Estimation of Seasonality Index 15OCT2011

8/3/2019 Estimation of Seasonality Index 15OCT2011

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Time Series with Cyclical and Seasonal

Variations


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Seasonal Effect

Seasonal effect is defined as the repetitive and predictable

pattern data behaviour in a time-series around the trend line.

To measure the seasonal effect the time period must be less thanone year, such as, days, weeks, months or quarters.


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Seasonal effect

Description of seasonal effect provides better understanding of

the seasonal component.

Seasonal effect can be eliminated from the time-series. This

process is called deseasonalizing or seasonal adjusting.


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Seasonal Adjusting

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Seasonal Index

Method of simple averages

Ratio-to-moving average method

Ratio to Trend method


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Method of simple averages

Average the unadjusted data by months (or quarters, if quarterlydata is given).

Add the data for each month (or quarter) and calculate theaverage by diving the monthly (quarterly) totals by number of

years. Calculate the average of monthly averages.

Seasonal index for month i is the ratio of monthly average ofmonth i to the average of monthly averages times 100.


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Example: Use the method of simple averages to calculate seasonal index and findthe forecast for October 2005

Month 2006 2007 2008

Jan 15 23 25

Feb 16 22 25

Mar 18 28 35

Apr 18 27 36

May 23 31 36

June 23 28 30

July 20 22 30

Aug 28 28 34

Sep 29 32 38

Oct 33 37 47

Nov 33 34 41

Dec 38 44 53


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Month 2006 2007 2008 Monthly Total Monthly Average Percentage Average

of Monthly

Averages

Jan 15 23 25 6321 70

Feb 16 22 25 63

21 70Mar 18 28 35 81

27 90

Apr 18 27 36 8127 90

May 23 31 36 9030 100

June 23 28 30 81 27 90

July 20 22 30 7224 80

Aug 28 28 34 9030 100

Sep 29 32 38 9933 110

Oct 33 37 47 11739 130

Nov 33 34 41 10836 120

Dec 38 44 53 13545 150

90 30 1200


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Month 2006 2006

deseasonalized

data

2007 2007 deseasonalized

data

2008 2008

deseasonalized

data

Jan 1521.428571

2332.857143

2535.714286

Feb 16 22.857143 22 31.428571 25 35.714286

Mar 1820

2831.111111

3538.888889

Apr 1820

2730

3640

May 2323

3131

3636

June 2325.555556

2831.111111

3033.333333

July 2025

2227.5

3037.5

Aug 2828

2828

3434

Sep 2926.363636

3229.090909

3834.545455

Oct 3325.384615

3728.461538

4736.153846

Nov 3327.5

3428.333333

4134.166667

Dec 3825.333333

4429.333333

5335.333333

Deseasonalized data = actual data / seasonality index


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Trend

Trend is calculated using regression on

deseasonalized data.

Deseasonalized data is obtained by dividing theactual data with its seasonality index.


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Forecasting using method of averages in the presence of seasonality


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Forecast using Method of Averages

Let us assume that we want to forecast value for

2009 October ( t = 46).

Trend Component = 21.94 + 0.4352 x 46 = 41.97

Seasonality Index for October = 130

Forecasted value for October 2005 = 41.97 x 1.3 =

54.56


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Ratio to Moving Average Method

A moving average smoothes the data of their

variations.

In a multiplicative time series model, the ratio to

moving average results in Seasonal and random

error component.


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The ratio to moving average

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Ratio to Moving Average - Steps

Compute a moving average (based on the number of seasons,

that is n is the equal to the number of seasons).

Center the moving averages by averaging every consecutive pair.

For each data point, divide the original series value by thecorresponding moving average and multiply by 100. This gives

ratio to moving average.

For each season average all data corresponding to the season.

This will result in seasonal index. The seasonal indexes are

adjusted so that the mean is 100.


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Forecasting using Ratio to moving

average

Approach 1: Use moving average values to

get the trend equation using regression.

Approach 2: Deseasonalize the data by

dividing the actual data with seasonality

index. Derive the trend equation usingdeseasonalized data (using regression)


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Year Quarter1 2 3 4

2005 75 60 54 59

2006 86 65 63 80

2007 90 72 66 85

2008 100 78 72 93

Forecast the value for Q3 2009

Ratio to Moving Average Example


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Year Quarter Value 4Q MA Centered MA Ratio to MA

Seasonality

Index

Deseasonalized

Data

2005 Q1 75

Q2 60

Q3 54 62 63.375 85.2071006 84.68 63.76949

Q4 59 64.75 65.375 90.248566 100.49 58.71231

2006 Q1 86 66 67.125 128.119181 122.34 70.2959Q2 65 68.25 70.875 91.7107584 92.47 70.29307

Q3 63 73.5 74 85.1351351 84.68 74.39773

Q4 80 74.5 75.375 106.135987 100.49 79.60991

2007 Q1 90 76.25 76.625 117.455139 122.34 73.56547

Q2 72 77 77.625 92.7536232 92.47 77.86309

Q3 66 78.25 79.5 83.0188679 84.68 77.94048

Q4 85 80.75 81.5 104.294479 100.49 84.58553

2008 Q1 100 82.25 83 120.481928 122.34 81.73941

Q2 78 83.75

Q3 72

Q4 93

Ratio to Moving Average


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Forecast for 2009 Q3

t = 19

Trend = 57.12 + 2.10 x 19 = 97.02

Forecast = 97.02 x 84.68 / 100 = 82.15


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Recommended Readings

Amir D Aczel and J Sounderpandian, Complete Business

Statistics, The McGraw Hill, 2009.

Estimation of Seasonality Index 15OCT2011

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