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 your answers clearly in the
margin. Ambiguous responses
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 stochastic process cfw_Xt , t Z, be defined for some cons
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) + Z (t) :
Assume that this process is stationary.
1. M
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 distribution,
each value independent of the others. This is a basi
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
model for a time series. The goal of this activity is
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 is discrete, where the sum is over all possible values o
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 on the same topics. Typically in an fty
minute session
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
century. The gures are given below, along with the plot.
Deca
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
1977
1977
1977
1978
1978
1978
1978
1979
1979
1979
1979
Qu
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
1861-70
1871-80
Quarter
1
2
24.7 22.0
24.7 22.1
25.2 21.8
2
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 stochastic process
X (t) by
X (t) = Z (t)+0:3Z (t ! 1)+0:2Z (
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 (t) = ' 0 Z (t) + ' 1 Z (t ! 1) + " " " + ' q Z (t ! q
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 (t ! 3) :
1. The model X (t) is an MA(q) for which value
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 answer to three decimal places.
Part (a) Xt = Zt + 0.25
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 file earnings.csv contains monthly earnings of a company
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 contains the value, in $, for the daily closing price of
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 Zt1 )
Part (b) The variance of Xt
Part (c) The autocova
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 0.061
The mean of the observed series was x = 0:03; and c0
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 will consider the model
X (t) = 0:7X (t ! 1) + Z (t) ! 0
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 variance
" 2 and acf % (") :
X
1. Now consider the sto
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 tted values and therefore the forecasts.
1. Clickers ques
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: Suppose our data are from a process with a trend,
say
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 a sample,
N
1 X
x=
$
x (t)
N t=1
constitutes a useful
Operators on Random
Variables
Stat 443: Time Series and
Forecasting
Dr. Bruce Dunham
Department of Statistics
Lecture 5
Review
1. Let X be a random variable with variance Var(X )
and expectation E (X ) ; then Var(X ) can be written
(a) E (X )
!
2
(b)
E X2
Properties of Time Series
Stat 443: Time Series and
Forecasting
Dr. Bruce Dunham
Department of Statistics
Lecture 2
Review question
1. Have you registered your clicker device on the
course page on Connect?
(a) Yes!
(b) No, not yet.
(c) Not sure.
(d) Click
Autoregressive Processes
Stat 443: Time Series and
Forecasting
Dr. Bruce Dunham
Department of Statistics
Lecture 9
1. Let Z (t) be a white noise process with mean 0.
Is the process
X (t) = Z (t) ! 0:7Z (t ! 1) + 0:2Z (t ! 2)
invertible?
(a) Yes
(b) No
(c)
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 as usual Z (t) denotes white noise with
mean zero. By
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 lectures on the same topics. Typically in an fty
minute sessio
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) = 0.5X (t 1) + Z (t) 0.8Z (t 1) + 0.4Z (t 2)
has been tt
Activity Solution: Sample Mean Special Case
Recall from the previous activity that
Var (x) =
2
X
N
1+2
N 1
X
!
k
N
1
k=1
(k)
where the data are N observations from a stationary process with variance
2
X and acf ( ) :
1. Now consider the stochastic process
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
1
1+2
0
k=0
k = 1
otherwise.
and the spectral density i
Selena Shao
Assignment Time Series - WeBWorK 4 due 03/07/2017 at 09:00pm PST
STAT443-202 2016W2
Part (c) What is the predicted value for the month after the
one you predicted in (a)?
Suppose Zt is white noise with mean zero. After suitable
pre-processing,
UNIVERSITY OF BRITISH COLUMBIA
Department of Statistics
Stat 443: Time Series and Forecasting
Assignment 2: Time Series Models
Sample Solutions and Grading Scheme
1. Suppose cfw_Xt is a stationary process with autocovariance (h). Define the process cfw_Y
Selena Shao
Assignment Time Series - WeBWorK 1 due 01/17/2017 at 09:00pm PST
STAT443-202 2016W2
B. There had been a change in the trend in the underlying process.
C. There had been a change in the random error (i.e.,
noise) in the underlying process.
D
Let Zt denote a white noise process with mean zero and variance 0.22 . Define the stochastic process Xt by
Xt = Zt 0.8Zt1 0.1Zt2 + 0.1Zt3
Provide the following, each to two decimal places.
Part (a) Var(Xt )
Part (b) Cov(Xt , Xt+1 )
Part (c) The autocorrel