y_t\")\n",
    "plot!(plt_2, title = \"y_t\", xlim = (0, T), legend = :bottomleft)\n",
    "\n",
    "plot(plt_1, plt_2, layout = (2, 1), size = (800,500))"
   ]
  },
  {
   "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": 6,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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"
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function simulate_likelihood(amf, Xit, Yit)\n",
    "    # Get size\n",
    "    I, T = size(Xit)\n",
    "\n",
    "    # Allocate space\n",
    "    LLit = zeros(I, T-1)\n",
    "\n",
    "    for i in 1:I\n",
    "        LLit[i, :] = loglikelihood_path(amf, Xit[i, :], Yit[i, :])\n",
    "    end\n",
    "\n",
    "    return LLit\n",
    "end\n",
    "\n",
    "# Get likelihood from each path x^{i}, Y^{i}\n",
    "LLit = simulate_likelihood(amf, Xit, Yit)\n",
    "\n",
    "LLT = 1 / T * LLit[:, end]\n",
    "LLmean_t = mean(LLT)\n",
    "\n",
    "plot(seriestype = :histogram, LLT, label = \"\")\n",
    "plot!(title = \"Distribution of (I/T)log(L_T)|theta_0\")\n",
    "vline!([LLmean_t], linestyle = :dash, color = :black, lw = 2, alpha = 0.6, label = \"\")"
   ]
  },
  {
   "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": 7,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The pdf at +/- 1.175 sigma takes the value: 1.0001868966924388\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Probability of dL being larger than 1 is approx: 0.7600052842019751\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Fraction of dlogL being nonnegative in the sample is: 0.7593364864864864"
     ]
    }
   ],
   "source": [
    "normdist = Normal(0, F)\n",
    "mult = 1.175\n",
    "println(\"The pdf at +/- $mult sigma takes the value: $(pdf(normdist,mult*F))\")\n",
    "println(\"Probability of dL being larger than 1 is approx: \"*\n",
    "        \"$(cdf(normdist,mult*F)-cdf(normdist,-mult*F))\")\n",
    "\n",
    "# Compare this to the sample analogue:\n",
    "L_increment = LLit[:,2:end] - LLit[:,1:end-1]\n",
    "r,c = size(L_increment)\n",
    "frac_nonegative = sum(L_increment.>=0)/(c*r)\n",
    "print(\"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": 8,
   "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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"
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xgrid = range(-1,  1, length = 100)\n",
    "println(\"The pdf at +/- one sigma takes the value: $(pdf(normdist, F)) \")\n",
    "plot(xgrid, pdf.(normdist, xgrid), label = \"\")\n",
    "plot!(title = \"Conditional pdf f(Delta y_(t+1) | x_t)\")"
   ]
  },
  {
   "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": 9,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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"
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create the second (wrong) alternative model\n",
    "amf2 = AMF_LSS_VAR(A = 0.9, B = 1.0, D = 0.55, F = 0.25) # parameters for θ_1 closer to θ_0\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[:, end]\n",
    "LLmean_t2 = mean(LLT2)\n",
    "\n",
    "plot(seriestype = :histogram, LLT2, label = \"\")\n",
    "vline!([LLmean_t2], color = :black, lw = 2, linestyle = :dash, alpha = 0.6, label = \"\")\n",
    "plot!(title = \"Distribution of (1/T)log(L_T) | theta_1)\")"
   ]
  },
  {
   "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": 10,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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"
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(seriestype = :histogram, LLT, bin = 50, alpha = 0.5, label = \"True\", normed = true)\n",
    "plot!(seriestype = :histogram, LLT2, bin = 50, alpha = 0.5, label = \"Alternative\",\n",
    "      normed = true)\n",
    "vline!([mean(LLT)], color = :black, lw = 2, linestyle = :dash, label = \"\")\n",
    "vline!([mean(LLT2)], color = :black, lw = 2, linestyle = :dash, label = \"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we’ll plot the histogram of the difference in log likelihood ratio"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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"
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LLT_diff = LLT - LLT2\n",
    "\n",
    "plot(seriestype = :histogram, LLT_diff, bin = 50, label = \"\")\n",
    "plot!(title = \"(1/T)[log(L_T^i | theta_0) - log(L_T^i |theta_1)]\")"
   ]
  },
  {
   "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 loglikeklihood 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/zreferences.html#bansal-yaron-2004), [[HHL08]](https://lectures.quantecon.org/zreferences.html#hhl-2008), [[Han07]](https://lectures.quantecon.org/zreferences.html#hansen-2007))  \n",
    "- **benefits of eliminating aggregate fluctuations** in representative agent macro models (e.g., [[Tal00]](https://lectures.quantecon.org/zreferences.html#tall2000), [[Luc03]](https://lectures.quantecon.org/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 $\n",
    "\n",
    "Here is code that accomplishes these tasks"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The (min, mean, max) of additive Martingale component in period T is\n",
      "\t (-1.7175606267647656, 0.004161887563984208, 1.9856009482425967)\n",
      "The (min, mean, max) of multiplicative Martingale component in period T is\n",
      "\t (0.1604218891617355, 1.00516580788869, 6.509179739744747)\n"
     ]
    }
   ],
   "source": [
    "function simulate_martingale_components(amf, T = 1_000, I = 5_000)\n",
    "    # Get the multiplicative decomposition\n",
    "    @unpack A, B, D, F, ν, lss = amf\n",
    "    ν, H, g = multiplicative_decomp(A, B, D, F, ν)\n",
    "\n",
    "    # Allocate space\n",
    "    add_mart_comp = zeros(I, T)\n",
    "\n",
    "    # Simulate and pull out additive martingale component\n",
    "    for i in 1:I\n",
    "        foo, bar = simulate(lss, T)\n",
    "        # Martingale component is third component\n",
    "        add_mart_comp[i, :] = bar[3, :]\n",
    "    end\n",
    "\n",
    "    mul_mart_comp = exp.(add_mart_comp' .- (0:T-1) * H^2 / 2)'\n",
    "\n",
    "    return add_mart_comp, mul_mart_comp\n",
    "end\n",
    "\n",
    "# Build model\n",
    "amf_2 = AMF_LSS_VAR(A = 0.8, B = 0.001, D = 1.0, F = 0.01, ν = 0.005)\n",
    "\n",
    "amc, mmc = simulate_martingale_components(amf_2, 1_000, 5_000)\n",
    "\n",
    "amcT = amc[:, end]\n",
    "mmcT = mmc[:, end]\n",
    "\n",
    "println(\"The (min, mean, max) of additive Martingale component in period T is\")\n",
    "println(\"\\t ($(minimum(amcT)), $(mean(amcT)), $(maximum(amcT)))\")\n",
    "\n",
    "println(\"The (min, mean, max) of multiplicative Martingale component in period T is\")\n",
    "println(\"\\t ($(minimum(mmcT)), $(mean(mmcT)), $(maximum(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": 13,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
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"
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(seriestype = :histogram, amcT, bin = 25, normed = true, label = \"\")\n",
    "plot!(title = \"Histogram of Additive Martingale Component\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here’s a histogram of the multiplicative martingale component"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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"
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(seriestype = :histogram, mmcT, bin = 25, normed = true, label = \"\")\n",
    "plot!(title = \"Histogram of Multiplicative Martingale Component\")"
   ]
  },
  {
   "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",
    "Here is some code that tackles these tasks"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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"
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Mtilde_t_density(amf, t; xmin = 1e-8, xmax = 5.0, npts = 5000)\n",
    "\n",
    "    # Pull out the multiplicative decomposition\n",
    "    νtilde, H, g =\n",
    "        multiplicative_decomp(amf.A, amf.B, amf.D, amf.F, amf.ν)\n",
    "    H2 = H*H\n",
    "\n",
    "    # The distribution\n",
    "    mdist = LogNormal(-t * H2 / 2, sqrt(t * H2))\n",
    "    x = range(xmin,  xmax, length = npts)\n",
    "    p = pdf.(mdist, x)\n",
    "\n",
    "    return x, p\n",
    "end\n",
    "\n",
    "function logMtilde_t_density(amf, t; xmin = -15.0, xmax = 15.0, npts = 5000)\n",
    "\n",
    "    # Pull out the multiplicative decomposition\n",
    "    @unpack A, B, D, F, ν = amf\n",
    "    νtilde, H, g = multiplicative_decomp(A, B, D, F, ν)\n",
    "    H2 = H * H\n",
    "\n",
    "    # The distribution\n",
    "    lmdist = Normal(-t * H2 / 2, sqrt(t * H2))\n",
    "    x = range(xmin,  xmax, length = npts)\n",
    "    p = pdf.(lmdist, x)\n",
    "\n",
    "    return x, p\n",
    "end\n",
    "\n",
    "times_to_plot = [10, 100, 500, 1000, 2500, 5000]\n",
    "dens_to_plot = [Mtilde_t_density(amf_2, t, xmin=1e-8, xmax=6.0) for t in times_to_plot]\n",
    "ldens_to_plot = [logMtilde_t_density(amf_2, t, xmin=-10.0, xmax=10.0) for t in times_to_plot]\n",
    "\n",
    "# plot_title = \"Densities of M_t^tilda\" is required, however, plot_title is not yet\n",
    "# supported in Plots\n",
    "plots = plot(layout = (3,2), size = (600,800))\n",
    "\n",
    "for (it, dens_t) in enumerate(dens_to_plot)\n",
    "    x, pdf = dens_t\n",
    "    plot!(plots[it], title = \"Density for time (time_to_plot[it])\")\n",
    "    plot!(plots[it], pdf, fillrange = [[0], pdf], label = \"\")\n",
    "end\n",
    "plot(plots)"
   ]
  },
  {
   "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/zreferences.html#tall2000), [[Luc03]](https://lectures.quantecon.org/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": 16,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(4.54129843114712, 0.24220854072375247, 0, 0.25307727077652764)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Uu(amf, δ, γ)\n",
    "    @unpack A, B, D, F, ν = amf\n",
    "    ν_tilde, H, g = multiplicative_decomp(A, B, D, F, ν)\n",
    "\n",
    "    resolv = 1 / (1 - exp(-δ) * A)\n",
    "    vect = F + D * resolv * B\n",
    "\n",
    "    U_risky = exp(-δ) * resolv * D\n",
    "    u_risky = exp(-δ) / (1 - exp(-δ)) * (ν + 0.5 * (1 - γ) * (vect^2))\n",
    "\n",
    "    U_det = 0\n",
    "    u_det = exp(-δ) / (1 - exp(-δ))  * ν_tilde\n",
    "\n",
    "    return U_risky, u_risky, U_det, u_det\n",
    "end\n",
    "\n",
    "# Set remaining parameters\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": 17,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0809878812017448"
      ]
     },
     "execution_count": 17,
     "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 - 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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  "download_nb_path": "https://lectures.quantecon.org/",
  "filename": "multiplicative_functionals.rst",
  "filename_with_path": "time_series_models/multiplicative_functionals",
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  "title": "Multiplicative Functionals"
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