STAT 758, Spring 2012
Key solution for Home Work 2
Prepared by Tracy Backes
ACF, stationarity
2.1 Let cfw_Xt be a sequence of uncorrelated random variables, each with mean 0 and
variance 2 . For each of the following processes, nd its representation in t
STAT 758, Spring 2012
Key solution for Home Work 1
Prepared by Tracy Backes
Dierencing, backshift operator
All notations are from lectures.
1.1 Show that the dierence operators and 12 are commutative, that is
12 = 12 .
Solution: Let us apply the operators
STAT 758: Homework #6
Due on Wednesday, 11 April, 2012
Zaliapin, 1:00pm
Tracy Backes
1
Tracy Backes
STAT 758 (Zaliapin): HW #6
Problem #1
We assume below that Zt W N (0, 2 ), B is a backshift operator.
6.1 For the model (1 B )(1 0.2B )Xt = (1 0.5B )Zt :
a
STAT 758, Spring 2012
Home Work 6
SARIMA, Second-order forecasting
2
We assume below that Zt is a white noise with mean 0 and variance Z .
Problem 1
For the model (1 B )(1 0.2 B )Xt = (1 0.5 B )Zt :
a) Classify the model as an ARIMA(p, d, q ) process (i.e
STAT 758, Spring 2012
Home Work 1
Due date: Feb. 6
Dierencing, backshift operator
All notations are from lectures.
1.1 Show that the dierence operators and 12 are commutative, that is 12 =
12 .
1.2 Show that the dierence operator 7 eliminates a linear tr
STAT 758, Spring 2012
Home Work 2 (due Feb 13)
ACF, stationarity
2.1 Let cfw_Xt be a sequence of uncorrelated random variables, each with mean 0 and
variance 2 . For each of the following processes, nd its representation in terms
of lagged X -values (i.e
STAT 758, Spring 2012
Home Work 3 (due Feb 13)
ACF, iid sequence, white noise, random walk
3.1 Give two examples (specify distributions) of each: a) iid sequence,
b) white noise,
c) random walk.
3.2 Give example of weakly but not strictly stationary stoch
STAT 758, Spring 2012
Home Work 4
MA(q )
Below we assume that Zt W N (0, 2 ).
4.1 Consider MA(1) process Xt = a Zt + b Zt1 . Find the white noise Wt such that
the process Xt is presented as Xt = Wt + Wt1 with Wt W N (0, 2 ).
4.2 Find acvf and acf for MA(1
STAT 758, Spring 2012
Home Work 5 (due Mar 7)
MA(q ), invertibility
Below we assume that Zt W N (0, 2 ), B is a backshift operator.
5.1 Find the operator inverse to
a) 1 + 2B ,
b) 1 0.3B ,
c) 2 + 0.6B .
5.2 Examine invertibility for the following processe
STAT 758, Spring 2012
Key solution for Home Work 3
Prepared by Tracy Backes
ACF, iid sequence, white noise, random walk
3.1 Give two examples (specify distributions) of each:
a) iid sequence:
(a.1) Xt iid N (0, 1), t Z,
(a.2) Xt iid Uniform([0, 1]), t Z
b
STAT 758, Spring 2012
Key solution for Home Work 4
Prepared by Tracy Backes
MA(q )
Below we assume that Zt W N (0, 2 ).
4.1 Consider MA(1) process Xt = a Zt + b Zt1 . Find the white noise Wt such that
the process Xt is presented as Xt = Wt + Wt1 with Wt W
STAT 758, Spring 2012
Solution key for Home Work 5
Prepared by Tracy Backes
MA(q ), invertibility
Below we assume that Zt W N (0, 2 ), B is a backshift operator.
5.1 Find the operator inverse to
a) 1 + 2B
1
=
1 + 2B
(2)i B i
i=0
b) 1 0.3B
1
=
1 0.3B
0.3i