1148 lines
287 KiB
Plaintext
1148 lines
287 KiB
Plaintext
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Load Packages"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 90,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"TaskLocalRNG()"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"using Printf, Statistics, StatsBase, Random, Distributions\n",
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"include(\"jlFiles/printmat.jl\")\n",
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"Random.seed!(678) #set the random number generator to this starting point"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 91,
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"metadata": {},
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"outputs": [],
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"source": [
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"using Plots\n",
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"\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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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Introduction\n",
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"\n",
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"This exam explores how autocorrelation ought to change how we test statistical hypotheses."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Task 1\n",
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"\n",
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"Code a function for simulating $T$ observations from an AR(1) series\n",
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"\n",
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"$\n",
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"y_t = (1-\\rho)\\mu + \\rho y_{t-1} + \\varepsilon_t \\sigma\n",
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"$\n",
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"where $\\varepsilon_t$ is N(0,1).\n",
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"\n",
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"That is, generate $y_1,y_2,...,y_T$ from this formula.\n",
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"\n",
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"To make also the starting value ($y_0$) random, simulate $T+100$ data points, but then discard the first 100 values of $y_t$.\n",
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"\n",
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"Generate a single \"sample\" using `(T,ρ,σ,μ) = (500,0,3,2)`. Calculate and report the average (mean) and the first 5 autocorrelations (hint: `autocor()`) of this sample. Redo a 2nd time, but with `ρ=0.75`."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 92,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"SimAR1 (generic function with 1 method)"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"function SimAR1(T,ρ,σ,μ)\n",
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" y = fill(NaN, T + 100)\n",
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" e = rand(Normal(0, 1), T + 100)\n",
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" y[1] = 1\n",
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" for i = 2:T + 100\n",
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" y[i] = (1-ρ)μ + ρ*y[i-1] + e[i]*σ\n",
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" end\n",
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" return y[101:end]\n",
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"end"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 93,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"average from one sample with ρ=0 1.986\n",
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"\n",
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"autocorrelations with ρ=0\n",
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"\n",
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" 1 -0.016\n",
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" 2 0.027\n",
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" 3 0.004\n",
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" 4 0.004\n",
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" 5 0.008\n",
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"\n"
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]
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}
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],
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"source": [
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"y1 = SimAR1(500,0,3,2)\n",
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"\n",
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"printmat(\"average from one sample with ρ=0\", mean(y1))\n",
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"\n",
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"printmat(\"autocorrelations with ρ=0\")\n",
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"printmat(1:5, autocor(y1)[2:6])"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 94,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"average from one sample with ρ=0.75 3.187\n",
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"\n",
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"autocorrelations with ρ=0.75\n",
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"\n",
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" 1 0.766\n",
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" 2 0.580\n",
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" 3 0.495\n",
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" 4 0.441\n",
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" 5 0.372\n",
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"\n"
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]
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}
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],
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"source": [
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"y2 = SimAR1(500,0.75,3,2)\n",
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"\n",
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"printmat(\"average from one sample with ρ=0.75\", mean(y2))\n",
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"\n",
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"printmat(\"autocorrelations with ρ=0.75\")\n",
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"printmat(1:5, autocor(y2)[2:6])"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Task 2\n",
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"\n",
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"Do a Monte Carlo simulation. Use the parameters `(T,ρ,σ,μ) = (500,0,3,2)`.\n",
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"\n",
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"1. Generate a sample with $T$ observations and calculate the average. Repeat $M=10,000$ times and store the estimated averages in a vector of length $M$. (The rest of the question uses the symbol $\\mu_i$ to denote the average from sample $i$.)\n",
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"\n",
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"2. What is average $\\mu_i$ across the $M$ estimates? (That is, what is $\\frac{1}{M}\\sum\\nolimits_{i=1}^{M}\\mu_i$?) _Report_ the result.\n",
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"\n",
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"3. What is the standard deviation of $\\mu_i$ across the $M$ estimates? Compare with the theoretical standard deviation (see below). _Report_ the result.\n",
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"\n",
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"4. Does the distribution of $\\mu_i$ look normal? _Plot_ a histogram and compare with the theoretical pdf (see below).\n",
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"\n",
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"\n",
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"## ...basic stats (the theoretical results)\n",
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"\n",
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"says that the sample average of an iid (\"independently and identically distributed\") data series is normally distributed with a mean equal to the true (population) mean $\\mu$ and a standard deviation equal to $s=\\sigma_y/\\sqrt{T}$ where $\\sigma_y$ is the standard deviation of $y$.\n",
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"\n",
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"To compare with our simulation results, you could estimate $\\sigma_y$ from a single simulation with very many observations (say 10'000)."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 95,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Average across the simulations: 2.001\n",
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"\n",
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"Std across the samples (with ρ=0) and in theory:\n",
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"simulations theory\n",
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" 0.134 0.133\n",
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"\n"
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]
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}
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],
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"source": [
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"M = 10_000\n",
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"\n",
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"(T,ρ1,σ,μ) = (500,0,3,2)\n",
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"\n",
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"μi1 = fill(NaN, M)\n",
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"σi1 = fill(NaN, M)\n",
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"\n",
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"# Monte Carlo simulation\n",
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"for i = 1:M\n",
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" y = SimAR1(T, ρ1, σ, μ)\n",
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" μi1[i] = mean(y)\n",
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" σi1[i] = std(y)\n",
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"end\n",
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"\n",
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"# Theoretical results\n",
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"y1 = SimAR1(10_000,ρ1,σ,μ)\n",
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"σy1 = std(y1)\n",
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"s1 = σy1/sqrt(T)\n",
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"\n",
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"printmat(\"Average across the simulations:\", mean(μi1))\n",
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"println(\"Std across the samples (with ρ=0) and in theory:\")\n",
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"printmat([\"simulations\", std(μi1)], [\"theory\", s1])"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 96,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"<circle clip-path=\"url(#clip552)\" style=\"fill:#009af9; stroke:none; fill-opacity:0\" cx=\"1677.31\" cy=\"1140.04\" r=\"2\"/>\n",
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" 156.081,1140.04 172.358,1140.04 322.857,1140.04 413.301,1140.04 489.77,1140.04 572.836,1140.04 649.417,1140.04 690.468,1140.04 731.519,1140.03 755.3,1140 \n",
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|
" 779.08,1139.74 802.861,1138.38 826.642,1132.4 837.209,1125.97 847.776,1115.17 858.343,1097.84 868.909,1071.31 874.193,1053.66 879.476,1032.59 884.76,1007.74 \n",
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" 890.043,978.805 895.326,945.545 900.61,907.815 905.893,865.591 911.177,818.996 920.443,727.692 929.709,626.849 938.975,521.153 948.241,416.837 952.874,367.442 \n",
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" 957.507,321.149 962.14,278.893 966.773,241.561 971.406,209.966 976.039,184.813 980.673,166.678 985.306,155.98 990.357,153.081 995.408,159.398 1000.46,174.755 \n",
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|
" 1005.51,198.727 1010.56,230.657 1015.61,269.693 1020.66,314.815 1025.72,364.89 1035.82,475.052 1045.92,590.529 1056.02,702.645 1066.13,804.684 1071.21,850.751 \n",
|
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|
|
" 1076.29,892.844 1081.37,930.803 1086.45,964.604 1091.53,994.331 1096.61,1020.16 1101.7,1042.34 1106.78,1061.17 1116.94,1090.08 1127.1,1109.57 1137.26,1122.15 \n",
|
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|
|
" 1147.43,1129.92 1169.9,1137.53 1192.37,1139.52 1214.84,1139.95 1237.32,1140.03 1280.08,1140.04 1322.84,1140.04 1411.77,1140.04 1486.63,1140.04 1578.02,1140.04 \n",
|
|||
|
|
" 1663.93,1140.04 1801.57,1140.04 1822.76,1140.04 \n",
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|
" \"/>\n",
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|
"</svg>\n"
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|
]
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|||
|
|
},
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|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "display_data"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"histogram(μi1,bins=0:0.1:4,normalize=true,legend=false,title=\"Histogram of 10000 averages with ρ=0\")\n",
|
|||
|
|
"plot!(μi1->pdf(Normal(μ, s1), μi1))"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {},
|
|||
|
|
"source": [
|
|||
|
|
"# Task 3\n",
|
|||
|
|
"\n",
|
|||
|
|
"Redo task 2, but now use `ρ=0.75` (the other parameters are unchanged)."
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 97,
|
|||
|
|
"metadata": {},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"name": "stdout",
|
|||
|
|
"output_type": "stream",
|
|||
|
|
"text": [
|
|||
|
|
"Average across the simulations: 1.989\n",
|
|||
|
|
"\n",
|
|||
|
|
"Std across the samples (with ρ=0.75) and in theory:\n",
|
|||
|
|
"simulations theory\n",
|
|||
|
|
" 0.534 0.206\n",
|
|||
|
|
"\n"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"M = 10_000\n",
|
|||
|
|
"\n",
|
|||
|
|
"(T,ρ2,σ,μ) = (500,0.75,3,2)\n",
|
|||
|
|
"\n",
|
|||
|
|
"μi2 = fill(NaN, M)\n",
|
|||
|
|
"σi2 = fill(NaN, M)\n",
|
|||
|
|
"\n",
|
|||
|
|
"# Monte Carlo simulation\n",
|
|||
|
|
"for i = 1:M\n",
|
|||
|
|
" y = SimAR1(T, ρ2, σ, μ)\n",
|
|||
|
|
" μi2[i] = mean(y)\n",
|
|||
|
|
" σi2[i] = std(y)\n",
|
|||
|
|
"end\n",
|
|||
|
|
"\n",
|
|||
|
|
"# Theoretical results\n",
|
|||
|
|
"y2 = SimAR1(10_000,ρ2,σ,μ)\n",
|
|||
|
|
"σy2 = std(y2)\n",
|
|||
|
|
"s2 = σy2/sqrt(T)\n",
|
|||
|
|
"\n",
|
|||
|
|
"printmat(\"Average across the simulations:\", mean(μi2))\n",
|
|||
|
|
"println(\"Std across the samples (with ρ=0.75) and in theory:\")\n",
|
|||
|
|
"printmat([\"simulations\", std(μi2)], [\"theory\", s2])"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 98,
|
|||
|
|
"metadata": {},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"data": {
|
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|
|
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|
|||
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" 200.032,1140.04 215.892,1140.04 362.538,1140.04 450.666,1140.04 525.177,1140.04 606.118,1140.04 680.737,1139.89 720.737,1138.95 760.737,1133.81 783.909,1124.85 \n",
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" 807.081,1106.08 818.667,1090.86 830.253,1070.35 841.839,1043.38 853.425,1008.83 863.721,970.98 874.018,925.877 884.314,873.317 894.61,813.46 904.907,746.913 \n",
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" 915.203,674.786 925.499,598.711 935.795,520.818 944.824,453.025 953.853,387.714 962.882,326.899 971.911,272.587 980.94,226.67 989.969,190.819 994.483,177.104 \n",
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" 998.998,166.379 1003.51,158.748 1008.03,154.285 1012.95,153.081 1017.87,155.708 1022.79,162.136 1027.71,172.291 1032.64,186.055 1037.56,203.271 1042.48,223.744 \n",
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" 1146.19,907.058 1156.09,953.253 1166,992.626 1176.95,1028.63 1187.89,1057.45 1198.84,1079.97 1209.79,1097.19 1231.69,1119.45 1253.58,1130.88 1295.25,1138.45 \n",
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" 1336.92,1139.83 1423.57,1140.04 1496.51,1140.04 1585.57,1140.04 1669.28,1140.04 1803.4,1140.04 1824.04,1140.04 \n",
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" \"/>\n",
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"</svg>\n"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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|
"source": [
|
|||
|
|
"histogram(μi2,bins=0:0.1:4,normalize=true,legend=false,title=\"Histogram of 10000 averages with ρ=0.75\")\n",
|
|||
|
|
"plot!(μi2->pdf(Normal(μ, s2), μi2))"
|
|||
|
|
]
|
|||
|
|
},
|
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|
|
{
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|
|
"cell_type": "markdown",
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|
|
"metadata": {},
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|
|
"source": [
|
|||
|
|
"# Task 4\n",
|
|||
|
|
"\n",
|
|||
|
|
"You decide to test the hypothesis that $\\mu=2$. Your decision rule is \n",
|
|||
|
|
"\n",
|
|||
|
|
"- reject the hypothesis if $|(\\mu_i-2)/s|>1.645$ with $s=\\sigma_y/\\sqrt{T}$\n",
|
|||
|
|
"\n",
|
|||
|
|
"With this decision rule, you are clearly assuming that the theoretical result (definition of $s$) is correct.\n",
|
|||
|
|
"\n",
|
|||
|
|
"Estimate both $\\mu_i$ and $\\sigma_y$ from each sample.\n",
|
|||
|
|
"\n",
|
|||
|
|
"In what fraction of the $M$ simulation do you reject your hypothesis when $\\rho=0$ and when $\\rho=0.75$? For the other parameters, use `(T,σ,μ) = (500,3,2)` (same as before)."
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
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|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 99,
|
|||
|
|
"metadata": {},
|
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|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"name": "stdout",
|
|||
|
|
"output_type": "stream",
|
|||
|
|
"text": [
|
|||
|
|
"Frequency of rejections:\n",
|
|||
|
|
" with ρ=0 with ρ=0.75 \n",
|
|||
|
|
" 0.098 0.536\n",
|
|||
|
|
"\n"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"(T,σ,μ) = (500,3,2)\n",
|
|||
|
|
"\n",
|
|||
|
|
"rejection1 = 0\n",
|
|||
|
|
"rejection2 = 0\n",
|
|||
|
|
"\n",
|
|||
|
|
"# Count how many times we reject the hypothesis\n",
|
|||
|
|
"for i = 1:M\n",
|
|||
|
|
" # rejections for ρ = 0\n",
|
|||
|
|
" s1 = σi1[i]/sqrt(T)\n",
|
|||
|
|
" if abs((μi1[i] - 2)/s1) > 1.645\n",
|
|||
|
|
" rejection1 += 1\n",
|
|||
|
|
" end\n",
|
|||
|
|
"\n",
|
|||
|
|
" # rejections for ρ = 0.75\n",
|
|||
|
|
" s2 = σi2[i]/sqrt(T)\n",
|
|||
|
|
" if abs((μi2[i] - 2)/s2) > 1.645\n",
|
|||
|
|
" rejection2 += 1\n",
|
|||
|
|
" end\n",
|
|||
|
|
"end\n",
|
|||
|
|
"\n",
|
|||
|
|
"println(\"Frequency of rejections:\")\n",
|
|||
|
|
"printmat([\"with ρ=0 \", rejection1 / (M)], [\"with ρ=0.75 \", rejection2 / (M)])"
|
|||
|
|
]
|
|||
|
|
}
|
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],
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