MAT3379 (Winter 2015)
Assignment 1
Due date (Assignment 1): Wednesday, 28 January 2015
Based on lectures 1-4
2
Q1. (4 points) Let cfw_Zt be an IID sequence of normal random variables with mean 0 and variance Z . Let a, b, c be
constants. Which of the fol

MAT3379 (Winter 2015)
Assignment 5
Due date (Assignment 5): April 13, 2015
Q1. Assume that Z = (Z1 , Z2 ) is a random vector with the mean vector 0 and the covariance matrix
=
11
21
12
22
.
A=
a11
a21
a12
a22
.
Let A be a deterministic matrix dened by
Con

MAT3379 (Winter 2015)
Assignment 3
Due date (Assignment 3): March 23, 2015
Q1. (Practical Question).
(Refer to R Data Examples Part 4)
We have already tted AR(4) to US unemployment data. We estimated parameters using the Yule-Walker
procedure.
Predict t

MAT3379 (Winter 2015)
Assignment 4
Due date (Assignment 4): April 8, 2015
Q1. (Theoretical Question - 5 points). GARCH models.
Consider a GARCH model
2
2
Xt = t Zt ,
t = 0 + 1 Xt1 ,
where Zt are i.i.d. random variables with mean 0 and variance 1.
2
(a) Co

MAT3379 (Winter 2016)
Assignment 4 - SOLUTIONS
The following questions will be marked: 1a), 2, 4, 6, 7a
Total number of points for Assignment 4: 20
Q1. (Theoretical Question, 2 points). Yule-Walker estimation for AR(p) models.
2
Assume that Zt are i.i.d r

MAT 3379
Introduction to Time Series Analysis
Winter 2016
Instructor:
Rafal Kulik
Department of Mathematics and Statistics
585 King Edward Av.
Office 203D
Phone: 562-5800 Ext. 3526
Email: [email protected]
Schedule:
Monday
LEC
13:00-14:30 MRT250
Wednesday

Q1. (4 points) Consider the sequence
Xt = Zt Zt1 + Zt3 ,
2
where Zt are i.i.d random variables with mean 0 and variance Z
.
(a) Compute E[Xt Xt+h ] for h = 0, 1, 3. (3 points)
(b) Is the sequence Xt stationary? Why? (1 point)
Solution to Q1:
(a)
4
2
E[Xt2

MAT3379 (Winter 2016)
Assignment 1 - SOLUTIONS
Q1. Let cfw_Zt be an IID sequence of normal random variables with mean 0 and variance 2 . Let a, b, c be constants.
Which of the following processes are stationary? Evaluate mean and autocovariance function.

Formula Sheet.
For the stationary M A() process
Xt =
j Ztj
j=0
we have
V ar(Xt ) =
2
j2
j=0
,
Cov(Xt , Xt+h ) = X (h) =
2
j j+h
j=0
and
X (h) = X (h) = X (|h|).
Backward and dierence operator
BXt = Xt1 , (1 B)Xt = Xt Xt1 .
ARM A(p, q) process
(B)Xt =

MAT 3379, Summer 2016
Assignment 2
Due in class on June 15, 2016
1. Use R to simulate the following models in R and estimate the parameters and compare them with
your models.
(i)
(1 0.5B + .06B 2 )Xt = (1 B)Zt .
(ii).
(1 0.2B 2 )Xt = Zt .
(iii).
(1 0.036B

MAT 3379, Summer 2016 (May 2-July 25)
An Introduction to Time Series Analysis
Lectures: Monday 16:00-17:30 and Wednesday 16:00-17:30 in VNR 2095.
Instructor: M. Zarepour, office: Room 207D, Mathematics and Statistics department, 585 King Edward Ave., Tel:

MAT 3379, Summer 2016
Solution to Assignment 3
2.1 .
We solve this problem for a general time series firs (stationarity is not required). To minimize
g(a, b) = E(Xn+h aXn b)2
solve for a and b
2E(Xn+h aXn b) = 0
2E(Xn (Xn+h aXn b) = 0.
We get
E(Xn+h ) = a

MAT 3379, Summer 2016
Solution to Assignment 2
2.3. In an M A() process Xt =
P
i=0 i Zti
We have
X (h) = 2
X
i i+h .
i=0
Therefore
1.25, if
0.18 if
X (h) =
0.4 if
0
if
h = 0,
|h| = 1
|h| = 2,
|h| > 2.
This result is identical for Y (h).
2.4. (a). Let Xt

MAT 3379, Summer 2016
Solution to Assignment 1
1.2. (a) Since the conditional expectation forces X1 , . . . , Xn to be a constant leL f (X1 , . . . , Xn ) = c
let
E(Xn+1 c)2 |X1 , . . . , Xn ) = g(c)
0
Let g (c) = 0 to solve for c to find the minimum as f

MAT3379 (Winter 2016)
Assignment 2 - SOLUTIONS
Total number of points for Assignment 2: 19
Q1. (Theoretical Question) For the following processes, compute autocovariance function.
(a) AR(2) - use the recursive method;
(b) ARMA(1, 1) - you may use the line

MAT3379 (Winter 2016)
Assignment 3 - SOLUTIONS
Total number of points for Assignment 3: 10
Q1. (Theoretical Question - 5 points). Durbin-Levinson procedure and PACF for AR(p) models. Assume that Zt
2
are i.i.d random variables with mean 0 and variance Z
.

MAT3379 (Winter 2016)
Assignment 4
Due date (Assignment 4): March 18, 2016
Note: You have to drop off your assignment in the lobby of KED585 building. There will be a box marked
MAT3379. It is advised that you make a copy of your assignment.
Note: I will

MAT 3379 (Winter 2015)
FINAL EXAM
18 April 2015
Professor: R. Kulik
Time: 180 minutes
Student Number:
Family Name:
First Name:
This is closed book exam. You are allowed to use one double-sided A4 sheet of notes. Only
non programmable calculators are permi

MAT3379 (Winter 2015)
Assignment 2
Due date (Assignment 2): February 23, 2015
Q1. (Theoretical Question - 4 points). Yule-Walker procedure for AR(p) models. Assume that Zt are i.i.d random
2
variables with mean 0 and variance Z .
(a) Apply the Yule-Walker

MAT 3379 (Winter 2015)
Midterm
25 February 2015
Professor: R. Kulik
Time: 75 minutes
Student Number:
Family Name:
First Name:
This is closed book exam. You are allowed to use one double-sided A4 sheet of notes. Only
non programmable calculators are permit

MAT 3379 - Winter 2015
Introduction to Time Series Analysis
Study Guide for Midterm
THIS IS THE FINAL VERSION !
1
Topics
1. Evaluate covariance function in simple models - see Q1 in Assignment 1;
2. Check if a process is causal and stationary - see Q4 in

MAT 3379 (Winter 2015)
FINAL EXAM
18 April 2015
Professor: R. Kulik
Time: 180 minutes
Student Number:
Family Name:
First Name:
This is closed book exam. You are allowed to use one double-sided A4 sheet of notes. Only
non programmable calculators are permi

MAT3379 (Winter 2016)
Assignment 7 - data project
Due date (Assignment 7): April 13, 2016
Note: You have to drop off your assignment in the lobby of KED585 building.
Q1.
Find an univariate financial data set.
Remove trend if necessary. Compute the station

MAT 3379 (Winter 2016)
MIDTERM EXAM (PRACTICE)
29 February 2016
Professor: R. Kulik
Time: 70 minutes
Student Number:
Family Name:
First Name:
This is closed book exam. You are allowed to use one double-sided A4 sheet of notes. Only
non programmable calcul

MAT 3379
Introduction to Time Series Analysis
Study Guide for Final Exam (Winter 2016)
You will be allowed to have one A4 sheet (double-sided) of notes.
1
Topics
1. Evaluate covariance function in simple models - see Q1, Q2, Q3 in Assignment 1; Final prac

MAT 3379 (Winter 2016)
FINAL EXAM (PRACTICE)
15 April 2016 (180 minutes)
Professor: R. Kulik
Student Number:
Name:
This is closed book exam. You are allowed to use one double-sided A4 sheet of notes. Only
non programmable calculators are permitted.
There

MAT3379 (Winter 2016)
Assignment 6 - SOLUTIONS
The following questions will be marked: Q1
Total number of points for Assignment 6: 5
Q1. Assume that Z = (Z1 , Z2 ) is a random vector with the mean vector 0 and the covariance matrix
11 12
=
.
21 22
Let A b

MAT 3379 (Winter 2016)
MIDTERM EXAM
29 February 2016
Professor: R. Kulik
Time: 70 minutes
Student Number:
Family Name:
First Name:
This is closed book exam. You are allowed to use one double-sided A4 sheet of notes. Only
non programmable calculators are p

MAT3379 (Winter 2016)
Assignment 5 - SOLUTIONS
The following questions will be marked: Q1
Total number of points for Assignment 5:
Q1. (Theoretical Question). GARCH models. (6 points)
Consider a GARCH model
2
Xt = t Zt ,
t2 = 0 + 1 Xt1
,
where Zt are i.i.

MAT 3379 - Winter 2016
Introduction to Time Series Analysis
Study Guide for Midterm
You will be allowed to have one A4 sheet (one-sided) of notes. Midterm
date: Monday, Febraury 29,
1
Topics
1. Evaluate covariance function in simple models - see Q1, Q2, Q