Accreditation Commission for Acupuncture and Oriental Medicine
DKFJAL
MATH 200

Spring 2009
Solutions to HW 4
Exercise 5.1
X t +1 = Z t +1 + Z t ,
X t + h = Z t + h + Z t + h 1
^ X t +1 = E ( X t +1  X t , X t 1 ,.) = Z t ^ X t + h = E ( X t + h  X t , X t 1 ,.) = 0, h 2
^ et +1 = X t +1  X t +1 = Z t +1
Var(et + h ) = 2
^ et + h = X t +
Accreditation Commission for Acupuncture and Oriental Medicine
DKFJAL
MATH 200

Spring 2009
Solutions to HW 6
Exercise 6.1 (a)
X t = X t 1 + Z t , or (1  B ) X t = Z t ,
Then
2 2 ( B) = Z ( B ) ( B 1 ) = Z
i.e.,
( B) =
1 1 B
1 , (1  B)(1  B 1 )
f ( ) =
2 2 Z 1 1 = . e i = Z 2 2 (1  e i )(1  ei ) 2 (1 + 2  2 cos )
(
)
(b)
f ( ) =
f (
Accreditation Commission for Acupuncture and Oriental Medicine
DKFJAL
MATH 200

Spring 2009
Solutions to HW 7
Exercise 12.1 VAR(1) model, X t = X t 1 + Z t , where = i.e.,
1 0.5 , 0.2 0.7
X t , 1 1 0.5 X t 1, 1 Z t , 1 . = + X t , 2 0.2 0.7 X t 1, 2 Z t , 2
The model is nonstationary if there are roots of  ( B )  = I  B  = 0 incide
Accreditation Commission for Acupuncture and Oriental Medicine
DKFJAL
MATH 200

Spring 2009
Solutions to HW 8
Exercise 12.4
All pure MA processes, whether univariate or multivariate, are stationary. The model is invertible if all the roots of  ( B )  = I + B  = 0 are outside of the unit circle.
0.4 B 1 0 0.6 B 0.4 B 1 + .6 B ( B) = I + B = +
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
Matlab Solutions Time Series (2DD23) Exercise 2.5
a.
12 X t
Week 2
= X t  X t12 = (a + bt + St + t )  (a + b(t  12) + St12 + t12 ) = a  a + bt  bt + 12b + St  St12 + t  t12 = 0 + 0 + 12b + 0 + t  t12
We can see that 12b is constant, and t 
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
Matlab Solutions Time Series (2DD23) Exercise 3.1
The formula of the autocorrelationfunction is: (k) =
(k) . (0)
Week 3
(k) = Cov(X t , X t+k ) = Cov(Z t + 0.7Z t1  0.2Z t2 , Z t+k + 0.7Z t+k1  0.2Z t+k2 ) Now use the formula: Cov(X 1 + X 2 , Y )
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
Matlab Solutions Time Series (2DD23) Exercise 3.4
X t  = 0.7(X t1  ) + Z t = 0.7(0.7(X t2  ) + Z t1 ) + Z t = 0.7(0.7(0.7(X t3  ) + Z t2 ) + Z t1 ) + Z t = 0.73 (X t3  ) + 0.72 Z t2 + 0.7Z t1 + Z t
Week 4
=
i=0
0.7i Z ti
Now we can compute
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
Matlab Solutions Time Series (2DD23) Exercise 4.4
Consider the AR(2) process 1 2 X t1 + X t2 + Z t 3 9
Week 5
Xt =
In exercise 3.6 it is shown that the autocorrelationfunction is: 16 21 2 3
k
(k) =
+
5 1  21 3
k
Now we can compute the partial autoc
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
Matlab Solutions Time Series (2DD23) Solution to the Matlab Exercise
(D + C sin(2 t + 2 ) sin(1 t + 1 ) Use: 2 cos A cos B = cos(A + B) + cos(A  B) (see exercise 2.3, page 26. So
Week 8
(1)
1 1 1 1 sin A sin B = cos(A ) cos(B ) = cos(A+B)+ cos(AB) =
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
Matlab Solutions Time Series (2DD23) Exercise 6.2b
X t = Z t + 0.5Z t1  0.3Z t2 The power spectral density function is: 1
Week 9
f () =
2
k=1
(k) cos k + (0) =
1
2
k=0
(k) cos k  (0)
The autocovariance function of a MA(q) process has a cutoff aft
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
Matlab Solutions Time Series (2DD23) Exercise 8.1
X t = Z 1,t + 11 Z 1,t1 + 12 Z 2,t1 Yt = Z 2,t + 21 Z 1,t1 + 22 Z 2,t1
Week 10
X Y (0) = Cov(X t , Yt ) = Cov(Z 1,t + 11 Z 1,t1 + 12 Z 2,t1 , Z 2,t + 21 Z 1,t1 + 22 Z 2,t1 ) = Cov(11 Z 1,t1 , 21
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
Matlab Solutions Time Series (2DD23) Exercise 10.2
The special case of the linear growth model is described by the following equations: X t = t + n t t = t1 + t1 t = t1 + wt
Week 12
(1) (2) (3)
Please notice that there is a small difference with the "r
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H1 A time series is a collection of observations cfw_xt made sequentially through time. When the variation of some quantity over a region of space is studied, t is a spatial variable.
Examples of time series: 1) Lynx data: a periodic oscillation with an
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H2 HW1 (due Jan. 17). Exercise 3.1 from the textbook and
Problem 1
For the autocovariance function,
(t1 , t2 ) = E X t1 X t2  (t1 ) (t2 ) .
Problem 2 Problem 3
cfw_
(t1 , t2 ) = E X t1  (t1 ) X t2  (t2 )
cfw_
, show that
Show that a strictly stati
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H3
For MA(q) processes,
2 q h j =0 j j + h , 0 h q, (h) = h > q. 0,
qh j =0 j j + h , 0 h q, (h) = 1 + 2 + + 2 q 1 h > q. 0,
The ACF of a MA(q ) process "cuts off" after the point q. This is a benchmark property for MA processes.
AR(1):
X t = X t 1
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H4 HW2 (due Jan. 24). Exercise 3.9 from the textbook and
Problem 1 Let
cfw_Zt ~ IID(0, 2 ) , and let c
be a constant. Consider the process
X t = Z1 cos(ct ) + Z 2 sin(ct ) .
Find the mean and autocovariance function and determine whether the process is s
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H6 HW3 (due Jan. 31). Exercise 4.4 from the textbook and
Problem 1 Explore the correlation structure of the following models,
X t = .7 X t 1  .5 X t 2 + Zt , X t = Zt + .7 Zt 1  .3Zt 2 ,
by simulating their realizatins of 500 observations and comput
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H7
Given a set of observations cfw_ X 1 ,., X n from a stationary time series, the ACVF is estimated by the sample autocovariance function defined as
^ (h) =
1 nh ( X t +h X n )( X t  X n ) , n t =1
where X n =
1 n Xn . n t =1
This also leads to estim
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H8 HW4 (due Feb. 7). Exercises 5.1, 5.2 from the textbook and
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 (1) and
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H10
Fitting an appropriate
ARMA(p, q) model,
X t = 1 X t 1 + + p X t  p + Z t + 1Z t 1 + + q Z t q , cfw_Z t ~ WN (0, 2 ) ,
to an observed time series data set ( x1 , . , xn ) involves
determining the order (p, q ) (model identification), estimatin
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H9
cfw_ X t is called an autoregressive integrated moving average (ARIMA) process of order ( p, d , q ) , denoted as cfw_ X t ~ ARIMA( p, d , q ) , where d 1 is an integer,
if its dorder difference Yt = (1  B ) X t is a casual ARMA( p, q ) process,
d
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H11 HW5 (due Feb. 28)
Problem 1
Consider the general ARIMA(1, 2, 1) model. (a) (b) Convert the model to the equivalent ARMA(3, 1) form.
^ Find the forecasts, X t + h , h 1 , for the model.
Problem 2
Analyze the Glacial Varve Series (use data file "data/va
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H12 Spectral Analysis 1. A function that satisfies the equation,
f ( x) = f ( x + kp) ,
is called periodic with period Virtually any periodic function
all x ,
k = 0, 1 , 2, . ,
p , if p is the smallest number such that the equation holds for all x .
f ( x
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H13 HW6 (due Mar. 6). Exercises 6.1, 6.2 from the textbook.
The spectral density function or simply the spectrum of a stationary time series,
f ( ) =
1 2
h =
h e i h ,
in frequency domain.
(1)
is the counterpart of the covariance function Here Since
h
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H14
The secondorder properties of a time series are completely described by its ACVF or equivalently, under mild conditions (a sufficient condition is
h ,
h =
 h  < ), by its Fourier transform,
which is called the spectral density function or the spe
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H15 HW7 (due Mar. 20). Exercises 12.1, 12.2 (a, d) from the textbook
In many cases, at each time t, several related quantities are observed and, therefore, we want to study these quantities simultaneously by grouping them to form a vector. By doing so we
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H16
The discussion of bivariate processes is readily extended to the general multivariate case. If we have m processes, cfw_ X t ,1, cfw_ X t , 2 , .,cfw_ X t , m , each having zero mean, we define the covariance matrix at lag h by
( h) = [ ij ( h)], i
Accreditation Commission for Acupuncture and Oriental Medicine
DJAOI
MATH 200

Spring 2009
H17 HW8 (due Mar. 27). Exercises 12.4, 12.5 from the textbook and
Problem 1
Consider VARMA model X t = 1 X t 1 + Z t + 1 Z t 1 , where 1 =
0.7 0.7 0.6 0.4 and 1 = 0.2 0.4 . Is the model stationary and invertible? 0.7 0.7
Problem 2
Find a statespace r