Time series, forecasting, and index numbers

Time Series, Forecasting, and Index
Numbers

Slide 1

Shakeel Nouman
M.Phil Statistics
Time Series, Forecasting, and Index Numbers By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

12

Slide 2

Time Series, Forecasting, and Index Numbers

• Using Statistics
• Trend Analysis
• Seasonality and Cyclical Behavior
• The Ratio-to-Moving-Average Method
• Exponential Smoothing Methods
• Index Numbers
• Summary and Review of Terms


12-1 Using Statistics

Slide 3

A time series is a set of measurements of a variable that are ordered
through time. Time series analysis attempts to detect and
understand regularity in the fluctuation of data over time.
Regular movement of time series data may result from a tendency
to increase or decrease through time - trend- or from a tendency to
follow some cyclical pattern through time - seasonality or cyclical
variation.
Forecasting is the extrapolation of series values beyond the region
of the estimation data. Regular variation of a time series can be
forecast, while random variation cannot.


The Random Walk

Slide 4

A random walk:
Zt- Zt-1=at
or equivalently:
Zt= Zt-1+at
The difference between Z in time t and time t-1 is a random error.
R andom W alk
15

10

5

Z

0

There is no evident regularity in a
random walk, since the difference in the
series from period to period is a random
error. A random walk is not
forecastable.

a
0

10

20

30

40

50

t


A Time Series as a
Superposition of Cyclical
Functions

Slide 5

A Time Series as a Superposition of Two Wave
Functions and a Random Error (not shown)
z

4
3
2
1
0
-1
0

10

20

30

40

50

t

In contrast with a random walk, this series exhibits obvious
cyclical fluctuation. The underlying cyclical series - the
regular elements of the overall fluctuation - may be analyzable
and forecastable.

12-2 Trend Analysis: Example
12-1

Slide 6

The following output was obtained using the template.

Zt = 696.89 + 109.19t

Note: The template contains forecasts for t = 9 to t = 20 which corresponds to years 2002 to 2013.

12-2 Trend Analysis: Example
12-1

Slide 7

Straight line trend.

Example 12-2

Slide 8

The forecast

for t = 10
(year 2002)
is 345.27

Observe that the forecast model is Zt = 82.96 + 26.23t

12-3 Seasonality and Cyclical
Behavior
G ro s s E arning s : A nn ual

Monthly S ale s of S untan Oil

Monthly Numbers of Airline P as s engers

20

100

11000

P a s s e ng er s

E a rnin g s

S a le s

200

Slide 9

15

10

10000
9000
8000
7000

0

6000

5
J FMAMJ J AS OND J FMAMJ J ASOND J FMAMJ J A

80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95

Time

t
86 87

88

89 90 91

92 93

94 95

Ye ar

When a cyclical pattern has a period of one year, it is
usually called seasonal variation. A pattern with a period
of other than one year is called cyclical variation.


Time Series Decomposition
•

•
•

Slide 10

Types of Variation
Trend (T)
Seasonal (S)
Cyclical (C)
Random or Irregular (I)
Additive Model
Zt = Tt + St + Ct + It
Multiplicative Model
Zt = (Tt )(St )(Ct )(It)


Estimating an Additive Model
with Seasonality

Slide 11

An additive regression model with seasonality:

Zt=0+ 1 t+ 2 Q1+ 3 Q2 + 4 Q3 + at
where
Q1=1 if the observation is in the first quarter, and 0
otherwise
Q2=1 if the observation is in the second quarter,
and 0 otherwise
Q3=1 if the observation is in the third quarter, and
0 otherwise


12-4 The Ratio-to-MovingAverage Method

Slide 12

A moving average of a time series is an average of a fixed
number of observations that moves as we progress down the
series.
Time, t:
Series, Zt:

1
15

2
12

3
11

4
18

6
7
8
9 10
11
12
13
16
14
17 20 18
21
16
14
Five-period
15.4 15.6 16.0 17.2 17.6 17.0 18.0 18.4 17.8 17.6

14
19

3
11

4
18

14
19

moving average:

Time, t:
Series, Zt:

1
15

2
12

5
21

5
6
7
8
9 10
11
12
21
16
14
17 20 18
21
16
(15 + 12 + 11 + 18 + 21)/5=15.4
(12 + 11 + 18 + 21 + 16)/5=15.6
(11 + 18 + 21 + 16 + 14)/5=16.0

13
14

. . . . .
(18 + 21 + 16 + 14 + 19)/5=17.6


Comparing Original Data and
Smoothed Moving Average
Original Series and Five-Period Moving Averages

•

Slide 13

Moving Average:
– “Smoother”
– Shorter
– Deseasonalized

20

Z

– Removes seasonal and
irregular components

15

– Leaves trend and
cyclical components

10

0

5

10

15

t

Z

t  TSCI  SI
MA TC


Ratio-to-Moving Average
•

Slide 14

Ratio-to-Moving Average for Quarterly Data
Compute a four-quarter moving-average
series.
Center the moving averages by averaging
every consecutive pair and placing the average
between quarters.
Divide the original series by the corresponding
moving average. Then multiply by 100.
Derive quarterly indexes by averaging all data
points corresponding to each quarter. Multiply
each by 400 and divide by sum.


Ratio-to-Moving Average:
Example 12-3
Ratio
to Moving
Average
*
*
93.3
100.9
110.2
93.8
90.1
107.5
106.4
100.7
90.5
95.0
113.9
97.9
*
*

Four-Quarter Moving Averages
Actual
Smoothed
Actual
Smoothed

170

160

Sales

Quarter
1998W
1998S
1998S
1998F
1999W
1999S
1999S
1999F
2000W
2000S
2000S
2000F
2002W
2002S
2002S
2002F

Simple
Centered
Moving
Moving
Sales Average
Average
170
*
*
148
*
*
141
*
151.125
150
152.25
148.625
161
150.00
146.125
137
147.25
146.000
132
145.00
146.500
158
147.00
147.000
157
146.00
147.500
145
148.00
144.000
128
147.00
141.375
134
141.00
141.000
160
141.75
140.500
139
140.25
142.000
130
140.75
*
144
143.25
*

Slide 15

150
Moving Average
Length:
4

140

MAPE:
MAD:
MSD:

130
0

5

10

7.816
10.955
152.574

15

T
ime


Seasonal Indexes: Example 12-3
Year

Winter

1998
1999
2000

Quarter
Spring
Summer

110.2
106.4
2002

Sum
Average

93.8
100.7
113.9

93.3
90.1
90.5
97.9

330.5
110.17

292.4
97.47

273.9
91.3

Slide 16

Fall

100.9
107.5
95.0
303.4
101.13

Sum of Averages = 400.07
Seasonal Index = (Average)(400)/400.07

Index

110.15

Seasonal
97.45

91.28

101.11


Deseasonalized Series: Example
12-3

Original and SeasonallyAdjusted Series
170
160

Sales

Seasonal
Deseasonalized
Quarter
Sales
Index (S)
Series(Z/S)*100
1998W
170
110.15
154.34
1998S
148
97.45
151.87
1998S
141
91.28
154.47
1998F
150
101.11
148.35
1999W
161
110.15
146.16
1999S
137
97.45
140.58
1999S
132
91.28
144.51
1999F
158
101.11
156.27
2000W
157
110.15
142.53
2000S
145
97.45
148.79
2000S
128
91.28
140.23
2000F
134
101.11
132.53
2002W
160
110.15
145.26
2002S
139
97.45
142.64
2002S
130
91.28
142.42
2002F
144
101.11
142.42

Slide 17

150
140
130
1992W

1992S

1992S

1992F

t
Original

Deseasonalized - - -

Time Series, Forecasting, and Index Numbers By Shakeel Nouman M.Phil Statistics Govt. College
University Lahore, Statistical Officer

The Cyclical Component:
Example 12-3
Trend Line and Moving Averages
180

Sales

170

160
150
140
130
120
1992W

1992S

1992S

t

1992F

Slide 18

The cyclical component is
the remainder after the
moving averages have been
detrended. In this example,
a comparison of the moving
averages and the estimated
regression line:

Z  155.275  11059 t
.
illustrates that the cyclical
component in this series is
negligible.


Example 12-3 using the
Template

Slide 19

The centered moving average, ratio to moving average, seasonal index, and deseasonalized
values were determined using the Ratio-to-Moving-Average method.

This displays just
a partial output
for the quarterly
forecasts.


Template

Slide 20

Graph of the quarterly Seasonal
Index

Template

Slide 21

Graph of the Data and the quarterly Forecasted values

Template

Slide 22

The centered moving average, ratio to moving average, seasonal index, and deseasonalized
values were determined using the Ratio-to-Moving-Average method.

This displays just
a partial output
for the monthly
forecasts.


Template

Slide 23

Graph of the monthly Seasonal Index


Template

Slide 24

Graph of the Data and the monthly Forecasted values

Forecasting a Multiplicative
Series: Example 12-3

Slide 25

The forecast of a multiplicative series :
ˆ
Z = TSC
Forecast for Winter 2002 (t = 17) :
Trend : z = 152.26 - (0.837)(17) = 138.03
ˆ
S = 1.1015
C  1 (negligibl e)
ˆ
Z = TSC
= (1)(138.03)(1.1015) = 152.02


Multiplicative Series: Review

Slide 26

Z  ( Trend )( Seasonal ( Cyclical )( Irregular )
 TSCI
MA  ( Trend )( Cyclical )
 TC
Z
TSCI

 SI
MA
TC
S = Average of SI (Ratio - to - Moving Averages)
Z
TSCI

 CTI (Deseasonalized Data)
S
S


12-5 Exponential Smoothing
Methods

Slide 27

Smoothing is used to forecast a series by first removing sharp
variation, as does the moving average.
Weights Decline as we go back in
Time
Weights Decline as We Go Back in Time and Sum to 1

W e ig ht

Weight

0.4
0.3
0.2
0.1
0.0
-15

-10

-5

Exponential smoothing is a forecasting
method in which the forecast is based in
a weighted average of current and past
series values. The largest weight is
given to the present observations, less
weight to the immediately preceding
observation, even less weight to the
observation before that, and so on. The
weights decline geometrically as we go
back in time.

0

Lag

-10

Lag

0


The Exponential Smoothing
Model

Slide 28

Given a weighting factor: 0 < w < 1:
2
3

Z t 1  w ( Z t )  w (1  w )( Z t 1 )  w (1  w ) ( Z t  2 )  w (1  w ) ( Z t  3 ) 

Since
2
3

Z t  w ( Z t 1 )  w (1  w )( Z t  2 )  w (1  w ) ( Z t  3 )  w (1  w ) ( Z t  4 ) 
2
3

(1  w ) Z t  w (1  w )( Z t 1 )  w (1  w ) ( Z t  2 )  w (1  w ) ( Z t  3 ) 

So


Z t 1  w(Z t )  (1  w)(Z t )


Z t 1  Z t  (1  w)(Z t - Z t )

Example 12-4

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

Z

925
940
924
925
912
908
910
912
915
924
943
962
960
958
955
*

w=.4

w=.8

925.000
925.000
931.000
928.200
926.920
920.952
915.771
913.463
912.878
913.727
917.836
927.902
941.541
948.925
952.555
953.533

925.000
925.000
937.000
926.600
925.320
914.664
909.333
909.867
911.573
914.315
922.063
938.813
957.363
959.473
958.295
955.659

Exp o n e ntial S m o othing: w=0 .4 and w=0 .8
960
950

w = .4

Day

Slide 29

940
930
920
910
0

5

10

15

Day

Original data:
Smoothed, w=0.4: ......
Smoothed, w=0.8: -----


Example 12-4 – Using the
Template

Slide 30


12-6 Index Numbers

Slide 31

An index number is a number that measures the relative
change in a set of measurements over time. For example: the
Dow Jones Industrial Average (DJIA), the Consumer Price
Index (CPI), the New York Stock Exchange (NYSE) Index.
Value in period i
Index number in period i: = 100
Value in base period
Changing the base period of an index:
Old index value
New index value: = 100
Index value of new base


Index Numbers: Example 12-5

Pric e and Index (1982=100) of Natural Gas Price

Index
Index
Year Price 1984-Base 1991-Base

121
121
133
146
162
164
172
187
197
224
255
247
238
222

100.0
100.0
109.9
120.7
133.9
135.5
142.1
154.5
162.8
185.1
210.7
204.1
196.7
183.5

64.7
64.7
71.1
78.1
86.6
87.7
92.0
100.0
105.3
119.8
136.4
132.1
127.3
118.7

250

Original
P ric e

1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997

Slide 32

Index (1984)
150

Index (1991)
50
1985

1990

1995

Ye ar


Consumer Price Index –
Example 12-6
Consumer Price index (CPI): 1967=100

Slide 33

Example 12-6:

450

CPI

350

250

150

50
1950 1955 1960 1965 1970 1975 1980 1985 1990 1995

Ye a r

Salary
Adjusted Salary =
100
CPI
Adjusted
Year
Salary Salary
1980
29500 11953.0
1981
31000 11380.3
1982
33600 11610.2
1983
35000 11729.2
1984
36700 11796.8
1985
38000 11793.9


Example 12-6: Using the
Template

Slide 34


Name
Religion
Domicile
Contact #
E.Mail
M.Phil (Statistics)

Shakeel Nouman
Christian
Punjab (Lahore)
0332-4462527. 0321-9898767
sn_gcu@yahoo.com
sn_gcu@hotmail.com
GC University, .
(Degree awarded by GC University)

M.Sc (Statistics)
Statitical Officer
(BS-17)
(Economics & Marketing
Division)

GC University, .
(Degree awarded by GC University)

Livestock Production Research Institute
Bahadurnagar (Okara), Livestock & Dairy Development
Department, Govt. of Punjab


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