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Time Series Analysis 367 Solution: Year Index Three- Three-yearly No. yearly Total Moving Total Average 1914 87 i 1915 62 196 65.33 1916 47 133 44.33 1917 24 116 38.67 1918 45 126 42.00 1919 57 198 66.00 1920 96 250 83.33 1921 97 277 92.33 1922 84 260 86.67 1923 79 240 80.00 1924 77 236 78.67 1925 80 249 83.00 1926 92 278 92.67 1927 106 311 103.67 1928 113 Problem 7: Calculate the five-yearly moving averages of students studying in a commerce college as shown by the following figures. Year: 1991 92 93 94 95 96 97 98 99 2000 No. of students: 332 317 357 392 402 405 410 427 405 438 Draw a graph to represent the moving averages.

Solution: No. of Five-yearly Five-yearly Students Total Moving Average Year 332 1800 360.00 1991 317 1873 374.60 1992 357 1966 393.20 1993 392 2036 407.20 1994 402 2049 409.80 1995 405 2085 417.00 1996 410 1997 427 1998 405 1999 438 2000 CU IDOL SELF LEARNING MATERIAL (SLM)

368 Business Mathematics and Statistics Problem 8: The following table shows the yearly available supplies of all cereals per adult equivalent of population in India for a number of years. Draw a graph to represent the data: Year: 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 Supplies in Kg.: 214 208 204 204 190 203 195 180 191 176 178 179 Calculate three-year moving averages and plot them on the same chart. What is the deviation of the actual value from the moving average for the year 19867 Solutions: Year Supply Three Yearly Three Yearly Total Moving Average 1981 214 1982 208 626 208.67 1983 204 616 205.33 1984 204 598 199.33 1985 190 597 199.00 1986 203 588 196.00 1987 195 578 192.67 1988 180 566 188.67 1989 191 547 182.33 1990 176 545 181.67 1991 178 533 177.67 1992 179

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Time Series Analysis 369 Problem 9: Calculate four-yearly moving averages for the following data: Year Bank Four Yearly Two Yearly Four Yearly Clearances Total Total of Four Moving 1986 1987 53 Yearly Totals Average 0 1988 79 70.500 1989 76 0 0 74.375 1990 66 274 564 79.875 1991 69 290 595 86.125 1992 94 305 639 90.000 1993 105 334 689 1994 87 355 720 79 365 Problem 10: Calculate five-yearly moving averages for the following time series: Year: 1 2 34 5 67 8 9 Annual figure: 78 67 107 142 152 155 160 177 155 Year Annual Fig. Five Yearly Five Yearly 1 78 Total Moving Average

2 67 3 107 546 109.2 4 142 623 124.6 5 152 716 143.2 6 155 786 157.2 7 160 799 159.8 8 177 9 155 CU IDOL SELF LEARNING MATERIAL (SLM)

370 Business Mathematics and Statistics Problem 11: Calculate four-yearly centered moving averages for the production in million tonnes for the following data. Year Production Five-yearly Fived-yearly Total Moving Average 1990 65 316 63.2 1991 62 319 63.8 1992 61 320 64 1993 63 326 65.2 1994 65 323 64.6 1995 68 317 63.4 1996 63 1997 67 1998 60 1999 59 Solution: Year Production Four-yearly Two-yearly total Four-yearly Total of Moving Average Four-yearly Total 1990 68 254 1991 62 251 505 63.125 1992 61 259 508 63.500 1993 63

1994 65 263 651 64.500 1995 68 258 621 65.250 1996 63 258 521 65.250 1997 67 249 507 63.375 1998 60 1999 59 CU IDOL SELF LEARNING MATERIAL (SLM)

Time Series Analysis 371 Problem 12: Fit a straight line trend by the method of least squares to the following data: X (Years): 1991 1992 1993 1994 1995 1996 1997 92 (Production): 80, 90 92 83 94 99 Years XY XY X2 1991 1 80 80 1 1992 2 90 180 4 1993 3 92 276 9 1994 4 83 332 16 1995 5 94 470 25 1996 6 99 594 36 1997 7 92 644 49 28 630 2576 140 Substituting the values N = 7, 6Y = 630, 6X = 28 in the two normal equations: (1) we get 630 = 7a + 28b (2) i.e., 90 = a + 4b ... 2576 = 28a + 140b i.e., 92 = a + 5b... Subtracting (1) from (2) we get

b = 2 and a = 90 – 4b = 90 – 8 = 82 The equation to the line of the best fit: Y = 82 + 2x By putting X= 1, 2, 3, 4, 5, 6, 7 in the equation we obtain trend values 84, 86, 88, 90, 92, 92, 94, 96 respectively. x = 1, Y = 82 +2=84 x = 12 Y = 82 + 4 = 86 x = 3, Y = 82 +6=88 x = 4, Y = 82 +8=90 x = 5, Y = 82 +10=92 x = 6, Y = 82 +12=94 x = 7, y = 82 + 14 = 9 6 CU IDOL SELF LEARNING MATERIAL (SLM)

372 Business Mathematics and Statistics Problem 13: The following are the annual profits in thousands of rupees of a certain business. Year: 1991 1992 1993 1994 1995 1996 1997 Profit in thousands of rupees: 60 72 71 65 80 85 95 Use the method of least squares to fit a straight line to the above data. Plot the above figures and draw the line. Also make an estimate of the profits ~ 1998. Solution: Let the equation to the line of best fit be Y = a + sx, where Y represents profits in thousands of rupees and X the years. To find the values of the constants a and b we prepare the following table: Year Y X XY X2 1991 1992 60 –3 –180 9 1993 72 –2 –144 4 1994 75 –1 –75 1 1995 1996 65 0 00 1997 80 1 80 1 85 2 170 4 95 3 285 9 Total 528 0 136 28 From the table, we have N = 5, 6X2 = 28, 6Y = 528, 6XY = 136. Substituting these values in the two simplified equations we get: ¦ Y 532

a = N 7 = 76 And, 136 ¦XY 28 = 4.86 b = ¦X2 Therefore the equation to the line of best fit is: = 76 + 4.86X By putting, X = –3, –2, –1, 0, +1, +2, +3 in this equation, we get the trend values as follows: 61.42, 66.28, 71.14, 76.00, 80.86, 85.72, 90.58. Putting X = –3, y = 76 + (4.86) (–3) = 76 – 14.58 = 61.42 CU IDOL SELF LEARNING MATERIAL (SLM)

Time Series Analysis 373 Putting X = –2, Y = 76 + (4.86) (–2) = 76 – 9.72 = 66.28 Putting X = –1, y = 76 + (4.86) (–1) = 76 – 4.86 = 71.14 Putting X = 0, y = 76 + (4.86) (0) = 76.00 Putting X = 1, y = 76 + (4.86) (1) = 80.86 Putting X = 2, y = 76 + (4.86) (2) = 76 + 9.7285.72 Putting X = 3, y = 76 + (4.86) (3) = 76 + 14.58 = 90.58 By putting X = 4, we the profits for 1958 are: = 76 + (4.86) (4)

= 76+19.44 = 95.44 Profits for 1998 = ` 95.44 (in thousands) Problem 14: Given below are the production figures (in thousand maunds) of a sugar factory. Obtain the secular trend by fitting a straight line. Year: 1990 1991 1992 1993 1994 1995 1996 Production: 12 10 14 11 13 15 16 Also plot the trend values to indicate the trend line. CU IDOL SELF LEARNING MATERIAL (SLM)

374 Business Mathematics and Statistics Solution: Let the equation to the line of best fit be Y = a + bX where Y represents the sugar production in thousand maunds. To find the values of the constants a and b we prepare the following table. In this table, X stands for the deviation of the years from 1993. Year Y X XY X2 1990 12 –3 36 9 1991 10 –2 –20 4 1992 14 –1 –14 1 1993 11 00 0 1994 13 +1 +13 1 1995 15 +2 +30 4 1996 16 +3 +41 9 91 0 21 28 From the table, we have N = 7, 6X2 = 28, 6Y = 91, 6XY = 21. Substituting these values in the two simplified equation we get: ¦ Y 21 a = N 31 = 13 And, z XY 21 b = ¦X2 28 = 0.75

Therefore the equation to the line of best fit is: Y = 13 + 0.75X By putting X = –3, –2, –1, 0, +1, + 2, +3 in this equation, we get the following trend values: 10.75, 11.50, 12.25, 13.00, 13.75, 14.50, 15.25. These trend values may be plotted on the graph and the line of best fit may be drawn. 16.6 Summary An orderly set of numbers written in the order corresponding to time is called a time series. For instance, a series of terms showing the profits of a business concern over a number of years is a time series because the terms of series representing profits are written in relation to time. A CU IDOL SELF LEARNING MATERIAL (SLM)

Time Series Analysis 375 careful scrutiny of a time series along with a detailed analysis of condition in the past enables one to forecast future changes and estimates. The following is a time series showing the production of cereals in India in million tonnes. Net Production (million tonnes)* of Cereals in India Year Production 1951 40.01 1952 40.59 1953 45.36 1954 53.44 1955 51.59 1956 50.33 1957 52.67 1958 49.35 1959 57.29 1960 56.76 1961 60.66 1962 62.08 1963 58.43 A detailed analysis of the above time series enables us to find the different types of movements influencing its course. z General Trend z Seasonal Changes

z Cyclical Fluctuations z Irregular Variations z General Trend: The time series of the net productions of cereals in India shows that there is certainly a gradual increase in the production of cereals even though the increase is not regular. This gradual increase is called the “General Trend” or secular trend. z Seasonal Fluctuations: Eliminating seasonal influences is no easy task. The occurrence of seasons is responsible for these fluctuations. Natural phenomena are always unavoidable beyond any measure of control. CU IDOL SELF LEARNING MATERIAL (SLM)

376 Business Mathematics and Statistics ¦ Cyclical Fluctuations: These are due to the occurrence of business or trade cycles in the economic world. For instance; there may be a period of depression followed by a boom after a few years. ¦ Irregular Variations: In many time series of economic data, we observe sudden upward or downward movements in the trend. Such movements are irregular and are brought about by unforeseen events and sudden changes in economic activities and business conditions. A general trend can be measured by the following methods: ¦ Graphical method-freehand curve. ¦ Method of moving averages. ¦ Method of least squares. Graphical Method: This method consists in drawing the curve of the time series on graph paper. Method of Moving Averages: Moving Averages are calculated on the basis of a given period, such as, 3-yearly or 4-yearly or 5-yearly. Method of Least Squares: Instead of drawing an observation-trend-line for a time series freehand, it is better to fit a straight line trend more accurately using the mathematical method of least squares. This straight line so drawn is called the ‘Line of Best Fit’. If the equation to the line of best fit is written as Y = a + bx, then the values of the constants a and b are determined by using the following two normal equations: 6Y = Na + b6x

6XY = a6X + b6x2 Where X represents the year (time). (iv) represents the actual values of the terms in the series. N is the number of terms in the time series. The graph that represents a time series is called a ‘Historigram’. It is so called, simply because it represents a historical series which is another name for a time series. CU IDOL SELF LEARNING MATERIAL (SLM)

Time Series Analysis 377 Parabolic Trend Method: The straight line trend shows the increase or decrease in a time series by a constant amount. However, in certain cases the straight line trend may not be adequate and appropriate. In such a case the trend is indicated by a non linear curve and not by a straight line. The equation of the parabola is given by Y = a + bX + CX2 where the values of a, b, and c can be determined by solving the following normal equations: 6Y = Na + b6x + c6X2 6XY = b6x + b6x2 + c6X2 6X2Y = a6X2+ b6X3 + c6X4 When trends are plotted on a semi-logarithmic chart and arc represented by a non-linear curve, then an upward curve would indicate the increase at different rates. The rate of ioncrease depends on the slope-steeper the slope, higher the rate of increase. Using the logarithmic procedure we can calculate the exponential trends. 16.7 Key Words/Abbreviations General trend, Seasonal, Cyclical, Irregular, Variation, Fluctuation, Moving Averages, Least Sequences Method. 16.8 Learning Activity

(E) Calculate five-yearly moving averages of students in a commerce college as shown by the following figures: Year No. of Students 1 1402 2 1405 3 1410 4 1427 s 1405 6 1438 7 1445 8 1447 CU IDOL SELF LEARNING MATERIAL (SLM)

378 Business Mathematics and Statistics 9 1480 10 1482 11 1482 12 1500 5 The following are the index numbers of annual production of a certain commodity. Assuming a five-yearly cycle, find the trend values. Year Production 1991 325 1992 310 1993 301, 1994 315 1995 323 1996 345 1997 335 1998 325 1999 333 2000 349

2001 365 11. Draw a graph to represent the following data showing the number of students in a college: Year No. of Students 1985 705 1986 685 1987 703 1988 687 1989 705 1990 689 1991 715 1992 685 1993 725 1994 751 Calculate the five-yearly moving averages of the above data and plot them on the same chart. CU IDOL SELF LEARNING MATERIAL (SLM)

Time Series Analysis 379 4. The following series relate to the profits of a commercial concern for eight years. Year Profits (`) 1995 15420 1996 14470 1997 15520 1998 21020 1999 26120 2000 31950 2001 35370 2002 35670 Find the trend of profits. Assume a three year cycle and ignore decimals. 5. Find the trend of bank clearances by moving averages (assume a five-yearly cycle). Years Bank Clearances (` in crores) y 53 I 79 y2

y 3 76 y4 66 y 5 69 y6 94 y 7 105 y8 87 y 9 79 y 10 104 y 11 97 y 12 92 y 13 101 16.9 Unit End Questions (MCQ and Descriptive) A. Descriptive Type: Short Answer Type Questions 11. Explain clearly what is meant by time series analysis. Indicate fully the importance of such analysis in business. CU IDOL SELF LEARNING MATERIAL (SLM)

380 Business Mathematics and Statistics ? Describe the various types of fluctuations influencing a time series. Indicate fully the procedure of estimating the secular trend. ? Explain what is meant by long term trend of a time series. Use the method of moving averages (three-yearly) to determine the trend in the following series showing index numbers for values of imports into India during 1984-1998. 87, 62, 47, 24, 45, 57, 96, 97, 84, 79, 77, 80, 92, 106 and 113. ? What are the components of a time series? Explain the importance of the moving average method in the analysis of time series. ? Distinguish between regular and irregular fluctuations in a time series. ? Distinguish between seasonal variations and cyclical fluctuations. ? Explain how a “growth factor”, a “decline factor”, a “seasonal factor II and a “cyclical factor” affect a variable over a period of time. ? Write a short essay on “Components of a Time Series”. ? Explain the meaning and use of a moving average. ? Discuss the different methods of determining secular trend in a time series with their relative merits and demerits. ? ’ Write a note on Trend in a Time Series . ? Discuss any two methods of determining the secular trend in a time series with their merits and demerits. ? What is a “historical series”? What are its main components? Discuss briefly anyone method of determining the general trend present in a given historical series. ? Calculate the five-yearly moving averages of students in a commerce college as shown by the following figures.

Year No. of Students 1 402 2 405 3 410 4 427 5 405 CU IDOL SELF LEARNING MATERIAL (SLM)

Time Series Analysis 381 6 438 7 445 8 447 9 480 10 482 11 482 12 500 = Explain the “moving averages method II to find the secular trend in a time series. Mention its uses and limitations. = (a) What is a time series? What is meant by the analysis of time series? Discuss the practical utility of the analysis of a time series. The following are the index numbers of the annual production of a certain commodity. Assuming a five-yearly cycle, find the trend values. Year Index Num 1991 225 1992 210 1993 201 1994 215

1995 223 1996 245 1997 235 1998 225 1999 233 2000 249 2001 265 ? Explain the method of moving averages in finding the trend in a time series data. ? What is a “secular trend”? State the important method of finding the secular trend, explaining in detail anyone of them. ? (a) What is a time series? State the various components of a time series? What is the purpose of such an analysis? Determine the trend component for the following time series using the five-yearly moving averages. CU IDOL SELF LEARNING MATERIAL (SLM)

382 Business Mathematics and Statistics Year Index Number (in lakh) 1988 1989 80 1990 79 1991 98 1992 77 1993 64 1994 51 1995 50 1996 54 1997 53 1998 61 1999 123 2000 119 2001 120 98 ? Calculate four-yearly centered moving averages production in million pounds for the following data: Year Prod 1990 68

1991 62 1992 61 1993 63 1994 65 1995 68 1996 63 1997 67 1998 60 1999 59 CU IDOL SELF LEARNING MATERIAL (SLM)

Time Series Analysis 383 B. Multiple Choice/Objective Type Questions: 1. A time series is also called __________. (a) Value series (b) Historical series (c) biographical series (d) None of these. 2. General Trend in a time series is also called _____________. (a) Secular Trend (b) Fine Trend (c) Peculiar Trend (d) None of these 3. The graph that represent a time series is called a __________. (a) Biographical (b) Historical (c) Historigram (d) All the above … Parabolic Trend method is the straight line trend shows the increase or decrease in a time series by __________ amount. (a) Varying (b) Constant (c) Recurring (d) All the above Answers: (1) (b); (2) (a); (3) (c); (4) (b) 16.10 References References of this unit have been given at the end of the book.

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384 Business Mathematics and Statistics REFERENCES ( Dr. A.B. Rao, “Business Mathematics”, Himalaya Publishing House Pvt. Ltd. ( Dr. A.B. Rao, “Business Statistics”, Himalaya Publishing House Pvt. Ltd. ( Dr. A.B. Rao, “Decision-making”, Himalaya Publishing House Pvt. Ltd. ( Dr. A.B. Rao, “Operation Research”, Jaico Publishing House Pvt. Ltd. ( Dr. A.B. Rao, “Quantitative Techniques in Business”, Jaico Publishing House Pvt. Ltd. ( Lenin Jothi, “Business Mathematics”, Himalaya Publishing House Pvt. Ltd., Mumbai. ( http://data.conferenceworld.in/SGTB/P726-729.pdf ( https://www.inc.com/encyclopedia/annuities.html ( https://blog.myrank.co.in/combination-properties-of-%E2%81%BFcr/ ( https://blog.myrank.co.in/applications-of-permutation-and-combination/ ( https://www.hindawi.com/journals/mpe/2010/723402/ ˆˆ

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