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) =
0Z
(t) +
1Z
(t
1) +
+
for some constants 0 ; 1 ; :
STATISTICS 443
Time Series and Forecasting
Dr. Bruce Dunham
Department of Statistics
Lecture 1
! Welcome to the course!
! Lecturer: Dr. Bruce Dunham, Department of
Statistics
email: b.dunham@stat.ubc.ca. ODce: ESB 3118
! Class times: M/W/F 12noon (ESB 101
AR Processes and
Stationarity, the
Yulecfw_Walker Equations
Stat 443: Time Series and
Forecasting
Dr. Bruce Dunham
Department of Statistics
Lecture 10
1. Let Z (t) be a white noise process with mean 0.
Is the process X (t) deMned by
X (t) = 1:2X (t ! 1) +
MA(1) Processes and
Invertibility
Stat 443: Time Series and
Forecasting
Dr. Bruce Dunham
Department of Statistics
Lecture 8
1. Consider now the process
X (t) = Z (t) + 5Z (t 1) .
where Z (t) is white noise with zero mean and
variance 2. Then (1) is
(a) 0.
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)
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 is 0.8, the variance of Y
is 0.4, the variance of Z is
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
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: 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
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: 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 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: 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 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: 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: 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: 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
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 be written
X (t) = # + S (t) + " (t) ;
where # is the un
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
Autocorrelation
Stat 443: Time Series and
Forecasting
Dr. Bruce Dunham
Department of Statistics
Lecture 4
1. Recall that for a set of N paired observations
(x1; y1) ; : : : ; (xN ; yN ) we deCne the sample correlation coe3cient to be
PN
E
E
i=1 (xi ! x) (
Activity Solution: Cross-Correlation
This activity aims to help you understand cross-correlation, as applied
to two stochastic process and also the sample version in the context of two
independent AR(1) samples.
For a bivariate stochastic process cfw_X (t
Activity Solution: The CrossCorrelogram
Recall for a bivariate stochastic process cfw_X (t) , Y (t) we dened the
crosscovariance function at lag k to be
XY (k) = Cov (X (t) , Y (t + k)
= E (X (t) X ) (Y (t + k) Y ) .
The crosscorrelation at lag k is dene
Activity Solution: Box
Jenkins Forecasting
Application
Recall the lh data, comprising 48 observations of the level of luteinizing
hormone in the blood of a woman, taken at ten minute intervals. Earlier
analysis suggested an AR(1) model, though cases could
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
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
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
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 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
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
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 processes that produce realworld
time series. No mention w
Activity solution: Estimation of seasonal effects using
smoothing
The following data are the quarterly energy consumption figures (in MWe) in the UK
for the years 19751979, where yt is the moving average of order 4 of original series xt and
m
t is the mo