PSTAT 174/274: Homework # 1.
1. (a) Dene a time series.
(b) Give examples of time series likely to be stationary and examples of time series
likely to be nonstationary.
2. Evaluate ACF function X (k), k = 0, 1, . . . for
(a) Xt = Zt + Zt1 , Z W N (0, Z
174/274 HW3 Solution
1. For AR models, ACF does not have cut-os. Therefore model (C), (D) and (E) do not
yield this pattern of ACF.
A MA(q) model Xt = Zt + 1 Zt1 + + q Ztq has the general ACF is of the form
X (k) =
k + 1 k+1 + + qk q
, k = 1, 2, , q
PSTAT 174/274: Homework 3
Note: Zt denotes white noise: cfw_Zt W N (0, 2 ), 2 = Z .
1. You are given a stationary time series model where the ACF is zero except at lags
1, 6, 7, 8. Which of the following yields this pattern of ACF?
(A)Xt = Zt 1 Zt1 2 Z
WEEK 1: Examples of Time Series. Autocorrelation function. Stationarity.
0. Introduction to Time Series. (Based on 1.1-1.2, [BD].)
[BD]: P. J. Brockwell and R. A. Davis, Introduction to Time Series and Forecasting, 2002
Times series is a data obtained fro
3000 observations of a Gaussian MA(1) with coefficient 0.6 and WNV=1.0
were generated using ITSM. The option Transform>Subsequence allows
you to pick out the independent subsamples 1-200, 301-500, , 2701-2900,
and get independent estimates for each of
Solution for HW2
AR(2) Process because the pacf cuts off after lag 2 (Good enough for full credit)
Just in case the students want to know how to find the parameters:
! 2 ! 1
! 1 = ! ; ! =
# GETTING R: Go to "http:/cran.stat.ucla.edu" and look for the right
# version for your computer.
# BASIC R OPERATIONS: Simple operations and lists
# 1. Define a variable and look at it
x <- 12
# 2. Mathematical operations on a variable
# Prior to running the code below, follow instructions in problems 1 and 2
# of lab 4 to download data set and move file to R's working directory
# Read the data into R and plot the time series
wine.csv = read.table("monthly-australian-wine-sales-th.csv",
# US Population Data #
# Load the tseries package
# Read the USPOP Data
# By default, header = FALSE. What happens when header = TRUE?
uspop <- read.table("uspop.txt",header=TRUE)
# What happens
# Simulate data from an MA(2) process with positive coefficients
y <- arima.sim(model = list(ma=c(0.45,0.55), n = 100)
# Plot data and ACF
windows(width=10, height = 5)
# Simulate data from an MA(2) process with positive
Solutions to selected problems in
Brockwell and Davis
This document contains solutions to selected problems in
Peter J. Brockwell and Richard A. Davis, Introduction to Time Series and Forecasting, 2nd Edition, Springe
174/274 HW1 Solution
1. (a) Time series are data obtained from observations collected sequentially over time.
(b) Stationary: White noise cfw_Zt . Nonstationary: Random Walk Xt = Z1 + + Zt , where
Zt s are i.i.d white noise.
X (k) = Cov(Xt , Xt+k )
PSTAT 174/274: Homework # 2.
Note: cfw_Zt W N (0, Z ) denotes white noise.
1. You are given PACF for a stationary process:
11 = 0.60, 22 = 0.36, kk = 0 for k 3.
What time series model could have this PACF?
2. Determine which of the the following ARMA p
174/274 HW4 Solution
1. Appendix B includes all the information that students need for this homework in terms of
statistics. Students should save it for future reference.
2. In order to construct an approximate 95% condence interval for the sample mean, w
PSTAT 174/274: Homework 5
1. The sunspot numbers cfw_Xt , t = 1, 2, . . . , 100 have sample autocovariances (0) =
1382.2, (1) = 1114.4, (2) = 591.73, and (3) = 96.216. Use these values to nd the
Yule-Walker estimates of 1 , 2 and in the model:
Yt = 1 Yt
PSTAT 174/274: Homework 4
1. Review Appendix B of [BD]. (Posted on Gaucho Space)
2. Suppose that in a sample of size 100 from an MA(1) process with mean , = 0.6,
and 2 = 1, we obtain x100 = 0.157. Construct an approximate 95% condence
interval for . Are t
PSTAT 174/274: Homework 6
For this assignment, please review material of diagnostic checking in the textbook or in 12 of the
lecture notes, Lecture 12. In particular, recall notations: Xt denotes the observed time series,
Zt the white noise generating ARM
PSTAT 174/274: Homework 7
For this assignment, Xt denotes the observed time series, Zt the white noise generating he process
cfw_Xt , B the back shift operator, n the sample size, xt is the observed value of Xt in a sample.
1. Show that for cfw_Xt follow
PSTAT 174/274: Lab 1
Getting Started with R
A complete R tutorial can be found at http:/cran.r-project.org/doc/manuals/R-intro.html
1. Changing the working directory:
First, download the uspop.txt data le from Gauchospace to your local compute
PSTAT 174/274: Lab 2
Problem 1. Consider an MA(2) process, given by
Xt = Zt + 1 Zt1 + 2 Zt2
where Zt N (0, 2 ).
1. Is Xt stationary? Calculate and plot the ACF of Xt .
2. Is Xt invertible? Derive constraints on 1 and 2 such that Xt is an inv
PSTAT 174/274: Lab 7
Useful R commands:
To estimate parameters of an AR model:
ar(data, aic = TRUE, order.max = NULL, method = c(".")
To estimate parameters of an MA or ARMA model:
arima(data, order = c(p, 0, q), method = c(".")
To compare mo