Time series, forecasting, and index numbers

Slide 1

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

Shakeel NoumanM.Phil Statistics

Time Series, Forecasting, and Index Numbers

Slide 2

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

Time Series, Forecasting, and Index Numbers12


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.

12-1 Using Statistics


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.

5 0403020100

15

10

5

0

t

Z

Random Walk

a

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.

The Random Walk


Slide 5

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

5 04 03 02 01 00

4

3

2

1

0

-1

t

z

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.

A Time Series as a Superposition of Cyclical

Functions


Slide 612-2 Trend Analysis: Example

12-1

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.


Slide 712-2 Trend Analysis: Example

12-1

Straight line trend.


Slide 8

Example 12-2

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

The forecast for t = 10

(year 2002) is 345.27


Slide 9

888786

11000

10000

9000

8000

7000

6000 t

Pas

s en g

ers

Monthly Numbers of Airline Pas sengers

89 90 91 92 93 94 9595949392919089888786858483828180

20

15

10

5

Ye ar

Ea

r nin

gs

G ro s s E arning s : Annual

AJJMAMFJDNOSAJJMAMFJDNOSAJJMAMFJ

200

100

0

Time

Sal

es

Monthly S ale s of S untan Oil

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.

12-3 Seasonality and Cyclical Behavior


Slide 10

• Types of VariationTrend (T)Seasonal (S)Cyclical (C)Random or Irregular (I)

• Additive ModelZt = Tt + St + Ct + It

• Multiplicative ModelZt = (Tt )(St )(Ct )(It)

Time Series Decomposition


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

Estimating an Additive Model with Seasonality


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: 1 2 3 4 5 6 7 8 9 10 11 12 13 14Series, Zt: 15 12 11 18 21 16 14 17 20 18 21 16 14 19

Five-periodmoving average: 15.4 15.6 16.0 17.2 17.6 17.0 18.0 18.4 17.8 17.6

Time, t: 1 2 3 4 5 6 7 8 9 10 11 12 13 14Series, Zt: 15 12 11 18 21 16 14 17 20 18 21 16 14 19

(15 + 12 + 11 + 18 + 21)/5=15.4 (12 + 11 + 18 + 21 + 16)/5=15.6

(11 + 18 + 21 + 16 + 14)/5=16.0 . . . . .

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

12-4 The Ratio-to-Moving- Average Method


Slide 13

151050

20

15

10

t

Z

Original Series and Five-Period Moving Averages • Moving Average:– “Smoother”– Shorter– Deseasonalized

– Removes seasonal and irregular components

– Leaves trend and cyclical components

SITC

TSCIMA

tZ

Comparing Original Data and Smoothed Moving Average


Slide 14

• Ratio-to-Moving Average for Quarterly DataCompute 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


Slide 15

151050

170

160

150

140

130

Sale

s

Time

Actual SmoothedActual Smoothed

MSD:MAD:MAPE:

Length:Moving Average

152.574 10.955 7.816

4

Four-Quarter Moving Averages Simple Centered Ratio Moving Moving to Moving

Quarter Sales Average Average Average1998W 170 * * *1998S 148 * * *

1998S 141 * 151.125 93.31998F 150 152.25 148.625 100.91999W 161 150.00 146.125 110.21999S 137 147.25 146.000 93.81999S 132 145.00 146.500 90.11999F 158 147.00 147.000 107.52000W 157 146.00 147.500 106.42000S 145 148.00 144.000 100.72000S 128 147.00 141.375 90.52000F 134 141.00 141.000 95.02002W 160 141.75 140.500 113.92002S 139 140.25 142.000 97.92002S 130 140.75 * *2002F 144 143.25 * *

Ratio-to-Moving Average: Example 12-3


Slide 16

Quarter Year Winter Spring Summer Fall

1998 93.3 100.91999 110.2 93.8 90.1 107.52000 106.4 100.7 90.5 95.0

2002 113.9 97.9

Sum 330.5 292.4 273.9 303.4Average 110.17 97.47 91.3 101.13

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

SeasonalIndex 110.15 97.45 91.28 101.11

Seasonal Indexes: Example 12-3


Slide 17

Seasonal Deseasonalized

Quarter Sales Index (S) Series(Z/S)*100

1998W 170 110.15 154.341998S 148 97.45 151.871998S 141 91.28 154.471998F 150 101.11 148.351999W 161 110.15 146.161999S 137 97.45 140.581999S 132 91.28 144.511999F 158 101.11 156.272000W 157 110.15 142.532000S 145 97.45 148.792000S 128 91.28 140.232000F 134 101.11 132.532002W 160 110.15 145.262002S 139 97.45 142.642002S 130 91.28 142.422002F 144 101.11 142.42

Original Deseasonalized - - -

1992F1992S1992S1992W

170

160

150

140

130

t

Sal es

Original and Seasonally Adjusted Series

Deseasonalized Series: Example 12-3


Slide 18

180

170

160

150

140

130

1201992F1992S1992S1992W

t

Sal e s

Trend Line and Moving Averages 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: illustrates that the cyclical component in this series is negligible.

. .Z t 155 275 11059

The Cyclical Component: Example 12-3


Slide 19

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

Example 12-3 using the Template

This displays just a partial output for the quarterly

forecasts.


Slide 20Example 12-3 using the

Template

Graph of the quarterly Seasonal Index



Template

Graph of the Data and the quarterly Forecasted values


Slide 22

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

Example 12-3 using the Template

This displays just a partial output for the monthly

forecasts.



Template

Graph of the monthly Seasonal Index



Template

Graph of the Data and the monthly Forecasted values


Slide 25

152.02=)(1.1015)(1)(138.03 = =ˆ

e)(negligibl 1 C1.1015 = S

138.03=)(0.837)(17-152.26=z :Trend

:17)=(t 2002 for WinterForecast

=ˆ

:series tivemultiplica a offorecast The

TSCZ

TSCZ

Forecasting a Multiplicative Series: Example 12-3


Slide 26

Z Trend Seasonal Cyclical IrregularTSCI

MA Trend CyclicalTC

ZMA

TSCITC

SI

ZS

TSCIS

CTI

( )( ( )( )

( )( )

S = Average of SI (Ratio - to - Moving Averages)

(Deseasonalized Data)

Multiplicative Series: Review


Slide 27

Smoothing is used to forecast a series by first removing sharp variation, as does the moving average.

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. 0-5-10-15

0.4

0.3

0.2

0.1

0.0

Lag

Wei

ght

Weights Decline as We Go Back in Time and Sum to 1

Weights Decline as we go back in Time

-10 0

Wei

ght

Lag

12-5 Exponential Smoothing Methods


Slide 28

Given a weighting factor: < < :

Since

So

0 1

1 1 1 12

2 13

3

1 1 2 12

3 13

4

1 1 1 12

2 13

3

w

Zt w Zt w w Zt w w Zt w w Zt

Zt w Zt w w Zt w w Zt w w Zt

w Zt w w Zt w w Zt w w Zt

( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( )

Zt 1 w(Zt ) (1 w)(Zt )

Zt 1 Zt (1 w)(Zt - Zt )

The Exponential Smoothing Model


Slide 29

Day Z w=.4 w=.8

1 925 925.000 925.0002 940 925.000 925.0003 924 931.000 937.0004 925 928.200 926.6005 912 926.920 925.3206 908 920.952 914.6647 910 915.771 909.3338 912 913.463 909.8679 915 912.878 911.57310 924 913.727 914.31511 943 917.836 922.06312 962 927.902 938.81313 960 941.541 957.36314 958 948.925 959.47315 955 952.555 958.29516 * 953.533 955.659

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

Day151050

960

950

940

930

920

910

w= .

4

Expo ne ntial S mo othing: w=0 .4 and w=0 .8

Example 12-4


Slide 30Example 12-4 – Using the

Template


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.

Index number in period : = 100 Value in period Value in base period

Changing the base period of an index:

New index value: = 100 Old index valueIndex value of new base

i i

12-6 Index Numbers


Slide 32

Index IndexYear Price 1984-Base 1991-Base

1984 121 100.0 64.71985 121 100.0 64.71986 133 109.9 71.11987 146 120.7 78.11988 162 133.9 86.61989 164 135.5 87.71990 172 142.1 92.01991 187 154.5 100.01992 197 162.8 105.31993 224 185.1 119.81994 255 210.7 136.41995 247 204.1 132.11996 238 196.7 127.31997 222 183.5 118.7

Year

Pr i

ce

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

199519901985

250

150

50

Original

Index (1984)

Index (1991)

Index Numbers: Example 12-5


Slide 33

Example 12-6:

AdjustedYear Salary Salary1980 29500 11953.01981 31000 11380.31982 33600 11610.21983 35000 11729.21984 36700 11796.81985 38000 11793.9

Adjusted Salary = SalaryCPI

100

CPI

Ye a r

4 5 0

3 5 0

2 5 0

1 5 0

5 01 9 9 51 9 9 01 9 8 51 9 8 01 9 7 51 9 7 01 9 6 51 9 6 01 9 5 51 9 5 0

Consumer Price index (CPI): 1967=100

Consumer Price Index – Example 12-6


Slide 34Example 12-6: Using the

Template


Slide 35

M.Phil (Statistics)

GC University, . (Degree awarded by GC University)

M.Sc (Statistics) GC University, . (Degree awarded by GC University)

Statitical Officer(BS-17)(Economics & Marketing Division)

Livestock Production Research Institute Bahadurnagar (Okara), Livestock & Dairy Development

Department, Govt. of Punjab

Name Shakeel NoumanReligion ChristianDomicile Punjab (Lahore)Contact # 0332-4462527. 0321-9898767E.Mail [email protected] [email protected]


mailto:[email protected]

mailto:[email protected]

Education

Time series, forecasting, and index numbers