Time Series Analysis

INTRODUCTION

1.1 TIME SERIES AND ITS IMPORTANCE Statistics is a tool of all sciences indispensable to search an intelligent judgment and has become a recognized discipline in its own right. There is hardly any field where statistical tools are not applicable. Time series is a statistical series based on time. Analysis of  time series is a statistical device which can be used to understand, interpret, and evaluate changes in economic phenomena over time. According to Patters on “A time series consists of  statistical data which are collected, recorded, observed over successive increments.”

Wessel and Wellet says that when quantitative data are a re arranged in the order of  their occurrence, the resulting statistical series is called a time series. Time series helps in comparing and planning a process. It will enable us to apply the scientific procedure of  ‘holding other things constant’ as we examine one variable at a time. One should not be led

to believe that by time series series one can foretell with complete accuracy the course of future events. This could be possible only if the influence of the various va rious forces which affect these series such as climate, customs and traditions, growth and decline factors and complex forces which produce business cycle would have been regular in their operations. Time series have a unique important place in the field of Business Statistics The present study of time series and its analysis is divided into seven chapters. The first chapter is the introductory chapter which contains time series and its importance in the day to day life. The second chapter concerned with time series and its analysis analysis which includes components, mathematical models, uses and applications of time series. The third chapter is about the first component i.e., secular trend and its analysis. Various method for measuring trend such as graphic method, method of semi-average, method of curve fitting by principles of least squares and method of moving average are included in this chapter. The fourth chapter is dealing with second component i .e., seasonal movement and it is followed by

measurement of seasonal seasonal variations by different methods methods such as

method of simple average, ratio to trend method, ratio to moving average method and link relative method. In the fifth chapter

we are discussing discussing

the third component i.e., cyclical

variations. Residual method and reference cycle analysis method for measuring cyclical variations are included in this chapter.

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Sixth chapter is regarding with the fourth component, irregular variations and its measurement. Seventh chapter consists of time series analysis on a sample time series data. It includes trend and seasonal analysis of some selected banks in BSE and BANKEX. The study concludes with a time series model for BANKEX using ACF and PACF functions. TIME SERIES

2.1 INTRODUCTION A statistical series based on time is a time series and an analysis of such a series is called time series analysis. A time series is defined as “ a set of observations of a

variable recorded at successive intervals or points of time. ” These are statistical data arranged chronologically, usually in relation to regular time intervals. According to Ya-lun Chou “A time series may be defined as a collection of readings belonging to different time periods, of  some economic variable or composite of variables.”

In a time series there exists two variables - one is the time and the other is the value or quantity which depends on time. Hence time is called independent variable and value or quantity is called dependent variable. Time may be a year, half-year, month, week , day, hour etc and the value or quantity may be production, price, sales, export, import, yield of  crops, temperature, income etc. Mathematically a time series is defined by the functional relationship Y t=f(t). Where Yt is the value of the variable under consideration at time ‘ t ’. Thus if the values of a phenomenon or variable at times t 1 , t2 , t3 ,..., tn are Y1 , Y2 ,Y3 ,...,Yn respectively, then the given series constitute a time series t : t1 , t2 , t3 ,..., tn Y : Y1 , Y2 , Y3 ,..., Yn If the data is segregated by time (days, months, years, etc.) the value v alue of  the variable under consideration changes from time to time. These fluctuations are affected not by a single force but are due to the net effect of multiplicity of forces pulling it up and down and if these forces were in a state of equilibrium then the series would remain constant. The analysis of time series helps in knowing the real behavior of the past and planning the future operations. 2.2 COMPONENTS OF A TIME SERIES The various forces affecting the values of a phenomenon in a time series, series, can be broadly classified into the following four categories, commonly known a s the

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components of a time series, some or all of which are present (in a given time series) in varying degrees. The components of time series are the following. a) Secular trend or Long term trend (T) b) Seasonal movements (S) c)

Cyclical changes (C)

d)

Irregular variations or Random variations (R) The value of the time series Y t at any time t is regarded as the resultant of the combined impact of above components. Although it is a simple matter to classify the factors affecting time series into these four groups for analytical purposes, the actual application of the classification frequently presents serious problems. Seasonal variations are by no means always so uniform in amplitude and timing that their identification can be made with certainty. Consequently, the investigator is often hard put to distinguish seasonal influences from cyclical or random factors. Another difficulty arises because the four components of time series data are not mutually independent of one another. An exceedingly severe seasonal influence may aggravate or even precipitate a change in the cyclical movement. Conversely, cyclical influence may seriously affect the seasonal. A very rapidly rising trend virtually eliminates seasonal and cyclical variations. In practice the cycle itself is so erratic and is so interwoven with irregular movements that it is impossible to separate them. In the analysis of a time series into its component fluctuations, therefore, trend and seasonal movements are usually measured directly, while cyclical and irregular fluctuations are left together after the elements have been removed.

2.3 MATHEMATICAL MODELS FOR TIME SERIES In a time series analysis it is assumed that there exist two models commonly for the decomposition of a time series into its components. a) ADDITIVE MODEL According to the additive model the decomposition of time series is done on the assumption that the effects of various components are additive in nature or in other words, Yt = T + S + C + I Where

Yt is the time series value and T , S , C and I stands for trend, seasonal

variations, cyclical variations and irregular variations respectively. In this model S , C and 3

I are absolute quantities and can have positive or negative values. The model assumes that the four components of the time series are independent of each other and none has any effect on the remaining three components. It means that the trend does not affect S, C or I. It is also assumed that S does not affect C and C does not affect S. In actual practice this hypothesis does not hold good as these factors affect each other. A sharply rising or falling trend may completely wipe out the effects of seasonal or cyclical variations. Similarly powerful seasonal changes may intensify the changes in the cyclical variations. b)

MULTIPLICATIVE MODEL According to the multiplicative model the decomposition of a time series is done on the assumption that the effects of the four components of a time series ( T , S , C and I) are not independent of each other. In facts the model presumes that their effects are interdependent. According to the multiplicative model Yt = T × S × C × I In this model T , S , C and I are not absolute amounts as in case of the Additive model. They are relative variations and are expressed as rate or indices fluctuating above or below unity. It is presumed that the geometric mean of all S , C and I would be unity. The multiplicative model can be expressed in terms of the logs. Thus if we take logs of the multiplicative model, we get log Yt = log T + log S + log C + log I It means that the multiplicative model is the additive model if we take into account the logs of the given time series values. Most of the series in the field of  Economics and Business adhere to the multiplicative model as the effect of various factors affecting such time series are not independent of each other. In addition to the above mentioned additive and multiplicative model there are many types of mixed models like, Yt = T C S + I Yt = T C + C I Yt = T + S C I

2.4 USES OF TIME SERIES The time series analysis is of great importance not only to business man or an economist but also to people working in various disciplines in natural, social and physical sciences. Some of its points are enumerated below.

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a) It enables us to study the past behavior of the phenomenon under consideration i.e., to determine the type and nature of variations in the data. b) 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 decisions . c)

It helps to compare the actual current performance of accomplishments with the expected ones (on the basis of past performances) and analyse the causes of such variations, if any.

d) It enables us to estimate or predict or forecast the behaviour of the phenomenon in future which is very essential for business planning. e) It helps us to compare the changes in the values of different phenomenon at different times or places etc. 2.5 APPLICATION Analysis of time series is a statistical device which can be used to understand, interpret, and evaluate changes in economic phenomena over time. So time series have a unique important place in the field of Economics and Business Statistics since the series relating to prices, consumption and production of various commodities ; money in circulation ; bank deposits and bank clearings ; sales and profits in a departmental store ; agricultural industrial production, national income and foreign exchange reserves, prices and dividends of shares in a stock exchange market, etc., are all time series spread over a long period of time. SECULAR

TREND

The smooth, regular and gradual movement of a time series which shows the growth or decline over a long period of time preferably in years is called the secular trend. Secular trend is also known as long term trend or simply trend. Sudden changes and frequent fluctuations have no plac e in secular trend. The trend movement may be upward or downward. This is true for most of series of Business and Economic Statistics. According to Simpson and Kafka “Trend, also called secular or long term

trend, is the basic tendency of series to gr ow or decline over a period of time.” The concept doesn’t include short term oscillations, but rather the long time. It should not

be inferred that all the series must show an upward or downward trends. We might come across certain series whose values fluctuate round a constant reading which doesn’t change with time.

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If the time series values plotted on graph cluster more or less around a straight line, then the trend exhibited by the time series is termed as linear trend otherwise non linear trend. In a linear trend, time series values increase or decrease more or less by a constant absolute amount, i.e., the rate of growth ( or decli ne ) is constant. In an economic and business phenomenon, the rate of growth or decline is not constant nature throughout but varies considerably in different sectors of time. Usually, in the beginning the growth is slow, then rapid which is further accelerated for quite sometime after which it becomes stationary

or stable for some period and

finally retards slowly. Also it is not necessary that the increase or decline should be in the same direction throughout the given period. It may be possible that different tendencies of increase, decrease or stability are observed in different sections of time. However the overall tendency may be upward, downward or stable. Such tendencies are the result of the forces which are more or less constant for a long time or which change very gradually and continuously over a long period of time such as the change in the population, habits, customs of the people in the society and so on. They operate in an evolutionary manner and don’t reflect sudden changes.

3.1 TREND ANALYSIS In the study of time series analysis, the measurement of trend is very important. It helps us to forecast future results. There are four methods of  measurements Trend can be studied and measured by the following methods. 3.1.1

GRAPHIC METHOD OR TREND BY INSPECTION. In this method the time series data are plotted on a graph paper against

time and then a curve is drawn as usual by joining the points plotted on the graph paper. Then a smooth free hand curve is drawn through the scatter of the plotted points in order to fit the data best and to channelize their pattern of movements over time. The smooth free hand curve thus drawn is known as trend line. Smoothing of the curve eliminates other components such as regular and irregular fluctuations. This trend line can be used to obtain the value at particular time points such as year, month, days etc. Merits

i.

Graphic method is very simple and flexible since it does not involve any complex mathematical techniques and can be used to describe all types of trend, linear and non-linear. 6

Demerits

i.

The model is very subjective, i.e. the bias of the person, handling the data plays a very important role and as such different curves will be obtained by different persons for the same data. As s uch ‘ trend by inspection’ should be attempted only by skilled and experienced statistician and this limits the utility and popularity of the method.

ii.

It does not enable us to measure trend 3.1.2

METHOD OF SEMI-AVERAGES In this method the data of time series are divided into two equal parts. If the

number of observations in the series is even, then the whole series may be divided exactly into two equal parts. If the number of observations is odd then the whole series can be divided into two equal parts by leaving the observation that comes in the middle of the whole series. Then average value of each part is considered to be the representative value of that part and is plotted on a graph paper at the midpoint of the respective period. So two points are obtained on the graph paper and these two points are joined to give the trend line. Merits

i.

This method is re adily comprehensible as compared to the ‘method of least squares’ and ‘method of moving average’.

ii.

As compared with graphic methods, the obvious advantage of this method is its objectivity in the sense that everyone who applies it would get the same results. Demerits

i.

This method assumes linear relationship between the plotted points, which may not exist always.

ii.

In this method trend is affected by higher or extreme values since we use arithmetic mean. This is the limitation of the arithmetic mean on which it is based. It is therefore necessary that to avoid the undue influence of bigger items. However it would be better to if instead of the semi-average method we use the moving average method in which such abnormalities are taken care of. 3.1.3

METHOD OF CURVE FITTING BY PRINCIPLES OF LEAST SQUARES. The principle of least squares 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 be fitted is obtained either by a scrutiny of the graphical plot of the values over time or by theoretical understanding of the mechanism of the variable change. An examination of  7

the plotted data often provides an adequate basis for deciding upon the trend to use. For deciding about the type of trend to be fitted to a given set of data, the following points may be helpful. 

When the time series is found to be increasing or decreasing by absolute amounts, the straight line trend is used



The logarithmic straight line is used as an expression of the secular movement, when the series is increasing or decreasing by a constant percentage rather than a constant absolute amount. In this case, the data plotted on a semi-logarithmic scale will give a straight line graph.



Second degree curve fitted to logarithms may be tried for fitting if the data plotted on a semi-logarithmic scale is not a straight line. The trend

may be

linear or

non-linear. If the

trend is

linear

the graph will show a straight line trend and the trend values may be obtained by applying the equation for straight line. i.e. Yt = a + bt Where

‘Yt’ is the dependent variable, ‘a’ and ‘b’ are two

unknown

constants and ‘t’ is the unit of time. Principles of least squares consist in minimizing

the sum of squares of the deviations between the given values of Y t corresponding to n different values of t, i.e. Z = ∑ ( Yt -a - bt)

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is minimum.

By applying the method of least squares we get two normal equations for estimating ‘a’ and ‘b’. The normal equations are, ∑ Yt = na + b∑t ∑ t Yt = a∑t + b∑t

2

With the values of ‘a’ and ‘b’, the line gives the desir ed trend line. Merits

i.

Because of its mathematical or analytical character, this method completely eliminates the element of subjective judgment or personal bias on the part of the investigator.

ii.

Unlike the method of moving averages this method enables us to compute the trend values for all the given time periods in the series.

iii.

The trend equation can be used to estimate or predict the values of the variable for any period t in future or even in the intermediate periods of the given series.

iv.

The curve fitting by the principle of least squares is the only technique which enables us to obtain the rate of growth per annum, for yearly data, if linear trend is fitted. 8

Demerits

i.

This method is quite tedious and time consuming as compared with other methods. It is rather difficult for a non-mathematical person to understand and use.

ii.

The addition of even a single new observation necessitates all calculations to be done afresh.

iii.

Future predictions based on this method are based only on the long term variations, i.e. trend and completely ignore the cyclical , seasonal , irregular variations.

iv.

The most serious limitation of this method is the determination of the trend curve to be fitted, i.e. whether we should fit a linear or parabolic trend or some other more complicated trend curve.

v.

It cannot be used to fit growth curve like Modified Exponential curve and Logistic curve to which most of the economic and business time series data conform. 3.1.4

METHOD OF MOVING AVERAGE Method of moving average is a simple method used for the measurement

of trend. It consist in measurement of trend by smoothing out the fluctuations of the data by means of a moving average. Moving average of period ‘m’ is a series of  successive averages of ‘m’ terms at a time. Thus the first average is the mean of the first ‘m’ terms,

second is the mean of the ‘m’ terms from second term to ‘m + 1’

terms, and third is the m ean of the ‘m’ terms from third to ‘ m+ 2’ terms and so on. If  ‘m’ is odd =2k+1 say, moving average is placed against the mid -value of the time interval it covers, i.e., against t= k+1 and if ‘m’ is even =2k say, it is placed between

the two middle values of the time interval it covers, i.e., between t = k and t = k+1. In the latter case the moving average does not coincide with an original time period. In this method, the main problem lies in determining the period of the moving average which will completely eliminate the oscillatory movements affecting the series. It has been established mathematically that if the fluctuations are regular and periodic then the moving average completely eliminates the oscillatory movements provided, i ) the period of moving average is exactly equal to (or a multiple of ) the period of  oscillation and ii ) the trend is linear. The

main purpose of moving average is to obtain trend values so that

all types of fluctuations are eliminated or in any case reduced to the minimum. Thus

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the period of moving average is selected in such a way that it will achieve these objectives. Hence the period of moving average is equivalent to the period of cycle. Merits

i.

The method is simple and objective.

ii.

If the moving average period coincides with the period of cycle in the time series, then cyclical variations are considerably reduced and in some cases when the cyclic period is uniform they are completely eliminated.

iii.

This method is used not only for determining trend values but also for measuring seasonal, cyclical and irregular variations. Demerits

i.

Moving average method of finding the trend values does not result in setting up a fundamental relationship between the values of ‘t’ and ‘Y t’. As such method is not

helpful in forecasting, which is the main purpose of time series analysis. ii.

Under this method there is no trend values for some year in the beginning and some in the end.

iii.

In the case of non-linear trend the values obtained by moving average are biased in one direction or other

iv.

The selection of the period of moving average is difficult.

SEASONAL VARIATIONS

The term seasonal variations refers to systematic, though not necessarily regular intrayear movements in economic time series. Short-term fluctuations observed in a time series data particularly in a specified period usually within a year are called seasonal variations. It is the periodic movement of a time series data caused by rhythmic forces which operate in a regular and periodic manner over a span of less than one year. There are many seasons within a year, where the business activities are brisk or slack. The seasonal variation are attributed to the following two causes. i.

Climate and weather conditions -Natural forces such as the various seasons or weather conditions and climatic changes are the most important factor causing seasonal variations. Changes in the weather conditions such as rainfall, temperature, etc act on different products and industries differently.

ii.

Customs, traditions and habits -Though nature is primarily responsible for seasonal variations in time series, customs, traditions and habits of the people in the society also have their impacts. 10

4.1 MEASUREMENT OF SEASONAL VARIATIONS Most of the phenomena in economics and business show seasonal pattern. When data expressed annually there is no seasonal variations. However, monthly or quarterly data

frequently exhibit

strong seasonal

movements. The study

and

measurement of seasonal patterns constitute a very important part of analysis of a time series. To obtain a static description of a pattern of seasonal variations it will be desirable to first free the data from the effects of trend, cycles and irregular variations. Once these components have been

eliminated we can calculate a measure

of seasonal variations which is usually referred to as a ‘seasonal index’. Thus the measures of seasonal variations are called seasonal indexes (percent). Seasonal variation is measured in terms of an index, called a seasonal index. It is an average that can be used to compare an actual observation relative to what it would be if there were no seasonal variation. An index value is attached to each period of the time series within a year. This implies that if

monthly data are

considered there are 12 separate seasonal indices, one for each month. There can also be a further 4 index values for quarterly data. A seasonal index is a number that indicates the seasonality for a given time period. For example, a seasonal index for observed value in a particular month would indicate the way in which the value in that month is affected by the seasonal pattern in the data. Seasonal indices are used in seasonal adjustments. A specific seasonal index refers to the seasonal changes during a particular year. Seasonal patterns are exhibited by most of the business and economic phenomena and their study is necessitated by the following reasons. i.

To isolate the seasonal variations , i.e., to determine the effect of seasons on the size of the variable.

ii.

To eliminate them, i.e., to study as to what would be the value of the variable if there is no seasonal variations. The determination of seasonal effect is of paramount importance in planning i) business efficiency or ii)a production program. Moreover, the isolation and elimination of seasonal factor from the data is necessary to study the effect of cycles. Obviously for the study of seasonal variations the data must be given for parts of  year, i.e., monthly, quarterly, weekly, daily or hourly. For computing seasonal index we can use any method which should be designed to meet the following criteria:

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i.

It should measure only the seasonal forces in the data. It should not be influenced by the forces of trend or cycle that may be present.

ii.

It should modify the erratic fluctuations in the data with an acceptable system of  averaging.

iii.

It should recognize slowly changing seasonal patterns that may be present and modify the index to keep up with those changes. The following are some of the method more popularly used for measuring seasonal variations. 4.1.1

METHOD OF SIMPLE AVERAGES. This is the simplest of all the method of measuring seasonality and consist in

the following steps. i.

Arrange the data by years and months (or quarters if quarterly data is given).

ii.

Compute the average x i ( i=1, 2, …,12) for the i month for all years. [ i

th

th

month, i=1, 2,

…,12 represents January, February, …,December respectivel y.]

iii.

Compute the average x of the monthly averages, i.e., x = * ∑ xi ] /12

iv.

Seasonal indices for different months are obtained by expressing monthly averages as percentages of

x.

Thus seasonal index for i

th

month is

=

[ xi / x ] × 100

If instead of monthly averages, we use monthly totals for the years, the results remains the same. Merit

i.

Method of simple average is the simplest of all methods of measuring seasonality. Demerit

i.

Method of simple average is based on the basic assumpti on that the data don’t contain

any trend

and cyclic

components and consists in

eliminating irregular

components by averaging the monthly (or quarterly ) values over years. Since most of  the economic time series have trends, these assumptions are not in general true and as such these method, though simple, is not of much practical utility.

4.1.2

‘RATIO TO TREND’ METHOD.

Ratio to trend method is also known as percentage to trend method and this method is relatively simple and yet an improvement over the method of simple averages. This method assumes that seasonal variations for a given month is a 12

constant fraction of trend. The ratio to trend method presumably isolates the seasonal factor in the following manner. Trend is eliminated when the ratios are computed, in effect T×S×C×I

=S × C × I

T Random elements are supposed to disappear when the ratios are averaged. A careful selection of the period of years used in the computation is expected to cause the influences of prosperity or depression to offset each other and thus remove cycle. For series that are not subject to cyclical or random influences and for which trend can be computed accurately. Computation of seasonal index by this method involves the following steps. i.

Obtain the trend values by the least square method by fitting a mathematical curve, straight line or second degree polynomial, etc.

ii.

Express the original data as the percentage of the trend values. Assuming the multiplicative model, these percentage will contain the seasonal, cyclical and irregular variations.

iii.

The

cyclical

and

irregular

components

are

then

wiped

out

by

averaging

the

percentages for different months ( quarters ),if the data are monthly ( quarterly ) , thus leaving us with indices of seasonal variations. Either arithmetic mean or median can be used for averaging, but median is preferred to arithmetic mean since the latter gives undue weight age to extreme values which are not primarily due to seasonal swings. If  there are a few abnormal values modified mean ( which consists in calculating arithmetic mean after dropping out

the extreme or abnormal values ) may be used

with advantage. iv.

Finally these indices, obtained in step (iii), are adjusted to a total 1200 for monthly data or 400 for quarterly data by multiplying them by a constant factor ‘k’. Where ‘k’ is given by, k=

1200

, for monthly data.

Total of the indices k =

400

, for quarterly data.

Total of the indices Merits

i.

It is simple to compute and easy to understand.

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ii.

Compared with the method of monthly averages this method is certainly a more logical procedure for measuring seasonal variations.

iii.

For it has a ratio to trend value for each month for which data are available. Thus there is no loss of data as occurs in the case of moving average. It has an advantage over the moving average procedure too. Demerits

i.

The main defect of ratio to trend method is that if there are pronounced cyclical swings in the series , the trend (whether a straight line or a curve ) can never follow the actual data as closely as a twelve month moving average does. In consequence as seasonal index computed by the ratio to moving average method may be less biased than the one calculated by the ratio to trend method.

4.1.3

‘RATIO TO MOVING AVERAGE’ METHOD.

The ratio to moving average method also known as the percentage of moving average method is the most widely used method of measuring seasonal variations. The steps necessary for determining seasonal by this method are: i.

Calculate the centered 12 month moving average of the data. These moving average values will give estimates of the combined effects of trend and cyclical variations.

ii.

Express the original data ( except for 6 months in the beginning and 6 months at the end ) as percentages of the centered moving average values. Using multiplicative model, these percentage would then represent the seasonal and irregular components.

iii.

The preliminary seasonal indices are now obtained by eliminating the irregular or random component by averaging these percentages. For averaging

we can use either

arithmetic mean or median ( preferably median ). iv.

The sum of these indices = S (say) will not in general, be 1200. Finally an adjustment is done to make the sum of the indices 1200 by multiplying throughout by a constant factor, ( 1200 /S ). i.e., by expressing the preliminary seasonal indices as percentage of  their arithmetic mean. The result gives the desired indices of the seasonal variations. Merits

i.

Of all the methods of measuring seasonal variations, the ratio to the moving average method is the most satisfactory, flexible and widely used method.

ii.

These indices don’t fluctuate so much as the indices by the ratio to trend method. Demerits

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i.

This method does not completely utilize the data, e.g., in the case of 12-month moving average seasonal indices cannot be obtained for the first six months and for the last six months. 4.1.4

'LINK RELATIVE’ METHOD.

Link relative method is also known as Pearson’s method is based on averaging the link relatives. Link relative is the value of the season expressed as percentage of  the preceding season. Here the word ‘season’ refers to time period, it would mean

month for monthly data and quarter for quarterly data , etc. Thus for monthly data: Link relative for any month =

Current month’s figure

× 100

Previous month’s figure

The steps involved in this method may be summed up as follows: i.

Translate the original data into link relatives ( L.R) as explained above.

ii.

As in the case of ‘ratio to trend’ method average the li nk relatives for each month

(quarter ) if the data are monthly (quarterly ). Mean or median may be used for averaging. iii.

Convert the average ( mean or median ) link relatives into chain relatives on the base of the first season. Chain relative ( C.R ) for any season is obtained on multiplying the link relative of that season by the chain relative of the preceding season and dividing by 100. Thus for monthly data, the chain relatives for the first season (month) , i.e.,

C.R. for February = ( L.R. of Feb. × C.R. of Jan.)/100 = L.R of Feb.

C.R. for March =

( since C.R. of Jan. = 100 )

( L.R. of March × C.R. of Feb.) /100

Similarly, C.R. for Dec. =

(L.R. of Dec. × C.R. of Nov). /100

Now by taking this December value as a base, a new chain relative for January can be obtained as: (L.R. of Jan. × C.R. of Dec)/100 iv.

This adjustment is done by subtracting a ‘correction factor’ from each chain relative. If 

we write, d =

( second (new) C.R. for Jan.    100)/12

then, assuming linear tre nd, the correction factor for February, March, …, December is d, 2d, 3d, …,11d respectively.

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v.

Finally adjust the corrected chain relatives to total 1200 by expressing the corrected chain relatives as percentages of their arithmetic mean. The resultant gives the adjusted monthly indices of seasonal variations. Merit

i.

This method utilizes data more completely than moving average method. There is only one less link relative while a 12 month moving average result in cut of six months at each end. Demerits

i.

The link relatives averaged together contain both the trend and cyclic components. Although the trend is subsequently eliminated the method is effective only if the growth is of constant rate.

ii.

This method is not so simple as the moving average method and the actual computation of the link relative method are much less. CYCLICAL VARIATIONS

The term ‘cycle’ refers to the recurrent variations in time series that

usually last longer than a year and are regular neither in amplitude nor in length. The oscillatory movements in time series with period of oscillation more than one year are termed as cyclic fluctuations. It moves like the pendulum of a clock and it is a never ending process. The cyclic movements in time series are referred to as the “four phase cycle”

called

‘Business

cycle’

composed

of

prosperity, decline, depression

and

improvement (recovery). Each phase changes gradually into the phase which follows it in the order given. The movements will have some sort of regularity, even though not perfect. According to Arthur Burns and Miller, “ Business cycles are a type

fluctuations found in

the aggregate economic activity of motions that organize their

work mainly in business expenses: a cycle consist of expansions occurring at about the same time in many economic activities followed by similarly general recessions, contractions and revivals which merge into the expansion phase of the next cycle, this sequence of changes is recurrent but not periodic. In duration business cycles vary from more than one year to ten or twelve years and they are not divisible into shorter cycles of similar character with amplitudes approximating their own.” The following diagram would illustrate a cycle.

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Phases of Business Cycle Prosperity

Improvement Decline

Normal Depression

Most of the time series relating to economics and business show some kind of  cyclical or oscillatory variations. Cyclical fluctuations are long-term movements that represents consistently recurring rises and

declines in activity. The study of cyclical variations is

extremely useful in framing suitable policies for stabilizing the level of business activity, i.e., for avoiding periods of booms and depressions as both are bad f or economy. Particularly depression which brings about a complete disaster and shatters the economy. Although measuring cyclical variations have the great importance, they are the most difficult type of  economic fluctuations to measure because of the following reasons. 

Business cycle do not show regular periodicity, they differ widely in timing, amplitude, pattern which make their study very tough and tedious.



Business cyclical variations are mixed with erratic, random or irregular forces which make it impracticable to isolate separately the effect of cyclical and irregular forces. Business cycles are distinguished from seasonal variations in the following aspects.



The cyclical variations are of a longer durations than a year. A business cycle may be of any duration but normally the

period of business cycle is two to ten years.

Moreover, they don’t ordinarily exhibit regular periodicity as successive cycles vary

widely in timing, amplitude and pattern. 

The fluctuations in a business cycle result from a different set of causes. The period of  prosperity, decline, depression and improvement viewed as four phases of a business cycle are generated by factors other than weather, social customs, and those which create seasonal patterns.

5.1 MEASUREMENT OF CYCLICAL VARIATIONS Business cycles are perhaps the most important type of fluctuations in economic data. Despite the importance of business cycles, they are the most difficult type of fluctuation to measure. This is because successive cycles vary so widely in 17

timing, amplitude and pattern and because the cyclical rhythm is mixed up with irregular factors. Because of these reasons it is impossible to construct meaningful indexes as in the case of trend and seasonal. The various methods used for measuring cyclical variations are: I.

Residual method.

II.

Reference Cycle Analysis method or National Bureau method.

III.

Direct method.

IV.

Harmonic Analysis method. Although the four methods are available for measuring cyclical variations,

only the first two methods which are popular in use and are discussed below. RESIDUAL METHOD. Amongst all the methods of arriving at estimates of the cyclical movements of  time series, the residual method is most commonly used. This method consists of  eliminating seasonal variation and trend, thus obtaining the cyclical irregular movements. Symbolically : T× S× C× I

= T×C×I

and,

S T× C×I = C× I T Next the data are usually smoothed in order to obtain the cyclical movements, which are sometimes termed the ‘cyclical relatives’, since they are always percentages. It is

because the cyclical- irregular or the cyclical movements remain as residuals that this procedure is referred to as the residual method. Limitation

The residual method is based on the assumption that trend and seasonal variations can be accurately measured. If the trend ordinates perfectly depicted the pattern of secular change and if the seasonal index exactly reflected seasonal influences, the residual method would leave values reflecting only cyclical and irregular influences. Because such perfection is rarely encountered, the computed values almost always contain some trend and seasonal elements. 5.1.1

REFERENCE CYCLE ANALYSIS OR NATIONAL BUREAU METHOD. The National Bureau of Economic Research has developed a different

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method of analyzing cyclical variations which it has used in the study of more than 1000 specific time series. This method is of value in analyzing past cycles only. This procedure aims to answer two sets of questions: 1) Is there in a given series a pattern of change that repeats itself in successive cycles in business at large ? If so what are its characteristics ? 2) Is there in a given series a wave movement peculiar to that series ? If so what are its characteristics ? A procedure involving ‘reference dates’ had been designed by the National

Bureau of Economic Research as a device which allows one not only to compare each series with a standard set of dates and to observe the behavior of individual series during expansion and contraction of general business, but also to compare the results for the various individual series. The first step is the selection of the reference dates which are the dates of the peaks and troughs of the business cycles. The reference dates which cover a duration of over one year and not over ten or twelve years were chosen after examination of a large number of economic time series and after study of  the “contemporary” reports of observers of business scene.

The next step consists of processing the data of the individual series in order to obtain a cyclical pattern for each series for all series, enabling one to compare the results for the various series. The processing of each series proceeds as follows: i.

The data are adjusted for seasonal variation.

ii.

The seasonally adjusted data are divided into reference cycle segments, these segments corresponds to the intervals between adjacent reference troughs.

iii.

For each segment, the monthly value are expressed as percentages of the average of  all the values in the segment. These are “reference cycle relatives”. As a result of this

step, all series, no matter what the original limit, are in percentage form. This step eliminates inter- cycle trend, but not eliminates the intra-cycle trend. iv.

Each reference cycle segment is broken into nine stages. The nine stages are identified as follows: 

The three months centered on the initial trough.



The first third of the expansion period.



The second third of the expansion period.



The last third of the expansion period. 19



The three months centered on the peak.



The first third of the contraction period.



The second third of the contraction period.



The last third of the contraction period.



The three months centered on the terminal trough.

The nine stages of averages for each reference cycle segment serve to reduce the erratic movements in a series and give a reference cycle pattern for the particular series under consideration. Merits

i.

It has proved that this method is the simple and most accurate way of comparing the cyclical variations of individual series with those of general business

ii.

In addition, it is free of errors that might be introduced where secular trend improperly estimated. Demerits

i.

The National Bureau method of cycle analysis may seem more complicated than the residual method.

IRREGULAR VARIATIONS

Irregular variation is caused due to chance factors. Irregular variations are also called ‘erratic’, accidental, random, refers to such variations are in business activity which don’t repeat in a definite patterns. Apart from the regular variations, almost all

the series contain another factor called random variations or residual fluctuations which are not accounted by secular trend, seasonal variations and cyclic variations. These

variations

are

purely

random, unpredictable

and

are

due

to

irregular

circumstances which are beyond the control of human hand but at the same time are a part of our system such as earth quakes, wars, floods, strike etc. According

to

Patterson

“The

irregular

variations

in

time

series

is

composed of non recurring sporadic forces which are not attributed to trend, cyclical or seasonal factors.”

The first three components such as trend, seasonal variation and

cyclical variation act in such a way as to produce certain systematic effects. Irregular movements on the other hand are considered to be largely random, being the result of chance factors which are wholly unpredictable like those determining the fall of a coin. Quantitatively it is most impossible to separate out the irregular movements and the cyclical movements. Therefore while analyzing time series the trend and seasonal 20

variations are measured separately and the cyclical and irregular variations are left altogether. There are two reasons for recognizing irregular movements: i.

To suggest that on occasions it may be possible to explain certain movements in the data due to specific causes and to simplify further analysis.

ii.

To emphasis the fact that predictions of economic conditions are always subject to degree of error owing to the unpredictable erratic influences which may enter. 6.1 MEASUREMENT OF IRREGULAR VARIATIONS The irregular component in a time series represents the residue of  fluctuations after trend, cyclical and seasonal movements have been accounted for. Thus in multiplicative model, if the original data is divided by T, S and C we get I. TSCI  

 I 

TSC 

Similarly in additive model, if the three components such as trend, seasonal variation and cyclical variations are calculated then we can find the irregular variation by subtracting these components from the original data.

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