From 4e9531468899b8a90b933d36160382b576cc82fb Mon Sep 17 00:00:00 2001 From: Marcel Date: Mon, 17 Jan 2022 21:33:30 +0100 Subject: [PATCH] exam2 --- .gitignore | 3 +- Exam2.zip | Bin 405170 -> 0 bytes Exam2/.python-version | 1 + Exam2/Exam2.html | 14325 +++++++++++++++++++++++++++++++ Exam2/Exam2.ipynb | 608 +- Exam2/Exam2_Marcel_Mettler.pdf | Bin 0 -> 142131 bytes Exam2/Exam2_Marcel_Mettler.zip | Bin 0 -> 89356 bytes Exam2/file2.pdf | Bin 0 -> 143131 bytes 8 files changed, 14632 insertions(+), 305 deletions(-) delete mode 100644 Exam2.zip create mode 100644 Exam2/.python-version create mode 100644 Exam2/Exam2.html create mode 100644 Exam2/Exam2_Marcel_Mettler.pdf create mode 100644 Exam2/Exam2_Marcel_Mettler.zip create mode 100644 Exam2/file2.pdf diff --git a/.gitignore b/.gitignore index aba41e0..101eb67 100644 --- a/.gitignore +++ b/.gitignore @@ -1,3 +1,4 @@ /.DS_Store /.ipynb_checkpoints -.DS_Store \ No newline at end of file +.DS_Store +.venv \ No newline at end of file diff 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21^qbVEjU=*vJEW6sFarD1$U5LAWYeB#8O z!z3s{aK}qRUW-L}I?>9HaUCG)gD>pm-V!x4-!T~zGK0@A9C$a`|rOlSFKH$^bo zcp8oYYBO8IvdEGgji~eZ`12PzO?CR!3@Ln_Uxd|eHVgeOXQpuUj)vSp2jXcnc{?HR znRy?N0*g{5c{Od9(5r@;R5eutVTY*mtCco1AB17i)TzoHfwP60IoBK3;;^Fpyo5d*k@GooI56!Ad_d+16AoRPGn6dcP#4NKZpp&SeQB> zPl(R#e5q(znxn{jdo$v1DwZ*whJQZ|< z|bov>-{S$c9|=J}zW z%N$cOhOZ-|p~jLooSIsHFz9r0JDal;6I?ayw@y&3bPy2;v8Zx0_wPjq%^*eF1_(L` z>Pp>U+th2yXtaG&qIMc6jC!)lHYd@Fe=+VlL_`Z76QvRIt;JDIlPC(%1KDdyc{xci z?c5*j?bCRmpziMO$RSyOr4IeJ-&GVMy;F01lRRa66(3$m4GSSJ*h80A#@)ow=B5vN z15MI8cj(9u?+>KmLbkJ0y<`3E=vzlTS0{IbWD;|`F~L%XQVzS+M>0T&TjBbd$YOAM z%}Z>xHe~l*G!HPlowq!-wVLZ^;xqY{(d#Vx_d3X&d= zhk=(FPgg|QoevU<12d1Ov(Q@2Y0m|iu#NqNfc+?aE_>HVs5WS#m6#6d%3_UE`?U_r zDWKF(RtYl7u`Y%D?*r;royQH$pG~wz%h#!u57>;eW!v4{EIxQ1(-eYMocKyBV#tcS zER!iVKD=d(`uGCogFbe-MJ1-4%oQP8)WacG>;JJZ{`KS122e?GDJ2Y&C=wXB9N_#H;f;!Fd zxX#}S?;Et43t(b9e4)ztV%4z8=jc|2wWTZ;=hzd^?ffzwQoemjkj3?`;igD#uD1ae zNoZ!2{iIMjo>=O@J#{2kj6z^vUuKl!Fx)`UZSl6+8@vI*_X-u<1GoJB}W6a;{__8FoB|Y`kseYscqt>gwb8PZATGxS^=$Cw* z=}K6iCu+D#y2geChlI-$v(l`@b&>cQAqKt%??h-lKA6hmPnWH(?z5PKUYm@h*=8 zZk`-{-#QUA@Y~Mi700l57GUu{J!!oio?1r_+4F-Y6~CUD2a$xjo;x>u zm-RSSp1&Ze+_OZXKIw`jZ=%w;OCkSlMVD{3s+M+(-O;RIztHUo7n)0_Sn-Q4vUiP5 z`=Uy_qR7bZcRz~O?=O|P-B$RInI%YMhE>0{9fi%Oy6w-2`LXbn`W~3kZ|ddSqTBa& ze)R~dHyVEf>;>v?3 zPNMNmuda$>k!vIRBhyyL*4m%S*N;5{E;@N%Y{qhc)s~q#!2#bR3Q+0q6C@mfsjhml z@`#&*&lMMDA*^r5Go?=XBlLXkVpa)#szu|HPvA)5TDz^jpaU8wNG0X^z<=7L3#;gR zYU@;3uCh{}5H~;Uj|WtgPz%57a)H008OVvQ8dZZJ4(EV^_=HbIwzEuKB}$$vJmY#qiqi63AM| z;;7u?VV`#fIGM`zVZ8&O>F1n|($5msfV4E2gMAWD^r8c7snh|}`|n_ZxO8o?0+NZ- zcS0+KhUr5IAd&tS74DxsDIt;6{Mqor})SYv-3>Gr?ON`ln>iRy?To@lXfwDm}Ni!cSDB_OrtZ zog8vG=RY@MVpfx=BWSS$L^{^M3nN7E|DG0a__qvjT7D_Oeq`l5u)i~eD7^kn@nvC3 zVqtA+Zo03gs<)4?7pW1598ZSANDhZkOd5cB&hkFB$hllku)vB8A{!wu2@Ll_8na7B$P z3_f6KI2hGm;1?et{XLBT + + + +Exam2 + + + + + + + + + + + + + + + + + + + + + + +

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+
In [ ]:
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using Printf, Statistics, StatsBase, Random, Distributions
+include("jlFiles/printmat.jl")
+Random.seed!(678)          #set the random number generator to this starting point
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+ + + + +
+
TaskLocalRNG()
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In [ ]:
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+
+
using Plots
+
+gr(size=(480,320))
+default(fmt = :svg)
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+ +
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+
+ +
+
+
+
+

Introduction

This exam explores how autocorrelation ought to change how we test statistical hypotheses.

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+

Task 1

Code a function for simulating $T$ observations from an AR(1) series

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$ +y_t = (1-\rho)\mu + \rho y_{t-1} + \varepsilon_t \sigma +$ +where $\varepsilon_t$ is N(0,1).

+

That is, generate $y_1,y_2,...,y_T$ from this formula.

+

To make also the starting value ($y_0$) random, simulate $T+100$ data points, but then discard the first 100 values of $y_t$.

+

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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+
In [ ]:
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+
+
function SimAR1(T,ρ,σ,μ)
+    y = fill(NaN, T + 100)
+    e = rand(Normal(0, 1), T + 100)
+    y[1] = 1
+    for i = 2:T + 100
+        y[i] = (1-ρ)μ + ρ*y[i-1] + e[i]*σ
+    end
+    return y[101:end]
+end
+
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+ + + + +
+
SimAR1 (generic function with 1 method)
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In [ ]:
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y1 = SimAR1(500,0,3,2)
+
+printmat("average from one sample with ρ=0", mean(y1))
+
+printmat("autocorrelations with ρ=0")
+printmat(1:5, autocor(y1)[2:6])
+
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+
average from one sample with ρ=0     1.986
+
+autocorrelations with ρ=0
+
+     1        -0.016
+     2         0.027
+     3         0.004
+     4         0.004
+     5         0.008
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In [ ]:
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+
y2 = SimAR1(500,0.75,3,2)
+
+printmat("average from one sample with ρ=0.75", mean(y2))
+
+printmat("autocorrelations with ρ=0.75")
+printmat(1:5, autocor(y2)[2:6])
+
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+
average from one sample with ρ=0.75     3.187
+
+autocorrelations with ρ=0.75
+
+     1         0.766
+     2         0.580
+     3         0.495
+     4         0.441
+     5         0.372
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+

Task 2

Do a Monte Carlo simulation. Use the parameters (T,ρ,σ,μ) = (500,0,3,2).

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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 $i$.)

    +
  2. +
  3. 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.

    +
  4. +
  5. What is the standard deviation of $\mu_i$ across the $M$ estimates? Compare with the theoretical standard deviation (see below). Report the result.

    +
  6. +
  7. Does the distribution of $\mu_i$ look normal? Plot a histogram and compare with the theoretical pdf (see below).

    +
  8. +
+

...basic stats (the theoretical results)

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 $y$.

+

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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In [ ]:
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+
+
M = 10_000
+
+(T,ρ1,σ,μ) = (500,0,3,2)
+
+μi1 = fill(NaN, M)
+σi1 = fill(NaN, M)
+
+# Monte Carlo simulation
+for i = 1:M
+    y = SimAR1(T, ρ1, σ, μ)
+    μi1[i] = mean(y)
+    σi1[i] = std(y)
+end
+
+# Theoretical results
+y1 = SimAR1(10_000,ρ1,σ,μ)
+σy1 = std(y1)
+s1 = σy1/sqrt(T)
+
+printmat("Average across the simulations:", mean(μi1))
+println("Std across the samples (with ρ=0) and in theory:")
+printmat(["simulations", std(μi1)], ["theory", s1])
+
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+ + +
+
Average across the simulations:     2.001
+
+Std across the samples (with ρ=0) and in theory:
+simulations    theory
+     0.134     0.133
+
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In [ ]:
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+
+
histogram(μi1,bins=0:0.1:4,normalize=true,legend=false,title="Histogram of 10000 averages with ρ=0")
+plot!(μi1->pdf(Normal(μ, s1), μi1))
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+

Task 3

Redo task 2, but now use ρ=0.75 (the other parameters are unchanged).

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In [ ]:
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+
M = 10_000
+
+(T,ρ2,σ,μ) = (500,0.75,3,2)
+
+μi2 = fill(NaN, M)
+σi2 = fill(NaN, M)
+
+# Monte Carlo simulation
+for i = 1:M
+    y = SimAR1(T, ρ2, σ, μ)
+    μi2[i] = mean(y)
+    σi2[i] = std(y)
+end
+
+# Theoretical results
+y2 = SimAR1(10_000,ρ2,σ,μ)
+σy2 = std(y2)
+s2 = σy2/sqrt(T)
+
+printmat("Average across the simulations:", mean(μi2))
+println("Std across the samples (with ρ=0.75) and in theory:")
+printmat(["simulations", std(μi2)], ["theory", s2])
+
+ +
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+ +
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+ + +
+
Average across the simulations:     1.989
+
+Std across the samples (with ρ=0.75) and in theory:
+simulations    theory
+     0.534     0.206
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In [ ]:
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histogram(μi2,bins=0:0.1:4,normalize=true,legend=false,title="Histogram of 10000 averages with ρ=0.75")
+plot!(μi2->pdf(Normal(μ, s2), μi2))
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Task 4

You decide to test the hypothesis that $\mu=2$. Your decision rule is

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  • reject the hypothesis if $|(\mu_i-2)/s|>1.645$ with $s=\sigma_y/\sqrt{T}$
  • +
+

With this decision rule, you are clearly assuming that the theoretical result (definition of $s$) is correct.

+

Estimate both $\mu_i$ and $\sigma_y$ from each sample.

+

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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In [ ]:
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+
(T,σ,μ) = (500,3,2)
+
+rejection1 = 0
+rejection2 = 0
+
+# Count how many times we reject the hypothesis
+for i = 1:M
+    # rejections for ρ = 0
+    s1 = σi1[i]/sqrt(T)
+    if abs((μi1[i] - 2)/s1) > 1.645
+        rejection1 += 1
+    end
+
+    # rejections for ρ = 0.75
+    s2 = σi2[i]/sqrt(T)
+    if abs((μi2[i] - 2)/s2) > 1.645
+        rejection2 += 1
+    end
+end
+
+println("Frequency of rejections:")
+printmat(["with ρ=0 ", rejection1 / (M)], ["with ρ=0.75 ", rejection2 / (M)])
+
+ +
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+ + +
+
Frequency of rejections:
+ with ρ=0 with ρ=0.75 
+     0.098     0.536
+
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+ +
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+ +
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+
+ + + + + + diff --git a/Exam2/Exam2.ipynb b/Exam2/Exam2.ipynb index 3af9ea8..6a3f201 100644 --- a/Exam2/Exam2.ipynb +++ b/Exam2/Exam2.ipynb @@ -9,7 +9,7 @@ }, { "cell_type": "code", - "execution_count": 90, + "execution_count": 100, "metadata": {}, "outputs": [ { @@ -30,7 +30,7 @@ }, { "cell_type": "code", - "execution_count": 91, + "execution_count": 101, "metadata": {}, "outputs": [], "source": [ @@ -71,7 +71,7 @@ }, { "cell_type": "code", - "execution_count": 92, + "execution_count": 102, "metadata": {}, "outputs": [ { @@ -98,7 +98,7 @@ }, { "cell_type": "code", - "execution_count": 93, + "execution_count": 103, "metadata": {}, "outputs": [ { @@ -129,7 +129,7 @@ }, { "cell_type": "code", - "execution_count": 94, + "execution_count": 104, "metadata": {}, "outputs": [ { @@ -184,7 +184,7 @@ }, { "cell_type": "code", - "execution_count": 95, + "execution_count": 105, "metadata": {}, "outputs": [ { @@ -227,378 +227,378 @@ }, { "cell_type": "code", - "execution_count": 96, + "execution_count": 106, "metadata": {}, "outputs": [ { "data": { "image/png": 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