Paper No: 14 Module: 37 Principal Investigator & Co- Principal Investigator Paper Coordinator Content Writer Content Reviewer Development Team Prof. R.K. Kohli Prof. V.K. Garg &Prof.AshokDhawan Central University of Punjab, Bathinda Dr. Harmanpreet Singh Kapoor, Central University of Punjab, Bathinda Dr. Harmanpreet Singh Kapoor Central University of Punjab, Bathinda Prof. Kanchan Jain, Panjab University, Chandigarh Anchor Institute Central University of Punjab 1
Description of Module Subject Name Paper Name Module Name/Title Module Id EVS/SAES-XIV/37 Pre-requisites Basic Mathematics, Mod 35-36 Objectives Keywords To give method of measurement of seasonal, cyclic and irregular component with their examples including merits and demerits. Time series, seasonal, cyclic and irregular component, average, moving average 2
Module 37: 1. Learning Objectives Learning Objectives Introduction. Measurement of Seasonal Variation De-seasonalization Measurement of Cyclic Variation Measurement of Irregular Component Summary Suggested Readings Introduction to the Time series and its components of the time series were discussed in the module named as Introduction to the time series and its importance. Measurement of secular trend was discussed in the module named as Measurement of the secular trend. Now we will discuss the measurement of seasonal, cyclic and irregular variation. These types of variation will be discussed in this module with examples and merits / demerits of the measurement methods. 2. Introduction: Measurement of time series components means to identify the components, isolate the components and eliminate the components. The components of time series like seasonal, cyclic and irregular has already been discussed in other modules. One can visit the module Introduction to the time Series and its importance to recall about the components of time series. In this module, our main concern is, if one can detect the components of time series in the data then what to do next. One must apply some methods to measure or eliminate them. Hence in this module, measurement methods for seasonal, cyclic and irregular components are discussed one by one. In this module, first we will discuss methods of measurement for seasonal variation. As seasonal is a component of time series that prevails for a short time period so most of measurement methods for seasonality can only be applied on the data whose values are available after a short time periods like monthly, erly etc. 3
In continuation, we will discuss about the methods for the measurement of cyclic variations. As cyclic variation can only be observed on a long period data, hence it is generally evaluated by eliminating other components from the time series. Last component of time series is irregular variation and it is generally considered as an error term or residual term that cannot be completely removed from the data. In the next session, methods for the measurement of different components will be discussed in steps. As lot of mathematical calculations are required to find out these values, thus one should be aware of basic terms like sum, average, proportion etc. 3. Measurement of seasonal variation: Seasonal variation occurs due to seasonal factors and man-made effects on the variable in any time series. There are many methods available in the literature for the measurement of the seasonal variation. Some of the most useful and important methods are: 1. Method of simple average 2. Ratio to trend method 3. Ratio to moving average method 4. Link relative method Let us discuss these methods in details. 3.1 Method of simple average: This is the easiest and simplest method to measure the seasonal variations. To find out the simple average one has to follow the steps. Steps for calculating seasonal variation indices are: Arrange the observations in erly/ monthly/ yearly manner according to the requirement of study variable or availability of the data. Calculate the average values of observations for all ers / months/ years over different years. Now calculate the erly/ monthly/ yearly (seasonal) average values. To find out average of erly/ monthly values, divide total of erly/monthly values by 4 / 12 to get seasonal average values respectively. seasonal average = total of the season with respect to one year number of the partition in a season 4
Calculate seasonal indices for different years by expressing erly/ monthly (seasonal) average as percentage of total average of erly/ monthly (seasonal). Seasonal index for ith year = seasonal average total of the seasonal average 100 Note that total of seasonal indices for er must be equal to 4*100 i.e. 400 and for months, it must be 12*100 i.e.1200. Note that seasonal here represents er or month. There are four er and twelve months in a year. Next, we will discuss an example of sales of bicycle to compute the seasonal indices using simple average method. Example 1: Sale of bicycles (in lac.) from year 2012-2017 are given in the Table 1. Compute the seasonal indices using method of simple average (hypothetical data). Solution: Years 1st 2nd 3rd 4th 2012 14 10 16 12 2013 12 5 12 2 2014 10 11 8 12 2015 3 14 10 14 2016 10 11 8 13 2017 12 13 10 8 Table 1 Steps to compute seasonal indices by method of simple average for given example of sale of bicycle (in lac.) from year 2012-2017 are: 5
First arrange the er s data corresponding to year. As in 2012 year 1 st er, 2 nd er, 3 rd er and 4 th er values are given as 14, 10, 16, and 12 respectively. Similarly, we also have values for 2013, 2014, 2015, 2016 and 2017 years. Now calculate seasonal average for all year using formula for calculating seasonal average. Seasonal average for 2012 = (14 + 10 + 16 + 12)/4 = 13 Also calculate the same for 2013, 2014, 2015, 2016 and 2017. Total average for each er is calculated by using formula 14 + 12 + 10 + 3 + 10 + 12 Total average for Ist er = = 10.1666667 6 Similarly, calculate the same for other ers. Now compute the seasonal indices by given formula for calculating seasonal indices in method. seasonal indices for 2012 = 13/10.41666 = 124.79996 Also calculate the same for 2013, 2014, 2015, 2016 and 2017. Now calculate total of seasonal indices by summing all seasonal indices for 2012, 2013,.2017. It is equal to approximate 600. We know that for erly data total of seasonal indices must be 400. So there is need to do adjustment in them. To do adjustment in seasonal indices, multiply seasonal indices by correction factor k. 400 k = Total of Seasonal indicies k = 400/600 = 0.6666 Adjusted seasonal indices for 2012 = 124.799 0.6666=83.1991 Similarly calculate for 2013, 2014,...,2017. Now, calculate total of adjusted seasonal indices, and this total is equal to 399.96 approx. 400. The following table shows seasonal indices: Years 1st 2nd 3rd 4th Seasonal Average seasonal indices adjust SI 2012 14 10 16 12 13 124.7999601 83.19914138 2013 12 5 12 2 7.75 74.40004762 49.59953574 2014 10 11 8 12 10.25 98.40006298 65.59938598 2015 3 14 10 14 10.25 98.40006298 65.59938598 6
2016 10 11 8 13 10.5 100.8000645 67.19937101 2017 12 13 10 8 10.75 103.200066 68.79935603 Total Average 10.1666667 10.66666667 10.6666667 10.166667 10.41666667 600.0002642 399.9961761 Table 2 One can observe that last column shows seasonal indices and total of seasonal indices is 400. Merits and Demerits: This method is very simple but it is used less to calculate seasonal indices. This method is based on the assumption that data doesn't contain any trend and cyclic component. The major important factor by using this method is that irregular component eliminates by averaging seasonal indices that can be erly or monthly. This assumption is not really true in economic sector hence this method can't be used in all sectors in an efficient manner. 3.2 Ratio to Trend method: Ratio to trend method is an improvement over the method of simple average. To calculate the seasonal indices by this method one must has to follow some steps. These steps are: Firstly calculate the trend value by least square method. Method of least square was already discussed in the module Measurement of Secular Trend. then denote the data as a percentage of trend values. Assuming that the multiplicative model of the time series fits the data then this modified data contain seasonal, cyclic and irregular components. For removing the cyclic and irregular components one has to take average for all ers/months separately. Now the total of seasonal indices should be 1200 for months and 400 for ers. If this is not the case then there is need to do some modification in the seasonal indices by multiplying it with constant k. k = 1200 Total of Seasonal indicies ; k = 400 Total of Seasonal indicies for monthly and erly respectively. 7
Example 2: Price of a commodity from year 2013-2017 are shown in Table 3. Apply ratio to trend method for computing seasonal indices (hypothetical data). Solution: year(t) 1st quar 2nd quar 3 rd quar 4th quar 2013 30 40 36 34 2014 34 52 50 44 2015 40 58 54 48 2016 54 76 68 62 2017 80 92 86 82 Table 3 For computation of seasonal indices by ratio to trend method for the given example of price of a commodity from year 2013-2017, first calculate trend value by using method of least square method for fitting an linear trend. Following table will be helpful to learn about computation of linear trend values by least square method. This table s last column shows trend values. year(t) 1st 2nd 3rd 4th Average(y) x=t- 2015 x 2 xy y t 2013 30 40 36 34 35-2 4-70 32 2014 34 52 50 44 45-1 1-45 44 2015 40 58 54 48 50 0 0 0 56 2016 54 76 68 62 65 1 1 65 68 2017 80 92 86 82 85 2 4 170 80 Total 280 0 10 120 56 Table 4 Computation of linear trend by using method of least square. Let us suppose straight line equation as: 8
y = a + bx. The normal equations for estimating of the values of a and b are: y i = na + b x i ; x i y i = a x i + b x i 2. One can solve the above equations to find out the values of a and b by putting values of x i y i, x i 2 and y i, i = 1,2,, n and n represents number of observations. Hence, the straight line trend is: a = y i n = 280 5 = 56 ; b = xy x 2 = 120 10 = 12. y t = 56 + 12x. Here, yearly increment in trend equation is b i.e.12. The positive value of b implies that there is an increasing trend in the data. To compute erly increment divide b by 4 i.e. 12/4 which is 3. Computation of seasonal indices Trend values are computed for the ers using trend line equation. Using the trend line equation the value for the year 2013 is 32 which is in between of the half of second and third er value. So trend value for second and third er is 32-1.5 and 32+1.5 respectively. For first er and fourth er is 30.5-3 and 33.5+3 respectively. Similarly compute trend values for 2014, 2015,.., 2017. Trend eliminated values can be find by dividing original erly values to trend erly values. Trend eliminated value for 2013: For 1 st er = (30/27.5) 100 = 109.1 For 2 nd er = (40/30.5) 100 = 131.1 For 3 rd er = (36/33.5) 100 = 107.5 For 4 th er = (34/36.5) 100 = 93.1 Similarly one can find values for 2013, 2014,.,2017. All these values having no trend are known as seasonal indices. 9
Compute average of seasonal indices and total of average of seasonal indices. Total of average of seasonal indices is 403.08. computation of seasonal indices trend values trend eliminated value year(t) 1st 2nd 3rd 4th 1st 2nd 3rd 4th 2013 27.5 30.5 33.5 36.5 109.1 131.1 107.5 93.1 2014 39.5 42.5 45.5 48.5 86.1 122.4 109.9 90.7 2015 51.5 54.5 57.5 60.5 77.7 106.4 93.9 70.3 2016 63.5 66.5 69.5 72.5 85 114.3 97.8 58.5 2017 75.5 78.5 81.5 84.5 106 117.1 105.5 97 Total 436.9 591.3 514.6 445.6 average seasonal indices 89.28 118.26 102.92 89.12 adjusted seasonal indices 92.07 117.36 102.14 88.44 Table 5 Sum of average seasonal indices is greater than 400. There is a need for adjustment. For adjustment, multiply average seasonal indices by correction factor k = 400 403.08 = 0.9924. These new seasonal indices are known as adjusted seasonal indices and sum of adjusted seasonal indices is 400 for erly data. Merits and demerits: This method is an improvement over the method of simple average. This method is based on the assumption that seasonal variation is a constant factor for the trend. So in this method trend remains in the data and it removes cyclic,irregular components from the data. 3.3 Ratio to moving average method: 10
It is already discussed in the module Measurement of secular trend that moving average method is used to remove periodic movements from the data. Seasonal variation can be removed permanently from the erly/ monthly data by taking 4 ers/ 12 months moving average if there exist a constant pattern and intensity. To calculate seasonal indices by ratio to moving average method steps are: This method is similar to the ratio to trend method. There is only one difference between these two methods. Trend value can be found from original data in ratio to trend method whereas centered moving average value can be found from original data in ratio to moving average method. origial value ratio to moving average = moving average 100. Further step are same as the ratio to trend method after moving averages. Example 3: Sales of refrigerator (in thousand) from year 2009 to 2011 are shown in Table 6, compute seasonal indices by using ratio to moving average method (Hypothetical data). year 1st quar 2nd quar 3rd quar 4th quar 2009 35 34 40 29 2010 21 25 26 28 2011 30 35 36 22 Table 6 Solution: To compute seasonal indices by ratio to moving average method for the given example of sales of refrigerator, first calculate moving average and then compute ratio to moving average. Calculate moving average: Here, we are calculating moving average for even period i.e. 4. Add first four values and place it middle of the 2 and 3 er for year 2009. Next four, place it middle of 3 or 4 and so on until all observations are covered. To do centering of these values, add first two values and place it middle of these two values. Suppose we add 138 and 124, place it in front of 3 and similarly other also can find. Now, take average of these 2 -period moving Total by dividing 2. 6 th column of the following table shows 4-er moving average. 11
Year y t 4-quatrely moving totals 2 period moving totals 4- er moving average ratio to moving average 1 35 2 34 138 2009 3 40 262 32.75 122.1374046 124 4 29 239 29.875 97.07112971 115 1 21 216 27 77.77777778 101 2 25 201 25.125 99.50248756 100 2010 3 26 209 26.125 99.5215311 109 4 28 228 28.5 98.24561404 119 1 30 248 31 96.77419355 129 2 35 252 31.5 111.1111111 2011 123 3 36 4 22 12
Table 7 Ratio to moving average Ratio to moving average can be calculated by given formula for ratio to moving average in the method. ratio to moving average for 3rd er for year 2009 = (40/32.75) 100 = 122.137. Similarly, one can find ratio to moving average values for other years. These values of ratio to moving average values are also known as seasonal indices. Sum of average seasonal indices is 401.06, which is greater than 400. There is need for adjustment in seasonal indices. To do adjustment in seasonal indices, multiply average seasonal indices by correction factor k. K = 400/401.06 = 0.9973 For 1 st er, adjusted seasonal indices = 87.2755 0.9973 = 87.044 Similarly, one can find other adjusted seasonal indices. Year 1st 2nd 3rd 4th Total 2009 - - 122.1374 97.07113 2010 77.777 99.502 99.521 98.245 2011 96.77419 111.111 Total 174.55119 210.613 221.6584 195.31613 average(s.i) 87.275595 105.3065 110.8292 97.658065 401.06936 adjusted S.I. 87.0443147 105.027438 110.535503 97.3943882 400.001643 Table 8 From above table, one can see that sum of adjusted seasonal indices is 400. These are the seasonal indices for the given example. Merits and demerits: 13
Ratio to moving average method is the very flexible, easiest and widely used method to measure the seasonal variation because it removes trend and cyclic component from the indices. One drawback of this method is that there is some loss of information at the starting of this method. 3.4 Link relative method: Link relative method is based on the averaging the link relatives. Link relatives is a percentage value of one season with respect to previous season. Here season means time interval. link relative for any season = current season value previous month value 100. Steps included in this method: Calculate link relatives of the original data. Find the average of the link relatives for each year and every er/month. Convert the average link relatives into the chain relatives on the base of the preceding season. (L. R. for that season) (C. R. of preceding season) chain relatives for any season = 100 Total of chain relatives are not equal to 400 for ers and 1200 for months then there is a need for correction. This correction will be done by subtracting correction factor to all chain relatives. new C. R. for any season = d = L. R. for season value C. R. for last season 100 new C. R. for any season 100 number of season Note: number of season for months is 12, for ers is 4. 14
Example 4: Prices of a commodity from year 2001 to 2004 are shown in Table 9, compute seasonal indices by using Link relative method. year 2001 2002 2003 2004 1st quar 20 12 23 24 2nd quar 22 19 19 12 3rd quar 28 16 16 18 4th quar 23 10 13 15 Table 9 Solution: To compute seasonal indices by using link relative method, steps are: To compute link relatives for values given in Table 9 for any season i.e. er. Starting with first value, one cannot compute link relative for first observation because there is no value available preceding this value. So start with second value to compute link relative, then using 1 st er value with respect to 2002 link relative = (12/20) 100 = 60, then using 1 st er value with respect to 2003, link relative = (23/12) 100 = 191.6666. Similarly, one can find link relatives for other values till all values are covered. After finding link relatives for all values, take erly average of link relatives. These values are denoted as average link relative. Now, convert average link relative into chain relatives by using the formula given in the theory for calculating chain relatives for any season. First chain relative is always 100; second relative for second er = (85.2970 100)/100 = 85.2970; third chain relative for 3 rd er = (125.744 85.29705)/100 = 107.255; fourth chain relative for 4 th er = (104.160 107.255)/100 = 112.717. 15
Add all chain relatives, total must be 400. As the total of chain relatives is coming out to be 404.271 from Table 10. There is a need for adjustment. This adjustment can be done by subtracting correction factor d to all chain relatives. Then new C.R. for any season = 118.67 111.717/100 = 132.576 Correction factor d = (132.576-100)/4 = 8.144. Adjusted chain relative for first er is 100; adjusted chain relatives for second er = 85.29705-8.144 =77.15305; for third er= 107.255-2 8.144 = 90.9679; for Fourth er= 111.7179866-3 8.144 = 87.2859. Take average for all adjusted chain relatives, to compute seasonal indices. Average of adjusted chain relatives = (100+77.15305+90.967+87.2859)/4 = 88.8514 Seasonal indices can be calculated by using formula of seasonal indices. Seasonal indices for first er = (adjusted chain relatives/average of adjusted chain relatives 100 = (100/88.8514) 100 = 112.547; for 2 nd er = (77.153/88.8514) 100= 86.8334. Similarly, one can find for other 3 rd and 4 th er. Values link relatives year 1st 2nd 3rd 4th 1st 2nd 3rd 4th Total 2001 20 22 28 23-91.66666667 233.3333333 127.7777778 2002 12 19 16 10 60 86.36363636 57.14285714 43.47826087 2003 23 19 16 13 191.6666667 100 100 130 2004 24 12 18 15 104.3478261 63.15789474 112.5 115.3846154 Total 356.0144928 341.1881978 502.9761905 416.640654 Average 118.6714976 85.29704944 125.7440476 104.1601635 C.R. 100 85.29705 107.2559632 111.7179866 404.271 adjusted C.R. 100 77.15305 90.96796317 87.28598661 88.8517499 S.I. 112.5470236 86.83346141 102.381735 98.23777997 400 Table 10 16
From Table 10, sum of seasonal indices evaluates as 400. Merits and demerits: Link relatives are more complicated as compared to ratio to moving average method but these two methods are based on the same assumption and also have same results. The major benefit of this method over the ratio to moving average is due to less loss of information as compared to ratio to moving average method. This is the main reason that ratio to moving average method is widely used method in practice. 4 De-seasonalization: De-seasonalization is mostly used to remove the effect of seasonality in the study variable. It is also helpful for the interpretation of the data. Assuming multiplicative model, de-seasonalization can be done by divide trend value tothe seasonal value. deseasonlisation for multiplicative model deseasonlisation for additive model y T x C x S x I = = T x C x I S S Y S = T + C + I. Hence one can leave with trend, cyclic and irregular component in the data. 5 Measurement of cyclic variation: Cyclic variation exists in the data when tendency of the data increases and decreases in a given period but time period is not fixed for cyclic variation. Residual method is most commonly used for measuring the cyclic variation. For measurement of cycle variation first calculate seasonal and trend components then remove seasonal, trend component and irregular component. Irregular component is just like an error term like the previous knowledge which cannot be directly eliminated. To eliminate irregular component moving average method is used. This method of elimination of the irregular component is known as smoothing of irregular component. Steps for computation of cyclic variation are: First estimate trend (T) and seasonal values (S) of the given time series. a) Divide time series values (Y) by trend (T) and seasonal estimated value (S), get cyclic (C) and random component (R). 17
Y T S = TCSI T S = CI b) Now eliminate random component from second step by using moving average of 3 or 5months period and get cyclic component. There are some other methods that are available in the literature.some of them are: 1) References of cyclic analysis method. 2) Direct percentage variation method. 3) Fitting of sine function method or harmonic analysis. One can refer to the text in the references to know about them. In this module, we try our best to give you an understanding about the methods and techniques that are used to eliminate seasonal variation from the data. 6 Measurement of irregular component: Irregular component is the last component, also known as error term of the time series. An error term can't be eliminate fully from any time series because this happens due to natural forces. There are no methods available to measure this component in literature. But this component can be removed little bit by averaging of the indices. If there is multiplicative model of time series then it can be removed by dividing all other components to the irregular component. If there is additivemodel then it can be removed by subtracting all components to their regular component. 7 Summary In this module, the measurement of seasonal, cyclic and irregular variation are discussed. These types of variation are discussed in this module with examples and merits / demerits of various measurement methods in detail. 8 Suggested Readings Gupta, S. C. and Kapoor, V. K., Fundamentals of Applied Statistics, Sultan Chand & Sons, New Delhi, 2009. Gupta, S. P., Statistical Methods, Sultan Chand & Sons, New Delhi, 2012. 18
Gupta, S.C., Fundamental of Statistics, Himalaya Publishing House, Nagpur, 2016. Sharma, J. K., Business Statistics, Vikas Publishing House, 2014. Tsay, R. S., Time Series and Forecasting: Brief History and Future Research, Journal of the American Statistical Association, Vol.95, pp. 638-643, 2000. 19