Ch02_StatsAppendix.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": []
   },
   "source": [
    "# Review of Statistics\n",
    "\n",
    "This notebook shows some basic statistics needed for this course.\n",
    "\n",
    "It uses the `Statistics` package (built in) for descriptive statistics (averages, autocorrelations, etc) and the [Distributions.jl](https://github.com/JuliaStats/Distributions.jl)  package for statistical distributions (pdf, cdf, etc)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Load Packages and Extra Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Printf, Statistics, LinearAlgebra, DelimitedFiles, Distributions\n",
    "\n",
    "include(\"src/OlsGMFn.jl\")\n",
    "include(\"src/printmat.jl\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots, LaTeXStrings             #packages for plotting and LaTeX\n",
    "default(size = (480,320),fmt = :png)  #or :svg"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Distributions\n",
    "\n",
    "## Probability Density Function (pdf)\n",
    "\n",
    "The cells below calculate and plot pdfs and cdfs of some distributions often used in the lecture notes. The `Distributions.jl` package has many more distributions.\n",
    "\n",
    "### A Remark on the Code\n",
    "\n",
    "- Notice that the `Distributions.jl` package wants `Normal(μ,σ)`, where `σ` is the standard deviation. However, the notation in the lecture notes is $N(\\mu,\\sigma^2)$. For instance, $N(0,2)$ from the lectures is coded as `Normal(0,sqrt(2))`.\n",
    "- `pdf.(Normal(0,1),x)` calculates the pdf of a standard normal variable at each value in the array `x`. Notice the dot (`.`)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pr(x<=0.5)      0.691\n"
     ]
    },
    {
     "data": {
      "image/png": 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       "</defs>\n",
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\" />"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x  = -3:0.1:3\n",
    "xb = x[x.<=0.5]                  #pick out x values <= 0.5\n",
    "\n",
    "pdfx  = pdf.(Normal(0,1),x)       #calculate the pdf of a N(0,1) variable\n",
    "pdfxb = pdf.(Normal(0,1),xb)      #colour the area below this curve\n",
    "Prb = cdf(Normal(0,1),0.5)\n",
    "printlnPs(\"Pr(x<=0.5) \",Prb)\n",
    "\n",
    "p1 = plot( x,pdfx,                        #plot pdf\n",
    "           linecolor = :red,\n",
    "           linewidth = 2,\n",
    "           legend = nothing,\n",
    "           ylims = (0,0.4),\n",
    "           title = \"pdf of N(0,1)\",\n",
    "           xlabel = \"x\",\n",
    "           annotation = (1.1,0.3,text(\"the area covers $(round(Prb,digits=2))%\\n of the probability mass\",:left,8)) )\n",
    "plot!(xb,pdfxb,linecolor=:red,linewidth=2,legend=nothing,fill=(0,:red))    #plot area under pdf\n",
    "display(p1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5% quantile of N(0,1):     -1.645\n"
     ]
    }
   ],
   "source": [
    "printlnPs(\"5% quantile of N(0,1): \", quantile(Normal(0,1),0.05))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Expected Values and Variances\n",
    "\n",
    "The random variable used in the example below can only take two values in the vector `x = [x₁,x₂]`, with probabilities in `p = [p₁,1-p₁]`.   (`π` is already defined in one of the packages.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "       μ:     10.400\n",
      "      σ²:      0.540\n",
      "\n",
      "Check the variance against the theoretical result:     0.540\n"
     ]
    }
   ],
   "source": [
    "x = [9.5,11]\n",
    "p = [0.4,0.6]\n",
    "\n",
    "μ = sum(p.*x)               #or `dot(π,x)`\n",
    "printlnPs(\"μ: \",μ)\n",
    "\n",
    "σ² =  dot(p,(x.-μ).^2)  \n",
    "printlnPs(\"σ²: \",σ²)\n",
    "\n",
    "printlnPs(\"\\nCheck the variance against the theoretical result:\", p[1]*p[2]*(x[2]-x[1])^2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Expected Value of a function and Its Derivative"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "log(x) and its expected value\n",
      "                         x      log(x)  dlog(x)/dx\n",
      "state 1              9.500       2.251       0.105\n",
      "state 2             11.000       2.398       0.091\n",
      "expected value      10.400       2.339       0.097\n",
      "\n"
     ]
    }
   ],
   "source": [
    "Elnx = dot(p,log.(x))     #expected value of log(x)\n",
    "println(\"log(x) and its expected value\")\n",
    "Edlnx = dot(p,1.0./x)      #derivative of log(x) is 1/x\n",
    "\n",
    "xut = hcat([x;μ],[log.(x);Elnx],[1.0./x;Edlnx])\n",
    "printmat(xut;rowNames=[\"state 1\",\"state 2\",\"expected value\"],colNames=[\"x\",\"log(x)\",\"dlog(x)/dx\"],width=12)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Expected Value and Variance of Linear Combinations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.03"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "μ = [11.5, 9.5, 6]/100          #expected returns for 3 assets\n",
    "Σ = [166  34  58;               #covariance matrix\n",
    "      34  64   4;\n",
    "      58   4 100]/100^2\n",
    "Rf = 0.03                       #riskfree return"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Float64}:\n",
       " 0.25\n",
       " 0.68\n",
       " 0.07"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w = [0.25,0.68,0.07]      #portfolio weights summing to 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.005442940000000001"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ERp = dot(w,μ)\n",
    "VarRp = w'*Σ*w;          #or dot(w,Σ,w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean         0.098\n",
      "variance     0.005\n",
      "std          0.074\n",
      "\n"
     ]
    }
   ],
   "source": [
    "xut = [ERp,VarRp,sqrt(VarRp)]\n",
    "printmat(xut;rowNames=[\"mean\",\"variance\",\"std\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear Regressions\n",
    "\n",
    "using the included function `OlsGMFn(Y,X)` where `Y`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sample size:    (388,)\n"
     ]
    }
   ],
   "source": [
    "x = readdlm(\"Data/FFmFactorsPs.csv\",',',skipstart=1)\n",
    "\n",
    "                #yearmonth, market, small minus big, high minus low\n",
    "(ym,Rme,RSMB,RHML) = (x[:,1],x[:,2]/100,x[:,3]/100,x[:,4]/100)\n",
    "x = nothing\n",
    "\n",
    "printlnPs(\"Sample size:\",size(Rme))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[34m\u001b[1mOLS Results:\u001b[22m\u001b[39m\n",
      "\n",
      "            b       std\n",
      "c       0.007     0.002\n",
      "SMB     0.217     0.073\n",
      "HML    -0.429     0.074\n",
      "\n",
      "      R²:      0.134\n"
     ]
    }
   ],
   "source": [
    "Y = Rme                    #T-vector, to get standard OLS notation\n",
    "T = size(Y,1)              \n",
    "X = [ones(T) RSMB RHML]    #TxK matrix, regressors\n",
    "\n",
    "(b,_,_,V,R²) = OlsGMFn(Y,X)\n",
    "Stdb = sqrt.(diag(V))        #standard errors, assuming iid errors\n",
    "\n",
    "printblue(\"OLS Results:\\n\")\n",
    "xNames = [\"c\",\"SMB\",\"HML\"]\n",
    "printmat(b,Stdb,colNames=[\"b\",\"std\"],rowNames=xNames)\n",
    "\n",
    "printlnPs(\"R²: \",R²)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Test a Single Coefficient\n",
    "\n",
    "Suppose we want to test if the 2nd coefficient os equal to 0.3 THis can be done as follows. \n",
    "If the |t-stat|, is larger than 1.645, then you cannot reject the null hypothesis on the 10% significance level"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    |t| :      1.132\n"
     ]
    }
   ],
   "source": [
    "t = (b[2]-0.3)/Stdb[2]\n",
    "\n",
    "printlnPs(\"|t| : \",abs(t))    \n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Test Several coefficients\n",
    "\n",
    "or some linear combinations of them, then \n",
    "\n",
    "Since the estimator $\\hat{\\beta}_{_{k\\times1}}$ satisfies\n",
    "\n",
    "$\n",
    "\\hat{\\beta}-\\beta_{0} \\sim N(0,V_{k\\times k})  ,\n",
    "$\n",
    "\n",
    "we can easily apply various tests. Consider a joint linear hypothesis of the\n",
    "form\n",
    "\n",
    "$\n",
    "H_0: R\\beta=q,\n",
    "$\n",
    "\n",
    "where $R$ is a $J \\times k$ matrix and $q$ is a $J$-vector. To test this, use\n",
    "\n",
    "$\n",
    "(R\\beta-q)^{\\prime}(RVR^{\\prime}) ^{-1}(R\\beta\n",
    "-q)\\overset{d}{\\rightarrow}\\chi_{J}^{2}.\n",
    "$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Vector{Float64}:\n",
       " 0.3\n",
       " 0.0"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R = [0 1 0;\n",
    "     0 0 1]\n",
    "q = [0.3;0]               #β[2]=0.3 and β[3]=0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Compare with 10% critical value of Χ^2(2):     34.669               4.605\n"
     ]
    }
   ],
   "source": [
    "Q = (R*b-q)'*inv(R*V*R')*(R*b-q)\n",
    "\n",
    "printlnPs(\"Compare with 10% critical value of Χ^2(2): \",Q,\" \",quantile(Chisq(2),0.9))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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