1 Economic statistics

APPLIED STATISTICS

Economic statistics

Renu Kaul and Sanjoy Roy Chowdhury Reader, Department of Statistics, Lady Shri Ram College for Women Lajpat Nagar, New Delhi – 110024 04-Jan-2007 (Revised 20-Nov-2007)

CONTENTS Time series analysis

Components of Time series Models for Time series Determination of Trend Growth curve

Analysis of Seasonal Fluctuations Construction of seasonal indices

Method of simple averages Ratio to trend method

Ratio to moving average method

Link relative method (Pearson’s method) Measurement of cyclic movement

Measurement of random component Index numbers

Problems involved in computation of index numbers Calculation of index numbers

Price index numbers Quantity index number Value index numer

Index numbers based on average of price-relatives Link and chain indices

Tests for index numbers Consumer price index number

Construction of consumer price index number Limitations of index numbers

Practice session

Keywords

Time Series; Additive & multiplicative models; Trends; Growth curves, Price index number; Quantity index number; Value index number; Chain indices; Price relatives; Consumer price index

Time series analysis

A time series is a set of statistical data spread over time. Example: the daily production of milk in a milk plant, the weekly sales in a departmental store, the monthly publication of the Consumer Price Index, the quarterly statement of GNP, as well as the annual revenue of a firm. A number of factors influence the value of the variable under study. The aim of the time series analysis is to identity and isolates these influencing factors for forecasting purposes.

Time series analysis provides knowledge about the fluctuations in economic and business phenomena. The past trend is projected into future trend, to predict the changes, which are likely to occur in the economic activity.

Components of Time series

The factors influencing the movement of a time series are called the components of a time series. There are four components of a time series, viz.:

1. Trend [Tt]- Secular trend or Long term movement

2. Seasonal variations [St] - Periodic changes or short term fluctuations 3. Cyclic variations [Ct] - Periodic changes or short term fluctuations 4. Irregular movements [Rt]

At any given time, the value of time series may be obtained by the combination of some or all of these. The part of the time series, which can be explained may be attributed to components:

1. Secular Trend

2. Seasonal Variation and 3. Cyclical Variation.

While the part which can not be explained may be attributed to the:

4. Random Component.

Models for Time series

The principle objective of studying time series analysis is to identify the various components of the time series (some or all of which may be present in a time series) and to measure each one of them separately. However, the analysis of time series would depend on how these components have been combined. There are generally two methods of combining the effects of various components. In the first method, it is assumed that the various components operate independently of each other and the value of the time series is obtained by merely adding them i.e.

Ut = Tt + St + Ct + Rt (1)

where, Ut represents the value of the variable under study at time t, Tt is the trend value, St is the seasonal variation, Ct is the cyclic variation and Rt stands for the random component. The model considered is called the Additive Model. In this case, St will have some positive values and some negative, depending on the season of the year. Similarly Ct will have positive and negative values depending on whether we are above normal or below normal phase of the cycle and sum total of these values for any year/ cycle will be zero. Rt will also have positive and negative values and sum total of these values over a long period of time (∑Rt) will be zero. If the time series values are given annually, the St component will not appear. The very assumption that the various components operate independently of each other becomes the major drawback of this model. In reality the distinct components operate in conjunction with

each other and it is not possible to separate them. In fact, today’s value is affected by yesterday’s value and in turn affects the future value to a large extent.

In the second method, it is assumed that the various components operate proportionately to the general value of the series. Thus in this case, St, Ct and Rt, instead of assuming positive and negative values, take values below or above unity so that the geometric mean of these values in a year or in a cycle or over a long period of time is unity. Most of the business and economic time series follow the multiplicative model. In this case we take:

Ut = Tt . St . Ct . Rt (2)

However, if we take logarithms of both side in T-2, the above model reduces to:

log Ut = log Tt + log St + log Ct + log Rt (3) i.e. an additive model only.

NOTE: Sometimes, the components of a time series may be combined in a number of different ways such as:

Ut = Tt + St . Ct . Rt

Ut = Tt + St + Ct . Rt (4) Ut = Tt . Ct + St . Rt

These are known as Mixed models.

Determination of Trend

By the secular trend of a time series we mean the overall or persistent, smooth long term movement, which may be upward, downward or constant. This component of the time series is basically studied for predictive purposes. Also one may wish to study trend in order to isolate it and then eliminate its effect on the time series. The trend gets affected by: the introduction of new technology, population growth, market fluctuation, etc. Its duration is over several years.

There are mainly four different methods for the measurement of Trend.

1. Graphical Method

This method does not involve any mathematical computations. In this method we plot a free hand smooth curve based on time series values Ut and time t. The plotted curve depicts only the direction of trend and does not predict the future values. The given time series may have an increasing or decreasing trend. However, one may come across cases where time series values fluctuate around a constant term.

EXAMPLE - 1

The following table gives the production of cars (in thousands) by a car manufacturer in India over the years. Draw a graph to show the production trend.

Year (t) 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Production (Ut) 125 132 138 117 135 142 137 156 168 172 SOLUTION:

P rod u ction T ren d

1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0

1 9 9 1 1 9 9 2 1 9 9 3 1 9 9 4 1 9 9 5 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 Y ears

Production (No. of cars in thousands)

CONCLUSION:

The graph shows an upward trend over the years showing an increase in demand of cars.

2 Semi-average method

In this method, the time series data Ut are divided into two equal groups with respect to time.

The first group consists of the first half periods (years) and the second group consists of the remaining periods. If the time series data consists of an even number of periods, say 2n where n is an integer, then each group will consist of n periods. A simple arithmetic mean of Ut

values is computed for each group and is plotted against the mid point of the respective time period covered. In case n is even, none of the groups has a mid-year and thus the middle point of the two mid years will be the mid period where the average of each group is to be plotted.

The line joining the two points gives the required trend line. In case the time series consists of an odd number of periods (years), say 2n+1, then the value corresponding to the mid period is omitted to obtain two equal groups and again a straight line is plotted against the mid-values of each group. The semi average method does not depict the true trend lines. However, one can forecast the future value for any given year by obtaining corresponding point on the trend line. The semi-average method is affected by the extreme values and does not ensure the elimination of short term and cyclic variations.

EXAMPLE - 2

The following table gives the annual production of certain types of crops cultivated in a mountain region over a period of 15 years. Fit a trend line by the method of semi-averages.

Years 1990 1991 1992 1993 1994 1995 1996 1997 Production

(in ‘000 Kg.)

84.8 126.4 121.6 105.6 110.4 150.4 168.0 139.2 Years 1998 1999 2000 2001 2002 2003 2004 2005 Production

(in ‘000 Kg.)

126.4 166.4 155.2 147.2 161.6 166.4 168.0 170.2

SOLUTION

Since there are 16 observations, the average of first 8 observations is 126.00 and the average of last 8 observations is 157.70. The trend line is as shown below:

CONCLUSION:

The graph shows an upward trend. This implies that the production of crops is increasing over the period.

3 Method of Least Squares

In this method we fit a suitable mathematical model to the given time series data. The selection of the appropriate mathematical model is made either by plotting the given time series data against time or by studying the nature of the variable involved in the series. The constants of the mathematical model are estimated from the given series by the method of least squares so that the sum of squares of the deviations of actual values from the corresponding estimated values is minimum.

For example, consider a linear model

0 1

t b b t

U = + +ε (5)

where, b₀ and , are two constants and b₁ ε, is the random error term distributed N(0,σ²).

t t 2

U na b t

tU a t b t

= +

= +

∑ ∑

∑ ∑ ∑

t

t,

Let a and b be the estimates of b and . The normal equations for obtaining the values of a and b are:

0 b1

(6)

Thus the fitted model will be of the form:

= +a b (7)

ˆt

U

where, a, is the intercept which the estimated line cuts on the axis of U and b, is the slope of the trend line.

By this method

∑

(^Ut−^Uˆ^t)²is minimum and also this method gives

∑

(^Ut−^Uˆ^t) 0= ^. The trend line obtained by this method not only depicts the trend but also enables one to

orecast the future values for any given year.

f

ing trend over time are:

The most commonly used curves for fitt Linear Model: Ut = b0 + (b1 - t)

l:

Quadratic Mode Ut = b0 + (b1 - t) + (b2 - t²)

(b - t²) + (b - t³) Cubic Model:

U = b + (b - t) + 2 3

t 0 1

U = b + b ln(t)

r ln(U) = ln(b ) + (b t)

0 + (b1 / t)

ower Model U = b (t ) or ln(U) = ln(b ) + b ln(t) (8)

d), whereas in actual ituations Ut also depends on a number of other explanatory variables.

EXAMPLE - 3

t 0 1

Logarithmic Model:

Exponential Model U = b e_{t 0} ^{(b t)1} o _{0 l} Inverse Model Ut = b

Growth Model Ut = e^{(b + b t)0 l} or ln(U) = b + (b t)_{0 l}

b1

P _{t 0 0 l}

The estimates obtained by the method of least squares are unbiased and have minimum variance. However, they still are not optimum. This is because in the above models we

ssume that the variable U

a t depends only on the factor t (time perio s

The following table gives the production of oil by an oil producing company over a period of 0 years. Fit a linear trend by the method of least squares.

1

Year (t) 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Production (Ut)

(in ‘000Gallons)

159.8 203.8 212.6 187.2 232.3 246.0 223.7 284.9 211.9 268.8 270.3

SOLUTION

Consider the expression (6), i.e. ˆU_t = +a bt.

For estimating the constants by the method of least squares we construct the following table:

Years (t)

(in ‘000 gallons)

t tu_t t²

Va Production (U_t) Trend

lues

1990 159.8 -5 -799.2 25 182.1

1991 203.8 -4 -815.0 16 191.1

1992 212.6 -3 -637.9 9 200.2

1993 187.2 -2 -374.4 4 209.3

1994 232.3 -1 -232.3 1 218.3

1995 246.0 0 0.0 0 227.4

1996 223.7 1 223.7 1 236.5

1997 284.9 2 569.8 4 245.5

1998 211.9 3 635.8 9 254.6

1999 268.8 4 1075.2 16 263.6

2000 270.3 5 1351.5 25 272.7

TOTAL 2501.3 0 997.0 110

On solving the normal equations for estimating a and b we get:

ifferent years are shown in the he following graph shows th

a = 227.4 and b = 9.1

Hence the trend equation is: ˆU_t =227.4 9.1+ t

By giving different values of t, the estimated trend values for d st column of the above table.

la

T e trend of the production of oil.

Fitting of Trend Line by Least Square Method

150.0 170.0 190.0 210.0 230.0 250.0 270.0 290.0 310.0

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Years

Production ('000 gallons)

Production (Ut) ('000 gallons) Trend (' 000 gallons)

CONCLUSION

e increase in production from 1990 to 2000.

between them. The values now correspond to a me period in the given time series. Finally, draw a graph plotting the moving average values

the period of cycles and trend is linear. However, this method does not provide a umber of trend values for each end of the series and can not be used for forecasting urposes.

The upward trend shows th 4 Moving average method

Fitting of trend by the method of moving averages is based on computing a series of successive arithmetic averages over a fixed number of years. This method smoothes out the fluctuations of the given data with the help of a moving average. To obtain trend by moving average we consider the observations for the first few years (say n) of the series in a group and find their arithmetic average i.e. the arithmetic average of the first n values, n is called the period of the moving average. Enter the average value of this group against the mid point of the group. Next delete the 1^st observation from this group of n years and add the (n+1)^th observation to the group. Compute the average of this group. This gives the second mean.

Again enter the average value of the group against the mid point of the group. Repeat this procedure till we exhaust all the observations. When n is odd, the moving averages correspond to a time period in the given time series. However, when n is even they are placed between the two middle values. In this case, we further calculate a moving average of period two of these moving averages and place it

ti

against time. This gives the required trend.

The method of moving averages reduces the effect of extreme observations in the series. If the cyclical variations are regular both in period and amplitude then this method eliminates the fluctuations to a great extent provided the period of the moving average is equal to or a multiple of

n p

EXAMPLE – 4

The following table gives the consumption of rice in a particular village of West Bengal.

etermine the underlying trend by a 3 year moving average and a 5 year moving average and nt.

1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 D

comme Year

Consumption uintals)

104.0 109.2 104.0 114.4 127.4 119.6 109.2 114.4 104.0 117.0 (‘000 Q

Year 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 Consumption

000 Quintals)

111.8 124.8 119.6 136.5 149.5 148.2 133.9 149.5 139.1 161.2 (‘

SOLUTION

For fitting trend b the method of my oving averages, follow ng table has beeni constructed.

Year Consumption 3 year moving average

5 year moving average (in ‘000 quintals)

1970 104.0

1971 109.2 105.73

1972 104.0 109.20 111.80

1973 114.4 115.27 114.92

1974 127.4 120.47 114.92

1975 119.6 118.73 117.00

1976 109.2 114.40 114.92

1977 114.4 109.20 112.84

1978 104.0 111.80 111.28

1979 117.0 110.93 114.40

1980 111.8 117.87 115.44

1981 124.8 118.73 121.94

1982 119.6 126.97 128.44

1983 136.5 135.20 135.72

1984 149.5 144.73 137.54

1985 148.2 143.87 143.52

1986 133.9 143.87 144.04

1987 149.5 140.83 146.38

1988 139.1 149.93

1989 161.2 The following graph shows the trend.

F itting o f T rend by the M etho d o f M o ving Averages

170.0

100.0 110.0 120.0 130.0 140.0 150.0

1970 160.0

Consumption (in '000 quantals) 3 year moving average 5

1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 Y ear s

year moving average

It has been the trend depends on the value of n. The fitted trend show n f rice over the period.

CONCLUSION

oothness of observed that the sm

s a increase in the consumption o

Growth curve

ost of the time series relating to business and economic phenomena over long period of M

me do not exhibit growth which is at a constant rate and in a particular direction over long eriod of time, chronological series are not likely to show either a constant amount of change . The rate of growth is initially slow, then it picks up and erated, then becomes stable for some time after which it shows

rowth of an industry, where initially the growth slow during the period of experimentation, then the growth is rapid during the period of again slow and stable when a period of stability is ached.

e of the important growth curve, which are generally used to describe 2 Gompertz Curve

e

eters exceeds the number hnique of least squares to fit these curves fails.

a modified form

in which the amount of growth declined by a constant percentage, but the curve also approaches an upper limit called asymptote.

The gene of this curve

U =a+bct (9)

here, a, b, c, are the three constants or parameters to be determined, and Ut and t, are the

Fitting of Modified Exponential Curve

here are following two methods of fitting a Modified Exponential Curve:

1 2

hod of Three Selected Points

Consider the three ordinates corresponding to three equidistant points of time such tha t − = −t t t

stitut modified exponential curve we get:

ti p

or a constant ratio of change becomes faster and gets accel

retardation. The curves, which can be fitted to such data are called growth curves. These asymptotic growth curves are suitable for a spatially limited universe, in which a population grows and are also useful in describing the g

is

development and then the growth is re

The following are som

the measurements in a time series.

1 Modified Exponential Curve 3 Logistic or Pearl-Reed Curv

NOTE: In all the three growth curves mentioned above, the number of param usual tec

of variables and thus the

Modified Exponential Curve

This curve is of the exponential curve. This curve not only describes a trend

ral equation is:

t

w

variables.

T

Method of Three Selected Points Method of Partial Sums.

Met

1 2 3

U ,U ,U

1 2 3 2 2 1

Sub ing U , t ; U , t ; U , t in the general equation of t ,t ,t3 t

1 1 2 2 3 3

t2

U2 = +a bc U₁= +a bc^t1 U₃= +a bc^t3 (10) Thus, U₂−U₁=bc^t1[c^t2−t1−1] and U₃−U₂ =bc [c^{t2 t3−t2} −1] (11)

=> ^U3−^U2 =c^t2−t1

2 1

U −U (12)

=> ^{3 2 2 1}

U U

= ⎢⎡ − ⎥

1 (t t )

U −U ⎤ ⁻

2 1

c ⎣ ⎦ (13)

=> _{2 1} ^{t1 3 2}

2 1

U U

U U bc 1

U U

⎡ − ⎤

− = ⎢⎣ − − ⎥⎦

=>

(14)

2 1 1

t t

2 t

2 1 2 1

3 2 1 3 2

(U U ) U U

b .

(U 2U U ) U U

⎡ ⎤ −

− −

= − + ⎢⎣ − ⎥⎦ (15)

Similarly,

a=U₁−bc^t1

2

1 3 2

U1 (16)

=>

3 2

U U U

a U 2U

= −

− +

btained as:

t

ˆU a bct = + (17)

qual parts, each part containing n consecutive values of say t = 1,2,…,n, then t = n+1, n+2, …, 2n and finally t =

1, 2n+2, …, 3n.

Let be the partial sums of the three parts given by

= = + = +

(18) ubstituting the value of Ut in the above equations we get:

Once the values of a, b, c has been computed the fitted curve is o

Method of Partial Sums

In this method we divide the data into three e Ut

2n+

1 2 3

S ,S ,S

n 2n 3n

1=

∑

U , St 2 =

∑

U and St 3 =

∑

Ut

t 1 t n 1 t 2n 1

S S

n n

t 1

t 1

c 1 S (a bc ) na bc

= c 1

⎡ − ⎤

=

∑

+ = + ⎢⎣ − ⎥⎦ Similarly,

2n n

S₂ (a b ^{t n 1} c 1

c ) na bc ⁺ ⎡ − ⎤

t n 1_{= +} c 1

=

∑

+ ^{= +} ⎢⎣ − ⎥⎦

3n n

t 2n 1

bc ) na bc

c 1

+ 3

t 2n 1

c 1

S (a

= +

⎡ − ⎤

=

∑

+ = + ⎢⎣ − ⎥⎦ ⁽¹⁹⁾

=> S³−S₂ c^{2n 1+} −c^{n 1+} _n

= _{n 1} =c

2 1

S −S c ⁺ −c (20)

1 3 2 n

S S

c S S

⎡ − ⎤

= ⎢⎣ − ⎥⎦ (21)

=>

2 1

Also,

n

n 1 c 1

S S b[c ⁺ c]⎡ − ⎤

− = − ⎢ ⎥ (22)

2 1

c 1−

⎣ ⎦

=> _{2 1})³

2

(S S b ⎡c 1− ⎤. −

= ⎢ ⎥ (23)

3 2 1

c (S −2S +S )

⎣ ⎦

3 n

Also, ^{2 1} ₂

3 2 1

(S S )

c 1 (c 1)

(S 2S S ) . (c 1)

−

2 1

S S na c

− −

⎡ ⎤

− = + ⎢ ⎥⎦ − + − (24)

=>

⎣

S )2 2− 1

2 1

3 2 1

na (S S ) (S

S 2S S

= − −

− + (25)

=>

2

1 3 2

3 2 1

S S S a 1

n S 2S S

⎡ − ⎤

= ⎢ − + ⎥

get the fitted modified exponential curve.

t percentage.

⎣ ⎦ (26)

Substituting the values of a, b, c we

Gompertz Curve

The Gompertz curve named after Benfamin Gompertz, is a type of a mathematical model for a time series where growth is the slowest at the start and end of the time period. The Gompertz curve describes a trend in which the growth increments of the logarithms are declining by a constant percentage. Thus the natural values of the trend would show a declining ratio of increase, but the ratio does not decrease by either a constant amount or a constan

The general equation of the Gompertz curve is:

(

ct

ct

U

t

= ab

27)

where, a = the upper asymptote c = the growth rate b = constant (b and c can be negative)

The equation can also be written as:

be

t

= ae

₍₂₈₎

U

ing f Gompertz Curve

In the Gompertz curve, as in the case of modified exponential curve, the number of

ame least squares fails.

his can be written as:

31)

The ve tion of modified exponential curve.

We thu curve by the same principal as that used in modified exponential cur .e. b selected points and the method of partial sums.

Log c Cu

A logistic function or logistic curve models the S-shaped curve growth of some set P. The

init petition arises, the growth

ows and at maturity growth stops. The curve was first developed by P. F. Verhulst in 1838.

l J. Reed independently developed this curve in 1920. It is rl-Reed Curve. The Verhulst equation is a typical pplication of logistic equation and is a common model for population growth which states that the rate of reproduction is proportional to existing population, all else being equal and

that the rate of reprodu being

qual.

Fitt o

par ters exceeds the number of variables. Thus the method of Consider the equation of Gompertz curve

ct

U

t

= ab

(29)

Taking log on both sides we get:

t

g U = log a + c log bt (30)

lo

T

Y_t = +A Bct (

where, Y = log U ; A = log a; B = log b _{t t} abo equation is comparable to the equa can s fit a Gompertz

ve i y the method of three isti rve / Pearl-Reed Curve

ial stage of growth is approximately exponential, then as com sl

Further, Raymond Pearl and Lowel now frequently referred to as the Pea a

ction is proportional to amount of available resources all else e

The general form of the logistic curve is:

t a bt

U k ,

1 e

⁺

= + b>0

₍₃₂₎

where, k, a, b are the parameters of the curve and k = max (Ut)i.e. the maximum value which the variable can take over all values of time t.

logistic law of growth then their reciprocals follows modified exponential law:

If the given data follows

a bt

t k

1 1

.[1 e ] U

= + + (33)

a b t

t

1 = 1

or e

.(e )

U k+ k (34)

if

a

1 e b

A ; B= and C=e

k k

= we get (35)

t t

1 A BC

U = + (36)

or (37)

which is the equation of a modified exponential curve.

Fitting of Logistic Curve

There are several methods of fitting a logistic curve, viz:

6) Meth

Method of Three Selected Points equation of logistic curve:

t t = +A BC Y

1) Method of Three Selected Points 2) Yule’s Method

3) Hotelling’s Method 4) Nair’s Method

5) Method of Successive Approximations od of Sum of Reciprocals

7) Rhodes Method

We describe below some of the common used mly ethods.

Consider the

U

t

, b>0

+

^{a bt (38)}

k 1 e

⁺

=

t − =t Now

In this method we select three ordinates U ,U ,U corresponding to three equidistant points _{1 2 3} of time t ,t ,t such that _{1 2 3 3 2} t₂−t₁

a bt

k

e

⁺

+ =

t

1 U

⁽³⁹⁾

=> _{1 2 3}

2 3

a bt ;log 1 a bt ;log 1k a bt U

⎡ ⎤

⎡ ⎤ ⎡ ⎤

= + ⎢ − = +⎥ ⎢ − = +⎥

⎦ ⎣ ⎦ (40)

On simplification we get:

bt ;log 1 a bt ;log 1 a bt U

⎡ ⎤

⎡ ⎤ ⎡ ⎤

= + ⎢ − = +⎥ ⎢ − = +⎥

⎦ ⎣ ⎦ (40)

On simplification we get:

U1 U

⎣U ⎦ ⎣U

⎣ ⎦ ⎣

k k

log 1⎢⎢ −− ⎥⎥

2 1

2 1

2 1

U k U

k U U

log⎡ − . ⎤ b(t t )

= − (41)

similarly,

⎢ − ⎥

⎣ ⎦

3 2

3 2

2 3

k U U

log . b(t t )

U k U

⎡ − ⎤

= −

⎢ − ⎥

⎣ (42)

we

⎦ as t3-t2=t2-t1, get:

2 1

2 1

log .

U k U

k U U

⎡ − ⎤=

⎢ − ⎥

⎣ ⎦

3 2

3 2

log .

U k U

k U U

⎡ − ⎤

⎢ − ⎥

⎣ ⎦ (43)

After simplification we get:

2

2 1 3 1 2 3

2

U (U +U ) 2U U U−

= −

2 1 3

k U U U (44)

2 1

2 1 2 1

b log .

t t U k U

= 1 ⎢⎡k U− U ⎥⎤

− ⎣ − ⎦ and (45)

1 2 1

1 2 1 2 1

a log 1 log .

U t t U k U

t k U U

⎡ k ⎤ ⎡ − ⎤

= ⎢⎣ − −⎥⎦ − ⎢⎣ − ⎥⎦ (46)

Using the values of a, b and k we get the best fitted equation.

Yule’s Method

Again consider the logistic curve as:

a bt t

1 e k

U

+ + = (47)

Let us suppose the value of k is known, say k′ then

a bt k

1 e ⁺ ′

+ = (48)

Ut

=> _t

t

1 Y (say) U − =

⎜ ⎟

⎝ ⎠ (49)

=> Y = +a bt (50)

a bt log⎛ ′k

+ = ⎞

t

which is the equation of straight line having two parameters and two variables. Thus the normal equations by the method of least squares are:

Y =na b+ t

∑

^t

∑

⁽⁵¹⁾

(52)

otteling’s Method

Again consider the logistic curve as:

∑

tY^t =a

∑

t b+

∑

t²

Thus the values of a and b will give the best fitted curve.

H

a bt t

1 e k

U

+ + = (53)

differentiating with respect to t we get:

a bt t

2. U_t dt

b.e ⁺ = ^−k dU (54)

2

bU a_+bt t

dt = ^t e k dU −

(55)

=> ^{t t}

t

1 dU − ⎜b 1⎝ U ⎞

U dt k ⎟⎠

= ⎛ − (56)

urther, if the interval of differencing is not too large then we can approximate

F ^t

t

dU 1

U dt by Ut

1 ∆ Ut ∆t

>

= ^Ut _t

t

1 b

b+ U

U ∆t k

∆

L

= − (57)

et _t ^t

Ut ∆t U

1 ∆ b

Y ; A=-b; B=

= k (58)

=> _t (59)

A ’ c st u method and hence we can find ‘b’ and

‘k h n ai curve p gh mean o

m t

A

usines and ec nomic phenome l p ommonl rved due to social and religious custom change ond

on data. he dura ion of such fluctua a period of 12 months based on onthly, quarterly, weekly, daily or hourly data.

Y_t =A BU+

gain ‘A and ‘B’ an be e imated sing least square

’. Furt er ‘a’ ca be obt ned on assuming that the asses throu f Ut and ean of .

nalysi of seasonal fluctuations s

In many b s o na the seasona atterns are c y obse s, s in weather c

tions is within

itions highlighting the effect of

seasons T t

m

The study of seasonal variations is necessary for two reasons:

forecasting some future monthly or quarterly movements ations so as to study the effect of cycles. These fluctuations

peat themselves year after year.

al indices The different methods f

1 Metho ple avera

In this m we arrange by nd r q epending on whether monthly rterly data le) e co he a r each month/quarter

for all the years co th rag

1. One may be interested in

2. One may be interested in isolating and eliminating the effect of trend, seasonal variations, irregular fluctu

are regular in nature and tend to re Construction of season

or measuring seasonal variations are:

d of sim ges

ethod the data years a .

months o uarters (d or qua is availab

i.e

Next w mpute t verage fo e

. we mpute e ave x (i = 1,2, ...., n, _i n = 12 if ly data is av and qua ata is le) over all the years.

We then te the avera the s i.e.

month ailable n = 4 if rterly d availab

compu ge of all average ⁿ _i

i=1

=1

x x

n

∑

Finally, the seasonal index for the i^th month/quarter is computed as ^xi×100i.e. by expressing es as percen

or monthly data and 400 for quarterly data.

given time series is independent of the trend d cyclic variations.

x the respective averag tage of the overall average.

NOTE: The sum of the seasonal indice

his method is based on the assumptios is 1200 n that th f

e T

an

EXAMPLE – 5

The following table shows the monthly consumption of sugar in India for 4 years. Compute the seasonal indices by simple averages method.

Year Month

JAN FEB MAR APR MAY JUN 2001 9000 8250 7500 10500 11250 9250 2002 11250 10500 9750 12000 12000 8250 2003 12000 11250 10500 12750 11250 9750 2004 13500 12750 12600 14250 13500 10300 JUL AUG SEP OCT NOV DEC 2001 8900 9750 10250 13500 10875 11250 2002 9750 9000 9750 11400 10750 10500 2003 9000 9750 9000 11250 11225 10240 2004 11025 9900 10725 11813 11950 12600 SOLUTION

Months Year

2001 2002 2003 2004 total AVG. (Monthly) seasonal index JAN 9000 11250 12000 13500 45750 11437.50 105.97 FEB 8250 10500 11250 12750 42750 10687.50 99.02 MAR 7500 9750 10500 12600 40350 10087.50 93.47

Months Year

2001 2002 2003 2004 total AVG. (Monthly) seasonal index APR 10500 12000 12750 14250 49500 12375.00 114.66 MAY 11250 12000 11250 13500 48000 12000.00 111.19 JUN 9250 8250 9750 10300 37550 9387.50 86.98 JUL 8900 9750 9000 11025 38675 9668.75 89.59 AUG 9750 9000 9750 9900 38400 9600.00 88.95 SEP 10250 9750 9000 10725 39725 9931.25 92.02 OCT 13500 11400 11250 11813 47963 11990.63 111.10 NOV 10875 10750 11225 11950 44800 11200.00 103.77 DEC 11250 10500 10240 12600 44590 11147.50 103.29 TOTAL 518052.50 129513.13 1200

AVG. 43171.04 10792.76 100

EXAMPLE – 6

The quarterly consum p e

s in for e arter.

S I

RTE ption of

ach qu

electricity over a period of 7 years is given. Com ut the easonal dex

OLUT ON

QUA R YEAR

Jan - Mar Apr - Jun Jul - Sep Oct - Dec 2000 132.13 126.56 118.83 140.07 2001 111.75 107.46 101.24 118.40 2002 118.62 113.90 107.46 127.63 2003 101.64 97.35 91.41 107.75

2004 85.97 82.67 77.88 91.08

2005 91.25 87.62 82.67 98.18

2006 75.90 73.92 66.50 79.20

Quarterly Total 717.26 689.47 645.99 762.30 Quarterly Average 102.47 98.50 92.28 108.90 Seasonal Index 101.92 97.97 91.79 108.32

Average of Quarterly Average =100.54

2 Ratio to trend method

This method is applicable under the assumption that the seasonal variation for any given month/quarter is constant factor of the trend. In Ratio to trend method before computing the seasonal variations we first find the yearly averages/totals for all the years. We then fit a mathematical model, usually linear, quadratic or exponential etc. to the yearly averages/totals and obtain the trend values by the method of least squares. This gives annual trend values.

The monthly/quarterly trend values are obtained by suitably adjusting the trend equations.

Next, by assuming a multiplicative model, trend eliminated values are obtained by expressing

the given time series values as percentage of the trend values. These percentages contain the seasonal, cyclic and irregular var nd irregular variations can be

eliminated by averagin Finally, the seasonal

indices are adju a f 120 monthly data and 400 for quarterly data by multiplying them orr acto as:

iations. Now the cyclic a

g the percentages for different months/quarters.

sted to total o 0 for by a c ection f r c given

1200

Total d

c= or m ata)

of the in ices (f onthly d 400

Total of the indices

c= (for quarterly data)

EXAMPLE – 7

The following table gives the quarterly production of cotton fabric (in ‘000 mts.) by a manufacturer for the years 2001 to 2005. Compute the seasonal indices by ratio to trend method.

QUARTER YEAR

Jan - Mar Apr - Jun Jul - Sep Oct - Dec

2001 79 105 95 89

2002 89 137 132 116

2003 105 153 142 127

2004 142 201 179 164

2005 238 243 230 216

SOLUTIO

To obtaine tren s w t a l del g a

U bt, to the yearly averages.

pr-Jun Jul-Sep Oct-Dec total Average T = UT T² Yearly d e N

d the yearly d value e first fi inear mo iven by n expression

t = a +

The following table shows the yearly trend values obtained by the method of least squares.

The values of constants are: a = 149.4 and b = 33.3 Year Jan-Mar A

(t) (U) t-2003 Tren

Valu 2001 79 105 95 89 368 92 -2 -184 4 83 2002 89 137 132 116 474 119 -1 -119 1 116 2003 105 1 3 5 142 12 7 527 132 0 0 0 149 2004 142 201 179 164 686 172 1 172 1 183 2005 238 243 230 216 927 232 2 464 4 216 F

q t

rom t uarter

he abov ly incre

e cons ment is

tants th 33.3/4

e value

= 8.33.

of the Thus u

yearly sing the

increm above

ent = yearly

33.3 =>

trend va

the va lues th

lue of e quarte

the rly rend values are shown in the following table:

Quarterly Trend Value YEAR

Jan - Mar Apr - Jun Jul - Sep Oct - Dec 2001 70.5 78.8 87.2 95.5 2002 103.5 111.8 120.2 128.5 2003 136.5 144.8 153.2 161.5 2004 170.5 178.8 187.2 195.5 2005 203.5 211.8 220.2 228.5 The above table has been computed as follows:

The trend value for the second quarter for any s o tin the quarterly increment from the trend value of the corresponding year. Similarly, trend value for the third qu obtained by alf of the quarterly trend value of at year. The trend value for the first quarter is obtained by subtracting quarterly increment d quarter and finally the trend value for fourth quarter is Finally, the trend eliminated trend values obtained are as follows:

e e u

year i btained by subtrac g half of arter is adding h increment to the

th

from the trend value for the secon

obtained by adding quarterly increment to the trend value for the third quarter.

Tr nd Eliminat d Val es YEAR

Jan – Mar Apr – Jun Jul – Sep Oct – Dec 2001 112.0 133.2 109.0 93.2 2002 86.0 122.5 109.9 90.3 2003 76.9 105.6 92.7 78.6 2004 83.3 112.4 95.6 83.9 2005 116.9 114.7 104.5 94.5 Total 475.2 588.4 511.7 440.6 Average 95.03 117.68 102.33 88.11

Seaso al n 9 6 .5 l

Index

94.2 11 .76 101 3 87.42 Tota = 400

For computing the values of seasonal index the va e of th corre tion fa tor c = 1.007901 lu e c c EXAMPLE - 8

The data given in the following table shows the monthly production of wool (in .000 tons) by n

C state i

n

dustries from Jan 2001 to Dec. 2005. Compute the monthly seasonal indices by ratio to tre d method.

Year JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DE 2001 156.75 133.65 108.90 85.80 87.45 95.70 107.25 89.10 135.30 148.50 161.70 184.80 2002 185.31 1 2.53 138.45 142.71 74.55 83.07 48.99 123.54 157.62 27 08.74 215.13 238.56 2003 306.90 284.58 273.42 217.62 181.35 189.72 181.35 161.82 189.72 212.04 357.12 376.65 2004 305.76 274.56 180.96 168.48 124.80 184.08 234.00 168.48 212.16 271.44 377.52 411.84 2005 349.87 324.42 311.70 248.09 206.74 216.28 206.74 184.47 216.28 241.73 407.12 429.38

SOLUTION

We first fit a linear trend to the yearly averages by the method of least squares.

year (t) yearly total yearly average (U)

T = t-2003 UT TT Yearly Trend Values 2001 1494.90 124.58 -2.00 -249.15 4.00 127.54 2002 1789.20 149.10 -1.00 -149.10 1.00 167.72 2003 2932.29 244.36 0.00 0.00 0.00 207.89 2004 2914.08 242.84 1.00 242.84 1.00 248.06 2005 3342.81 278.57 2.00 557.14 4.00 288.23

Total 1039.44 0.00 401.73 10.00 where, a = 207.89; and b= 40.17.

Th tre a e

on T V s

us the nd equ tion is: UT = 207.89 + 40.17T and Monthly Incr ment = 3.35 M thly rend alue

Year JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC 2001 109.13 112.48 115.83 119.17 122.52 125.87 129.22 132.56 135.91 139.26 142.61 145.96 2002 149.30 152 65 156.00 159.35 1 2.69 1 6.04 169.39. 6 6 172.74176.08 179.43 182.7 186.8 13 2003 189 8 192 82 196.17 199.52 2 2.87 2 6.21.4 . 0 0 209.56212.91 216.26 219.60 222.9 226.5 30 2004 229.65 233.00 236.34 23 69 2 .04 2 .399. 43 46 249.73 253.08 256.43 259.78 263.13 266.47 2005 269.82 273 17 276.52 279.86 283.21 286.56 289.91 293.25 296.60 299.95 303.30 306.65.

i V

Trend Elim nated alues

Year JAN FEB MAR APR MAY JUN JUL AU SE OG P CT N V D C O E 2001 143.64 118.82 94.02 72.00 71.38 76.03 83.00 67.21 99.55 106.64 113.39 126.61 2002 124.1 113.02 88.75 89.56 45.82 50.03 28.922 71.52 89.51 116.33 117.70 128.17 2003 161.97 147.59 139.38 109. 7 89 39 92.00 8 .54 7 .000 . 6 6 87.73 96.56 160.18 66.441 2004 133 4 117.84 76.57 70.29 51.35 74.71 9 .70 6 .57.1 3 6 82.74 1 4.49 143.48 54.550 1 2005 129.67 118.76 112.72 88.65 73.00 75.48 71.31 62.91 72.92 80.59 4.23 0.0313 14

Total 692.53 616.03 511.44 429.56 330.94 368.25 363.47 344.21 432.45 504.60 668.97 715.80 Avg. 138.51 123.21 102.29 85.91 66.19 73.65 72.69 68.84 86.49 100.92 133.79 143.16 SI 139.01 123.65 102.66 86.2

(Adj)

3 66.43 73.92 72.96 69.09 86.80 101.29 134.28 143.68

Correction Factor = 1.0036 3 Ratio to moving averag od

This method is n c ting ing ave ges by nsidering n = 12 for monthly data and n = 4 for quarterly data. For mo y data e the su essive averages for the groups of size 12 and then take a 2-point mov g avera of thes average The resultant moving averages will give the est b d effec of tren and cyclical variations. Leaving the first 6 months and the last 6 months (as n = 12), convert the given

e meth

based o alcula mov ra co

nthl first calculat cc

in ge e s.

imates of the com ine ts d

data series as the percentages of the 2-point moving average values i.e. ((given data)/(2-point moving average)*100). These percentages would now represent seasonal variations along with random components. Further, the random component is eliminated by averaging these monthly percentages. Since the sum total of these seasonal indices is not equal to 1200 (for monthly data) and 400 (for quarterly data) so, finally adjusted seasonal indices are computed to make the sum e in 1200 400 by multiply ho correction factor c defined earlier.

EXAMPLE – 9

of th dices or ing them throug ut by a

The Airport aut s co data on the number of aircrafts, which could not fly on time from Delhi airport due to adverse weather conditions from 2001 to 2004. The number of aircrafts per mo the d are wn in ollo able. Compute seasonal indices by ratio to movi rag od.

Year 2001 2001 20 01 20 2001 2001

horitie llected nth for

ng ave perio

e meth sho t e fh wi g tn

2001 2001 2001 2001 01 2001 20 01

Month JAN FEB MAR R MAP AY JUN JUL AUG SEP OCT NOV DEC No. of

Aircrafts

95 81 66 52 53 58 65 54 82 90 98 112

Year 2002 2002 2002 2002 2002 2002 2002 2002 2002 2002 2002 2002 Month JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC

No. of Aircrafts

87 81 65 67 35 2339 74 98 58 101 112

Year 2003 2003 2003 2003 2003 2003 2003 2003 2003 2003 2003 2003 Month JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC

No. of Aircrafts

110 102 98 78 65 6568 68 76 58 128 135

Year 2004 2004 2004 2004 2004 2004 2004 2004 2004 20 2004 200404 Month JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC

No. of Aircrafts

98 88 58 54 40 59 75 54 68 87 121 132

SOLUTION

To compute the seasonal indices by 12 point to m ave ethod following two tables have been truct

Year Month No.

Aircrafts

Total 12-point Moving Average

2-point Moving Average

Ratio to Average ratio oving ra mge

c son ed:

of

Moving

2001 JAN 95

2001 FEB 81

2001 MAR 66

2001 APR 52

2001 MAY 53

Year Month No. of Aircrafts

Total 12-point Moving Average

2-point Moving Average

Ratio to Moving Average

2001 JUN 58

906 75.5

2001 JUL 65 75.17 86.47

898 74.83

2001 AUG 54 74.83 72.16

898 74.83

2001 SEP 82 74.79 109.64

897 74.75

2001 OCT 90 75.38 119.4

912 76

2001 NOV 98 75.25 130.23

894 74.5

2001 DEC 112 73.71 151.95

875 72.92

2002 JAN 87 71.17 122.25

833 69.42

2002 FEB 81 69.58 116.41

837 69.75

2002 MAR 65 69.42 93.64

829 69.08

2002 APR 67 69.42 96.52

837 69.75

2002 MAY 35 69.88 50.09

840 70

2002 JUN 39 70 55.71

840 70

2002 JUL 23 70.96 32.41

863 71.92

2002 AUG 58 72.79 79.68

884 73.67

2002 SEP 74 75.04 98.61

917 76.42

2002 OCT 98 76.88 127.48

928 77.33

2002 NOV 101 78.58 128.53

958 79.83

2002 DEC 112 81.04 138.2

987 82.25

2003 JAN 110 84 130.95

1029 85.75

2003 FEB 102 85.75 118.95

Year Month No. of Aircrafts

Total 12-point Moving Average

2-point Ratio to M A Moving Average

oving verage

1029 85.75

2003 MAR 98 85.5 114.62

1023 85.25

2003 APR 78 84.33 92.49

1001 83.42

2003 MAY 65 84.54 76.89

1028 85.67

2003 JUN 68 86.63 78.5

1051 87.58

2003 JUL 65 87.08 74.64

1039 86.58

2003 AUG 58 86 67.44

1025 85.42

2003 SEP 68 83.75 81.19

985 82.08

2003 OCT 76 81.08 93.73

961 80.08

2003 NOV 128 79.04 161.94

936 78

2003 DEC 135 77.63 173.91

927 77.25

2004 JAN 98 77.67 126.18

937 78.08

2004 FEB 88 77.92 112.94

933 77.75

2004 MAR 58 77.75 74.6

933 77.75

2004 APR 54 78.21 69.05

944 78.67

2004 MAY 40 78.38 51.04

937 78.08

2004 JUN 59 77.96 75.68

934 77.83

2004 JUL 75

2004 AUG 54

2004 SEP 68

2004 OCT 87

2004 NOV 121

2004 DEC 132

Year Month

2002 2003 2004 Seasonal

Indices

Adjusted Seasonal Indices 2001

JAN 122.25 130.95 126.18 126.46 127.02

FEB 116.41 118.95 112.94 116.10 116.61

MAR 93.64 114.62 74.60 94.29 94.70

APR 96.52 92.49 69.05 86.02 86.40

MAY 50.09 76.89 51.04 59.34 59.60

JUN 55.71 78.50 75.68 69.97 70.27

JUL 86.47 32.41 74.64 64.51 64.80

AUG 72.16 79.68 .44 67 73.09 73.42

SEP 109.64 98.61 81.19 96.48 96.91

OCT 1 9.40 127.48 1 93.73 113.54 114.04

NOV 130.23 128.53 161.94 140.23 140.85

DEC 151.95 138.20 173.91 154.69 155.37

TOTAL

1194.71 1200

EXAMPLE – 10 C = 1.004429

The following data gives the quarterly sale of umbrellas in a community market. Compute as al in tio ving ave age method.

Quarter 2003 2004

se on dices by ra to mo r

2001 2002

Jan-Mar 5876 622 632 32 6435 Apr-Jun 4701 4703 5065 5019 Jul-Sep 4231 4558 4643 4633 Oct-Dec 4623 5788 5980 5985 SOLUTION

Year ale

t)

quar ving

otal 4-qu moving total

4-quarter mo average

Ratio to average Qu

(U

arter S 4- ter T mo total

of two

atrer ving moving 2001 Jan-Mar 5876

Apr-Jun 4701

19431

Jul-Sep 4231 39208 4858.0 95.2

19777

Oct-Dec 4623 39556 4901.5 86.3

19779

2002 Jan-Mar 6222 39885 5090.4 92.4

Year Quarter Sale (Ut)

4-quarter moving total

Total of two 4-quatrer moving total

4-quarter movin average

Ratio to g average g movin

20106

Apr-Jun 4703 41377 5276.9 117.9

21271

Jul-Sep 4558 42652 5363.0 107.9

21381

Oct-Dec 5788 43124 5422.0 84.1

21743

2003 Jan-Mar 6332 43571 5459.8 92.8

21828

Apr-Jun 5065 43848 5494.4 115.2

22020

Jul-Sep 4643 44143 5499.3 108.7

22123

Oct-Dec 5980 44200 5506.4 84.3

22077

2004 Jan-Mar 6435 44144 5519.9 90.9

22067

Apr-Jun 5019 44139 5519.3 116.6

22072

Jul-Sep 4633

Oct-Dec 5985

Trend Eliminated Values

Year Jan-Mar Apr-Jun Jul-Sep Oct-Dec

2001 95.2 86.3

2002 92.4 117.9 107.9 84.1

2003 92.8 115.2 108.7 84.3

2004 90.9 116.6

Total 276.1 349.7 311.8 254.7 Average 92.0 16.6 1 103.9 84.9 Adj. Sea

In

1 1 al = 400

sonal dices

92.6 17.3 04.6 85.4 Tot Total of Average = 563 = alue 1.00

Link relative method (Pearson’s method)

s based on the concept of averaging the link relatives. Link relative for any 397.4 > the v of c = 64

4

This method i

season (month/quarter) is defined as:

th Value of the ith

-1) seasonth

season Lin ve for i

k relati season = i

V ×100

alue of the ( (60)

(link relativ e first season can not be defined)

First we co e given da ly/q ) fo rs of link relatives by the above e on. Next co aver rela r th/quarter over the years. Conv average k s n r a th the expression:

e for th

nvert th ta (month uarterly r all yea in terms xpressi

ert he

mpute relative

age link to chain

tives fo tives on

each mon basis of

t lin i el e

th th

e li e fo o lative of (i-1) season

Chain rel season

100 ative for ith Averag

= nk relativ r i seas n chain re×

(61)

wh t t season n to

No r seas ta ne a he

last season. This value in general will not be equal to 100 (as assumed by us) due to mainly trend. We thus ad by subtr ting c, 2c, …., 11c from

ebruary, March, …….., December values (assuming a linear trend). Thus relative of i season = chain relative of i season - (i-1) c × ere, the chain relative fo he firsr is take be 100.

w a new chain relative fo the first on is ob i d on the basis of ch in relative of t just the chain relatives for trend ac

F

th th

The adjusted chain

(62) where

New chain relative for first season -100

c= n (63)

and n = 12 for monthly data

F d se and 4 for quarterly data. l i a p s inally, the a justed asona ndices re com uted a :

i seasth

ex for ith Adju

= sted chain relative for on

×10 Adjusted seaso

nal ind season e 0

Av rage of adjusted chain relatives (64)

EXAMPLE - 11

The following table gives the quarterly production of coffee (in ‘000 kg) manufactured by XYZ coffee industry during the years 2001 – 2005. Compute the seasonal indices by the method of link relatives.

Year Jan-Mar Apr-Jun Jul-Sep Oct-Dec

2001 80.40 69.68 58.96 83.08

2002 93.80 75.04 58.96 96.48

2003 83.08 77.72 72.36 85.76

2004 91.12 83.08 67.00 93.80

2005 91.12 58.96 53.60 88.44

`

SOLUTION

The following table shows the calculations for computing seasonal indices by the method of link relatives.

l R v

Quarter y Link elati es

Year Jan-Mar Apr-Jun Jul- ep S Oct-Dec

2001 86.67 8 .62 4 140.91

2002 112.90 0.00 78.57 8 163.64

2003 86.11 3.55 93.10 9 118.52

2004 106.25 1.18 8 .65 9 0 140.00

2005 97.14 64.71 9 .91 0 165.00

Total 402.41 416.10 42 .84 728.06 7 Average 100. 0 83.22 8 .57 6 5 145.61

Chain Rela ves ti 100. 0 83.22 71.21 0 103.69 Adj. Chain Relatives

or t en )

100 82.14 69.05 100.45 (corrected f r d

Adj. Seasonal Indices 113.75 93.43 78.55 114.27 Total = 400 Correction Factor= 1.078719

EXAMPLE - 12

The data given in the following table represents the monthly consumption of milk (in ‘000 lts.) by ice-crèam industries from Jan. 2000 to Dec. 2005. Compute Seasonal indices

y the method of link relatives.

b

Consumption of Milk (in ‘000 lts.)

Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 2000 29.7 44.6 74.4 178.4 237.9 282.5 252.8 237.9 208.2 148.7 104.1 44.6 2001 37.2 59.5 119.0 208.2 267.7 297.4 252.8 208.2 178.4 104.1 101.1 86.2 2002 59.5 74.4 148.7 230.5 297.4 327.1 267.7 223.1 208.2 119.0 119.0 89.2 2003 116.0 133.8 185.9 282.5 371.8 416.4 297.4 208.2 233.5 148.7 178.4 119.0 2004 133.8 148.7 208.2 297.4 416.4 446.1 327.1 193.3 267.7 208.2 160.6 178.4 2005 83.3 119.0 203.0 258.0 420.0 356.9 282.5 267.7 223.1 133.8 136.8 74.4

SOLUTION

Link Relatives

Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 2000 150.0 166.77 240.0 133.3 118.7 89.5 94.1 87.5 71.4 70.0 42.9 2001 83.3 160.0 200.0 175.0 128.6 111.1 85.0 82.4 85.7 58.3 97.1 85.3 2002 69.0 125.0 200.0 155.0 129.0 110.0 81.8 83.3 93.3 57.1 100.0 75.0 2003 130.0 115.4 138.9 152.0 131.6 112.0 71.4 70.0 112.1 63.7 120.0 66.7 2004 112.5 111.1 140.0 142.9 140.0 107.1 73.3 59.1 138.5 77.8 77.1 111.1 2005 46.7 142.9 170.6 127.1 162.8 85.0 79.2 94.7 83.3 60.0 102.2 54.3 Total 441.5 804.4 1016.2 992.0 825.3 644.0 480.2 483.6 600.5 388.4 566.5 435.3 Avg. 88.3 134.1 169.4 165.3 137.6 107.3 80.0 80.6 100.1 64.7 94.4 72.5 CR 100.0 134.1 227.1 375.4 516.3 554.2 443.5 357.5 357.8 231.6 218.7 158.6 ACR 100.0 133.8 226.5 374.5 515.2 552.8 441.9 355.6 355.6 229.1 215.9 155.6 ASI 32.8 43.9 74.3 122.9 169.1 181.4 145.0 116.7 116.7 75.2 70.9 51.1120

0 Correction Factor = 0.3

CR = Chain Relatives ACR = Adjusted Chain relatives ASI = Adjusted Season Indices

Measurement of cyclic movement

A very simple method of measuring the cyclic variations of a time series is the residual g average of an appropriate period.

s may be eliminated in any order.

nent of a time ser

for

Ind Pri of t are lite a g ind

change in a variable or a g pro

or s wit

method. In this method, we divide the given time series values by trend values and then seasonal indices, by assuming a multiplicative model. The resultant will give us the cyclic and the irregular component. The irregular component is further eliminated by using movin NOTE: The component

Measurement of random component

There does not exist a sophisticated method of determining the random compo

ies generally the non-random components are determined and whatever is left unaccounted by these components constitutes the random component of the series.

ex numbers

ces of various commodities keep changing over a period of time. While the prices of some he commodities may increase, those of others may decrease. Also different commodities measured in different units. For example wheat and rice in kilograms, milk, oil etc. in rs, cloth in meters. Thus to have a general idea about the collective change in the prices of roup of related commodities, we have to reach at a single representative figure called an ex number. An index number is a statistical device designed to measure the relative

roup of variables over a period of time, over different places, fessions etc. The variable may refer to prices of commodities, their quantities consumed old or imported or exported, scores obtained by a student in different tests etc. The period h which the comparisons are made is called the base period. Index for the base period is