Tutorial_02_Stats.ipynb

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  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Stats\n",
    "\n",
    "This notebook loads some data, reports some simple descriptive statistics (means, standard deviations etc) and shows a number of useful plots (scatter plots, histograms, time series plots).\n",
    "\n",
    "Most of the descriptive stats use the standard package Statistics. The plots rely on the Plots package and the pdf and quantiles are from the [Distributions](https://github.com/JuliaStats/Distributions.jl) package.\n",
    "\n",
    "For more stat functions, see the [StatsBase](https://github.com/JuliaStats/StatsBase.jl) package. (Not used here.)\n",
    "\n",
    "Statistical calculations are often reported in tables. This notebook uses my own `printmat()` function. For more powerful alternatives, consider the [PrettyTables](https://github.com/ronisbr/PrettyTables.jl) package."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Load Packages and Extra Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "printyellow (generic function with 1 method)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Statistics, Printf, Dates, LinearAlgebra, DelimitedFiles, Distributions\n",
    "\n",
    "include(\"jlFiles/printmat.jl\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots\n",
    "\n",
    "#pyplot(size=(600,400))\n",
    "gr(size=(480,320))\n",
    "default(fmt = :svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Load Data from a csv File\n",
    "\n",
    "The following is a portion of MyData.csv:\n",
    "\n",
    "```\n",
    "date,Mkt-RF,RF,SmallGrowth\n",
    "197901,4.18,0.77,10.96\n",
    "197902,-3.41,0.73,-2.09\n",
    "197903,5.75,0.81,11.71\n",
    "197904,0.05,0.8,3.27\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "first four lines of x:\n",
      "197901.000     4.180     0.770    10.960\n",
      "197902.000    -3.410     0.730    -2.090\n",
      "197903.000     5.750     0.810    11.710\n",
      "197904.000     0.050     0.800     3.270\n",
      "\n"
     ]
    }
   ],
   "source": [
    "x = readdlm(\"Data/MyData.csv\",',',skipstart=1)  #reading the csv file\n",
    "                                                #skip 1st line\n",
    "println(\"\\nfirst four lines of x:\")\n",
    "printmat(x[1:4,:])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Creating Variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "first 4 obs of Rme and Re\n",
      "1979-01-01     4.180    10.190\n",
      "1979-02-01    -3.410    -2.820\n",
      "1979-03-01     5.750    10.900\n",
      "1979-04-01     0.050     2.470\n",
      "\n"
     ]
    }
   ],
   "source": [
    "ym  = round.(Int,x[:,1])             #yearmonth, like 200712\n",
    "(Rme,Rf,R) = (x[:,2],x[:,3],x[:,4])  #creating variables from columns of x\n",
    "Re  = R - Rf                         #do R .- Rf if R has several columns\n",
    "dN = Date.(string.(ym),\"yyyymm\") #convert to string and then to Julia Date\n",
    "\n",
    "println(\"first 4 obs of Rme and Re\")\n",
    "printmat([dN[1:4] Rme[1:4,:] Re[1:4,:]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Some Descriptive Statistics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Means and Standard Deviations\n",
    "\n",
    "The next few cells estimate means, standard deviations, covariances and correlations of the variables `Rme` (US equity market excess return) and `Re` (excess returns for a segment of the market, small growth firms). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "           Rme        Re\n",
      "mean     0.602     0.303\n",
      "std      4.604     8.572\n",
      "\n"
     ]
    }
   ],
   "source": [
    "μ = mean([Rme Re],dims=1)    #,dims=1 to calculate average along a column\n",
    "σ = std([Rme Re],dims=1)     #do \\sigma[Tab] to get σ\n",
    "\n",
    "printmat([μ;σ],colNames=[\"Rme\",\"Re\"],rowNames=[\"mean\",\"std\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Covariances and Correlations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "cov([Rme Re]): \n",
      "    21.197    28.426\n",
      "    28.426    73.475\n",
      "\n",
      "\n",
      "cor([Rme Re]): \n",
      "     1.000     0.720\n",
      "     0.720     1.000\n",
      "\n"
     ]
    }
   ],
   "source": [
    "println(\"\\n\",\"cov([Rme Re]): \")\n",
    "printmat(cov([Rme Re]))\n",
    "\n",
    "println(\"\\n\",\"cor([Rme Re]): \")\n",
    "printmat(cor([Rme Re]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## OLS\n",
    "\n",
    "A linear regression\n",
    "$\n",
    "y_t = x_t'b + u_t\n",
    "$,\n",
    "where $x_t=[1;R^e_{m,t}]$.\n",
    "\n",
    "Clearly, the first element of $b$ is the intercept and the second element is the slope coefficient.\n",
    "\n",
    "The code below creates a $Tx2$ matrix `x` with $x'_t$ on row $t$. $y$ is a T-element vector (or $Tx1$).\n",
    "\n",
    "The GLM package (not used here) has powerful regression methods. See https://github.com/JuliaStats/GLM.jl."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "OLS coefficients, regressing Re on constant and Rme\n",
      "       Calc 1    Calc 2\n",
      "c      -0.504    -0.504\n",
      "Rme     1.341     1.341\n",
      "\n",
      "      R2:      0.519\n",
      "no. of observations:    388    \n"
     ]
    }
   ],
   "source": [
    "c   = ones(size(Rme,1))         #a vector with ones, no. rows from variable\n",
    "x   = [c Rme]                   #x is a Tx2 matrix\n",
    "y   = copy(Re)                  #to get standard OLS notation\n",
    "\n",
    "b2  = inv(x'x)*x'y              #OLS according to a textbook\n",
    "b   = x\\y                       #also OLS, numerically more stable\n",
    "u   = y - x*b                   #OLS residuals\n",
    "R2  = 1 - var(u)/var(y)         #R2\n",
    "\n",
    "println(\"OLS coefficients, regressing Re on constant and Rme\")\n",
    "printmat([b b2],colNames=[\"Calc 1\",\"Calc 2\"],rowNames=[\"c\",\"Rme\"])\n",
    "printlnPs(\"R2: \",R2) \n",
    "printlnPs(\"no. of observations: \",size(Re,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "coeffs and t-stats\n",
      "         Coef    t-stat\n",
      "c      -0.504    -1.656\n",
      "Rme     1.341    20.427\n",
      "\n"
     ]
    }
   ],
   "source": [
    "Covb  = inv(x'x)*var(u)         #covariance matrix of b estimates\n",
    "Stdb  = sqrt.(diag(Covb))       #std of b estimates\n",
    "tstat = (b .- 0)./Stdb          #t-stats, replace 0 with your null hypothesis \n",
    "\n",
    "println(\"\\ncoeffs and t-stats\")\n",
    "printmat([b tstat],colNames=[\"Coef\",\"t-stat\"],rowNames=[\"c\",\"Rme\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Drawing Random Numbers and Finding Critical Values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Random Numbers: Independent Variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "mean and std of random draws: \n",
      "      series 1  series 2\n",
      "mean     0.140     0.150\n",
      "std      1.038     1.051\n",
      "\n",
      "covariance matrix:\n",
      "     1.078     0.057\n",
      "     0.057     1.104\n",
      "\n",
      "correlation matrix:\n",
      "     1.000     0.052\n",
      "     0.052     1.000\n",
      "\n"
     ]
    }
   ],
   "source": [
    "T = 100\n",
    "x = randn(T,2)    #T x 2 matrix, N(0,1) distribution\n",
    "\n",
    "println(\"\\n\",\"mean and std of random draws: \")\n",
    "μ = mean(x,dims=1)\n",
    "σ = std(x,dims=1)\n",
    "printmat([μ;σ],colNames=[\"series 1\",\"series 2\"],rowNames=[\"mean\",\"std\"])\n",
    "\n",
    "println(\"covariance matrix:\")\n",
    "printmat(cov(x))\n",
    "println(\"correlation matrix:\")\n",
    "printmat(cor(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Random Numbers: Correlated Variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "covariance matrix\n",
      "     1.097     0.766\n",
      "     0.766     2.539\n",
      "\n",
      "correlation matrix:\n",
      "     1.000     0.459\n",
      "     0.459     1.000\n",
      "\n"
     ]
    }
   ],
   "source": [
    "μ = [-1,10]          #vector of means\n",
    "Σ = [1 0.5;          #covariance matrix\n",
    "     0.5 2]\n",
    "\n",
    "T = 100\n",
    "x = rand(MvNormal(μ,Σ),T)'  #random numbers, T x 2, drawn from bivariate N(μ,Σ)\n",
    "\n",
    "println(\"covariance matrix\")\n",
    "printmat(cov(x))\n",
    "println(\"correlation matrix:\")\n",
    "printmat(cor(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Quantiles (\"critical values\") of Distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "5th percentile of N(0,1) and 95th of Chisquare(5)\n",
      "    -1.645    11.070\n",
      "\n"
     ]
    }
   ],
   "source": [
    "N05     = quantile(Normal(0,1),0.05)            #from the Distributions package\n",
    "Chisq05 = quantile(Chisq(5),0.95)\n",
    "\n",
    "println(\"\\n\",\"5th percentile of N(0,1) and 95th of Chisquare(5)\")      #lots of statistics functions\n",
    "printmat([N05 Chisq05])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Statistical Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
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       "  1758.22,515.835 1762.18,576.027 1766.26,716.828 1770.34,562.84 1774.03,500.947 1778.12,594.956 1782.07,808.073 1786.15,748.732 1790.11,486.909 1794.19,731.504 \n",
       "  1798.27,441.393 1802.23,555.396 1806.31,626.009 1810.26,493.928 1814.35,595.169 1818.43,556.034 1822.12,631.965 1826.2,577.941 \n",
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xTicksLoc = [Date(1980),Date(1990),Date(2000),Date(2010)]\n",
    "xTicksLab = Dates.format.(xTicksLoc,\"Y\")\n",
    "\n",
    "p1 = plot( dN,Rme,\n",
    "           linecolor = :blue,\n",
    "           legend = false,\n",
    "           xticks = (xTicksLoc,xTicksLab),\n",
    "           ylim = (-25,25),\n",
    "           title = \"Time series plot: monthly US equity market excess return\",\n",
    "           titlefont = font(10),\n",
    "           ylabel = \"%\" )\n",
    "display(p1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
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   ],
   "source": [
    "p1 = scatter( Rme,Re,\n",
    "              fillcolor = :blue,\n",
    "              legend = false,\n",
    "              xlim = (-40,40),\n",
    "              ylim = (-40,60),\n",
    "              title = \"Scatter plot: two monthly return series (and 45 degree line)\",\n",
    "              titlefont = font(10),\n",
    "              xlabel = \"Market excess return, %\",\n",
    "              ylabel = \"Excess returns on small growth stocks, %\",\n",
    "              guidefont = font(8) )\n",
    "plot!([-40;60],[-40;60],color=:black,linewidth=0.5)   #easier to keep this outside plot()\n",
    "display(p1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xGrid = -25:0.1:15\n",
    "pdfX  = pdf.(Normal(mean(Rme),std(Rme)),xGrid) #\"Distributions\" wants σ, not σ^2\n",
    "\n",
    "p1 = histogram( Rme,bins = -25:1:15,\n",
    "                normalized = true,     #normalized to have area=1\n",
    "                label = \"histogram\",\n",
    "                title = \"Histogram: monthly US equity market excess return\",\n",
    "                titlefont = font(10),\n",
    "                xlabel = \"Market excess return, %\",\n",
    "                legend = :topleft,\n",
    "                guidefont = font(8) )   #font size of axis labels\n",
    "plot!(xGrid,pdfX,linewidth=3,label=\"fitted N()\")\n",
    "display(p1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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