stat 497 lecture notes 3 stationary time series processes (arma processes or box-jenkins processes)...
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STAT 497LECTURE NOTES 3
STATIONARY TIME SERIES PROCESSES(ARMA PROCESSES OR BOX-JENKINS
PROCESSES)
1
AUTOREGRESSIVE PROCESSES
• AR(p) PROCESS:
or
where
tptptt aYYY 11
ttp aYB
.1 221
ppp BBBB
2
AR(p) PROCESS
• Because the process is
always invertible.• To be stationary, the roots of p(B)=0 must lie
outside the unit circle.• The AR process is useful in describing
situations in which the present value of a time series depends on its preceding values plus a random shock.
;11
p
jj
jj
3
AR(1) PROCESS
where atWN(0, )
• Always invertible.
• To be stationary, the roots of (B)=1B=0
must lie outside the unit circle.
t
c
ttt
B
ttt
aYBaYB
aYY
1111
2a
4
AR(1) PROCESS
• OR using the characteristic equation, the roots
of m=0 must lie inside the unit circle.
B=1 |B|<|1| ||<1 STATIONARITY CONDITION
5
AR(1) PROCESS
• This process sometimes called as the Markov
process because the distribution of Yt given
Yt-1,Yt-2,… is exactly the same as the
distribution of Yt given Yt-1.
6
AR(1) PROCESS
• PROCESS MEAN:
t
c
t
ttt
aYB
aYY
offunction abut mean process the
not is which
1
11
7
AR(1) PROCESS
• AUTOCOVARIANCE FUNCTION: K
kttkk
kttkttk
ktttk
kttkttk
YaE
YaEYYE
YaYE
YYEYYCov
1
1
1
,
8
AR(1) PROCESS tt YaE10
tt aY 1
12 tt aY
21 a
01 02
02
.1 where1 2
2
02
00
aa
9
AR(1) PROCESS
0
1 1,
k
k
kk k
.1,: 1 kACF kkk
When ||<1, the process is stationary and the ACF decays exponentially.
10
AR(1) PROCESS
• 0 < < 1 All autocorrelations are positive.• 1 < < 0 The sign of the autocorrelation
shows an alternating pattern beginning a negative value.
1,0
1,: 1
k
kPACF kk
11
AR(1) PROCESS
• RSF: Using the geometric series
t
it
iitt
aBB
aBaB
Y
22
0
1
1
11
0 1 2
0, iii
12
AR(1) PROCESS
• RSF: By operator method _ We know that
1 and 1 BBBB
0,
0
0
111
1
2212
11
221
j
BBB
BB
jj
13
AR(1) PROCESS
• RSF: By recursion
tttt
tttt
tttttt
ttttt
aaaY
aaaY
aaYaaY
YYaYY
122
33
1232
122
12
1 where
14
THE SECOND ORDER AUTOREGRESSIVE PROCESS
• AR(2) PROCESS: Consider the series satisfying
15
tttt aYYY 2211
t
c
t
tttt
aYBB
aYYY
212
21
221121
11
1
where atWN(0, ).2a
AR(2) PROCESS
• Always invertible.• Already in Inverted Form.• To be stationary, the roots of
must lie outside the unit circle. OR the roots of the characteristic equation
must lie inside the unit circle.16
01 221 BBB
0212 mm
AR(2) PROCESS
17
stationary be toprocess AR(2)for condition required the
22
112,1 1
2
4
m
2
1
121121
221221
mmmm
mmmm
AR(2) PROCESS
• Considering both real and complex roots, we have the following stationary conditions for AR(2) process (see page 84 for the proof)
18
1
1
1
2
12
21
AR(2) PROCESS
• THE AUTOCOVARIANCE FUNCTION: Assuming stationarity and that at is independent of Yt-k, we have
19
kttkk
ktttt
kttk
YaE
YaYYE
YYE
2211
2211
AR(2) PROCESS
20
2
2211
2211
2211
0
a
tt
tttt
tt
YaE
YaYYE
YYE
AR(2) PROCESS
21
1201
11201
12211
11
tt
tttt
tt
YaE
YaYYE
YYE
2
011 1
AR(2) PROCESS
22
0211
20211
22211
22
tt
tttt
tt
YaE
YaYYE
YYE
02
222
21
2 1
AR(2) PROCESS
23
2
1222
22
0
20
2
222
21
202
11
222110
11
1
11
a
a
a
AR(2) PROCESS
ACF: It is known as Yule-Walker Equations
24
0,2211 kkkk
0,2211 kkkk
ACF shows an exponential decay or sinusoidal behavior.
AR(2) PROCESS
• PACF:
25
2,0
1
1
221
212
2
11
k
kk
PACF cuts off after lag 2.
AR(2) PROCESS• RANDOM SHOCK FORM: Using the Operator
Method
26
tt
B
aYBB
2
211
1 BB
2,
2
2211
213
13
22
12
11
jjjj
The p-th ORDER AUTOREGRESSIVE PROCESS: AR(p) PROCESS
• Consider the process satisfying
27
tptptt aYYY 11
where atWN(0, ).2a
tt
B
pp aYBB
p
11
provided that roots of all lie outside the unit circle
01 1 ppBB
AR(p) PROCESS
• ACF: Yule-Walker Equations
• ACF: tails of as a mixture of exponential decay or damped sine wave (if some roots are complex).
• PACF: cuts off after lag p.
28
0,11 kpkpkk
0,11 kpkpkk
MOVING AVERAGE PROCESSES
• Suppose you win 1 TL if a fair coin shows a head and lose 1 TL if it shows tail. Denote the outcome on toss t by at.
• The average winning on the last 4 tosses=average pay-off on the last tosses:
29
up shows tail,1
up shows head ,1ta
321 41
41
41
41
tttt aaaa MOVING AVERAGE PROCESS
MOVING AVERAGE PROCESSES
• Consider the process satisfying
30
.,0~ where
1
2
1
11
at
t
B
qtqttt
WNa
aBB
aaaY
q
MOVING AVERAGE PROCESSES
• Because , MA processes
are always stationary.
• Invertible if the roots of q(B)=0 all lie outside
the unit circle.
• It is a useful process to describe events
producing an immediate effects that lasts for
short period of time.31
q
ii
jj
1
2
0
2 1
THE FIRST ORDER MOVING AVERAGE PROCESS_MA(1) PROCESS
• Consider the process satisfying
32
.,0~ where
1
2
1
1
at
t
B
ttt
WNa
aBaaY
tYE:Mean Process The
2220:Variance Process The aatYVar
MA(1) PROCESS
• From autocovariance generating function
33
122
111
2
12
1
BB
BB
BBB
a
a
a
1,0
1,
0,12
22
k
k
k
a
a
k
MA(1) PROCESS
• ACF
34
1,0
1,1 2
k
kk
ACF cuts off after lag 1.
General property of MA(1) processes: 2|k|<1
MA(1) PROCESS
• PACF:
35
4
2
2111 11
1
6
22
4221
21
22 11
11
1
1
112
2
k,
k
k
kk
MA(1) PROCESS
• Basic characteristic of MA(1) Process:– ACF cuts off after lag 1.– PACF tails of exponentially depending on the sign
of .– Always stationary.– Invertible if the root of 1B=0 lie outside the unit
circle or the root of the characteristic equation m=0 lie inside the unit circle.
INVERTIBILITY CONDITION: ||<1.
36
MA(1) PROCESS
• It is already in RSF.• IF:
37
tt
tti
ii
tt
tt
aYBB
aYB
aYB
aBY
22
0
1
11
1
1= 2=2
MA(1) PROCESS
• IF: By operator method
38
1,
111
12
21
j
BBB
BB
jj
THE SECOND ORDER MOVING AVERAGE PROCESS_MA(2) PROCESS• Consider the moving average process of order 2:
39
.,0~ where
1
2
2212211
2
at
t
B
tttt
WNa
aBBaaaY
tYE:Mean Process The
22
222
12
0:Variance Process The
aaa
tYVar
MA(2) PROCESS• From autocovariance generating function
40
22
22
1211211
22
21
2
122
2
12
1
BBBB
BB
BBB
a
a
a
2,0
2,
1,1
0,1
22
22
1
22
21
2
k
k
k
k
a
a
a
k
MA(2) PROCESS• ACF
• ACF cuts off after lag 2.• PACF tails of exponentially or a damped sine
waves depending on a sign and magnitude of parameters. 41
2,0
2,1
1,1
1
22
21
2
22
21
21
k
k
k
k
MA(2) PROCESS
• Always stationary.• Invertible if the roots of
all lie outside the unit circle.OR if the roots of
all lie inside the unit circle.
42
01 221 BB
0212 mm
MA(2) PROCESS
• Invertibility condition for MA(2) process
43
11
1
1
2
12
21
MA(2) PROCESS
• It is already in RSF form.• IF: Using the operator method:
44
2,
111
1
2211
221
221
j
BBBB
BB
jjj
The q-th ORDER MOVING PROCESS_ MA(q) PROCESS
45
.,0~ where
1
2
1
11
at
t
BB
qtqttt
WNa
aBB
aaaY
q
tYE:Mean Process The
2222
12
0:Variance Process The
aqaa
tYVar
Consider the MA(q) process:
MA(q) PROCESS
• The autocovariance function:
• ACF:
46
qk
qkqkqkkak
,0
,,2,1,12
qk
qkq
qkqkk
k
,0
,,2,1,1 22
1
1
THE AUTOREGRESSIVE MOVING AVERAGE PROCESSES_ARMA(p, q)
PROCESSES• If we assume that the series is partly
autoregressive and partly moving average, we obtain a mixed ARMA process.
47
qtqttptptt aaaYYY 1111
.,0~ and where 2attt WNaYY
tqtp aBYB
qpARMA
:,
ARMA(p, q) PROCESSES• For the process to be invertible, the roots of lie outside the unit circle.• For the process to be stationary, the roots of lie outside the unit circle.• Assuming that and share
no common roots,Pure AR Representation:Pure MA Representation:
48
0Bq
0Bp 0Bp 0Bq
B
BBaYB
q
ptt
B
BBaBY
p
qtt
ARMA(p, q) PROCESSES
• Autocovariance function
• ACF
• Like AR(p) process, it tails of after lag q.• PACF: Like MA(q), it tails of after lag p.
49
1,11 qkpkpkk
1,11 qkpkpkk
ARMA(1, 1) PROCESSES
• The ARMA(1, 1) process can be written as
50
tt
tttt
aBYB
aaYY
1111
.,0~ where 2at WNa
•Stationary if ||<1.
•Invertible if ||<1.
ARMA(1, 1) PROCESSES
• Autocovariance function:
51
kttkttk
kttkttktt
ktttt
kttk
YaEYaE
YaEYaEYYE
YaaYE
YYE
11
11
11
ARMA(1,1) PROCESS
• The process variance
52
2220
110
1
212
12
aaa
aaY
tttt
tt
tatatY
ta
YaEYaE
ARMA(1,1) PROCESS
53
20
11
0
101
212
a
aaY
tttt
ttt
YaEYaE
2
2
0
22200
121
1
aa
ARMA(1,1) PROCESS
• Both ACF and PACF tails of after lag 1.
54
1,
1,
11
11
k
kk
kk
kkk
ARMA(1,1) PROCESS
• IF:
55
0,
111
1
1
221
j
BBBB
BB
jj
ARMA(1,1) PROCESS
• RSF:
56
0,
111
1
1
221
j
BBBB
BB
jj
AR(1) PROCESS
57
AR(2) PROCESS
58
MA(1) PROCESS
59
MA(2) PROCESS
60
ARMA(1,1) PROCESS
61
ARMA(1,1) PROCESS (contd.)
62
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