31 Measurement of Seasonal, Cyclic and Irregular Variation

Dr. Harmanpreet Singh Kapoor

Module 37: Measurement of Seasonal, Cyclic and Irregular Variation

  • Learning Objectives
  • Introduction.
  • Measurement of Seasonal Variation
  • De-seasonalization
  • Measurement of Cyclic Variation
  • Measurement of Irregular Component
  • Summary
  • Suggested Readings

 

  1. Learning Objectives

 

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.

 

  1. 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, quarterly etc.

 

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.

 

  1. 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 quarterly/ monthly/ yearly manner according to the requirement of study variable or availability of the data.

Calculate the average values of observations for all quarters / months/ years over different years. Now calculate the quarterly/ monthly/ yearly (seasonal) average values. To find out average of quarterly/ monthly values, divide total of quarterly/monthly values by 4 / 12 to get seasonal average values respectively.

 

 

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 quarterly 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 quarters/months separately.
  • Now the total of seasonal indices should be 1200 for months and 400 for quarters. If this is not the case then there is need to do some modification in the seasonal indices by multiplying it with

 

 

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 quarter 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 1st quarter, adjusted seasonal indices = 87.2755 × 0.9973 = 87.044

Similarly, one can find other adjusted seasonal indices

 

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:

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.

 

 

 

 

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. quarter. 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 1st quarter value with respect to 2002 link relative = (12/20)×100 = 60, then using 1st quarter 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 quarterly 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 quarter = (85.2970×100)/100 =

85.2970; third chain relative for 3rd quarter = (125.744×85.29705)/100 = 107.255; fourth chain relative for 4th quarter = (104.160×107.255)/100 = 112.717.

 

 

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 to all chain relatives.

 

Then new C.R. for any season = 118.67× 111.717/100 = 132.576 Correction factor = (132.576-100)/4 = 8.144.

 

Adjusted chain relative for first quarter is 100; adjusted chain relatives for second quarter = 85.29705-8.144 =77.15305; for third quarter= 107.255-2×8.144 = 90.9679; for Fourth quarter= 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  quarter  =  (adjusted  chain  relatives/average  of  adjusted  chain relatives×100 = (100/88.8514) × 100 = 112.547; for 2nd quarter = (77.153/88.8514)×100= 86.8334. Similarly, one can find for other 3rd and 4th quarter.

 

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.

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.

  1. a) Divide time series values (Y) by trend (T) and seasonal estimated value (S), get cyclic (C) and random component (R).
  1. 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.
  • 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.