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