In the following, Z (t) denotes a white noise process with mean zero and
variance 2 unless otherwise stated. For questions 1, 2, 3, and 5, circle your
answers clearly. If you make a mistake, indicate
Selena Shao
Assignment Time Series - WeBWorK 6 due 03/28/2017 at 09:00pm PDT
STAT443-202 2016W2
the acf is
LINK for DATA SET (Download the csv file here)
Consider the process
(k) =
Xt = Zt + 0.25Zt1
Activity Solution: Autoregressive Processes
Suppose Z (t) is white noise with mean zero and variance # 2 : We have
seen that a process X (t) is said to be a moving average (MA) process of
order q if
X
Activity Solution: YuleWalker Equations
The i.i.d. sequence Z (t) has mean zero and variance # 2 : Suppose we
dene the stochastic process X (t) by
X (t) = 1:30X (t ! 1) ! 0:22X (t ! 2) ! 0:10X (t ! 3)
Activity Solution: Introduction to Stochastic Processes
The process Z (t) will be used in future to denote a purely random sequence of i.i.d. variables so all values of Z are from the same distributio
Activity Solution: Model Fitting using Acf and Pacf
We have seen how useful the sample autocorrelation function (acf) and
partial autocorrelation function (pacf) can be for determining the appropriate
Activity Solution: Variance and Covariance of
Random Variables
The mean (expectation, or expected value) of a random variable X
is denoted E (X). Formally, it is dened as
X
E (X) =
xP (X = x)
x
if X i
Activity Overview
The inclass activities created for the course are useful tools to enhance
student learning, and in my experience are far more eective than even the
most polished traditional lectures
Activity Solution: The Sample Autocorrelation
Recall the death rate data gives the death rates, per thousand people,
for England and Wales during year-quarters of four decades of the nineteenth
centur
Activity Solution: Smoothing for Seasonals
The following data are the quarterly energy consumption gures (in MWe)
in the UK for the years 19751979.
Year
1975
1975
1975
1975
1976
1976
1976
1976
1977
19
Activity Solution: Seasonal Eects
The death rates, per thousand people, for England and Wales during
yearquarters of four decades of the nineteenth century are given below.
Decade
1841-50
1851-60
1861
Activity Solution: Moving Average Processes
The i.i.d sequence Z (t) has mean zero and variance # 2 : Recall we dened
a stochastic process X (t) by
X (t) = Z (t) + 0:3Z (t ! 1) + 0:2Z (t ! 2) + 0:1Z (
Activity Solution: Properties of the Sample Mean
Suppose we would like to estimate the mean ! of our process X (t) using
some data x(1); : : : ; x(N ). We would like to know to what extent the mean
of
UNIVERSITY OF BRITISH COLUMBIA
Department of Statistics
Stat 443: Time Series and Forecasting
Assignment 4: Time Series Analysis in the Frequency Domain
Sample Solutions and Grading Scheme
1. Let a st
Let Zt denote white noise with standard deviation 1.69. In
each of the following cases, derive the spectral density function
for the process Xt , and evaluate it for the given frequency ,
giving your
UNIVERSITY OF BRITISH COLUMBIA
Department of Statistics
Stat 443: Time Series and Forecasting
Assignment 1: Exploratory Data Analysis
The assignment is due on Tuesday, January 24 at 12:30pm.
1. The fi
UNIVERSITY OF BRITISH COLUMBIA
Department of Statistics
Stat 443: Time Series and Forecasting
Assignment 3 : Analysis in the Time Domain
Sample Solutions and Grading Scheme
1. The data file acma.txt c
Let Zt denote a white noise process with mean zero and variance 22 . Define the stochastic process Xt by
Xt = 0.4Xt1 + Zt + 0.8Zt1 .
Provide the following, each to three decimal places.
Part (a) E(Xt
Activity Solution: Model Fitting
1. Clickers question: The rst 200 terms of a time series gave the following
results:
k
acf rk
pacf #kk
^
1
2
3
4
5
0.80 0.67 0.52 0.39
0.31
0.80 0.085 0.112 0.046 0.06
Activity Solution: MA Representations
This activity should help you appreciate how to obtain the MA representation of an ARMA model, and also why that representation can be useful
in forecasting. We w
Activity Solution: Sample Mean Special Case
Recall from the previous activity that
Var (") =
x
"2
X
N
1+2
N !1 #
X
k=1
1!
k
N
$
% (k)
!
where the data are N observations from a stationary process with
Activity: Exponential Smoothing
This activity aims to help you understand a forecasting method known
as exponential smoothing, and appreciate how the parameter in exponential
smoothing can aect the tt
Activity Solution: Holt and HoltWinters Method
This activity extends the method known as exponential smoothing to
cover non-stationary eects such as trends and seasonal variation.
1. Clickers question
Moving Average
Processes
Stat 443: Time Series and
Forecasting
Dr. Bruce Dunham
Department of Statistics
Lecture 7
1. The i.i.d sequence Z (t) has mean zero and variance # 2: Recall we deIned a stocha
Operators on Time Series
Stat 443: Time Series and
Forecasting
Dr. Bruce Dunham
Department of Statistics
Lecture 3
1. Suppose we assume a seasonal e=ect is additive,
in that a model for the data can b
Some Stochastic Models
for Time Series
Stat 443: Time Series and
Forecasting
Dr. Bruce Dunham
Department of Statistics
Lecture 6
1. Let X; Y; and Z be random variables. Suppose
that the variance of X
Chapter 3
Estimation, Model Fitting and
Prediction for Time Series
3.1
Introduction
In the previous chapter we described in some detail the family of mathematical models that are used to model the pro
ARMA, ARIMA, and
SARIMA Models
Stat 443: Time Series and
Forecasting
Dr. Bruce Dunham
Department of Statistics
Lecture 11
1. Consider the process
9
1
X (t) = X (t ! 1) ! X (t ! 2) + Z (t) ;
10
5
where
Activity Overview
The in
class activities created for the course are useful tools to enhance
student learning, and in my experience are far more eective than even the
most polished traditional lecture
Activity Solution: BoxJenkins Forecasting
This activity aims to help you understand how to apply BoxJenkins
forecasting methods once a model has been tted to a time series. The
following model
X (t) =