1148 lines
287 KiB
Plaintext
1148 lines
287 KiB
Plaintext
{
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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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"image/png": 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}
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],
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"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))"
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]
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},
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{
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"metadata": {},
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"source": [
|
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"# Task 3\n",
|
||
"\n",
|
||
"Redo task 2, but now use `ρ=0.75` (the other parameters are unchanged)."
|
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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": 97,
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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": [
|
||
"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",
|
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"\n"
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]
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}
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],
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"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])"
|
||
]
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},
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{
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"cell_type": "code",
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"execution_count": 98,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"histogram(μi2,bins=0:0.1:4,normalize=true,legend=false,title=\"Histogram of 10000 averages with ρ=0.75\")\n",
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]
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},
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{
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"source": [
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"# Task 4\n",
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"\n",
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"You decide to test the hypothesis that $\\mu=2$. Your decision rule is \n",
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"\n",
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"- reject the hypothesis if $|(\\mu_i-2)/s|>1.645$ with $s=\\sigma_y/\\sqrt{T}$\n",
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"\n",
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"With this decision rule, you are clearly assuming that the theoretical result (definition of $s$) is correct.\n",
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"\n",
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"Estimate both $\\mu_i$ and $\\sigma_y$ from each sample.\n",
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"\n",
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"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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]
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},
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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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"Frequency of rejections:\n",
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" with ρ=0 with ρ=0.75 \n",
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" 0.098 0.536\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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"(T,σ,μ) = (500,3,2)\n",
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"\n",
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"rejection1 = 0\n",
|
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"rejection2 = 0\n",
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"\n",
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"# Count how many times we reject the hypothesis\n",
|
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"for i = 1:M\n",
|
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" # rejections for ρ = 0\n",
|
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" s1 = σi1[i]/sqrt(T)\n",
|
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" if abs((μi1[i] - 2)/s1) > 1.645\n",
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" rejection1 += 1\n",
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" end\n",
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"\n",
|
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" # rejections for ρ = 0.75\n",
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" s2 = σi2[i]/sqrt(T)\n",
|
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" if abs((μi2[i] - 2)/s2) > 1.645\n",
|
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" rejection2 += 1\n",
|
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" end\n",
|
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"end\n",
|
||
"\n",
|
||
"println(\"Frequency of rejections:\")\n",
|
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"printmat([\"with ρ=0 \", rejection1 / (M)], [\"with ρ=0.75 \", rejection2 / (M)])"
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]
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}
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