STA457 Fall 2001
Test #1 - Solutions
(5) 1. True or False
a. A white noise process is a weakly stationary process.
b. A random walk is a weakly stationary process.
c. There exists a stationary process which satis es (3) 6= (;3)
d. Assume the population To
Solutions to Assignment #1
1. (a), (b) See the plots on the following pages.
(c) Periodogram of original data: maximum near 0, generally decreasing as frequency increases. Correlogram: decays very slowly. (This is quite consistent with w
STA457 Practice Problems - Week 5
1) Consider the simple moving average filter with weights aj = (2q+1)-1 , -1 q 1.
a) Let Xt = c0 + c1t . Show that
j = q
X t j = X t .
b) Let Zt be independent random variables with mean 0 and variance 2. Show that
Assignment #1 STA457H1S/2202H1S
due Wednesday February 3, 2016
Instructions: Students in STA457S do problems 1 through 3; those in STA2202S do all 4 problems.
1. Daily Japanese yen/US dollar exchange rates (JPY/USD) from Dec. 1, 1978 to Apr.29, 2005
University of California, Riverside
Jess Gonzalo U. Carlos III de Madrid
A sequence of uncorrelated random variables is called a white noise
process. cfw_ a : E ( a ) =
(normally = 0)
1. [4%] Define the DurbinWatson test and explain why we tend to reject the null hypothesiswhenthevalueoftheteststatisticissmall. Answer: (1) [3%]Assumethedatageneratingprocessof cfw_ y t followsthemodel , , | | 0, 1, .Thestatistica
A Little Book of R For Time Series
May 27, 2015
How to install R
1.1 Introduction to R . . . . .
1.2 Installing R . . . . . . . .
1.3 Installing R packages . . .
1.4 Running R . . . . . . . .
1.5 A brief introduction
Evaluating Wiener filter coefficients in R
Consider the signal + noise model
Yt = Xt + Nt
where cfw_Xt and cfw_Nt are uncorrelated stationary processes with spectral densities fX ()
and fN (), respectively. In class, we showed that if
Distribution of the sample mean of stationary
Suppose that cfw_Xt : t = 0, 1, 2, is a second-order stationary process with mean =
n to be the sample
E(Xt ) and autocovariance function (s) = Cov(Xt , Xt+s ). We define X
STA457 Practice Problems - Week 6
1) Determine which of the following ARMA processes are causal and which of them are
invertible. In each case Zt denote white noise.
a) X t + 0.2 X t 1 0.48 X t 2 = Z t .
b) X t + 1.9 X t 1 + 0.88 X t 2 = Z t + 0.2 Z t 1 +
STA457 Practice Problems - Week 4
1) Let cfw_Xt be the AR(1) process given by:
X t = X t 1 + Z t
where Zt is WN(0, 2)
a) Find the autocovariance function and the autocorrelation function for this process when
= 0.9 . Is the process stationary?
STA457 Practice Problems - Week 3
1) Question 2.7 on page 31 from the text book. Use Minitab for graphs and calculations.
2) Let cfw_Xt be the MA(2) process given by:
Xt = Zt + Zt-2 where Zt is WN(0, 2)
a) Find the autocovariance function and the autocorr
20 SIMPLE DESCRIPTIVE METHODS OF ANALYSIS
values. For example, by not joining successive points in Figs 1.4 and 1.5
we drew the readers attention to the gaps and jumps in the series which
are an intrinsic feature of the data. .
Another choice which can af
The Shape Parameter of a Two-Variable Graph
WILLIAM S. CLEVELAND, MARYLYN E. McGILL, and ROBERT McGILL*
The shape parameter of a two-variable graph is the ratio of the horizontal and vertical distances spanned by the data. For at
least 70 years this param
1. See solutions for mid term practice posted before.
2. Steps of time series analysis taught in class:
See the slides of Lecture 3 Box-Jenkins Approach.
Step 1: Understand the problem of interest, collect the data, graph the collected data to get
Midterm practice questions (July 2015)
1. Discuss the steps of time series analysis discussed in class.
2. Describe the components of classical decomposition in time series analysis.
3. Define a weakly stationary time series.
4. Describe th
University of Toronto at Mississauga STA457H5F - Fall 2009 Term Test #3
Aids Allowed: Any calculator without text keyboard 1. (6 marks) This question is about the stochastic process: yt = t (1+t-1)
a. (1 mark) What
University of Toronto at Mississauga STA457H5F - Fall 2009 Term Test #4
1. (7 marks) Refer to your R output for the time series of US dishwasher sales. a. (2 marks) Do you think the best model is probably (i) AR(p) or (ii) MA(q) or (iii) noise or (iv) som
STA457 Fall 2001 Test #1 - Solutions
(5) 1. True or False a. A white noise process is a weakly stationary process. b. A random walk is a weakly stationary process. c. There exists a stationary process which satis es (3) 6= (;3) d. Assume the population To
STA457 Fall 2001 Test #3
(5) 1. True or False a. If fXtg is strictly stationary then the joint distribution of X1 X2 X3 equals the joint distribution of Xk Xk+1 Xk;1 b. A stationary AR(4) process can be expressed as an MA(1) process. c. A stationary MA(4)
Unit root tests
In building a time series model, the statistician must often decide whether or not to difference
the data before fitting a stationary time series model; the implications of not differencing
when it is appropriate or of over-
Some theory for AIC in autoregressive models
Suppose that we want to model a time series x1 , , xn using an AR(p) model for 0 p r
and select the order p by minimizing Akaikes Information Criterion (AIC)
AIC(p) = n ln(b 2 (p) + 2p
where b 2 (p
Assignment #1 STA457H1S/2202H1S
due Friday February 3, 2017
Instructions: Students in STA457 do problems 1 through 4; those in STA2202 do all 5
1. Daily Canadian/U.S. dollar exchange rates ($US/$CAN) from Jan. 2, 1997 to Dec. 29,
2000 are given
Solutions to Assignment #1
1. The plots for parts (a), (b), (d) are shown on the following page.
(c) Periodogram of original data: maximum near 0, generally decreasing as frequency increases. Correlogram: decays very slowly. (This is quite consistent with