TIME SERIES ANALYSIS

INTRODUCTION

when a nation, state, an institution or a business unit etc., intend to study the behaviour of some element, such as price of a product, exports of a product, investment, sales, profit etc., as they have behaved over a period of time, the information shall have to be collected for a fairly long period, usually at equal time intervals. Thus, a set of any quantitative data collected and arranged on the basis of time is called ‘Time Series’. Depending on the research objective, the unit of time may be a decade, a year, a month, or a week etc. Typical time series are the sales of  a firm in successive years, monthly production figures of a cement mill, daily closing price of shares in Bombay stock market, hourly temperature of a patient.

Usually, the quantitative data of the variable under study are denoted by y1, y2, ...yn and the corresponding time units are denoted by t1, t2......tn. The variable ‘y’ shall have variations, as you will see ups and downs in the values. These changes account for the behaviour of that variable.

Instantly it comes to our mind that ‘time’ is responsible for these changes, but this is not true. Because, the time (t) is not the cause and the changes in the variable (y) are not the effect. The only fact, therefore, which we must understand is that there are a number of causes which affect the variable and have operated on it during a given time period. Hence, time becomes only the basis for data analysis.

Forecasting any event helps in the process of decision making. Forecasting is possible if we are able to understand the past behaviour of that particular activity. For understanding the past behaviour, a researcher needs not only the past data but also a detailed analysis of the same. Thus, in this unit we will discuss the need for analysis of time series, fluctuations of time series which account for changes in the series over a period of time, and measurement of trend for forecasting.

DEFINITION AND UTILITY OF TIME SERIES ANALYSIS

“A time series consists of statistical data which are collected, recorded over successive increments”. “When quantitative data are arranged in the order of their occurrence, the resulting statistical series is called a time series”. The analysis of time series is of great utility not only to research workers but also to economists, businessmen and scientists etc., for the following reasons: 

1) It helps in understanding past behaviour of the variables under study. 

2) It facilitates in forcasting the future behaviour with the help of the changes that have taken place in the past. 

3) It helps in planning future course of action. 

4) It helps in knowing current accomplishment. 

5) It is helpful to make comparisons between different time series and significant conclusions drawn therefrom. Thus we can say that the need for time series analysis arises in research because:

• we want to understand the behaviour of the variables under study, 
• we want to know the expected quantitative changes in the variable under study, and 
• we want to estimate the effect of various causes in quantitative terms. 

In a nutshell, the time series analysis is not only useful for researchers, business research institutions, but also for Governments for devising appropriate future growth strategies.

COMPONENTS OF TIME SERIES

If you are informed that the price of one kilogram sunflower oil was Rs. 0.50 in the year 1940 and in the year 1980 it was Rs. 30 and in the year 2004 it is reported to be Rs. 70, and if you are asked this question: shall sunflower oil be sold again in the future for either Rs. 0.50 or Rs. 30 per kg? Surely, your answer would be ‘No’.

Another question: Shall sunflower oil be sold again in future for Rs. 60 per kg? No doubt, your answer would be ‘Yes’. Have you ever thought about how you answered the above two questions? Probably you have not! The analysis of these answers shall lead us to arrive at the following observations:

– There are several causes which affect the variable gradually and permanently. Therefore we are prompted to answer ‘No’ for the first question

– There are several causes which affect the variable for the time being only. For this reason we are prompted to answer ‘Yes’ for the second question.

The causes which affect the variable gradually and permanently are termed as “Long-Term Causes”. The examples of such causes are: increase in the rate of capital formation, technological innovations, the introduction of automation, changes in productivity, improved marketing etc. The effect of long term causes is reflected in the tendency of a behaviour, to move in an upward or downward direction, termed as ‘Trend’ or ‘Secular Trend’. It reveals as to how the time series has behaved over the period under study.

The causes which affect the variables for the time being only are labelled as “Short-Term Causes”. The short term causes are further divided into two parts, they are 

‘Regular’ and ‘Irregular’. 

Regular causes are further divided into two parts, namely 

‘cyclical causes’ and 

‘seasonal causes’. 

The cyclical variations are also termed as business cycle fluctuations, as they influence the variable. A business cycle is composed of prosperity, recession, depression and recovery. The periodic movements from prosperity to recovery and back again to prosperity vary both in time and intensity. The seasonal causes, like weather conditions, business climate and even local customs and ceremonies together play an important role in giving rise to seasonal movements to almost all the business activities. For instance, the yearly weather conditions directly affect agricultural production and marketing.

It is worthwhile to say that the seasonal variations analysis will be possible only if the season-wise data are available. This fact must be checked first. For analysing the seasonal effects various methods are available. Among them seasonal index by ‘Ratio to Moving Average Method’ is the most widely used. However, if collected data provides only yearly values, there is no possibility of obtaining seasonal variations. Therefore, the residual amount after eliminating trend will be the effect of irregular or random causes.

Irregular causes are also termed as ‘Erratic’ or ‘Random’ causes. Random variations are caused by infrequent occurrences such as wars, strikes, earthquakes, floods etc. These reasons either go very deep downwards or very high upwards.

The foregoing paragraphs have, in a way, led us to enumerate the components of the time series. These components form the basis for ‘Time Series Analysis’.

Long-term causes : Secular Trend or Trend (T) 

Short-term causes :   Regular : Cyclical (C) : Seasonal (S)

Irregular or Random : Erratic (I)

 DECOMPOSITION OF TIME SERIES

Decomposition and analysis of a time series are one and the same thing. The original data or observed data ‘O’ is the result of the effects generated by the long-term and short-term causes, namely, (1) Trend = T, (2) cyclical = C, (3) seasonal = S, and (4) Irregular = I. Finding out the values for each of the components is called decomposition of a time series. Decomposition is done either by the Additive model or the Multiplicative model of analysis. Which of these two models is to be used in analysis of time series depends on the assumption that we might make about the nature and relationship among the four components.

Additive Model: It is based on the assumption that the four components are independent of one another. Under this assumption, the pattern of occurrence and the magnitude of movements in any particular component are not affected by the other components. In this model the values of the four components are expressed in the original units of measurement. Thus, the original data or observed data, ‘Y’ is the total of the four component values, that is,

Y = T + S + C + I

where, T, S, C and I represent the trend variations, seasonal variations cyclical variations, and erratic variations, respectively.

Multiplicative Model: It is based on the assumption that the causes giving rise to the four components are interdependent. Thus, the original data or observed data ‘Y’ is the product of four component values, that is :

Y = T × S × C × I

In this model the values of all the components, except trend values, are expressed as percentages.

In business research, normally, the multiplicative model is more suited and used more frequently for the purpose of analysis of time series. Because, the data related to business and economic time series is the result of interaction of a number of factors which individually cannot be held responsible for generating any specific type of variations.

According to multiplicative model

Y = T × S × C × I 

Thus, 79 (1 year and 1 quarter) = 100 × 82/100 ×120/100

130 (2 year and 1 quarter) = 100 × 120/100 ×108/100

Thus each quarterly figure (Y) is the product of the T, S, and CI. Such a synthetic composition looks like an actual time series and has encouraged use of the model as the basis for the analysis of time series data.

PRELIMINARY ADJUSTMENTS

Before we proceed with the task of analysing a time series data, it is necessary to do relevant adjustments in the raw data. They are:

1) Calender variations: As we are aware, all the calender months do not have the same number of days. For instance, the production in the month of February may be less than other months because of fewer days and if we take the holidays into account the variation is greater. Therefore, adjustments for calender variations have to be made.

2) Price changes: As price level changes are inevitable, it is necessay to convert monetary values into real values after taking into consideration the price indices. 

3) Population changes: Population grows constantly. This also calls for adjustment in the data for the population changes. In such cases, if necessary, per capita values may be computed (dividing original figures by the total population).

 METHODS OF MEASUREMENT OF TREND

The effect of long-term causes is seen in the trend values we compute. A trend is also known as ‘secular trend’ or ‘long-term trend’ as well. There are several methods of isolating the trend of which we shall discuss only two methods which are most frequently used in the business and economic time series data analysis. They are: Free Hand Method, and Method of Least Square.

1 Free Hand Method

In this method, the first requirement is that a graph is drawn of the original data. After plotting the data on the graph paper, without the help of any numerical calculations, a free hand straight line is drawn through the graph ensuring that it passes (as closely as possible) through the middle of the entire graph of the time series. This is, thus, the easiest and quickest method of estimating secular trend. Even though the straight line is drawn on personal judgments, it requires a careful inspection of the overall behaviour of movements in that time series graph.

Though this method is very simple, it does not have a common acceptance because it gives varying trend values for the same data when efforts are made by different persons or even by the same persons at different times. It is to be noted that free-hand method is highly subjective and therefore, different researchers may draw different trend lines from the same data set. Hence, it is not advisable to use it as a basis for forecasting, particularly, when the time series is subject to very irregular movements. Let us consider an illustration to draw a trend line by free-hand method.

2 Least Square Method

This is also known as straight line method. This method is most commonly used in research to estimate the trend of time series data, as it is mathematically designed to satisfy two conditions. They are:

Illustration 2

The decision making body of a fertilizer firm producing fertilizers wants to predict future sales trend for the years 2006 and 2008 based on the analysis of its past sales pattern. The sales of the firm for the last 7 years, for this purpose, are given below:

Years      Sales (in ’000 tonnes) 

1998        70 

1999        75 

2000        90 

2001        98 

2002        85

 2003       91 

2004       100

Solution: To find the straight line equation (Yc = a + bx) for the given time series data, we have to substitute the values of already arrived expression, that is:

LET US SUM UP

This unit has introduced you to the concept of time series and its analysis with a view to making more accurate and reliable forcasts for the future.

A set of quantitative data arranged on the basis of TIME are referred to as ‘Time Series’. The analysis of time series is done to understand the dynamic conditions for achieving the short-term and long-term goals of institution(s). With the help of the techniques of time series analysis the future pattern can be predicted on the basis of past trends.

The quantitative values of the variable under study are denoted by y1, y2, y3...... and the corresponding time units are denoted as x1, x2, x3...... . The variable ‘y’ shall have variations, you will see ups and downs in the values. There are a number of causes during a given time period which affect the variable. Therefore, time becomes the basis of analysis. Time is not the cause and the changes in the values of the variable are not the effect.

The causes which affect the variable gradually and permanently are termed as Long-term causes. The causes which affect the variable only for the time being are termed as Short-term causes. The time series are usually the result of the effects of one or more of the four components. These are trend variations (T), seasonal variations (S), Cyclical variations (C) and Irregular variations (I).

When we try to analyse the time series, we try to isolate and measure the effects of various kinds of these components on a series.

We have two models for analysing time series:

1) Additive model, which considers the sum of various components resulting in the given values of the overall time series data and symbolically it would be expressed as: Y = T + C + S + I.

2) The multiplicative model assumes that the various components interact in a multiplicative manner to produce the given values of the overall time series data and symbolically it would be expressed as : y = T × C × S × I.

The trend analysis brings out the effect of long-term causes. There are different methods of isolating trends, among these we have discussed only two methods which are usually used in research work, i.e. free hand and least square methods.

Long-term predictions can be made on the basis of trends, and only the least square method of trend computation offers this possibility.

KEY WORDS

Time Series : is the data on any variable accumulated at regular time intervals. 

Secular Trend : A type of variation in a time series, the long-term tendency of a time series to grow or decline over a period of time. 

Seasonal Variation : Patterns of change in a time series within a year and the same changes tend to be repeated from year to year. 

Cyclical Variations : A type of variation in a time series, in which the values of variables vary up and down around the secular trend line. 

Irregular Variations : A type of element of a time series, refers to such variations in business activity which do not repeat according to a definite pattern and the values of variables are completely unpredictable.