Time Series Analysis | Business Statistics Notes | B.Com Notes Hons & Non Hons | CBCS Pattern



Meaning of Time Series:

One of the most important tasks of any businessman is to make estimates of future demand of his product so that he can adjust his production according to the future demand. For this purpose, it is necessary to gather information from the past. In this connection one usually deals with statistical data which are collected, observed or recorded at successive intervals of time. Such data are generally referred to as Time series.

In the words of Morris Hamburg,”A time series is a set of statistical observations arranged is chronological order.”

In the words of Wessel & Wellet,” When quantitative data are arranged in the order of their occurrence, the resulting statistical series is called a time series.”

From the above explanation we can say that the time series consists of data arranged chronologically.

Table of Contents

1. Meaning of Time Series

2. Significance of Time Series Analysis

3. Components of Time Series Analysis

4. Different Models of Time Series Analysis

5. Different Methods of Times Series Analysis

i)        Graphic method

ii)      Semi-average method

iii)    Moving average method

iv)    Method of least squares.

6. Seasonal Index method

i)        Method of simple averages

ii)      Ratio to trend method

iii)    Ratio to moving average

iv)    Link relative method.

6. Shifting of base and Deseasonalised Value



Utility of Time Series Analysis

The analysis of Time Series is of great significance not only to the economist and businessman but also to the scientist, geologist, biologist, research worker, etc., for the reasons given below:

a)      It helps in understanding past behaviors: By observing data over a period of time one can easily understanding what changes have taken place in the past, Such analysis will be extremely helpful in producing future behavior.

b)      It helps in planning future operations: Plans for the future cannot be made without forecasting events and relationship they will have. Statistical techniques have been evolved which enable time series to be analyzed in such a way that the influences which have determined the form of that series to be analyzed in such a way that the influences which have determined the form of that series may be ascertained.

c)       It helps in evaluating current accomplishments: The performance can be compared with the expected performance and the cause of variation analyzed. For example, if expected sale for 1995 was 10,000 refrigerators and the actual sale was only 9,000, one can investigate the cause for the shortfall in achievement. Time series analysis will enable us to apply the scientific procedure for such analysis.

d)      It facilitates comparison: Different time series are often compared and important conclusions drawn there from. However, one should not be led to believe that by time series analysis one can foretell with 100percnet accuracy the course of future events.

Components of Time Series Analysis

The four components of time series are: (FACTORS RESPONSIBLE FOR TREND IN TIMES SERIES)

1. Secular trend

2. Seasonal variation

3. Cyclical variation

4. Irregular variation

Secular trend: A time series data may show upward trend or downward trend for a period of years and this may be due to factors like increase in population, change in technological progress, large scale shift in consumer’s demands etc. For example, population increases over a period of time, price increases over a period of years, production of goods on the capital market of the country increases over a period of years. These are the examples of upward trend. The sales of a commodity may decrease over a period of time because of better products coming to the market. This is an example of declining trend or downward trend. The increase or decrease in the movements of a time series is called Secular trend. Examples of Trend or secular trend: Increase in demand of two wheeler, decrease on death rate due to advancement of medical science, increase in food production due to increase in population.

Seasonal variation: Seasonal variations are short-term fluctuation in a time series which occur periodically in a year. This continues to repeat year after year. The major factors that are responsible for the repetitive pattern of seasonal variations are weather conditions and customs of people. More woolen clothes are sold in winter than in the season of summer .Regardless of the trend we can observe that in each year more ice creams are sold in summer and very little in winter season. The sales in the departmental stores are more during festive seasons that in the normal days. Examples of seasonal variation: sale of woolen clothes during winter, decline in ice-cream sales during winter, demand of TV during international games.

Cyclical variations: Cyclical variations are recurrent upward or downward movements in a time series but the period of cycle is greater than a year. Also these variations are not regular as seasonal variation. There are different types of cycles of varying in length and size. The ups and downs in business activities are the effects of cyclical variation. A business cycle showing these oscillatory movements has to pass through four phases-prosperity, recession, depression and recovery. In a business, these four phases are completed by passing one to another in this order. It has four important characteristics: i) Prosperity ii) Decline iii) Depression iv) Improvement. Examples of cyclical variation: Recession, Boom, Depression, Recovery, balancing of demand and supply.

Irregular variation: Irregular variations are fluctuations in time series that are short in duration, erratic in nature and follow no regularity in the occurrence pattern. These variations are also referred to as residual variations since by definition they represent what is left out in a time series after trend, cyclical and seasonal variations. Irregular fluctuations results due to the occurrence of unforeseen events like floods, earthquakes, wars, famines, etc. Examples of irregular variation: Flood, fire, strike, lockout, earthquake, hot wave in winter, rain in desert.


Unit 1: Statistical Data and Descriptive Statistics

a. Nature and Classification of data

b. Measures of Central Tendency

c. Measures of Variation

d. Skewness, Moments and Kurtosis

Unit 2: Probability and Probability Distributions

a. Theory of Probability

b. Probability distributions:Binomial-Poisson-Normal

Unit 3: Simple Correlation and Regression Analysis

a. Correlation Analysis

b. Regression Analysis

Unit 4: Index Numbers

Unit 5: Time Series Analysis

UNIT 6: Sampling Concepts, Sampling Distributions and Estimation


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Different models of time series

Times series model are of two types. One is multiplicative model and other one is additive model.

Multiplicative Model: In Traditional time series analysis, it is ordinarily assumed that there is a multiplicative relationship between the components of time series.    Symbolically, Y=T X S X C X I

                                Where T= Trend

                                S= Seasonal component

                                C= Cyclical component

                                I= Irregular component

                                Y= Result of four components.

Additive Model: Another approach is to treat each observation of a time series as the sum of these four components Symbolically, Y=T + S+ C + I

Different methods of measuring trend and seasonal variation

The following four methods are commonly used for measuring trends:

i)        Graphic method

ii)       Semi-average method

iii)     Moving average method

iv)     Method of least squares.

Again, the following methods are commonly used for measuring seasonal variation:

i)        Method of simple averages

ii)       Ratio to trend method

iii)     Ratio to moving average

iv)     Link relative method.

i) Graphic method: This is the simplest method of studying trend. The procedure of obtaining a straight line trend is:

a) Plot the time series on a Graph.

b) Examine the direction of the trend based on the plotted information.

c) Draw a straight line which shows the direction of the trend.

The trend line thus obtained can be extended to predict future values.


i) This method is simplest method of measuring trend.

ii) This method is very flexible. I can be used regardless of whether the trend is a straight line or curve.


i)  This method is highly subjective because it depends on the personal judgement of the investigator.

ii)  Since this method is subjective in nature it cannot be used for predictions.

ii) Semi-average method: - Under this method, the given data is divided into two parts. After that an average of each part is obtained which gives two points. Each point is plotted at the mid-point of the class interval covered by the respective part and then the two points are joined by a straight line which gives the required trend line.


i) This method is simple to understand as compared to the moving average method and the method of least square.

ii) This is an objective method of measuring trend as everyone who applies this method gets the same result.


i) It is affected by extreme values.

ii) This method assumes straight relationship between the plotted points whether this exist or not.

iii) Method of moving average: - Under this method the average value for a certain time span is secured and this average is taken as the trend value for the unit of time falling at the middle of the period covered in the calculation of the average. While using this method it is necessary to select a period for moving average.


i) This method is simple to understand and apply.

ii) It is particularly effective if the trend of a series is very irregular.

iii) It is a flexible method of measuring trend because all figures are not changed if a few figures are added to the data.


i) Trend values cannot be computed for all years.

ii) No there is no hard and fast rule for selecting the period of moving average.

iii) This method is not appropriate if the trend situation is not linear.

iv) Method of Least Square: This method is most commonly used method of measuring trend. It is a mathematical method and a trend line is fitted to the data in such a manner that the following two conditions are satisfied:-

i) the sum of deviation of the actual values from their respective mean is zero.

ii) the sum of square of the deviations of the actual and compute values is least from this line. That is why this method is called method of least square.

The straight line trend is represented by the equation: Y = a + bx

Where, y = denotes the trend values

                a = represents the intercept on y axis.

                b= represents slope of the trend line.


i) This is a mathematical method of measuring trend.

ii) Trend values can be obtained for all the given time periods in the series.


i) This method is more tedious and time consuming.

ii) This method cannot be used to fit the growth curves.

Measurement of Seasonal Variations

Almost every business show seasonal patterns. If data are expression annually there is no seasonal variation but if data are expressed monthly or quarterly then data show strong seasonal movements. In order to analyse seasonal variation, it is necessary to assume that seasonal pattern is superimposed on a series of values. Before attempting to measure seasonal variation certain preliminary decision like whether weekly, quarterly or monthly indexes are required. This will be decided in the light of the nature of the problem and the type of data available. To obtain a static description of a pattern of seasonal variation it will be desirable to eliminate the effect of trend, cycles and irregular variation from the seasonal data then calculate the seasonal variation in the form of index which is known as seasonal indexes (percent).

The following methods are commonly used for measuring seasonal variation:

1) Method of simple averages: This is the simplest method of obtaining a seasonal index. The following steps are followed while calculating seasonal index:

a) Arrange the unadjusted data by years and months or quarters.

b) Find the monthly or quarterly total.

c) Divided each total by the number of years for which data are given.

d) Obtain an average of monthly averages by dividing the total of monthly averages by 12.

e) Taking the average of monthly averages as 100, compute the percentage of various monthly averages as follows:

Seasonal Index = (Monthly average /Average of monthly average)*100

Main advantage of this method is that is the simplest of all methods of measuring seasonal variation. But it is useful only when there are no definite trend exists. This method assumes that there is no trend component in the series i.e., 0 = CSI.

2) Ratio-to-trend Method: This method of calculating a seasonal index is relatively simple and an improvement over the method of simple average. This method assumes that seasonal variation for a given month is constant fraction of trend. The ratio-to-trend method presumably eliminates the seasonal factor in the following manner:



a) This method is simple to compute and easy to understand.

b) This method is useful when time series data are for a short period.

c) There is no loss of data in this method.


a) This method cannot be used when there is a pronounced cyclical swing in the series, the trend – whether a straight line or a curve – can never follow the actual data as closely as a 12-month moving average does.

3) Ratio to moving average: The ratio to moving average method is the most widely used method of measuring seasonal variations. This method assumes that all the four components of a time series i.e., TSCI are present and, therefore, widely used for measuring seasonal variations. However, the seasonal variations are not completely eliminated if the cycles of these variations are not of regular nature. Further, some information is always lost at ends of the time series.


a) The index obtained by the ratio-to-moving average method ordinarily does not fluctuate so such as the index based on the straight-line trends.

b) This method is more flexible as compared to other method of seasonal index.


a) Seasonal index cannot be calculated for each month for which data are available.

4) Link relative method: Amongst all methods of measuring seasonal variation, link relative method is the most difficult one. This method is based on the assumption that the trend is linear and cyclical variations are of uniform pattern. The link relatives are percentages of the current period (quarter or month) as compared with the previous period. With the computations of the link relatives and their average, the effect of cyclical and the random components are minimized. Further, the trend gets eliminated in the process of adjustment of chain relatives.

Uses of Seasonal Index

a) A seasonal index is employed to adjust original data in order to yield deseasonalised data that permit the study of short-run fluctuations of a series not associated with seasonal variations.

b) A seasonal index may be used for economic forecasting and managerial control. Management usually benefits from examining the seasonal pattern of its own business, patterns that directly influence its employment, production, purchase, sales and inventory policies.

c) Seasonal index helps in diversification. It benefits not only the firm but also society at large.

Limitations of Seasonal Index

a) No technique can measure seasonal variations precisely because various methods are based on unrealistic assumptions.

b) Seasonal Index only shows an average pattern during a number of years which may have no significance for a particular period.

Shifting of trend origin and deseasonalised data

Shifting: Shifting of trend origin means replacing the origin with new base. Shifting can be done by using the following formula: Y = a + b (X + k)

Where k is the number of time units shifted. If the origin is shifted forward in time, k is positive and if shifted backward in time, k is negative.

Deseasonalised Value: The value which shows how things would have been or would be if there were no seasonal fluctuations is called Deseasonalised data. In order to obtain Deseasonalised data, the effect of seasonal variations have to be removed. For this purpose, the actual data is divided by the appropriate seasonal indices.      

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