9879HW07_Chen_Weiyi.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <font color=\"blue\">  Weiyi Chen"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# MTH9879 Homework 7\n",
    "\n",
    "Assigned: March 31, 2015.\n",
    "Due: April 14, 2015 by 6pm. \n",
    "\n",
    "Late homework **will not be accepted**.\n",
    "\n",
    "$$\n",
    "\\newcommand{\\supp}{\\mathrm{supp}}\n",
    "\\newcommand{\\E}{\\mathbb{E}}\n",
    "\\newcommand{\\Eof}[1]{\\mathbb{E}\\left[ #1 \\right]}\n",
    "\\def\\Cov{{ \\mbox{Cov} }}\n",
    "\\def\\Var{{ \\mbox{Var} }}\n",
    "\\newcommand{\\1}{\\mathbf{1} }\n",
    "\\newcommand{\\PP}{\\mathbb{P} }\n",
    "%\\newcommand{\\Pr}{\\mathrm{Pr} }\n",
    "\\newcommand{\\QQ}{\\mathbb{Q} }\n",
    "\\newcommand{\\RR}{\\mathbb{R} }\n",
    "\\newcommand{\\DD}{\\mathbb{D} }\n",
    "\\newcommand{\\HH}{\\mathbb{H} }\n",
    "\\newcommand{\\spn}{\\mathrm{span} }\n",
    "\\newcommand{\\cov}{\\mathrm{cov} }\n",
    "\\newcommand{\\sgn}{\\mathrm{sgn} }\n",
    "\\newcommand{\\HS}{\\mathcal{L}_{\\mathrm{HS}} }\n",
    "%\\newcommand{\\HS}{\\mathrm{HS} }\n",
    "\\newcommand{\\trace}{\\mathrm{trace} }\n",
    "\\newcommand{\\LL}{\\mathcal{L} }\n",
    "%\\newcommand{\\LL}{\\mathrm{L} }\n",
    "\\newcommand{\\s}{\\mathcal{S} }\n",
    "\\newcommand{\\ee}{\\mathcal{E} }\n",
    "\\newcommand{\\ff}{\\mathcal{F} }\n",
    "\\newcommand{\\hh}{\\mathcal{H} }\n",
    "\\newcommand{\\bb}{\\mathcal{B} }\n",
    "\\newcommand{\\dd}{\\mathcal{D} }\n",
    "\\newcommand{\\g}{\\mathcal{G} }\n",
    "\\newcommand{\\p}{\\partial}\n",
    "\\newcommand{\\half}{\\frac{1}{2} }\n",
    "\\newcommand{\\T}{\\mathcal{T} }\n",
    "\\newcommand{\\bi}{\\begin{itemize}}\n",
    "\\newcommand{\\ei}{\\end{itemize}}\n",
    "\\newcommand{\\beq}{\\begin{equation}}\n",
    "\\newcommand{\\eeq}{\\end{equation}}\n",
    "\\newcommand{\\beas}{\\begin{eqnarray*}}\n",
    "\\newcommand{\\eeas}{\\end{eqnarray*}}\n",
    "\\newcommand{\\cO}{\\mathcal{O}}\n",
    "\\newcommand{\\cF}{\\mathcal{F}}\n",
    "\\newcommand{\\cL}{\\mathcal{L}}\n",
    "\\newcommand{\\BS}{\\text{BS}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font color = \"red\">Homework is to be done by each student individually.  To receive full credit, you must email a completed copy of this iPython notebook to Yu Gan (yugan323@gmail.com), Fubo Shi (fubo.shi.baruch@gmail.com), and Tai-Ho Wang (tai-ho.wang@baruch.cuny.edu) by the due date and time.  All R-code must run correctly and solutions must be written up neatly in Markdown/LaTeX format.\n",
    "\n",
    "<font color=\"blue\">If you encounter problems with Markdown/LaTeX or iPython notebook, please contact your TAs Yu Gan and/or Fubo Shi.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 0. (0 points) \n",
    "Read Sections 4, 5 and 6 of Bouchaud, Farmer and Lillo."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Long memory"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. (6 points)\n",
    "\n",
    "Consider the long memory process\n",
    "\n",
    "$$\n",
    "X_T=\\sum_{i=1}^T\\,\\epsilon_i\n",
    "$$\n",
    "\n",
    "with autocovariance function $\\gamma(\\tau)=\\E[\\epsilon_i\\,\\epsilon_{i-\\tau}] \\sim \\tau^{-\\alpha}$.  Show that as $\\tau \\to \\infty$,\n",
    "\n",
    "$$\n",
    "\\Var[X_{t+\\tau}-X_t] \\sim \\tau ^{2\\,H}\n",
    "$$\n",
    "\n",
    "where the Hurst exponent $H$ is given by\n",
    "\n",
    "$$\n",
    "H=1-\\frac{\\alpha}{2}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=\"blue\"> Answer:\n",
    "\n",
    "<font color=\"blue\">\n",
    "Given the long memory process, the delta is \n",
    "\n",
    "\\begin{equation}\n",
    "X_{t+\\tau}-X_t = \\sum_{i= t + 1}^{t + \\tau}\\epsilon_i\n",
    "\\end{equation}\n",
    "\n",
    "Therefore the variance of delta is\n",
    "\n",
    "\\begin{equation}\n",
    "\\Var\\left[X_{t+\\tau}-X_t\\right] = \\Var\\left[\\sum_{i= t + 1}^ {t + \\tau} \\epsilon_i\\right] = \\sum_{i= 1}^ {\\tau}\\sum_{j= 1}^ {\\tau}\\E\\left[\\epsilon_i\\epsilon_j\\right]\n",
    "\\end{equation}\n",
    "\n",
    "As $\\tau \\to \\infty$, every item goes to $\\tau^{-\\alpha}$ since given $\\E[\\epsilon_i\\,\\epsilon_{i-\\tau}] \\sim \\tau^{-\\alpha}$,\n",
    "\n",
    "\\begin{equation}\n",
    "\\Var\\left[X_{t+\\tau}-X_t\\right] = \\sum_{i= 1}^ {\\tau}\\sum_{j= 1}^ {\\tau}\\E\\left[\\epsilon_i\\epsilon_j\\right] \\sim \\tau^{2-\\alpha} = \\tau ^{2\\,H}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## MSFT data from Alejandro Ca&ntilde;ete"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Set up R environment"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%load_ext rpy2.ipython"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Loading required package: xts\n",
       "Loading required package: zoo\n",
       "\n",
       "Attaching package: ‘zoo’\n",
       "\n",
       "The following objects are masked from ‘package:base’:\n",
       "\n",
       "    as.Date, as.Date.numeric\n",
       "\n",
       "trying URL 'http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/MSFT130311.rData_.zip'\n",
       "Content type 'application/zip' length 71919 bytes (70 KB)\n",
       "opened URL\n",
       "==================================================\n",
       "downloaded 70 KB\n",
       "\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "library(highfrequency)\n",
    "download.file(url=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/03/MSFT130311.rData_.zip\", destfile=\"MSFT130311.zip\")\n",
    "unzip(zipfile=\"MSFT130311.zip\")\n",
    "load(\"MSFT130311.rData\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. (6 points)\n",
    "\n",
    "(a) Generate the series *tradeSign* of trade signs from the *msft130311.inet* data frame.\n",
    "\n",
    "(b) Fit an AR(1) model to the *tradeSign*  data to give the forecast:\n",
    "    $$\n",
    "    \\hat \\epsilon^{(1)}_t=\\phi_1\\,\\epsilon_{t-1}\n",
    "    $$\n",
    "\n",
    "(c) How does $\\phi_1$ relate to the autocorrelation coefficients of *tradeSign*?  What is the mean-squared error of this AR(1) forecast?\n",
    "\n",
    "(d) Fit an $AR(p)$ model to the *tradeSign* data, allowing R to find the best value of $p$ using the default AIC criterion.  This generates the forecast:\n",
    "    $$\n",
    "    \\hat \\epsilon^{(2)}_t=\\sum_{k=1}^p\\,\\phi_k\\,\\epsilon_{t-k}.\n",
    "    $$\n",
    "\n",
    "(e) What value of $p$ is selected by R?  What is the mean-squared error of this AR(p) forecast?\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%%R\n",
    "# (a) Generate the series tradeSign of trade signs from the msft130311.inet data frame.\n",
    "tradeSign = sign(msft130311.inet$signed.shares)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "ar(x = tradeSign, order.max = 1)\n",
       "\n",
       "Coefficients:\n",
       "    1  \n",
       "0.821  \n",
       "\n",
       "Order selected 1  sigma^2 estimated as  0.3236\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "# (b) Fit an AR(1) model to the tradeSign data to give the forecast:\n",
    "ar(tradeSign, order.max = 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Autocorrelations of series ‘tradeSign’, by lag\n",
       "\n",
       "    0     1     2     3     4     5     6     7     8     9    10    11    12 \n",
       "1.000 0.821 0.712 0.643 0.591 0.552 0.518 0.491 0.474 0.455 0.434 0.419 0.403 \n",
       "   13    14    15    16    17    18    19    20    21    22    23    24    25 \n",
       "0.395 0.387 0.375 0.364 0.352 0.340 0.331 0.324 0.310 0.300 0.293 0.285 0.279 \n",
       "   26    27    28    29    30    31    32    33    34    35    36    37    38 \n",
       "0.275 0.270 0.265 0.257 0.250 0.246 0.241 0.230 0.217 0.207 0.202 0.200 0.199 \n",
       "   39    40    41    42 \n",
       "0.193 0.189 0.180 0.175 \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "# (c) How does ϕ1 relate to the autocorrelation coefficients of tradeSign? \n",
    "#     What is the mean-squared error of this AR(1) forecast?\n",
    "acf(tradeSign)\n",
    "acf(tradeSign,plot=F)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font color=\"blue\"> $\\phi_1$ equals to the first order autocorrelation coefficient of *tradeSign* and the mean-squared error is 0.3236."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "ar(x = tradeSign)\n",
       "\n",
       "Coefficients:\n",
       "      1        2        3        4        5        6        7        8  \n",
       " 0.6962   0.0375   0.0422   0.0186   0.0236   0.0048   0.0002   0.0302  \n",
       "      9       10       11       12       13       14  \n",
       " 0.0112  -0.0080   0.0154  -0.0127   0.0237   0.0199  \n",
       "\n",
       "Order selected 14  sigma^2 estimated as  0.3116\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "ar(tradeSign)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font color=\"blue\"> p is selected as 14 and the mean-square error is 0.3116."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3. (8 points)\n",
    "\n",
    "(a) Bin the AR(p) forecast data from the previous problem by forecast sign with bin-breaks from -1.01 to 1.01 in steps of 0.02.\n",
    "\n",
    "(b) Plot the average actual order sign in each bin against the average forecast order sign.  What is the slope of the resulting graph? (Hint: use the R function *lm*).\n",
    "\n",
    "(c) If the expected order sign is $\\hat \\epsilon$, what is the probability $p$ that the next trade will be a market buy?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "\n",
    "# (a) Bin the AR(p) forecast data from -1.01 to 1.01 in steps of 0.02.\n",
    "\n",
    "num_order = 14\n",
    "ar_forecast = ar(tradeSign)\n",
    "\n",
    "tradeSign_copy = rep(0, length(tradeSign))\n",
    "for (i in 1 : num_order)\n",
    "    tradeSign_copy[i] = tradeSign[i]\n",
    "\n",
    "for (i in (num_order+1) : length(tradeSign))\n",
    "    for (j in 1 : num_order)\n",
    "        tradeSign_copy[i] = tradeSign_copy[i] + tradeSign[i-j] * ar_forecast$ar[j]\n",
    "hist(tradeSign_copy, breaks = seq(-1.01,1.01,by=0.02))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = actual ~ forecast)\n",
       "\n",
       "Coefficients:\n",
       "(Intercept)     forecast  \n",
       "  -0.009251     0.983831  \n",
       "\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "\n",
    "#(b) Plot the average actual order sign in each bin against the average forecast order sign.\n",
    "\n",
    "num_bins = 101\n",
    "actual   = rep(0, num_bins)\n",
    "forecast = rep(0, num_bins)\n",
    "\n",
    "for (i in 1 : num_bins){\n",
    "    count = 0\n",
    "    for (j in 1 : length(tradeSign))\n",
    "        if (((-1.01 + 0.02 * (i-1) < tradeSign_copy[j] )) && (tradeSign_copy[j] <= (-1.01 + 0.02 * i))) {\n",
    "            count       = count       + 1\n",
    "            actual[i]   = actual[i]   + tradeSign[j]\n",
    "            forecast[i] = forecast[i] + tradeSign_copy[j]\n",
    "        }\n",
    "    actual[i]   = actual[i]   / count\n",
    "    forecast[i] = forecast[i] / count\n",
    "}\n",
    "\n",
    "# What is the slope of the resulting graph?\n",
    "\n",
    "lm(actual~forecast)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font color=\"blue\"> (c)\n",
    "\n",
    "The expected order sign is given, \n",
    "\n",
    "\\begin{equation}\n",
    "\\hat \\epsilon = 1 \\times p + (-1) \\times (1-p) = 2p-1\n",
    "\\end{equation}\n",
    "\n",
    "So the probability p that the next trade will be a market buy is\n",
    "\n",
    "\\begin{equation}\n",
    "p = \\frac{\\hat \\epsilon + 1}{2}\n",
    "\\end{equation}"
   ]
  }
 ],
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