Unit – 3
Time Series and Index Number
A time series is set of data collected at successive point in a time or over successive period of time.
A time series is a collection of observations made sequentially through time.
The interval between observations can be any time interval (hours within days, weeks, months, years, etc.).
Some examples of time series are:
 Malaria incidence or deaths over calendar years
 Daily maximum temperatures
 Hourly records of babies born at a maternity hospital
 Monthly unemployment,
 Weekly measures of money supply,
 Daily closing prices of stock indices, and so on
An analysis of a single sequence of data is called univariate timeseries
Analysis.
An analysis of several sets of data for the same sequence of time periods is called multivariate timeseries analysis or, more simply, multiple timeseries analysis.
It helps us to understand past behaviour of time series data.
With the help of time series analysis, we can compare the actual performance and analyse the cause of variation.
Fluctuation in a time series is mainly due to four basic components.
1 Secular trend or trend (T)
2 Seasonal variation (S)
3 Cyclical variation or cyclic fluctuation (C)
4 Irregular or random moments (I)
Secular trend or trend (T)
Trend is the phenomenon of long term changed in a recorded data series, generally, in the same direction throughout the span of the series.
A sequence plot of time series (the time series value plotted vertically with respect to time itself on the horizontal axis) will usually reveal the presence of trend as a gentle upward or downward “drift” of the data path.
Upward sloping trend paths in a real value time series may be indicative of growth phenomenon, a downward sloping path suggest contraction.
In a moneyvalue time series an upward sloping path may represent some combination of real growth and inflation; a downward sloping trend path might indicate contraction with deflation.
Trend is usually the result of longterm factors such as changes in the population, demographics, technology, or consumer preferences.
Seasonal Variation
This is the pattern of variation within time series which repeat itself year to year.
Seasonality may be associated with agricultural functions, seasonal weather pattern, custom and convention, or religious or secular holidays.
It is important to remember that a seasonable pattern in one time series may or may not resemble that in another time series.
Fans and airconditioned sales are high in the summer month, agricultural sales are high at harvest time, RAIN CAOTS, UMBERELLA SALES HIGH IN MONSOON
Cyclic Components:
Any regular pattern of sequences of values above and below the trend line lasting more than one year can be attributed to the cyclical component. Usually, this component is due to multiyear cyclical movements in the economy.
Cyclic variations are recurrent upward or downward movements in a time series but the period of cycle is greater than a year. Also, these variations are not regular as seasonal variation.
A business cycle showing these oscillatory movements has to pass through four phasesprosperity, recession, depression and recovery. In business, these four phases are completed by passing one to another in this order
Irregular Variation
Irregular variations are fluctuations in time series that are short in duration, erratic in nature and follow no regularity in the occurrence pattern. These variations are also referred to as residual variations since by after trend, cyclical and seasonal variations. Irregular fluctuations result due to the occurrence of unforeseen events like: FLOODs, EARTHQUAKES, WARS, and FAMINES etc.
How the component relates to the original series
A model that expresses the time series variable Y in terms of the components T (trend), C (cycle), S (seasonal) and I (irregular).
Additive components model & multiplicative components model.
 Additive model: Y = T + C + S + I 2.
 Multiplicative model: Y = T *C * S * I
The principal methods of measuring trend fall into following categories:
1. Free Hand Curve methods
2. Method of Averages
3. Method of least squares
Freehand curve
 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 smootha 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 goodfit 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 timeconsuming 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) Semiaverages  

 
 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 threeyearly moving averages, determine the trend and shorttermerror.  

 
Sol.  The moving average calculation for the first 3 years is:  
 Moving average (year 13) =
 
 Similarly, the moving average calculation for the next 3 years is:  
 Moving average (year 24) =  

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

Table: Calculation of Trend and Shortterm Fluctuations
 

 
 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
 
 3.5 Method of least Squares and curve fitting  

 
 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.
This method gives the line which is the line of best fit. This method is applicable to give results either to fit a straightline 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 = Yaxis 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 straightline 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
 
 ,  
 
 
 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 thousand quintals
 
 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 nonlinear equation , the values of constants a, b, and c can be determined by solving three normal equations.  

 

 

 
 Example : The prices of a commodity during 19992004 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
 

 
 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 shortterm 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
 
 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 . Then the seasonal index for a particular month is given byFurther, 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 . The resulting figures are the adjusted seasonal indices, as given below:
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
 
 We have . Further, since the sum of terms in the last row of the table is 400, no adjustment is needed. These terms are the seasonal indices of respective quarters.
 

 
 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 and  
 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  
 Origin: Ist January 1984, unit of X:1quarter (i.e., 3 months).  
 Shifting origin to 15th Feb. 1984, we get , origin Iquarter, unit of X=1 quarter  

 
 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  
 Trend equation if Y= 224 + 48X, origin: Ist July 1997, unit of X = 1 year
 
 The quarterly trend equation is , origin: Ist July 1997, unit of X = 1 quarter.
 
 Shifting the origin to III quarter of 1997, we get  

 
 Table of Quarterly Trend Values
 

 
 Ratio to Trend Values
 
 Note that the Grand Average . Also check that the sum of indices is 400.  

 
 Remarks: If instead of multiplicative model we have an additive model, then Y = T + S + R or S + R = YT. Thus, the trend values are to be subtracted from the Y values. Random component is then eliminated by the method of simple averages.
 
Index numbers are intended to measure the degree of economic changes over time. These numbers are values stated as a percentage of a single base figure. Index numbers are important in economic statistics. In simple terms, an index (or index number) is a number displaying the level of a variable relative to its level (set equal to 100) in a given base period
Simple Index Number: A simple index number is a number that measures a relative change in a single variable with respect to a base. These type of Index numbers are constructed from a single item only.
This ratio is then finally converted to a percentage 100 Value in Base Year Value in Period Index for any Period x i i = i.e. Simple Price Index for period i = 1,2,3 ... Will be 100 0 0 x p pP i i = …………(61) Similarly, Simple Quantity Index for period i = 1,2,3 ... Will be 100 0 0 x q qQ i i = …………(62)
Composite Index Number: A composite index number is a number that measures an average relative change in a group of relative variables with respect to a base. A composite index number is built from changes in a number of different items.
Price index Numbers: Price index numbers measure the relative changes in prices of a commodity between two periods. Prices can be either retail or wholesale. Price index number is useful to comprehend and interpret varying economic and business conditions over time.
Quantity Index Numbers: These types of index numbers are considered to measure changes in the physical quantity of goods produced, consumed or sold of an item or a group of items.
Methods of constructing index numbers: There are two methods to construct index numbers: Price relative and aggregate methods (Srivastava, 1989)
In aggregate methods, the aggregate price of all items in a given year is expressed as a percentage of same in the base year, giving the index number.
Weighted aggregative index numbers: These index numbers are the simple aggregative type with the fundamental difference that weights are assigned to the various items included in the index.
Characteristics of index numbers:
 Index numbers are specialised averages.
 Index numbers measure the change in the level of a phenomenon.
 Index numbers measure the effect of changes over a period of time.

 
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  
 
 simple Price Index for Period will be  
 ……..(61)  
 Similarly, Simple Quantity Index for period will be  
 ……..(61)  

 
 Example  
 Given are the following pricequantity 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)
 
 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 61, 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 the actual price/quantity for the period may easily be obtained as:  
 ……..(63)  
 And 
 …..(64)  
 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:  
 …………….(65)  
 And 
 …………….(66)  

 
 Example  
 Given are the following pricequantity 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)
 
 (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.  MarshallEdgeworth Index  
 The MarshallEdgeworth Index uses the average of the base period and given period Quantities/prices as the weights, and is expressed as  
 …………….. (617)  
 …………….. (617)  

 
2.  Dorbish and Bowley Index  
 The Dorbish and Bowley Index is defined as the arithmetic mean of the Laspeyre’s And Paasche’s indices.  
 …………….. (619)  
 …………….. (620)  


 
3.  Fisher’s Ideal Index  
 The Fisher’s Ideal Index is defined as the geometric mean of the Laspeyre’s and Paasche’s indices.  
 …………….. (621)  
 …………….. (622)  

 
 3.11 INDEX OF WEIGHTED AVERAGE OF PRICE/QUANTITY RELATIVES  
 An alternative system of assigning weights lies in using value weights. The value weight for any single commodity is the product of its price and quantity, that is, If the index of weighted average of price relatives is defined as  
 …………….. (623)  
 Then can be obtained either as  
(i)  The product of the base period prices and the base period quantities denoted as that is, or  
(ii)  The product of the base period prices and the given period quantities denoted As that is,  
 When is , the index of weighted average of price relatives, is expressed as  
 …………….. (624)  
 It may be seen that is the same as the Laspeyre’s aggregative price index.Similarly, When is , the index of weighted average of price relatives, is expressed as  
 ……….. (625)  

 
 It may be seen that is the same as the Paasche’s aggregative price index.If the index of weighted average of quantity relatives is defined as  
 …….. (626)  
 Then v can be obtained either as  
(i)  The product of the base period quantities and the base period prices denoted as that is, , or  
(ii)  The product of the base period quantities and the given period prices denoted As that is,  
 When is , the index of weighted average of quantity relatives, is expressed as  
 …….. (627)  
 It may be seen that is the same as the Laspeyre’s aggregative quantity index.Similarly, When is , the index of weighted average of quantity relatives, is expressed as  
 …….. (628)  
 It may be seen that is the same as the Paasche’s aggregative quantity index.  

 
 Example 65  
 From the data in Example 6.2 find the:  
(a)  Index of Weighted Average of Price Relatives, using (i) as the value weights(ii) as the value weights  

 
(b)  Index of Weighted Average of Quantity Relatives, using (i) as the value weights(ii) as the value weights
 
Sol.:  Calculations for Index of Weighted Average of Price Relatives (Base Year = 1980)
 

 
(a)  Index of Weighted Average of Price Relatives, using  
 (i) as the value weights  


 


 


 

 
 (ii) as the value weights  


 


 


 
 Calculations for Index of Weighted Average of Quantity Relatives (Base Year = 1980)
 

 
(b)  Index of Weighted Average of Quantity Relatives, using  
 (i) as the value weights  


 


 


 
 (ii) as the value weights  


 


 


 

 
 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
 

 
 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
 

 
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
 

 
 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 Fixedbase Index into Chainbase Index  
 As shown in Table 6.8.3(i), to convert fixedbase index numbers into chainbase 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:  
 

 

Conversion of Fixedbase Index into Chainbase Index
 

 
 Conversion of Chainbase Index into Fixedbase Index  
 As shown in Table , to convert fixedbase index numbers into chainbase index Numbers, the following procedure is adopted:  
 The first year's index is taken what the chain base index is; but if it is to form the base it is taken equal to 100 In subsequent years, the index number is obtained by following formula:  
 

Conversion of chainbase Index into fixed base Index
 

 