A set of ordered observations of a quantitative variable taken at successive points in time is known as “Time Series”. In other words, arrangement of statistical data in chronological order, i.e. in accordance with occurrence of time, is known as ‘Time series’. Time could be in terms of years, months, days or hours, is simply a device that enables one to relate all phenomenon to a set of common, stable reference points.

“A time series may be defined as a collection of readings belonging to different time periods, of some economic variable or composite of variables.”

Objective of the Analysis of Time Series

The analysis of time series has developed in the main as a result of investigations into the nature and causes of those fluctuations in economic activity called trend cycles. Analysis of time series has attempted to test the plausibility or otherwise of these theories. At the same time, such analysis may suggest new hypothesis for statistical theories to work on. A large number of individual time series reveal cyclical fluctuations, and the relationship between these series can be investigated.

An analysis of time series may enable them to set up a model showing how the economy works and indicating the main forces which determine whether we have boom or depression. Thus one of the objects of the analysis of time series is to forecast how these series will behave in future on the basis of how they behaved in the past.

Mathematical Relationship

It is defined by the functional relationship



Components of Time Series

The various forces at work, affecting the values of a phenomenon in a time series, can be broadly classified into the following four categories, commonly known as the components of a time series:

  1. Secular trend or long term trend
  2. Periodic changes or short term fluctuations

      2.1 Seasonal variations
      2.2 Cyclic variations

      3. Random or irregular movements

Main Problems in the Analysis of Time Series

The main problems in the analysis of time series are:

  1. To identify the forces or components at work, the net effect of whose interaction is exhibited by the movement of a time series, and
  2. To isolate, study, analyse and measure them independently, i.e., by holding other things constant.

Basic Assumptions in the Analysis of Time Series

The immediate objective of the analysis of time series is to break down the series into the main components which reflect the secular trend, the periodic movements and the erratic movements. The method of analysis depends very largely on the hypothesis as to how components of the series are combined and interact. The simplest hypothesis is to assume that the separate influences have values which are additive and independent of each other. The latter assumption means that the seasonal influence will be the same irrespective of which phase of the cycle obtains. Thus we have



Measurement of Trend

Trend can be measured by following methods:

  1. Graphic method
  2. Method of semi-averages
  3. Method of curve fitting by Principle of least squares
  4. Method of moving averages

Graphical Method

A free hand smooth curve obtained on plotting the values zt against t enables us to form an idea about the general ‘trend’ of the series. Smoothing of the curve eliminates other components, regular and irregular fluctuations. This method does not involve any complex mathematical techniques and can be used to describe all type of trend, linear and non-linear.

Method of Semi-Averages

The whole data is divided into two parts with respect to time, e.g. if we are given zt for t from 1881-1992, i.e. over a period of 12 years, the two equal parts will be the data from 1881 to 1886 and 1887 to 1992. In case of odd number of years the two parts are obtained by omitting the value corresponding to the middle year, e.g. for the data from 1881 to 1992, the value corresponding to middle year 1886 being omitted. Next we compute the arithmetic mean for each part and plot these two averages against the mid values of the respective time periods covered by each part. The line obtained on joining these two points is the required trend line and may be extended both ways to estimate intermediate or future values.

Principle of Least Squares

It is the most popular and widely used method of fitting mathematical functions to a given set of data. The method yields very correct results if sufficiently good appraisal of the form of the function to the fitted is obtained either by a scrutiny of the graphical plot of the values over time or by a theoretical understanding of the mechanism of the variable change. The various types of curves that may be used in describing data are

  • A straight line: zt = a + bt
  • Second degree parabola: zt = a + bt + ct2
  • Kth-degree polynomial: zt = a0 + a1t + a2t2 + ….. + aktk
  • Exponential curves: zt = abt
  • Second degree curve fitted to logarithms: zt = abtct^2    
  • Growth curves: zt = a + bc^t

Time Series in Correlation and Regression

Very often the only data available are in the form of time series, and special care is needed in correlating data which are in this form. It may happen that two variables exhibit a high degree of correlation. Over time not because they are related in any way but because other factors have produced persistent trends causing both series to rise together or the one to rise and the other to fall steadily.

The removal of trend from a series is quite straightforward. It we regard the variable as being compounded as follows


Similarly if we are interested in the regression of a dependent variable Z on an independent variable Y, where the data are in the form of time series, the regression should be performed in terms of trend free data, so that it will be


Measurement of Seasonal Variations

When data are expressed annually there is no seasonal variation. but monthly or quarterly data frequently exhibit strong seasonal movements and considerable interest attaches to devising a pattern of average seasonal variation. It may be desired to compare the seasonal patterns of different series, but more often we may want to know the extent to which we should discount the most recently available statistics for seasonal factors.

Example of Measurement of Seasonal Variations

Average earnings in Australia were $143.60 per week in the March quarter 1974-75 and then rose to $156.30 in the June quarter. Was this due to an underlying upward tendency or simply because the June quarter is usually seasonally higher than the March quarter? If we knew how much the June quarter is usually above or below the March quarter for seasonal reasons.

Uses of Time Series

The time series analysis is of greater importance not only to businessman or an economist but also to people working in various disciplines in natural, social and physical sciences. Some of its uses are enumerated below:

  • It enables us to study the past behavior of the phenomenon under consideration, i.e., to determine the type and nature of the variations in the data.

The segregation and study of the various components is of paramount importance to a businessman in the planning of future operations and in the formulation of executive and policy decision.

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