STAT 420 Introduction to Time Series
Homework 1
1. Read Chapter 1 in the book, and answer the following question:
(a) Is the Marketing time series on page 3 continuous or discrete? give your reason.
Answer: Discrete. The observations are only taken every
STAT 420
H10
X1, . , X t ,
predict the value it will assume at some specific future time point, X t + h .
The prediction problem: from the observed values of a time series at past points,
X t +h ,
We refer to X
In forecasting
t is called the forecast orig
STAT 420 Introduction to Time Series
Homework 1 (Solution)
1. Read Chapter 1 in the book, and answer the following question:
(a) Is the Marketing time series on page 3 continuous or discrete? give your reason.
Answer: Discrete. The observations are only t
STAT 420
H7
Given a set of observations
cfw_ X1,., X n from a stationary time series, the ACVF is estimated
by the sample autocovariance function defined as
h =
1 nh
( X X n )( X t X n ) ,
n t=1 t+h
1 n
where X n = X n .
n t =1
This also leads to estim
STAT 420
H8
HW3 (due Feb. 11)
Problem 1
Suppose that in a sample of size 100, you obtain
1 = 0.432
and
2 = 0.145 .
Assuming that
the data were generated from an MA(1) model, construct approximate 95% CIs for
Based on these two CIs, are the data consisten
STAT 420
H9
Trend stationary models
In practice, most time series are non-stationary. Real TS data often exhibit time trend and/or cyclic features
that are beyond the capacity of stationary ARMA models. Several examples we have considered came
from clearl
Solutions to HW 5
Exercise 5.1
X t +1 = Zt +1 + Zt ,
X t +h = Zt +h + Zt +h1
X t +1 = E ( X t +1 | X t , X t 1,.) = Zt
X t +h = E ( X t +h | X t , X t 1,.) = 0, h 2
et+1 = X t+1 X t+1 = Zt+1
Var(et+h ) = 2
(
)
et+h = X t+h X t+h = Zt+h + Zt+h1 Var(et+h )
Solutions to HW 9
Problem 1
Xt
Let X t = X t1 , then the observation equation is X t = 1,0,0 X t , and the state equation is
X
t2
0.7 0.5 0.4
1
Xt = 1
0
0 X t + 0 Zt .
0
0
1
0
Problem 2
varve=read.table("data/varve.dat")
v=ts(varve)
lvarve =
Solutions to HW 6
Exercise 6.1
(a)
X t = X t 1 + Zt , or (1 B) X t = Zt ,
Then
( B) = Z2 ( B) ( B 1 ) = Z2
f ( ) =
f ( ) =
(b)
i.e.,
( B) =
1
1 B
1
,
(1 B)(1 B 1 )
2
Z2
1
1
.
e i = Z
=
2
2 (1 ei )(1 ei ) 2 (1 + 2 2 cos )
(
f ( )
X2
=
)
1 2
2 (1 + 2 2
STAT 420 Final Exam
Wed, May 4, 2011
Time: 120 minutes
Name:
Section:
Materials permitted:
Two pages of cheat sheets with double sides and calculators are allowed. BUT, books,
notes, laptop computers, phones, or any devices capable of wireless communicati
STAT 420 1st Midterm
Thursday, Feb 24, 2011
Time: 60 minutes (6:30-7:30pm)
Name:
Materials permitted:
One page of cheat sheet and calculators are allowed. BUT, books, notes, laptop computers, phones, or any devices capable of wireless communication are no
1. (20 pts) For a time series X1; = 0.5Xth1 + 0.1Xt_2 + Zt, where Z: N IID(O, a2)
(a) (4 pts) Classify the model and give the order of the model.
(b) (7 pts) Is the process {Xi} invertible and stationary?
(c) (9 pts) Determine the ac.f; at time lag I, 2