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