30 Measurement of Secular Trend

Dr. Harmanpreet Singh Kapoor

Module 36: Measurement of Secular Trend

 

  • Learning Objectives
  • Introduction.
  • Measurement of Trend
  • Summary
  • Suggested Readings

 

  1. Learning Objectives

 

This module is in the continuation of the module “Introduction to Time Series Analysis and its Importance”. In this module, we will discuss how to measure secular trend in the data using different methods. This module has a great importance in the study of time series.

 

  1. Introduction

 

Secular trend or simply trend is considered as the tendency of the observations. This tendency may be linear or non-linear and upward and downward type. There is a need to measure this component, so that one can remove the effect of factors from the time series that are responsible for this. For example, increase/decrease in the price of the commodity, shares, gold due to economy boom or decline help in understand the relation between factors and also in future prediction. So that one can forecast the values of the study variable in more exact manner through modelling as well as study of its characteristics.

 

Hence our main concern in this module is to give you a brief understanding about the measure of trend and try to make it as simple as possible but informative for non-mathematical background students.

 

In literature, there are many methods available for measurements of secular trend. Some of them are method of curve fitting by least square method and moving average method. These methods will be discussed one by one in this module along with their merits and demerits.

 

  1. Measurement of Trend

 

These are among the most important methods for Measurements of Trend.

 

1) Graphic Method

2) Method of semi averages

3) Method of curve fitting by principle of least square

4) Method of moving averages

 

We will now discuss these methods of measurement of trend in detail.

 

3.1)   Graphical method:

 

Graphical method, also known as ‘free hand curve fitting’, is the most simplest and flexible method for measurement trend component. This method is based on a free hand moving curve.

 

This smooth hand curve are plotted on the values of study variable value with respect to time interval of any time series. To draw smooth hand curve, first consider study variable value on the y- axis and time on the x-axis.

 

Now, plot each value of study variable against time value on the axis. After plotting, draw a curve in such a way that it presents the tendency of the original observations. Also remember that the curve should remain smooth between the values of the study variable. This free hand smooth curve eliminates the effect of other components of time series such as cyclic, seasonal, and irregular. This method can be used for all types of trend, linear and non-linear.

 

To obtain an appropriate trend line by using free hand curve method, one must follow some instructions.

 

These are:

 

  1. It should be smooth.
  2. Sum of square of the vertical deviations of the given points of the trend must be minimum as much as possible. It can be possible when sum of the vertical deviation of the given points above trend line is equal to the sum of the vertical deviation of the given points below trend line.
  • If the cycles are present in the trend line then number of cycle must be equal to both above and below the trend line, area of the each cycles must be almost same.

 

Merits:

 

a) It is simple and flexible method.

b) It does not need any mathematical calculation.

c) It is a time saving method.

 

Demerits:

 

  1. a) This method is subjective. The personal bias is present in the graphical method because different type of investigator handle data in different type according to their simplicity and investigate them. So investigators get different trend line for same type of data sets.
  2. b) To use this method in a proper manner one should be skilled and experienced who can remove this type of biasedness. This is one of the major restriction to use of this method.

 

Example 1:

 

Consider an example of sale of luxury cars in India (hypothetical data):

From the above graph one can see that the trend line lie between the points and area of curve below the line and above the line are the same.We will now discuss the second method of measuring trend that is method of semi average

 

3.2) Method of semi averages:

 

Method of semi averages eliminates some biasedness that prevail in the graphical method. In this method, we divide whole time series into two semi parts and take their average for obtaining trend line. For obtaining required trend line by using this method, one has to follow some steps. These are:

 

Steps:

 

a) First divide the whole time series data into two equal parts with respect to time. Two parts of data can be easily obtained in case of even number of years but in case of odd number of years two parts can be obtained by removing the value of middle year i.e. median.

b) Now compute averages of both two parts. This average value is known as semi- averages.

c) Now plot these semi-averages values against the mid-value of the respective time period covered by each part.

d) Now join these two points and get a trend line. This is the required trend line of the given time series data.

 

Merits:

 

1) It is easy to understand and apply as compared to other method of measuring trend.

2) The main advantage of this method is its objective method that is free from subjective biasedness. Hence it gives same trend values for all users.

3) This technique can be used for predicting both future and past estimates.

 

Demerits:

 

1) This method assumed linear relationship that may not exist in the values of study variable.

2) The predicted values by this method are not precise and reliable that fluctuate with a small change in the values of the study variable.

3) The use of arithmetic mean may also be questionable.

 

 

3.3)       Method of moving average:

 

Method of moving average method is very simple and flexible method for measuring the trend line by means of moving average of successive groups of the time series. This average process remove the fluctuation of the given time series.

 

For finding moving average of successive group of the given time series, first one has to assume constant value m which is known as period or extend of the moving average. Thus in the method moving average we take average of m observation at a time, i.e. we are taking moving average of m period. We start with 1st, 2nd, 3rd value and so on till all observations are covered.

 

Thus for any time series values        at time t where t=1, 2, 3…., the moving average of m periods are:

2nd moving average is               1+  2+⋯+  so on.

 

There are two cases according to the period of moving average:

 

  1. a) When Period (m) is odd: if moving period m is 2 + 1 (say), where k is any integer , moving average value placed against the middle value of the corresponding time interval. For example, if m=7 then 1st moving average placed against the middle period (i.e. 4th), 2nd moving average placed against 11th and so on.

 

Example 3: Fit a trend line for the following data by using moving average method for odd period ( 5 yearly and 7 yearly).

 

Solution:

 

To find trend line by using moving average method for 5-yearly of 7 yearly moving average, take average of first 5 or 7 values respectively and place them 3rd and 4th position as shown in the table respectively, so on.

 

First 5 values M.A. is 131.2. Place it against 3rd position in the 3rd column in the table, next take 5 values and it M.A. is 131.6. Place it against 4th position in the table and so on, till all values are

covered.

 

First 7 values M.A. is 131. Places it against 4th position in the table, next take 7 values. Its M.A. is 130.57. Place it against 5th position in the table and so on , till all values are covered.

  1. b) When period (m) is even: if moving average period m is 2 (say), where k is any integer, moving average value placed against the intermediate value of the corresponding time interval. In this case, moving average does not coincide with a period of the given time series. For placing M.A value against original value centering technique are used. Centering is a technique in which we take average of the two moving average value and placing them in between the corresponding time periods. This corresponding moving average are known as centering moving averages.

 

For example, if m=6 then 1st M.A. placed against between 3rd and 4th time interval; 2nd MA placed against between 9th and 10th time interval and so on. Now obtain central moving averages of 2 periods. These moving averages are placed against 3rd , 4th , 5th and so on time variable values.

 

If we use these values of moving average against given time interval then we get a trend curve. The basic problem to apply this method is to find period of moving average. Here period of moving average indicates the period of oscillatory movements in the series. So this method removes the effect of cyclic effect of the series. Thus the period of moving is equal to or multiple of the period

of the cyclic movements of the given series.

 

Example 3:  Fit a trend line for the following data by using of moving average method for  even perio yearly)

 

Solution:

 

To find trend line by using moving average method for 4-yearly moving average, take average of first 4 values and place it between the middle of two values position as shown in the table respectively. First 4 values M.A. is 3.5. Place it between 1990 and 1991 in the table, next 4 values M.A. is 3.75. Place it between 1991 and 1992 and so on till all the observations are covered.

 

To place M.A. values in front of time values take again 2- point M.A. Starting with two values 3.5 & 3.75, we get 2- point M.A. as 3.625 that is placed against 1991 and so on.

 

Merits:

 

  1. a) This method is very simple and flexible, easy to understand, does not include any typical mathematical calculation and complexity, as compared to other methods.
  2. b) This method removes all subjective characteristics.
  3. c) This method is not affected by adding a new observation in original data.
  4. d) This method is used for measurement of seasonal, cyclic and irregular fluctuations.

 

Demerits:

 

a) The main disadvantage of this method is that this method is not appropriate to provide trend value for all given time series values.

b) There is no functional relationship, hence this method cannot be used for forecasting future values.

c) The selection of period is not an easy task when there is no cyclic fluctuation in the time series. In this case we cannot find accurate period of moving average, cannot eliminate completely oscillatory movements from the given data.

 

3.4)       Method of curve fitting by the principle of least squares:

 

The principle of least square is a mathematical analysis device and most widely used method to fit a trend for a given time series. It is mostly used in cases where time series observation value’s relationship with time is very strong, this relationship shows the tendency of increment and decrement with an increase in the value of time then this method is most appropriate to get a reliable future forecast value. This method is used for all type of trends i.e. linear and non-linear.

 

In this module, we will discuss the fitting of linear, exponential and logistic trends explain by principle of least squares.

  • 1) Fitting of linear trend: consider a straight line curve between the given time series value and time given by the equation:

 

 

 

b) It requires more calculations and it is not suitable for non-mathematical person to apply it to find trend values.

c) The addition of even a single new observation in the given observation can change whole calculation of the method.

d) It has one more disadvantage that how to choose the type of trend curve. Trend curve may be liner, exponential, parabolic and quadratic. Thus selection of curve introduce some biasness.

 

4)   Summary

 

In this module, we discussed how to measure secular trend in the data using different methods. This module has a great importance in the study of time series. In literature, there are many methods available for measurements of secular trend. Some of them are method of curve fitting by least square method and moving average method. These methods are discussed in this module along with their merits and demerits.

 

5)   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.
  • 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.