IBA-JUWMBA

Course Instructor: Dr Swapan Kumar Dhar

Time Series:Time series is a collection of data recorded over a period of time – usually weekly, monthly, quarterly, or yearly. Some examples of time series are:(i) The Annual Production of Rice over the last 10 years(ii) The daily closing price of a share on a stock exchange for a week(iii) The monthly electric bills for 12 months.

The purpose of time series analysis is to describe the past movements and fluctuations to analyze their causes and interrelationship, to examine the causal factors operating in the present and to explain what significance the present combination of causal factors has in relation to the future.

Components of Time Series:

There are four components to a time series:(i) Trend or secular trend(ii) The seasonal variation(iii) The cyclical variation(iv) The irregular variation. Secular Trend: The smooth, regular and gradual movement of a time series which shows the increase or

decline over a long period of time is called the secular trend. Over a long period of time, if observed, most

time series reveal either an inclining or a declining tendency. This general tendency of a time series over

a fairly long period of time is termed a secular trend. This frequently happens with business and economic

time series. Inclining tendency is observed in case of population, agricultural production, money in

circulation etc. whereas declining tendency is inherent in time series relating to birth rate, death rate and

epidemic deaths etc. Due to advancement of medical science, facilities of health care and higher literacy,

birth rate, death rate and deaths due to epidemics are gradually decreasing.

Seasonal Variation: Another component of a time series is seasonal variation. Many sales, production and other series fluctuate with the season. For example, the sale of woolen electric fan rises in summer. The sales of clothing and shoes rise extremely before Eid and Durga Puja.For the following two reasons seasonal fluctuations take place:

(i)Natural Causes: Seasonal variation take place due to climatic changes. For example, during winter sale

of wool increases, during summer demand for ice cream, cold drinks, and electric fan etc. increases; in

rainy season demand for umbrella, rain coat etc, increases.

(ii)Rituals and Social Customs: Man made rituals, social customs and traditions are also responsible for

seasonal fluctuations of a time series. For example, just on the eve of new year the sale of greeting cards

increases to a great extent. In the beginning of an academic session sale of book, paper, uniforms etc.

increase.

Cyclical Variation: It is another component of a time series. A typical business cycle consists of a period

of prosperity followed by periods of recession, depression and recovery. Most of the business and

economic time series increase or decrease periodically with some amount of regularity. In general the

periodicity of this type of variation is more than one year. This periodic movement of a time series is

termed as cyclical fluctuations, for this happens due to business cycles. A business cycle has got four

phases namely prosperity or boom, recession, depression and recovery.

The time period between two successive booms or depressions is known as periodic time or length of a

cycle.

Note: Though seasonal fluctuations and cyclical variations both are periodic in nature, there is a

significant difference between the two types of movements. First, seasonal fluctuations take place within

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one year whereas in case of cyclical variations the time period is generally more than one year. Cyclical

variations take place in 3 to 10 years time period. Secondly, in case of seasonal variation the periodic

time remains the same but in case of cyclical fluctuations the periodic time does not remain the same.

Cyclical variations take place only with some rough regularity. Lastly, seasonal variations are mainly

attributed to climatic changes, and man made rituals and social customs whereas economic factors are

responsible for cyclical fluctuations. Increase or decrease in price, production, sales, demand etc. is some

of the economic phenomena which are responsible for cyclical fluctuations.

Irregular Variation: These variations are accidental or residual and are due to wars, floods, droughts, famines etc. There is no definite explanation for these variations. But these events influence the business activities to a great extent and cause irregular variation in time series data.

Methods of Measuring Secular Trend:The following methods are used to measure secular trend:

(i) Graphical Method

(ii) Least Squares Method

(ii) Semi-average Method

(iv) Moving Average Method

(i) Graphical method: Here time series values are plotted on a graph taking time variable along the X-

axis and the other variable along the Y-axis. The plotted points are then joined by straight lines or by a

free-hand smooth curve. The straight line is drawn through the plotted points in such a manner that half of

the points remain in one side of this straight line. The line indicates the nature of trend (rising or falling)

eliminating the effect of seasonality, cyclical variations and irregular fluctuations.

Example: Fit a trend line to the following data by the graphical method:

Year 1970 1971 1972 1973 1974 1975 1976 1977

Sales of a firm(in million Taka) 62 64 66 63.5 67 64.5 69 67

Solution: Required trend line by the freehand method is drawn in the following diagram:

Merits:

(i)It is the simplest way of measuring trend.

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(ii)This method can be used for measuring both type of trend-linear or non-linear.

Demerits:

(i)This method is subjective. This graphical form may vary from person to person.

(ii)It is not useful for forecasting purpose.

(iii)Only experienced and technically sound individuals should use this method.

(ii) Method of Least Squares: If the trend is linear i.e., the points on the graph paper follow a straight line pattern, then the equation of the straight line is taken as

where and

Example : The average yearly death in a certain city is given below. Fit a straight line (linear) trend by the method of least squares.

Tabulate the tend values. Also estimate the death rate for the year 1962.

Year 1954 1955 1956 1957 1958 1959 1960Number of death (yearly average)

940 912 1055 1002 977 961 888

Solution: Here the number of years is 7, i.e., odd. Then we choose the origin as the middle year.

We take 1957 as origin (i.e. and unit of as 1 year.Table: Calculations for fitting a straight line

Year Number of Death

Trend value

1954 -3 940 9 -2820 976.721955 -2 912 4 -1824 971.861956 -1 1055 1 -1055 967.001957 0 1002 0 0 962.141958 1 977 1 977 957.281959 2 961 4 1922 971.861960 3 888 9 2664 976.72Total 0 6735 28 -136 -

Let (1)be the equation of the straight line.

and .

Putting these values of and in equation (1), the required equation of straight-line (linear) trend becomes

(2)with origin – 1957 and unit – 1 year.Putting the values of in the trend equation (2) we get the corresponding trend values which are shown in the table.Again, the value of for the year 1962 is 5. Hence putting the equation (2), we have the estimate Death – rate for 1962 which is

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Example: Fit a straight-line trend by the method of least squares to the following data:

Year 1980 1981 1982 1983 1984 1985Sales (in tons) 210 225 275 220 240 235

Find also the trend values and estimate the sales in 1987.

Solution: Here the number of years is 6, i.e. even. We choose the origin at the middle of 1982 and

1983 and unit of year. Then the values of corresponding to the years 1982 and 1983 will be –1

and 1 and other values of are calculated accordingly.

Table: Calculations for a fitting a straight line

Year Sales (in tons) Trend values

1980 -5 210 25 -1050 145.971981 -3 225 9 -2025 181.251982 -1 275 1 -275 216.531983 +1 220 1 220 251.811984 +3 240 9 720 287.091985 +5 235 25 1175 322.37Total 0 1405 70 1235 -

Let (1)be the equation of the straight line.

and .

The trend equation is therefore (2)

with origin – middle of 1982-83 and unit - year.

The trend values are calculated by substituting the values of in equation (2) and are shown in the table.For the year 1987, the value of . (For one year the difference of is 2). Hence, the estimate for sales in 1987 is

(tons)

(iii) Semi-Average Method: In this method the given time series is broken up into two equal halves. If the

series contains odd number of observations, then the middle most observation is omitted. Suppose we

are given data for 15 years starting from 1960 to 1974. Then omitting the middle year 1967, the two equal

halves are from 1960 to 1966 and 1968 to 1974. If the series contains even number of observations then

it clearly contains two equal halves. Arithmetic means of the two halves are plotted on a graph against the

mid-time points of the respective two halves. Then these two points are joined by a straight line. This line

indicates the nature of secular trend.

Example: Draw a trend line by the Semi- Average Method using the following data:

Year 1973 1974 1975 1976 1977 1978

Production of Steel(in lakh tons) 253 260 255 263 259 264

Solution: The average production of steel for the first three years = lakh tons.

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The average production of steel for the last three years = lakh tons.

Thus we get two points 256 and 262 which are plotted against the respective middle years (mid-points)

1974 and 1977 of two parts 1973-75 and 1976-78. By joining these two points, the required trend line is

obtained as shown in the following figure.

Merits and Demerits of the Method:

Merits:

(i)This method is very simple in comparison to moving average method or least squares method for

determination of trend.

(ii)This method is objective. Everyone will get the same straight line from the same set of data.

Demerits:

(i)In this method it is assumed that the independent variable (time) and the other variable (data) are

linearly related. This assumption is not true in case of business and economic time series.

(ii)Since arithmetic averages are greatly affected by extreme values, the trend determined on the basis of

simple arithmetic averages is also likely to be influenced by extreme values present in the time series.

(iv) The Method of Moving AverageExample: From the following data calculate 3 yearly moving averages:

Year 196

0

196

1

196

2

1963 196

4

1965 1966 1967 196

8

196

9

197

0

197

1

Productio

n(lakh

tons)

17.2 17.3 17.7 18.9 19.2 19.3 18.1 20.2 25.3 24.9 23.2 24.3

Solution: Calculation of 3 yearly moving averages

Year Production(lakh tons) 3 yearly moving total 3 yearly moving average

1960 17.2 ----------------- ------------

1961 17.3 52.2 =17.40

1962 17.7 53.917.96

1963 18.9 55.8 18.60

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1964 19.2 57.419.13

1965 19.3 56.618.87

1966 18.1 57.619.20

1967 20.2 63.621.20

1968 25.3 70.423.47

1969 24.9 73.424.47

1970 23.2 72.424.13

1971 24.3 -------------------

Example: Calculate trend value by the 4-yearly moving average method for the following data:

Year 199

1

199

2

199

3

1994 199

5

1996 1997 1998 199

9

200

0

200

1

2002

Value 41 61 55 48 53 67 62 60 67 73 78 76

Solution: Calculation of 4-yearly moving averages:

Year Value 4-yearly totals 4-yearly moving

average

2-period moving

totals

Centered moving

average (Trend

Values)

1991

1992

1993

1994

1995

41

61

55

48

53

------

------

205

217

223

-------

-------

51.25

54.25

55.75

105.50

110.00

113.25

52.75

55.00

56.63

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1996

1997

1998

1999

2000

2001

2002

67

62

60

67

73

78

76

230

242

256

262

278

294

------

------

57.50

60.50

64.00

65.50

69.50

73.50

------

------

118.00

124.50

129.50

135.00

143.00

59.00

62.25

64.75

67.50

71.63

Example: From the following table find three yearly weighted moving average taking 1, 2, and 1 as

weights:

Year 1 2 3 4 5 6 7

Sales(Lakh Taka) 2 4 5 7 8 10 13

Solution: Calculation of 3-yearly weighted moving average:

Year Sales 3-year weighting moving total3-yearly weighted

moving average

2001 2 ---------------- ---------

2002 4 3.75

2003 5 5.25

2004 7 6.75

2005 8 8.25

2006 10 10.25

2007 13 ----------------- ----------

* Column 4 = Column 3 Total weight, where total weight = 1 + 2 + 1 = 4.

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Example: Determine trend and short-term variation from the following data by using 3 yearly moving

averages. Draw the graph of the original series and trend the values on the same paper.

Year 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Production('000 tons) 21 22 23 25 24 22 25 26 27 26

Solution:

Year Production3 yearly moving

total3 yearly moving average Short term variation

1996 21 ------- --------- -------

1997 22 66 22.00 0.00

1998 23 70 23.33 -0.33

1999 25 72 24.00 1.00

2000 24 71 23.67 0.33

2001 22 71 23.67 -1.67

2002 25 73 24.33 -0.67

2003 26 78 26.00 0.00

2004 27 79 26.33 0.67

2005 26 ------- ------ ------

Merits and Demerits of Moving Average Method:

Merits:

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(i)In comparison to the least squares method, this method is simple.

(ii)There is some inherent flexibility in this method. Addition of a few new values only increases the grand

value and this has no effect on previous calculations.

(iii)If the period of moving average is equal to the average period of cycles, then this method completely

eliminates the effect of cyclical fluctuations.

Demerits:

(i)In this method some M. As. are lost in the beginning and at the end of the series.

(ii)This method is not suitable for forecasting purpose.

(iii)If the time series reveals linear trend only then this method is applicableMeasurement of Seasonal Variation :(1) Method of Average Example: Compute the average seasonal variations by the method of averages for the following data:

Year Total production of paper (‘000 tons)Quarter

I II III IV1989 37 38 37 401990 41 34 25 311991 35 37 35 41

Solution:

Year Productions (‘000 tons)I II III IV Total

1989 37 38 37 40 -1990 41 34 25 31 -1991 35 37 35 41 -Total 113 109 97 112 431Average (Total 3) 36.33 32.33 37.33

143.66

Seasonal variation = (Average – Grand average)

1.75 0.41 -3.59 1.41 -0.02

Grand average = 143.66 4 = 35.92

The seasonal variations for the quarters I, II, III, IV are 1.75, 0.41, -3.59, 1.41 respectively. The total of seasonal variations for the 4 quarters must be approximately zero.

Example : Compute the seasonal indices by the method of averages from the data given in the previous example.

Solution: Here another method is applied. Seasonal Index

Year ProductionI II III IV Total

1989 37 38 37 40 -1990 41 34 25 31 -1991 35 37 35 41 -Total 113 109 97 112 431Average 37.67 36.33 32.33 37.33 143.66Seasonal Index 104.87 101.14 90.00 103.92 399.93

The seasonal indices for the quarters I, II, III and IV are 104.87, 101.14, 90.00, 103.92 and the total must be approximately 400.(2) Method of Moving average or Ratio to moving average method:

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It eliminates the trend, cyclical and irregular components from the original data. Here I refer trend, C to cyclical, S to seasonal and I to irregular variation. The numbers that result are called the typical seasonal index.Example: For the following data determine a quarterly seasonal index using the ratio to moving average method.

Year Total production of paper (’00,000 tons)Quarter

I II III IV1989 34 32 31 361990 37 34 33 411991 43 40 38 48

Solution:Year Production

(1)

4-quarter moving total

(2)

Four quarter moving average

(3)

Centered moving average

(4)

Ratio to moving average (%)

(5)

1990

I 34II 32

133 33.25III 31 33.62 92.21

136 34.00IV 36 34.25 105.11

138 34.50

1991

I 37 34.75 106.47140 35.00

II 34 35.62 95.45145 36.25

III 33 37.00 89.19151 37.75

IV 41 38.50 106.49157 39.25

43 39.87 107.85

1992

I162 40.50

II 40 41.37 96.69169 42.25

III 38IV 48

Note:

Calculation of Seasonal Index

Year Ratio to moving average (%)I II III IV Total

1990 - - 92.21 105.11 -1991 106.47 95.45 89.19 106.49 -1992 107.85 96.69 - - -Total 214.32 192.14 181.4 211.6 -Average (Total 2) 107.16 96.07 90.7 105.8 399.73Seasonal Index 107.23 96.14 90.76 105.87 400

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Grand average

The seasonal indices for the quarters I, II, III and IV are 107.23, 96.14, 90.76, 105.87 respectively.

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