Maharashtra Board Class 12 Maths Commerce Part II Chapter 4 Time Series PDF Download

Time Series

Let's Study

Uses of time series analysis.

Components of a time series.

Secular Trend

Seasonal Variation

Cyclical Variation

Irregular Variation

Mathematical Models

Additive Model

Multiplicative Model

Measurement of Secular Trend

Graphical Method

Method of Moving Averages

Method of Least Squares

Introduction

A manufacturing company wants to predict demand for its product for next year to make a production plan. An investor wants to know fluctuations in share prices so that he can decide if he should purchase or sell certain shares. These and many other situations involve a variable that changes with time. A variable observed over a period of time is called a time series. Analysis of time series is useful in understanding the patterns of changes in the variable over time. Let us now define a time series.

Definition

Time Series is a sequence of observations made on a variable at regular time intervals over a specified period of time.

Data collected arbitrarily or irregularly does not form a time series. Time series analysis involves the use of statistical methods to analyze time series data in order to extract meaningful statistics and understand important characteristics of the observed data.

Time Series Analysis helps us understand the underlying forces leading to a particular pattern in the time series and helps us in monitoring and forecasting data with help of appropriate statistical models.

Analysis of time series data requires maintaining records of values of the variable over time.

Some examples from day-to-day life may give a better idea of time series.

Monthly, quarterly, or yearly production of an industrial product.

Yearly GDP (Gross Domestic Product) of a country.

Monthly sales in a departmental store.

Weekly prices of vegetables.

Daily closing price of a share at a stock exchange.

Hourly temperature of a city recorded by the Meteorological Department.

4.1 Uses of Time Series Analysis

The main objective of time series analysis is to understand, interpret and assess chronological changes in values of a variable in the past, so that reliable predictions can be made about its future values. For example, the government may be interested in predicting population growth in near future for planning its welfare schemes, the agricultural ministry may be interested in predicting annual crop yield before declaring the MSP (minimum support price) of agricultural produce or an industrialist may be interested in predicting the weekly demand for his product for making the production schedule. Following are considered to be some of the important uses of time series analysis.

It is useful for studying the past behaviour of a variable.

In a time series, the past observations on a variable are arranged in an orderly manner over a period of time. By simple observation of such a series, one can understand the nature of changes that have taken place in values of the variable during the course of time. Further, by applying appropriate technique of analysis to the series, one can study the general tendency of the variable in addition to seasonal changes, cyclical changes, and irregular or accidental changes in values of the variable.

It is useful for forecasting future behaviour of a variable.

Analysis of a time series reveals the nature of changes in the value of a variable during the course of times. This can be useful in forecasting the future values of the variable. Thus, with the help of observations on an appropriate time series, future plans can be made relating to certain matters like purchase, production, sales, etc. This is how a planned economy makes plans for the future development on the basis of time series analysis of the relevant data.

It is useful in evaluating the performance.

Evaluation of the actual performances in comparison with predetermined targets is necessary to judge efficiency of the work. For example, the achievements of Five-Year Plans are evaluated by determining the annual rate of growth in the gross national product. Similarly, the national policy of controlling inflation and price rises is evaluated with the help of different price indices. All these are made possible by analysis of time series of the relevant variables.

It is useful in making a comparative study.

A comparative study of data relating to two or more periods, regions, or industries reveals a lot of valuable information that can guide management in taking a proper course of action. A time series itself provides a scientific basis for making comparisons between two or more related sets of data. Note that data are arranged chronologically in such a series and the effects of its various components are gradually isolated, analyzed, and interpreted.

Teacher's Note

Time series helps us see patterns in real life. For example, ice cream sales go up in summer every year in India.

Exam Trick

Remember: Time series = observations over time. Just like tracking your weight every week, time series tracks any data over time periods.

Points to Remember

Time series has four parts: trend, seasonal, cyclical, and irregular.

Trend means the general direction - up or down over many years.

Seasonal means patterns that repeat every year, like more blankets sold in winter.

Cyclical means changes that happen over many years, like business cycles.

4.2 Components of Time Series

A graphical representation of time series data shows continuous changes in its values over time, giving an impression of fluctuating nature of data. A close look of the graph, however, reveals that the fluctuations are not totally arbitrary, and a part of these fluctuations has a steady behavior and can be related to time. This part is the systematic part of the time series and the remaining part is non systematic or irregular. The systematic part is further divided in the following broad categories: (i) secular trend (T), (ii) seasonal variation (S), and (iii) cyclical variation (C). The non systematic part is also called (iv) irregular variation (I). Every time series has some or all of these components. Of course, only the systematic components of a time series are useful in forecasting its future values.

We now discuss the four components of a time series in detail.

4.2.1 Secular Trend (T)

The secular trend is the long term pattern of a time series. The secular trend can be positive or negative depending on whether the time series exhibits an increasing long term pattern or a decreasing long term pattern. The secular trend shows a smooth and regular long term movement of the time series. The secular trend does not include short term fluctuations but only consists of a steady movement over a long period of time. It is the movement that the series would take if there are no seasonal, cyclical or irregular variations. It is the effect of factors that are more or less constant for a long time or that change very gradually and slowly over time.

If a time series does not show an increasing or decreasing pattern, then the series is stationary around the mean.

The following table shows annual sales (in lakh Rs.) of a departmental store for years 2011 to 2018.

The following graph shows the above time series.

There are ups and downs in the graph, but the time series shows an upward trend in long run.

The following table shows production (in '000 tonnes) of a commodity during years 2001-2008.

The above graph shows a downward trend.

Teacher's Note

Secular trend is the main direction of data over many years. For example, the number of smartphones in India is going up every year - this is an upward trend.

Exam Trick

Remember: Trend = long-term direction. Ignore small up-and-down changes and look at the overall pattern over years.

Points to Remember

Secular trend shows long-term movement, not small changes.

Upward trend means the data is going up over many years.

Downward trend means the data is going down over many years.

4.2.2 Seasonal Variation (S)

Many time series related to financial, economic, and business activities consist of monthly or quarterly data. It is observed very often that these time series exhibit seasonal variation in the sense that similar patterns are repeated from year to year. Seasonal variation is the component of a time series that involves patterns of change within a year that repeat from year to year.

Several commodities show seasonal fluctuations in their demand. Warm clothes and woolen products have a market during the winter season. Fans, coolers, cold drinks and ice creams are in great demand during summer. Umbrellas and raincoats are in great demand during the rainy season. Different festivals are associated with different commodities and every festival season is associated with an increase in demand for related commodities. For example, clothes and firecrackers are in great demand during Diwali. Most of the seasonal variations in demand reflect changes in climatic conditions or customs and habits of people.

All the above examples have one year as the period of seasonal variation. However, the period of seasonal variation can be a month, a week, a day, or even an hour, depending on the nature of available data. For example, cash withdrawals in a bank show seasonal variation among the days of a month, the number of books borrowed by readers from a library show seasonal variation according to days of a week, passenger traffic at a railway station has seasonal variation during hours of a day, and the temperature recorded in a city exhibits seasonal variation over hours of a day, in addition to seasonal variation with changing seasons in a year.

Seasonal variation is measured with help of seasonal indices, which are useful for short term forecasting. Such short term forecasts are useful for a departmental store in planning its inventory according to months of a year. A bank manager can use such short term forecasts in managing cash flow on different days of a week or a month.

The following table shows quarterly sales (in lakh Rs.) of woolen garments in four consecutive years.

Figure shows a pattern that is repeated year after year. The values are lowest (in the year) in second quarter and highest (in the year) in fourth quarter of every year. Although the overall graph of the time series shows an increasing trend, the seasonal variation within every year is very clearly visible in the graph.

Teacher's Note

Seasonal variation happens every year at the same time. For example, in India, ice cream sales are always high in summer and low in winter every single year.

Exam Trick

Remember: Seasonal = same pattern repeats every year. Like how Diwali shopping happens at the same time every year, seasonal variation repeats yearly.

Points to Remember

Seasonal variation repeats every year at the same time.

Winter clothes sell more in winter, cold drinks sell more in summer.

Seasonal patterns help stores plan their inventory and stock.

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