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Time Series Analysis - Seasonal Indices Using Ratio to Moving Average Method
It is most widely used method known as the ratio to moving average method. It is better because of its accuracy. Also the seasonal indices calculated using this method are free from all the three components, namely, trend (T), cyclic (C) and Irregular variations (I).
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Time Series Analysis - Seasonal Indices Using Ratio to Moving Average Method
- 1. SUNDAR B. N. ASSISTANTPROFESSOR Seasonal Indices Using Ratio to Moving Average Method Production (in thousand units) Total Average Year I Quarter II Quarter III Quarter IV Quarter 2009 77.06 117.43 2010 98.18 99.12 89.69 125.35 2011 83 108.88 83.17 118.81 2012 96.48 99.5 Median 96.48 99.5 83.17 118.81 397.96 99.49 Indices 96.97 100.01 83.60 119.42 400.00 100.00
- 2. Components of TSA TrendEffect Seasonal Effect Cyclical Effect Irregular Effect
- 3. Seasonal Effect If valuesin a time series reflect seasonal variation with respect to a given period of time such as a quarter, a month or a year, the time series is called a time series with seasonal effect.
- 4. Components of SeasonalEffect i. Method of Simple Average ii. Ratio to Trend Method iii. Ratio to Moving Average Method iv. Method of Link Relatives
- 5. Ratio to Moving AverageMethod It is most widely used method known as the ratio to moving average method. It is better because of its accuracy. Also the seasonal indices calculated using this method are free from all the three components, namely, trend (T), cyclic (C) and Irregular variations (I).
- 6. Steps for obtainingseasonal indices using this Method 1) We arrange the data chronologically 2) If the cycle of oscillation is 1 year, we take the 12 months moving average of the 1st year, We enter the average value against the middle position, i.e., between the months of June and July. 3) We discard the value for the month of January of the first year and include the value for the month of January of the subsequent year. Then we calculate the average of these 12 values and enter it against the middle position, i.e., between July and August. We repeat the process of taking moving averages MA (1) and entering the value in the middle position, till all the monthly data are exhausted. 4) We calculate the centred moving average, i.e., MA(2), of the two values of the moving averages MA(1) and enter it against the first value, i.e., the month of July in the first year and subsequent values against the month of August, September, etc.
- 7. Steps for obtainingseasonal indices using this Method 5) After calculating the MA(1) and MA(2) values, we treat the original values (except the first 6 months in the beginning and the last 6 months at the end) as the percentage of the centred moving average values. For this we divide the original monthly values by the corresponding centred moving average, i.e., MA (2) values, and multiply the result by 100. We have now succeeded in eliminating the trend and cyclic variations from the original data. We now have to get rid of the data of irregular variations. 6) We prepare another two-way table consisting of the month-wise percentage values calculated in Step 5, for all years. The purpose of this step is to average the percentages and to eliminate the irregular variations in the process of averaging. 7) We find the median of the percentages or preliminary seasonal indices calculated in Step 5 month-wise and take the average of the month-wise median. Then we divide the median of each month by the average value and multiply it by 100. Generally, the sum of all medians is not 1200. Therefore, the average of all medians is not equal to 100. Hence, the seasonal indices are subjected to the same operation. We multiply the medians by the ratio of expected total of indices, i.e., 1200 to the actual total as follows:
- 8. Steps for obtainingseasonal indices using this Method ● We arrange the data chronologically ● Find Moving Average – If it is in month, enter 12 months averaged data between June to July. For next average discard January, then discard February, then March and so on repeat the procedure until all observations are exhaust. If it is Quarter data enter averaged data for II and III Quarter, then follow discard first and insert next element. ● Find Centred moving Average for Moving Average we found in the last example. For example Take CMA of June to July and enter in the July ● Find the Seasonal Relatives for each observations we divide the original monthly values by the corresponding centred moving average Formula ● Enter all the Seasonal Relatives Observations in their respected Months/Quarter ● Find the Median ● Find the Grand Mean of all the Median points
- 9. Problem #2 ● Obtain;Seasonal Indices by Ratio to Moving Average method from the following data. Production (in thousand units) Year I Quarter II Quarter III Quarter IV Quarter 2009 25 30 21 32 2010 27 28 25 34 2011 22 27 21 30 2012 24 25 20 33
- 10. Step I Arrangethe data Chronologically Year (1) Quarter (2) Productio n (3) 2009 I Quarter 25 II Quarter 30 III Quarter 21 1 Q IV Quarter 32 2 Q 2010 I Quarter 27 3 Q II Quarter 28 4 Q III Quarter 25 1 Q IV Quarter 34 Year (1) Quarter (2) Productio n (3) 2 Q 2011 I Quarter 22 3 Q II Quarter 27 4 Q III Quarter 21 1 Q IV Quarter 30 2 Q 2012 I Quarter 24 3 Q II Quarter 25 4 Q III Quarter 20 IV Quarter 33 Production (in thousand units) Year I Quarter II Quarter III Quarter IV Quarter 2009 25 30 21 32 2010 27 28 25 34 2011 22 27 21 30 2012 24 25 20 33
- 11. Step II FindMoving Average for Quarter Year (1) Quarter (2) Productio n (3) Average Moving MA (1) (4) 2009 I Quarter 25 II Quarter 30 27 III Quarter 21 1 Q 27.5 IV Quarter 32 2 Q 27 2010 I Quarter 27 3 Q 28 II Quarter 28 4 Q 28.5 III Quarter 25 1 Q 27.25 IV Quarter 34 Year (1) Quarter (2) Productio n (3) Average Moving MA (1) (4) 2 Q 27 2011 I Quarter 22 3 Q 26 II Quarter 27 4 Q 25 III Quarter 21 1 Q 25.5 IV Quarter 30 2 Q 25 2012 I Quarter 24 3 Q 24.75 II Quarter 25 4 Q 25.5 III Quarter 20 IV Quarter 33
- 12. Step III FindCentred moving Average for Moving Average Year (1) Quarter (2) Productio n (3) Average Moving MA (1) (4) Centred moving Average MA (2) (5) 2009 I Quarter 25 II Quarter 30 27 III Quarter 21 27.25 1 Q 27.5 IV Quarter 32 27.25 2 Q 27 2010 I Quarter 27 27.5 3 Q 28 II Quarter 28 28.25 4 Q 28.5 III Quarter 25 27.875 1 Q 27.25 IV Quarter 34 27.125 Year (1) Quarter (2) Productio n (3) Average Moving MA (1) (4) Centred moving Average MA (2) (5) 2 Q 27 2011 I Quarter 22 26.5 3 Q 26 II Quarter 27 25.5 4 Q 25 III Quarter 21 25.25 1 Q 25.5 IV Quarter 30 25.25 2 Q 25 2012 I Quarter 24 24.875 3 Q 24.75 II Quarter 25 25.125 4 Q 25.5 III Quarter 20 IV Quarter 33
- 13. Step IV Findthe Seasonal Relatives Year (1) Quarter (2) Productio n (3) Average Moving MA (1) (4) Centred moving Average MA (2) (5) Seasonal Relatives (3)/(5) 2009 I Quarter 25 II Quarter 30 27 III Quarter 21 27.25 77.06 1 Q 27.5 IV Quarter 32 27.25 117.43 2 Q 27 2010 I Quarter 27 27.5 98.18 3 Q 28 II Quarter 28 28.25 99.12 4 Q 28.5 III Quarter 25 27.875 89.69 1 Q 27.25 IV Quarter 34 27.125 125.35 Year (1) Quarter (2) Productio n (3) Average Moving MA (1) (4) Centred moving Average MA (2) (5) Seasonal Relatives (3)/(5) 2 Q 27 2011 I Quarter 22 26.5 83 3 Q 26 II Quarter 27 25.5 105.88 4 Q 25 III Quarter 21 25.25 83.17 1 Q 25.5 IV Quarter 30 25.25 118.81 2 Q 25 2012 I Quarter 24 24.875 96.48 3 Q 24.75 II Quarter 25 25.125 99.5 4 Q 25.5 III Quarter 20 IV Quarter 33
- 14. Step V SeasonalRelatives with their respected Observation in Quarter Year (1) Quarter (2) Production (3) Average Moving MA (1) (4) Centred moving Average MA (2) (5) Seasonal Relatives (3)/(5) 2009 I Quarter 25 II Quarter 30 27 III Quarter 21 27.25 77.06 1 Q 27.5 IV Quarter 32 27.25 117.43 2 Q 27 2010 I Quarter 27 27.5 98.18 3 Q 28 II Quarter 28 28.25 99.12 4 Q 28.5 III Quarter 25 27.875 89.69 1 Q 27.25 IV Quarter 34 27.125 125.35 Year (1) Quarter (2) Productio n (3) Average Moving MA (1) (4) Centred moving Average MA (2) (5) Seasonal Relatives (3)/(5) 2 Q 27 2011 I Quarter 22 26.5 83 3 Q 26 II Quarter 27 25.5 105.88 4 Q 25 III Quarter 21 25.25 83.17 1 Q 25.5 IV Quarter 30 25.25 118.81 2 Q 25 2012 I Quarter 24 24.875 96.48 3 Q 24.75 II Quarter 25 25.125 99.5 4 Q 25.5 III Quarter 20 IV Quarter 33 Production (in thousand units) Year I Quarter II Quarter III Quarter IV Quarter 2009 77.06 117.43 2010 98.18 99.12 89.69 125.35 2011 83 105.88 83.17 118.81 2012 96.48 99.5
- 15. Step VI Findthe Grand Mean Grand Mean = 96.48+99.5+83.17+118.81 =397.96/4 Grand Mean =99.49 Production (in thousand units) Year I Quarter II Quarter III Quarter IV Quarter 2009 77.06 117.43 2010 98.18 99.12 89.69 125.35 2011 83 108.88 83.17 118.81 2012 96.48 99.5 Median 96.48 99.5 83.17 118.81
- 16. Step V Findthe Median Production (in thousand units) Year I Quarter II Quarter III Quarter IV Quarter 2009 77.06 117.43 2010 98.18 99.12 89.69 125.35 2011 83 108.88 83.17 118.81 2012 96.48 99.5 Median 96.48 99.5 83.17 118.81
- 17. Step VII Findthe Seasonal Indices For Quarter I 96.48/99.49*100 =96.97 For Quarter II 99.5/99.49*100 =100.01 For Quarter III 83.17/99.49*100 =83.59 For Quarter IV 118.81/99.49*100 =119.42 Production (in thousand units) Total Average Year I Quarter II Quarter III Quarter IV Quarter 2009 77.06 117.43 2010 98.18 99.12 89.69 125.35 2011 83 108.88 83.17 118.81 2012 96.48 99.5 Median 96.48 99.5 83.17 118.81 397.96 99.49 Indices 96.97 100.01 83.60 119.42 400.00 100.00
- 18. 1) Bhardwaj, R.S. (2009). Business Statistics. Excel Books India. 2) Shukla, G. K.; Trivedi, Manish (2017). “Unit-13 SEASONAL COMPONENT ANALYSIS. IGNOU. References