A freehand curve drawn smoothly through the data values is often an easy and, perhaps, adequate representation of the data. The forecast can be obtained simply by extending the trend line. A trend line fitted by the freehand method should confirm to the following conditions:

(i)

The trend line should be smooth-a straight line or mix of long gradual curve.

(ii)

The sum of the vertical deviations of the observations above the trend line should equal the sum of the vertical deviations of the observations below the trend line.

(iii)

The sum of squares of the vertical deviations of the observations from the trend line should be as small as possible.

(iv)

The trend line should bisect the cycles so that area above the trend lines should be equal to the area below the trend line, not only the entire but as much as possible for each full cycle.

Example: fit a trend line to the following data by using the free hand method.

Sol.

Presents the graph of turnover from 1991 to forecast can simply by the trend.

Figure Graph of Sales turnover

Limitations of freehand method

(i)

This method is highly subjective because the trend line depends on personal

Judgement and therefore what happens to be a good-fit for one individual may not

Be so for another.

(ii)

The trend line drawn cannot have much value if it is used as a basis for

Predictions.

(iii)

It is very time-consuming to construct a freehand trend if a careful and

Conscientious job is to be done.

Method of Averages

The objective of smoothing methods into smoothen out the random variations due to irregular components of the time series and thereby provide us with an overall impression of the pattern of movement in the data over time. In this section, we shall discuss three smoothing methods.

(i) Moving averages

(ii) Weighted moving averages

(iii) Semi-averages

Moving Averages

The moving averages which serve as an estimate of the next period’s value of a variable

Given a period of length n is expressed as:

Moving average,

Where t = current time period

D = actual data which is exchanged each period

n = length of time period

Example: Using three-yearly moving averages, determine the trend and short-term-error.

Sol.

The moving average calculation for the first 3 years is:

Moving average (year 1-3) =

Similarly, the moving average calculation for the next 3 years is:

Moving average (year 2-4) =

A complete summary of 3-year moving average calculations is given in Table

Table: Calculation of Trend and Short-term Fluctuations

Year	Production Y	3years Moving Total	3-yealy Moving Average (Trend value )	Forecast Error )
1987 1988 1989 1990 1991 1992 1993 1994 1995 1996	21 22 23 25 24 22 25 26 27 26	- 66 70 72 71 71 73 78 79 -	- 22.00 23.33 24.00 23.67 23.67 24.33 26.00 26.33 -	- 0 -0.33 1.00 0.33 -1.67 0.67 0 0.67 -

Weighted Moving Averages

In moving averages, each observation is given equal importance (weight). However, different values may be assigned to calculate a weighted average of the most recent n values. Choice of weights is somewhat arbitrary because there is no set formula to determine them. In most cases, the most recent observation receives the most weightage, and the weight decreases for older data values.

A weighted moving average may be expressed mathematically as

Weighted moving average

Example: Vacuum cleaner sales for 12 months is given below. The owner of the

Supermarket decides to forecast sales by weighting the past three months as follows:

Weight applied

3

2

1.

6

Month

Last month

Two month ago

Three months ago

Forecast for the current month

Table 7.3: Weighted Moving Average

Month		Actual Sales		Three-month Weighted Moving Average
1	10		-
2	12		-
-3	13		-
4	16
5	19
6	23
7	26
8	30
9	28
10	18
11	16
12	14

3.5 Method of least Squares and curve fitting

• Linear tend

It is a mathematical method and with it gives a fitted trend line for the set of data in such a manner that the following two conditions are satisfied.

1. The sum of the deviations of the actual values of Y and the computed values of Y is zero.
2. The sum of the squares of the deviations of the actual values and the computed values is least.

This method gives the line which is the line of best fit. This method is applicable to give results either to fit a straight-line trend or a parabolic trend.

The method of least squares as studied in time series analysis is used to find the trend line of best fit to a time series data.

Secular Trend Line

The secular trend line (Y) is defined by the following equation:

Y = a + b X

Where, Y = predicted value of the dependent variable

a = Y-axis intercept i.e. the height of the line above origin (when X = 0, Y = a)

b = slope of the line (the rate of change in Y for a given change in X)

When b is positive the slope is upwards, when b is negative, the slope is downwards

X = independent variable (in this case it is time)

To estimate the constants a and b, the following two equations have to be solved simultaneously:

ΣY = na + b ΣX

ΣXY = aΣX + bΣX2

To simplify the calculations, if the midpoint of the time series is taken as origin, then the negative values in the first half of the series balance out the positive values in the second half so that ΣX = 0. In this case, the above two normal equations will be as follows:

ΣY = na

ΣXY = bΣX2

Example: Below are given the figures of production (in thousand quintals) of a sugar factory:

(a) Fit a straight-line trend to these figures.

(b) Plot these figures on a graph and show the trend line.

(c) Estimate the production in 2001.

Sol.

(a) Using normal equations and the sugar production data we can compute constants a and b as shown in Table 7.6:

Table : Calculations for Least Squares Equation

Year	Time period	Production			Trend values
1992 1993 1994 1995 1996 1997 1998	1 2 3 4 5 6 7	80 90 92 83 94 99 92	1 4 9 16 25 36 49	80 180 276 332 470 594 644	84 86 88 90 92 94 96
Total	28	630	140	2576

Therefore, linear trend component for the production of sugar is:

Fig.7.4: Linear Trend for Production of Sugar

b)

Plotting points on the graph paper, we get an actual graph representing production of sugar over the past 7 years. Join the point a = 82 and b = 2 (corresponds to 1993) on the graph we get a trend line as shown in Fig. 7.4.

(c)

The production of sugar for year 2001 will be

Parabolic Trend Model

The curvilinear relationship for estimating the value of a dependent variable y from an independent variable x might take the form

This trend line is called the parabola.

For a non-linear equation 

Example : The prices of a commodity during 1999-2004 are given below. Fit a parabola to these data. Estimate the price of the commodity for the year 2005.

Also plot the actual and trend values on a graph.

Exponential trend model

Logarithm y = aebx.
 

The equation is
y = aebx.
Taking log to the base e on both sides,
we get log y = log a + bx.
Which can be replaced as Y=A+BX,
where Y = logy, A = log a, B = b and X = x.

3.6. SEASONAL VARIATIONS

If the time series data are in terms of annual figures, the seasonal variations are absent. These variations are likely to be present in data recorded on quarterly or monthly or weekly or daily or hourly basis. As discussed earlier, the seasonal variations are of periodic in nature with period less than or equal to one year. These variations reflect the annual repetitive pattern of the economic or business activity of any society. The main objectives of measuring seasonal variations are:

(i)

To understand their pattern.

(ii)

To use them for short-term forecasting or planning.

(iii)

To compare the pattern of seasonal variations of two or more time series in a given period or of the same series in different periods.

(iv)

To eliminate the seasonal variations from the data. This process is known as depersonalisation of data.

Methods of Measuring Seasonal Variations

The measurement of seasonal variation is done by isolating them from other components of a time series. There are four methods commonly used for the measurement of seasonal variations. These methods are:

1.

Method of Simple Averages

2.

Ratio to Trend Method

3.

Ratio to Moving Average Method

4.

Method of Line Relatives

Note: In the discussion of the above methods, we shall often assume a multiplicative model. However, with suitable modifications, these methods are also applicable to the problems based on additive model.

Method of Simple Averages

This method is used when the time series variable consists of only the seasonal and random components. The effect of taking average of data corresponding to the same period (say 1st quarter of each year) is to eliminate the effect of random component and thus, the resulting averages consist of only seasonal component. These averages are then converted into seasonal indices, as explained in the following examples.

Example

Assuming that trend and cyclical variations are absent compute the seasonal index for each month of the following data of sales (in Rs. ‘000) of a company.

Sol.

Calculation Table

Year	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
1987	46	45	44	46	45	47	46	43	40	40	41	45
1988	45	44	43	46	46	45	47	42	43	42	43	44
1989	42	41	40	44	45	45	46	43	41	40	42	45
Total	133	130	127	136	136	137	139	128	124	122	126	134
	44.3	43.3	42.3	45.3	45.3	45.7	46.3	42.7	41.3	40.7	42.0	44.7
	101.4	99.1	96.8	103.7	103.7	104.6	105.9	97.7	94.5	93.1	96.1	102.3

In the above table, A denotes the average and S.I the seasonal index for a particular month of various years. To calculate the seasonal index, we compute grand average G, given by

Further, ΣS.I.=11998.9≠1200. Thus, we have to adjust these values such that their total is 1200. This can be done by multiplying each figure by 

Remarks: The total equal to 1200, in case of monthly indices and 400, in case of quarterly indices, indicate that the ups and downs in the time series, due to seasons, neutralise themselves within that year. It is because of this that the annual data are free from seasonal component.

Example

Compute the seasonal index from the following data by the method of simple averages.

Sol.

Calculation of Seasonal Indices

Year	IstQr	2ndQr	3rdQr	4thQr
1980 1981 1982 1983 1984 1985	106 84 90 76 80 104	124 114 112 94 104 112	104 107 101 91 95 102	90 88 85 76 83 84
Total	104 90	660 110	600 100	506 84.33
	93.67	114.49	104.07	87.77

We have

Merits and Demerits

This is a simple method of measuring seasonal variations which is based on the unrealistic assumption that the trend and cyclical variations are absent from the data. However, we shall see later that this method, being a part of the other methods of measuring seasonal variations, is very useful.

Ratio to Trend Method

This method is used when cyclical variations are absent from the data, i.e. the time series variable Y consists of trend, seasonal and random components.

Using symbols, we can write Y = T.S. R

Various steps in the computation of seasonal indices are:

(i)

Obtain the trend values for each month or quarter, etc. by the method of least squares.

(ii)

Divide the original values by the corresponding trend values. This would eliminate trend values from the data. To get figures in percentages, the quotients are multiplied by 100.

Thus, we have

(iii)

Finally, the random component is eliminated by the method of simple averages.

Example

Assuming that the trend is linear, calculate seasonal indices by the ratio to moving average method from the following data:

Quarterly output of coal in 4 years (in thousand tonnes)

Sol.

By adding the values of all the quarters of a year, we can obtain annual output for each of the four years. Fit a linear trend to the data and obtain trend values for each quarter.

From the above table, we get

Thus, the trend line is Y=240.5 – 3.6X, Origin: Ist January 1984, unit of X:6 months.

The quarterly trend equation is given by

Shifting origin to 15th Feb. 1984, we get

The table of quarterly values is given by

The table of Ratio to Trend Values

Note : Grand Average,

Example

Find seasonal variations by the ratio to trend method, from the following data:

Sol.

First, we fit a linear trend to the annual totals.

Now

The quarterly trend equation is

Shifting the origin to III quarter of 1997, we get

Table of Quarterly Trend Values

Ratio to Trend Values

Year	I	II	III	IV
1995 1996 1997 1998 1999 Total	109.1 86.1 77.7 85.0 106.0 463.9 92.78 91.10	131.1 122.4 106.4 114.3 117.2 591.4 118.28 117.35	107.5 109.9 93.9 97.8 105.5 514.6 102.92 102.11	93.2 90.7 79.3 85.5 97.0 445.7 89.14 88.44

Note that the Grand Average

Remarks: If instead of multiplicative model we have an additive model, then Y = T + S + R or S + R = Y-T. Thus, the trend values are to be subtracted from the Y values. Random component is then eliminated by the method of simple averages.

3.9

SIMPLE INDEX NUMBERS

A simple price index number is based on the price or quantity of a single commodity. To

Construct a simple index, we first have to decide on the base period and then find ratio of the value at any subsequent period to the value in that base period - the price/quantity relative. This ratio is then finally converted to a percentage

……..(6-1)

Similarly, Simple Quantity Index for period

……..(6-1)

Example

Given are the following price-quantity data of fish, with price quoted in Rs. Per kg and production inqtrs.

Construct:

(a)  the price index for each year taking price of 1980 as base,

(b)  the quantity index for each year taking quantity of 1980 as base.

Sol.:

Simple Price and Quantity Indices of Fish

(Base Year = 1980)

Year	Price (		Quantity		Price Index		Quantity Index
1980 1981 1982 1983 1984 1985		15 17 16 18 22 20		500 550 480 610 650 600		100.00 113.33 106.66 120.00 146.66 133.33		100.00 110.00 96.00 122.00 130.00 120.00

These simple indices facilitate comparison by transforming absolute quantities/prices into percentages. Given such an index, it is easy to find the percent by which the rice/quantity may have changed in a given period as compared to the base period. For example, observing the index computed in Example 6-1, one can firmly say that the output of fish was 30 per cent more in 1984 as compared to 1980.

It may also be noted that given the simple price/quantity for the base year and the index for the period

……..(6-3)

And

…..(6-4)

For example, with

3.10

COMPOSITE INDEX NUMBERS

The preceding discussion was confined to only one commodity. What about price/ quantity changes in several commodities? In such cases, composite index 183 numbers are used. Depending upon the method used for constructing an index,

Composite indices may be:

1. Simple Aggregative Price/ Quantity Index

2. Index of Average of Price/Quantity Relatives

3. Weighted Aggregative Price/ Quantity Index

4. Index of Weighted Average of Price/Quantity Relatives

SIMPLE AGGREGATIVE PRICE/ QUANTITY INDEX

Irrespective of the units in which prices/quantities are quoted, this index for given

Prices/quantities, of a group of commodities is constructed in the following three steps:

(i)

Find the aggregate of prices/quantities of all commodities for each period

(or place).

(ii)

Selecting one period as the base, divide the aggregate prices/quantities

Corresponding to each period (or place) by the aggregate of prices/ quantities

In the base period.

(iii)

Express the result in percent by multiplying by 100.

The computation procedure contained in the above steps can be expressed as:

…………….(6-5)

And

…………….(6-6)

Example

Given are the following price-quantity data, with price quoted in Rs per kg and

Production in qtls.

Find

(a) Simple Aggregative Price Index with 1980 as the base.

(b) Simple Aggregative Quantity Index with 1980 as the base.

Sol.:

Calculations for

Simple Aggregative Price and Quantity Indices

(Base Year = 1980)

	Prices		Quantities
Item	1980	1985	1980	1985
Fish Mutton Chicken	15 18 22	20 23 24	500 590 450	600 640 500
Sum 	55	67	1540	1740

(a) Simple Aggregative Price Index with 1980 as the base

(b) Simple Aggregative Quantity Index with 1980 as the base

Although Simple Aggregative Index is simple to calculate, it has two important limitations:

First, equal weights get assigned to every item entering into the construction of this index irrespective of relative importance of each individual item being different.

Improvements over the Laspeyre’s and Paasche’s Indices

To overcome the difficulty of overstatement of changes in prices by the Laspeyre's index and understatement by the Paasche's index, different indices have been developed to compromise and improve upon them. These are particularly useful when the given period and the base period fall quite apart and result in a greater divergence between Laspeyre's and Paasche's indices.

Other important Weighted Aggregative Indices are:

1.

Marshall-Edgeworth Index

The Marshall-Edgeworth Index uses the average of the base period and given period

Quantities/prices as the weights, and is expressed as

…………….. (6-17)

…………….. (6-17)

2.

Dorbish and Bowley Index

The Dorbish and Bowley Index is defined as the arithmetic mean of the Laspeyre’s

And Paasche’s indices.

…………….. (6-19)

…………….. (6-20)

3.

Fisher’s Ideal Index

The Fisher’s Ideal Index is defined as the geometric mean of the Laspeyre’s and

Paasche’s indices.

…………….. (6-21)

…………….. (6-22)

3.11 INDEX OF WEIGHTED AVERAGE OF PRICE/QUANTITY RELATIVES

An alternative system of assigning weights lies in using value weights. The value weight

…………….. (6-23)

Then

(i)

The product of the base period prices and the base period quantities denoted as

(ii)

The product of the base period prices and the given period quantities denoted

As

When

…………….. (6-24)

It may be seen that

Similarly, When 

……….. (6-25)

It may be seen that

If the index of weighted average of quantity relatives is defined as

…….. (6-26)

Then v can be obtained either as

(i)

The product of the base period quantities and the base period prices denoted as

(ii)

The product of the base period quantities and the given period prices denoted

As

When

…….. (6-27)

It may be seen that

Similarly, When

…….. (6-28)

It may be seen that

Example 6-5

From the data in Example 6.2 find the:

(a)

Index of Weighted Average of Price Relatives, using

(i) 

(ii)

(b)

Index of Weighted Average of Quantity Relatives, using

(i)

(ii)

Sol.:

Calculations for

Index of Weighted Average of Price Relatives

(Base Year = 1980)

Item
Fish Mutton Chicken	7500 10620 9900	9000 11520 11000	1000000 1357000 1080000	1200000 1472000 1200000
Sum 	28020	31520	3437000	3872000

(a)

Index of Weighted Average of Price Relatives, using

(i)

(ii) 

Calculations for

Index of Weighted Average of Quantity Relatives

(Base Year = 1980)

Item
Fish Mutton Chicken	7500 10620 9900	10000 13750 10800	9000000 1152000 1100000	1200000 1472000 1200000
Sum 	28020	34370	3152000	3872000

(b)

Index of Weighted Average of Quantity Relatives, using

(i)  

(ii)

Although the indices of weighted average of price/quantity relatives yield the same results as the Laspeyre's or Paasche's price/quantity indices, we do construct these indices also in situations when it is necessary and advantageous to do so. Some such situations are as follows:

(i)

When a group of commodities is to be represented by a single commodity in the

Group, the price relative of the latter is weighted by the group as a whole.

(ii)

Where the price/quantity relatives of individual commodities have been computed,

These can be more conveniently utilised in constructing the index.

(iii)

Price/quantity relatives serve a useful purpose in splicing two index series having

Different base periods.

(iv)

Depersonalizing a time series requires construction of a seasonal index, which also requires the use of relatives.

3.12 SPECIAL ISSUSES IN THE CONSTRUCTION OF INDEX NUMBERS

BASE SHIFTING. The need for shifting the base may arise either

(i) when the base period of a given index number series is to be made more recent, or

(ii) when two index number series with different base periods are to be compared, or

(iii) when there is need for splicing two overlapping index number series.

Whatever be the reason, the technique of shifting the base is simple:

Example

Reconstruct the following indices using 1997 as base:

Sol.:

Shifting the Base Period

Year	Index Number (1991 = 100)	Index Number (1997 = 100)
1991	100
1992	110
1993	130
1994	150
1995	175
1996	180
1997	200
1998	220

SPLICING TWO OVERLAPPING INDEX NUMBER SERIES

Splicing two index number series means reducing two overlapping index series with

Different base periods into a single series either at the base period of the old series

(one with an old base year), or at the base period of the new series (one with a recent

Base year). This actually amounts to changing the weights of one series into the

Weights of the other series.

1.

Splicing the New Series to Make it Continuous with the Old Series

Here we reduce the new series into the old series after the base year of the former. As

Shown in Table 6.8.2(i), splicing here takes place at the base year (1980) of the new

Series. To do this, a ratio of the index for 1980 in the old series (200) to the index of

1980 in the new series (100) is computed and the index for each of the following years in the new series is multiplied by this ratio

Splicing the New Series with the Old Series

Year	Price Index (1976=100) (Old Series)	Price Index (1980=100) (New Series)	Spliced Index Number [New Series (200/100)]
1976 1977 1978 1979 1980 1981 1982 1983 1984	100 120 146 172 200 -- -- -- --	-- -- -- -- 100 110 116 125 140	100 120 146 172 200 220 232 250 280

2.

Splicing the Old Series to Make it Continuous with the New Series

This means reducing the old series into the new series before the base period of the

Letter. As shown in Table , splicing here takes place at the base period of the

New series. To do this, a ratio of the index of 1980 of the new series (100) to the index

Of 1980 of the old series (200) is computed and the index for each of the preceding

Years of the old series are then multiplied by this ratio.

Splicing the Old Series with the New Series

Year	Price Index (1976=100) (Old Series)	Price Index (1980=100) (New Series)	Spliced Index Number [Old Series (100/200)]
1976 1977 1978 1979 1980 1981 1982 1983 1984	100 120 146 172 200 -- -- -- --	-- -- -- -- 100 110 116 125 140	50 60 73.50 86 100 110 116 125 140

3.13 CHAIN BASE INDEX NUMBERS

The various indices discussed so far are fixed base indices in the sense that either the base year quantities/prices (or the given year quantities/prices) are used as weights. In a dynamic situation where tastes, preferences, and habits are constantly changing, the weights should be revised on a continuous basis so that new commodities are included and the old ones deleted from consideration. This is all the more necessary in a developing society where new substitutes keep replacing the old ones, and completely new commodities are entering the market. To take care of such changes, the base year should be the most recent, that is, the year immediately preceding the given year. This means that as we move forward, the base year should move along the given year in a chain year after year

Conversion of Fixed-base Index into Chain-base Index

As shown in Table 6.8.3(i), to convert fixed-base index numbers into chain-base index

Numbers, the following procedure is adopted:

      The first year's index number is taken equal to 100

      For subsequent years, the index number is obtained by following formula: