THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 429, Fall 2012
Midterm 1 Solution, Oct 8th, in class.
1. (122=24 points) True or False. No need to explain.
(a) If a time series cfw_Xt is independent and identically distributed, then it is
STAT 429, Fall 2014
Midterm Exam, 10/20/14
Time Limit: 2 hours
Name (Print):
Instructor:
Stphane Guerrier
e
This exam contains 7 pages (including this cover page) and 4 problems. Check to see if any pages are missing.
Enter all requested information on th
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 429, Fall 2012
Practice Midterm 1
Print Your Name:
Student ID:
Registered in Section (circle one):
U3
G4
1. Please print your name and student ID number in the above space and circle the
secti
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 429, Fall 2012
Midterm 2, Nov 9th, in class.
I,
(print name), Student ID:
, Registered
in Section (circle one): U3, G4, will keep the information regarding this exam condential.
I will not dis
Notation and reminder
The autocorrelation function
is often denoted by (, ) and is expressed as:
(Xt , Xt+h ) = Corr (Xt , Xt+h ) =
Cov (Xt , Xt+h )
Xt Xt+h
for (weakly) stationary processes we have that
(Xt , Xt+h ) =
Cov (Xt , Xt+h )
(h)
= (h)
=
Xt Xt+h
Chapter 2
Part I
Statistics 429: Time Series Analysis
1 / 11
Quick Review of Regression
Data:
Response is a column vector y = (y1 , ., yT )T .
The explanatory variables (including a leading column of 1s for the
intercept) are collected as columns of a n p
Chapter 1
Statistics 429: Time Series Analysis
1 / 28
Introduction
Denition: Time Series (TS)
A TS is a stochastic process, a sequence of random variables (RV)
dened on a common probability space denoted as (Xt )t=1,.,T (i.e.
X1 , X2 , ., XT ). Note that
The Problem
The case of 1 = 2
The case of 1 = 2
Simulation results
Aggregated AR(1) processes
Problem 1
Group 10
Bingji Yi1 , Sida Li1 , Zhekai Pan1 , Rong Du2 , Umesh Rajan2 ,
Suk Oh Young2
1 Department
of Mathematics
University of Illinois at Urbana-Cha
Chapter 3
Part II
Statistics 429: Time Series Analysis
1/6
Causality of ARMA models
Denition (Def. 3.7 p.94)
An ARMA(p, q) process dened by (B)Xt = (B)Wt is causal, i
Xt can be expressed by
Xt =
j Wtj = (B)Wt ,
t = 0, 1, 2, ,
j=0
where (B) =
j
j=0 j B ,
a
Chapter 3
Part III
Statistics 429: Time Series Analysis
1/7
Forecasting
Property 3.3, p.109
Given X1 , . . . , XT , the Best Linear Predictor (BLP)
T
XT +m = 0 + T k Xk of XT +m (m 1), is found by solving
k=1
E
T
XT +m XT +m Xk = 0, k = 0, 1, ., T
where X
Chapter 3
Part I
Statistics 429: Time Series Analysis
1 / 13
Modeling paradigm:
Modeling objective
A common measure used to assess many statistical models is their
ability to reduce the input data to random noise. For example, we
often say that a regressi
Statistics 429: Time Series Analysis
Instructor: Stphane Guerrier
TA: Jianjun Hu
Fall 2015
Statistics 429: Time Series Analysis
1 / 15
General Information
Statistics 429: Time Series Analysis
2 / 15
General Information
Instructor
Stphane Guerrier
stephane
Chapter 1
Statistics 429: Time Series Analysis
1 / 28
Introduction
Denition: Time Series (TS)
A TS is a stochastic process, a sequence of random variables (RV)
dened on a common probability space denoted as (Xt )t=1,.,T (i.e.
X1 , X2 , ., XT ). Note that
STAT 429 Final Project: Problem 4
Allie Gardner
Frieder Philipps
Christian Truden
Joe Wu
Dan Zielinski
11/30/2014
Gardner, Philipps, Truden, Wu, Zielinski
STAT 429 Final Project: Problem 4
Problem setting
Our goal is to find the best predictive model for
Time Series Analysis
Model diagnosis and prediction
Andres M. Alonso
Carolina Garca-Martos
Universidad Carlos III de Madrid
Universidad Polit
ecnica de Madrid
June July, 2012
Alonso and Garca-Martos (UC3M-UPM)
Time Series Analysis
June July, 2012
1 / 51
1
Properties of the OLS estimator
The variance of can be computed as follow:
var = var
= XT X
XT X
1
= 2 XT X
1
XT y
= var + XT X
XT var (e) X XT X
1
XT X XT X
1
XT e
1
1
.
Remark:
In the case of homoscedastic and uncorrelated errors (i.e. = I)
we obtain th
Chapter 3
Part III
Statistics 429: Time Series Analysis
1/7
Forecasting
Property 3.3, p.109
Given X1 , . . . , XT , the Best Linear Predictor (BLP)
P
T
XT +m = 0 + T k Xk of XT +m (m 1), is found by solving
k=1
E
h
XT +m
i
T
XT +m Xk = 0, k = 0, 1, ., T
Chapter 3
Part I
Statistics 429: Time Series Analysis
1 / 13
Modeling paradigm:
Modeling objective
A common measure used to assess many statistical models is their
ability to reduce the input data to random noise. For example, we
often say that a regressi
THE UNIVERSITY OF ILLINOIS
Department of Statistics
FALL 2016 STATISTICS 429
Time Series Analysis
Homework 1: Due 5 PM Sept 22th 2016 in HW mailbox
1
1. Given three random variables X1 , X2 , and X3 such that = 0 and
2
2
0 1
3 0.5 compute:
= 0
1 0.5 1
(a)
THE UNIVERSITY OF ILLINOIS
Department of Statistics
FALL 2016 STATISTICS 429
Time Series Analysis
Homework 3: Due 5 PM Nov 8th 2016 in HW mailbox
1. Within this homework assignment, we will prove the consistency of an
estimator defined to be
= arg max ()
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 429, Fall 2012
Midterm 1 Solution, Oct 8th, in class.
1. (122=24 points) True or False. No need to explain.
(a) If a time series cfw_Xt is independent and identically distributed, then it is
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 429, Fall 2012
Midterm 2, Nov 9th, in class.
I,
(print name), Student ID:
, Registered
in Section (circle one): U3, G4, will keep the information regarding this exam confidential.
I will not d
THE UNIVERSITY OF ILLINOIS
Department of Statistics
FALL 2016 | STATISTICS 429
Time Series Analysis
Homework 3: Solutions
Thanks to James Joseph Balamuta
1
a
To simplify notation we denote that n = n and = 0 .
P
Let G be the -ball centered at 0 i.e. G = c
THE UNIVERSITY OF ILLINOIS
Department of Statistics
FALL 2016 | STATISTICS 429
Time Series Analysis
Homework 2: Solutions
Thanks to James Joseph Balamuta
1
To begin, we start with showing that the autocovariance function XY (h) = Y X (h). Note the subscri
THE UNIVERSITY OF ILLINOIS
Department of Statistics
FALL 2016 | STATISTICS 429
Time Series Analysis
Homework 1: Solutions
Thanks to James Joseph Balamuta
1
1
2
Given three random variables X1 , X2 , and X3 such that = 0 and = 0
2
1
0
3
0.5
1
0.5 compute:
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 429, Fall 2013
Practice Midterm 1 Solution
1. (82=16 points) True or False. No need to explain.
(a) If a time series cfw_Xt is stationary, then the difference sequence cfw_Xt is also station
Chapter 2
Part I
Statistics 429: Time Series Analysis
1 / 11
Quick Review of Regression
Data:
Response is a column vector y = (y1 , ., yT )T .
The explanatory variables (including a leading column of 1s for the
intercept) are collected as columns of a n p
Statistics 429: Time Series Analysis
Instructor: Stphane Guerrier
Fall 2014
Statistics 429: Time Series Analysis
1 / 10
General Information
Statistics 429: Time Series Analysis
2 / 10
General Information
Instructor
Stphane Guerrier
[email protected]
1