ISyE 6402, Spring-2016
Instructor : A.Shapiro
Homework # 7
1. Consider the following seasonal model:
3 Xt = 2 + at + 0.5at3 ,
at = 0.4at1 + et 0.2et1 ,
where et is a white noise process. Write this process in the form
Xt = + 1 Xt1 + . + p Xtp + et 1 et1 .
ISyE 6402, Spring-2016
Instructor : A.Shapiro
Homework # 7
1. Consider the following seasonal model:
3 Xt = 2 + at + 0.5at3 ,
at = 0.4at1 + et 0.2et1 ,
where et is a white noise process. Write this process in the form
Xt = + 1 Xt1 + . + p Xtp + et 1 et1 .
ISyE 6402, Spring-2016
Instructor : A.Shapiro
Homework # 4
In the following problems . a0 , a1 , ., at , ., is a white noise process with a2 , and kk
are partial autocorrelation coefficients of a (stationary) process Xt .
1. Let Xt be the stationary solut
1. First Data Set Analysis
Time Series Plot of C1
42.5
40.0
37.5
C1
35.0
32.5
30.0
27.5
25.0
1
15
30
45
60
75
90
105
120
135
150
Index
As shown above, this time series seems to have a constant mean. Also, the fact that its
autocorrelation and partial auto
ISyE 6402, Spring-2017
Instructor : A.Shapiro
Homework # 1
1. Let X and Y be random variables. Suppose E[X] = 2, Var[X] = 9, E[Y ] = 0,
Var[Y ] = 4 and Corr(X, Y ) = 1/4. Find: (i) Var(X + Y ), (ii) Cov(X, X + Y ), (iii)
Corr(X + Y, X Y ).
2. Suppose that
ISyE 6402, Spring-2017
Instructor : A.Shapiro
Homework # 2
1. Let at be a white noise process and Xt = a1 + . + at , t = 1, 2, . (Xt is called
random walk process). Find Var(Xt ) and Cov(Xt , Xs ) for 1 s < t. Is this process
(weakly) stationary?
2. Suppo
ISyE 6402, Spring-2017
Instructor : A.Shapiro
Homework # 4
In the following problems . a0 , a1 , ., at , ., is a white noise process with a2 , and kk
are partial autocorrelation coefficients of a (stationary) process Xt .
1. Let Xt be the stationary solut
ISyE 6402, Spring - 2016
Midterm Exam #1
Instructor : A.Shapiro
In all following questions at is a white noise process with a2 = 1.
Consider the following process
Xt = 0.4Xt1 + 0.21Xt2 + at 0.49at2 + 3.
1. [10] Is the process stationary (invertible)?
We c
ISyE 6402, Spring - 2017
Midterm Exam #1
Instructor : A.Shapiro
In all following questions at is a white noise process with a2 = 1.
Consider the following process
Xt = 0.9Xt1 0.2Xt2 + at 0.3at1 + 2.
1. [15] Write the corresponding characteristic equation
ISyE 6402, Spring - 2016
Midterm Exam #2
Instructor : A.Shapiro
Consider the following seasonal model
Xt = 0.5Xt2 + at ,
at = 0.4at1 + et + 2,
where et is a white noise process with e2 = 1.
1.[25] Is this process stationary?
We can write this process in t
ISyE 6402, Spring - 2017
Midterm Exam #2
Instructor : A.Shapiro
1. Consider the following process
Xt = 1.3Xt1 0.3Xt2 + at 0.4at1 0.6at2 + 2.
(i) Identify this process as a specific ARIMA model. Is this process stationary? Is
there parameter redundancy in
ISyE 6402, Spring - 2015
Midterm Exam #1
Instructor : A.Shapiro
In all following questions at is a white noise process with a2 = 1.
1. Consider the following AR(2) process
Xt = 3 + Xt1 0.6Xt2 + at .
(i) Is the process stationary?
This is an AR(2) process
ISyE 6402, Spring - 2015
Midterm Exam #2
Instructor : A.Shapiro
Name and student number:
Grade:
1. Consider the seasonal model
Xt = Xt2 + at ,
at = et + et1 + 1,
where et W N (0, e2 ) and | < 1. Find Var[Xt ] and X (1), X (2), X (3) in terms
of , , e2 .
W
ISyE 6402, Spring-2016
Instructor : A.Shapiro
Homework # 8
1. Consider the following transfer function model:
Yt = (1 0.6B)1 (2 + 0.4B)Xt1 + Nt ,
Xt = 1 + at 0.4at1 ,
Nt = 0.5Nt1 + et ,
where at W N (0, a2 ) and et W N (0, e2 ) are independent white noise
Time series analysis
January 9, 2017
Contents
1 Time series (random process)
2
2 Linear models
3
3 Model building
8
4 Forecasting
12
5 Nonstationary time series
14
6 Exponential smoothing
15
7 Seasonal ARIMA models
17
8 Linear transfer function models
18
ISyE 6402, Spring - 2017
Syllabus
Instructor : A.Shapiro
Textbook (suggested): There is no one text book for the class. The following is a good
reference book but is difficult to read. We may use this book later in class.
Introduction to Time Series and F
ISyE 6402, Spring-2016
Instructor : A.Shapiro
Homework # 3
In the following problems . a0 , a1 , ., at , ., is a white noise process with a2 = 1.
1. Consider the AR(1) model:
Xt = 3Xt1 + at .
Show that Xt =
P
i=1
1 i
3
at+i satisfies the above equation a
ISyE 6402, Spring-2016
Instructor : A.Shapiro
Homework # 1
1. Let X and Y be random variables. Suppose E[X] = 2, Var[X] = 9, E[Y ] = 0,
Var[Y ] = 4 and Corr(X, Y ) = 1/4. Find: (i) Var(X + Y ), (ii) Cov(X, X + Y ), (iii)
Corr(X + Y, X Y ).
2. Suppose that
ISyE 6402, Spring-2016
Instructor : A.Shapiro
Homework # 2
1. Let at be a white noise process and Xt = a1 + . + at , t = 1, 2, . (Xt is called
random walk process). Find Var(Xt ) and Cov(Xt , Xs ) for 1 s < t. Is this process
(weakly) stationary?
2. Suppo
ISyE 6402, Spring-2016
Instructor : A.Shapiro
Homework # 1
1. Let X and Y be random variables. Suppose E[X] = 2, Var[X] = 9, E[Y ] = 0,
Var[Y ] = 4 and Corr(X, Y ) = 1/4. Find: (i) Var(X + Y ), (ii) Cov(X, X + Y ), (iii)
Corr(X + Y, X Y ).
(i) Var(X + Y )
ISyE 6402, Spring-2016
Instructor : A.Shapiro
Homework # 8
1. Consider the following transfer function model:
Yt = (1 0.6B)1 (2 + 0.4B)Xt1 + Nt ,
Xt = 1 + at 0.4at1 ,
Nt = 0.5Nt1 + et ,
(1)
(2)
(3)
where at W N (0, a2 ) and et W N (0, e2 ) are independent
ISyE 6402, Spring-2016
Instructor : A.Shapiro
Homework # 9
1. Let Yt be a stationary process and define the bivariate process Xt1 = Yt , Xt2 = Ytd ,
where d 6= 0. Show that the process (Xt1 , Xt2 )0 is stationary and express its crosscorrelation function
ISyE 6402, Spring-2016
Instructor : A.Shapiro
Homework # 2
1. Let at be a white noise process and Xt = a1 + . + at , t = 1, 2, . (Xt is called
random walk process). Find Var(Xt ) and Cov(Xt , Xs ) for 1 s < t. Is this process
(weakly) stationary?
We have
ISyE 6402, Spring-2016
Instructor : A.Shapiro
Homework # 10
1. Let Xt be a (stationary) ARCH(1) process, i.e., Xt = t et , where et is an iid
2
with |1 | < 1.
N (0, 1) process and t2 = 0 + 1 Xt1
4
(i) Show that E[Xt ] is finite if and only if 12 < 1/3.
Si
ISyE 6402, Spring-2016
Instructor : A.Shapiro
Homework # 4
In the following problems . a0 , a1 , ., at , ., is a white noise process with a2 , and kk
are partial autocorrelation coefficients of a (stationary) process Xt .
1. Let Xt be the stationary solut
ISyE 6402, Spring-2016
Instructor : A.Shapiro
Homework # 3
In the following problems . a0 , a1 , ., at , ., is a white noise process with a2 = 1.
1. Consider the AR(1) model:
Xt = 3Xt1 + at .
Show that Xt =
P
1 i
3
at+i satisfies the above equation and i