Post on 08-Aug-2015
TIME SERIES ANALYSIS
Slides prepared by Edi Ariyanto, Andalas University
©2003 South-Western/Thomson Learning™
TIMES SERIES ANALYSIS
Time series represents a variable observed across time
Components of a time series
Long term Movement (T) Seasonal variation (S) Cyclical variation (C) Irregular activity (I)
LONG TERM MOVEMENT (T)
Long term Movement refers to general increases or decreases movement in long term period. Long term movement is useful to forecasting.
SEASONALITY (S)
Seasonal variation refers to periodic increases or decreases that occur within a calendar year in a time series. The key is that these movements in the time series follow the same pattern each year.
SEASONAL VARIATION
40 –
35 –
30 –
25 –
20 –
15 –
10 –
Po
wer
co
nsu
mp
tio
n (
mil
lio
ns
kw
h)
| | | | | | | | |Jan Jul Dec
1999
Jan Jul Dec
2000
Jan Jul Dec
2001
Figure 16.5
SEASONAL VARIATION
Figure 16.6
4 –
3 –
2 –
1 –Sa
les
of
Wil
dc
at
sa
ilb
oa
ts(m
illi
on
s o
f d
oll
ars
)
|July1998
|July1999
|July2000
|July2001
Linear trend
t
CYCLICAL VARIATION (C)
Cyclical variation describes a gradual cyclical movement about the trend; it is generally attributable to business and economic conditions
The length of the cycle is the period of that cycle and is measured from one peak to the next
CYCLICAL VARIATIONC
yclic
al a
ctiv
ity Z1
P1
V1
Z2P2
V2
t
Figure 16.7
TEXTILE EXAMPLE
4.0 –
3.5 –
3.0 –
2.5 –
2.0 –
1.5 –
1.0 –
Co
rpo
rate
tax
es
(mil
lio
ns
of
do
llar
s)
1 2 3
|1975
|1985
|1995
|2000
Figure 16.8
IRREGULAR ACTIVITY
Irregular activity consists of what is “left over” after accounting for the effect of any trend, seasonality, or cyclical activity
COMBINING COMPONENTS
Additive Structure
yt = TRt + St + Ct + It
Multiplicative Structure
yt = TRt • St • Ct • It
LINEAR TRENDS
Yt
t
(a) Increasing trend
Yt
t
(b) decreasing trend
Linear Trend
Y = a + bX
CURVILINEAR MODELS
Figure 16.4
Yt
tb2 < 0
(a)
Yt
tb2< 0(b)
CURVILINEAR MODELS
Figure 16.4
Yt
tb2 > 0
(c)
Yt
tb2 > 0
(d)
MEASURING TRENDLinear Trend
∑t = 1 + 2 + ... + T = T(T + 1)
2
∑t2 = 1 + 4 + ... + T2 =T(T + 1)(2T + 1)
6
t = =∑tT
T + 12
b1 =12B - 6(T + 1)A
T(T2 - 1)b0 = - b1
AT
T + 12
TREND LINE USING CODED DATA
12 –––
9 –––
6 –––
3 –––– | | | | | | | |
1994
(t =
1)
1995
(t =
2)
Year
2001
(t =
8)
Nu
mb
er o
f em
plo
yees
(th
ou
san
ds)
Yt
t
Figure 16.9
TREND LINE USING CODED DATA
12 –––
9 –––
6 –––
3 –––– | | | | | | | |
1994
(t =
1)
1995
(t =
2)
Year
2001
(t =
8)
Nu
mb
er o
f em
plo
yees
(th
ou
san
ds)
Yt
t
Figure 16.9
yt = TRt = b0 + b1t
EXCEL SOLUTION
Figure 16.10
QUADRATIC TREND
Figure 16.11
ILLUSTRATION OF QUADRATIC TREND LINES
Yt
Time (t)t = -
b1
2b2
Yt
Time (t)t = -
b1
2b2
A B
Figure 16.12
MEASURING CYCLICAL ACTIVITY
yt = TRt • Ct • It
Ct yt
yt
^
COMPLETE CYCLE
Yt
1 complete cycle
Trend
Ct > 1
Ct < 1
Time
Figure 16.13
TREND AND CYCLICAL ACTIVITY
t yt yt Ct
1 1.1 1.125 .9782 2.4 2.529 .9493 4.6 3.933 1.1694 5.4 5.337 1.0125 5.9 6.741 .8756 8.0 8.145 .9827 9.7 9.549 1.0168 11.2 10.953 1.022
yt
yt
^^
Table 16.1
CYCLICAL ACTIVITY
11.0 –
10.0 –
9.0 –
8.0 –
7.0 –
6.0 –
5.0 –
4.0 –
3.0 –
2.0 –
1.0 –
Nu
mb
er o
f em
plo
yees
(th
ou
san
ds)
|1994
|1995
|1996
|1997
|1998
|1999
|2000
|2001
t
yt = -.279 + 1.404t(trend line)
Actual yt
Yt
Figure 16.14
CYCLICAL COMPONENTS
1.15 –
1.10 –
1.05 –
1.00 –
.95 –
.90 –
Start End
Ct
t|1
|2
|3
|4
|5
|6
|7
|8
1994 1996 1998 2000
Figure 16.15
ADDITIVE SEASONAL VARIATION
100 units
100 units
100 units
Trend
Actual time series
|
Winter1999
|
Winter2000
|
Winter2001
t
Yt
2000 –
1500 –
1000 –
500 –
Un
its
sold
Figure 16.16
JETSKI SALES
700 –
600 –
500 –
400 –
300 –
200 –
100 –Sal
es (
ten
s o
f th
ou
san
ds
of
do
llar
s)Yt
TRt = 100 + 20t
Estimated sales using trend and seasonality
t|1
|2
|3
|4
|5
|6
|7
|8
|9
|10
|11
|12
|13
|14
|15
|16
|17
|18
|19
|20
Figure 16.17
HEAT PUMP SALES
Figure 16.18
100 units
250 units
180 units
Trend
Actual time series
|
Winter1999
|
Winter2000
|
Winter2001
t
Yt
2000 –
1500 –
1000 –
500 –
Un
its
sold
JETSKI SALES - MULTIPLICATIVE SEASON VARIATION
Figure 16.19
700 –
600 –
500 –
400 –
300 –
200 –
100 –Sal
es (
ten
s o
f th
ou
san
ds
of
do
llar
s)Yt
TRt = 100 + 20t
Estimated sales using trend and seasonality
t|1
|2
|3
|4
|5
|6
|7
|8
|9
|10
|11
|12
|13
|14
|15
|16
|17
|18
|19
|20
FOUR STEP PROCEDURE FOR DECOMPOSITION
1. Determine a seasonal index, St, for each time period
2. Deseasonalize the data, dt = TRt • Ct • It
3. Determine the trend component, TRt
4. Determine the cyclical component, Ct
CENTERED MOVING AVERAGES
Time Quarter t yt Moving Totals
1990 1 1 85 (1) 2632 2 41 (2) 2683 3 92 (3) 2704 4 45 and so on
1991 1 5 902 6 433 7 954 8 47
1992 1 9 92. . .. . .. . .
Table 16.2
SALES DATA FOR VIDEO-COMP
Year Quarter 1 Quarter 2 Quarter 3 Quarter 4
1998 20 12 47 601999 40 32 65 762000 56 50 85 1002001 75 70 101 123
Table 16.3
MOVING AVERAGES FOR VIDEO-COMP
Centered Ratio toMoving Moving Moving
Year Quarter t yt Total Average Average
1998 1 1 20 — —2 2 12 — — —3 3 47 139 37.25 1.264 4 60 159 42.25 1.42
1999 1 5 40 179 47.00 .852 6 32 197 51.25 .623 7 65 213 55.25 1.184 8 76 229 59.20 1.28
2000 1 9 56 247 64.25 .872 10 50 267 69.75 .723 11 85 291 75.13 1.134 12 100 310 80.00 1.25
2001 1 13 75 330 84.50 .892 14 70 346 89.38 .783 15 101 369 — —4 16 123 — — —
Table 16.3
SMOOTHING A TIME SERIES
120 –
100 –
80 –
60 –
40 –
20 –Sal
es (
nu
mb
er o
f u
nit
s)
Moving averages (no seasonality)
|1
|2
|3
|4
1998
|1
|2
|3
|4
1999
|1
|2
|3
|4
2000
|1
|2
|3
|4
2001
t
Yt
Quarters by year
Figure 16.20
RATIOS FOR EACH QUARTER
Quarter 1 Quarter 2 Quarter 3 Quarter 4
1.26 1.42.85 .62 1.18 1.28.87 .72 1.13 1.25.89 .78
Total 2.61 2.12 3.57 3.95Average 0.870 0.707 1.190 1.317
— —
— —
Table 16.5
DeseasonalizedValues
SeasonalYear t Yt Index (St) dt = —
Yt
St
1998 1 20 .852 23.472 12 .692 17.343 47 1.166 40.314 60 1.290 46.51
1999 1 40 .852 46.952 32 .692 46.243 65 1.166 55.754 76 1.290 58.91
2000 1 56 .852 65.732 50 .692 72.253 85 1.166 72.904 100 1.290 77.52
2001 1 75 .852 88.032 70 .692 101.163 101 1.166 86.624 123 1.290 95.35
DESEASONALIZING DATA
Table 16.6
TOTAL U.S. RETAIL TRADE
1997 1998 1999 2000Jan 134.738 139.935 146.613 158.691Feb 130.255 135.538 145.121 164.725Mar 148.497 151.118 165.736 183.875Apr 145.703 155.820 166.011 178.776May 156.603 162.797 173.496 190.753Jun 150.915 159.701 171.286 187.868Jul 153.200 161.541 172.364 182.891Aug 156.782 162.369 174.788 191.647Sep 149.407 155.747 169.809 183.229Oct 157.523 164.528 174.740 186.550Nov 161.925 169.914 185.347 198.706Dec 203.117 215.590 238.452 243.255
Table 16.7
SUMMARY OF RATIOS
Month (Period)
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec1997 0.993 1.013 0.964 1.013 1.036 1.2951998 0.888 0.857 0.952 0.979 1.018 0.994 1.000 1.001 0.954 1.002 1.029 1.2991999 0.878 0.864 0.981 0.976 1.014 0.992 0.990 0.996 0.959 0.980 1.032 1.3172000 0.871 0.899 0.996 0.963 1.022 1.003Average 0.879 0.837 0.977 0.973 1.018 0.996 0.994 1.004 0.959 0.998 1.033 1.304
Table 16.9
DESEASONALIZED DATA
•••••••
•••••••
•••••••
•••••••
•••
•••••
•••••
••••••
200 –
190 –
180 –
170 –
160 –
150 –
140 –|0
|10
|20
|30
|40
|50
Time (t)
Des
easo
nal
ized
Val
ues
(d
t)
Figure 16.21
CYCLICAL COMPONENTS
1 153.349 146.470 - .934(1) = 146.979 1.0433 —2 149.238 146.470 - .934(2) = 147.913 1.0090 1.0253 152.165 146.470 + .934(3) = 148.847 1.0223 1.0114 149.871 146.470 + .934(4) = 149.782 1.0006 1.0155 153.899 146.470 + .934(5) = 150.716 1.0211 1.0076 151.602 146.470 + .934(6) = 151.650 0.9997 1.010. . . . .. . . . .. . . . .
t dt dt —(= Ct • It)dt
dt
^^
3-MonthMoving
Average (Ct)
Table 16.12
PLOT OF CYCLICAL ACTIVITY
Figure 16.22
1.03 –
1.02 –
1.01 –
1.00 –
0.99 –
0.98 –
0.97 –|0
|10
|20
|30
|40
|50
Cyc
lical
Co
mp
on
ents
Month Feb Jan Jan JanYear 1997 1998 1999 2000
EXCEL PLOTS
Figure 16.23