1. Time Series Analysis

2. TIME – SERIES ANALYSIS : Introduction : Time – series has immense importance in the economic and business fields. Most of the economic and business data like prices, national income, population, imports, exports, production, consumption,sales, profits etc. are collected on the basis of time(such as days, months, years). Meaning : The set of data collected on the basis of time ( such as days, months, years ) is called as time – series. Under time – series, there are two types of variables (1) independent variables – which represent time and (2) dependent variables – which represent changes taking place in the value of data (sales etc) with the passage of time. YEARPRODUCTION 1975 50 TONNES 1976 52 TONNES 1977 53 TONNES 1978 55 TONNES 1979 60 TONNES TABLE – 1 YEARPRODUCTION 1931 27.9 TONNES 1941 31.9 TONNES 1951 36.1 TONNES 1961 43.9 TONNES 1971 54.5 TONNES TABLE – 2

3. Utility of time – series : To study the past behaviour of the data : With the help of time – series changes occurred in the past are studied. To forecast future behaviour : With the help of time – series anticipation of changes going to occur in the future become possible because studies about past prove to be very useful for forecasting future. estimation of trade-cycles : Cyclical fluctuations in a time-series give idea about the changes taking place in the business like boom, recession, depression and recovery. Comparison with other time-series : It is also important for the comparison of various time-series. Study of present variations : It is also helpful in studying the present variations in different economic variables like imports, exports etc. Universal utility : It also benefits all classes like businessmen, farmers, consumers, economists etc.

4. Time Series Components Trend Seasonal Cyclical Irregular

5. COMPONENTS OF TIME-SERIES : (1). Secular trend T : It refers to the general tendency of the data to grow or decline over a long period of time. It is of two types : (A) Linear trend – when long term rise or fall in a time-series takes place by a constant amount it is called linear trend. Y = a + bX (B) Parabolic trend – the trend is said to be parabolic when long term rise or fall in a time-series is not taking place at a definite rate. Y = a + bX + cX (2). Seasonal variations S: it refers to periodic variations in time-series which occur regularly within a period of 12 months. (3). Cyclical variations C : it refers to oscillatory variations in a time-series which have a duration anywhere between 2 to 10 years (4). Irregular variations I – it refers to htose short term variations which occur irregularly due to certain accidental causes. Prosperity Recession Recovery Cyclical variations Depression Trend Line PHASES OF BUSINESS CYCLE

6. Trend Component Overall Upward or Downward Movement Data Taken Over a Period of Years Sales Time Upward trend

7. Cyclical Component Upward or Downward Swings May Vary in Length Usually Lasts 2 - 10 Years Sales Time Cycle

8. Seasonal Component Upward or Downward Swings Regular Patterns Observed Within One Year Sales Time (Monthly or Quarterly) Winter

9. Irregular Component Erratic, unsystematic, ‘residual’ fluctuations Due to random variation or unforeseen events – Union strike – War Short duration & nonrepeating

10. Random or Irregular Component Erratic, Nonsystematic, Random, ‘Residual’ Fluctuations Due to Random Variations of – Nature – Accidents Short Duration and Non-repeating

11. DECOMPOSITION OF TIME-SERIES : Time-series is composed of four components viz. trend (T), seasonal variaions (S), cyclical variations (C) and irregular variations (I). Models of analysing time-series : (1). Additive model – this model is based on the assumption that time-series is the sum of the four components. O = T + S + C + I On the basis of additive model, the anaysis of various components is give below O – T = S + C +I O – T – S = C + I O – T – S – C = I (2). Multiplicative model – This model is based on the assumption that a time-series is the product of four components. O = T x S x C x I = TSCI It can be expressed in the following forms O/T = SCI O/TxS = CI O/TxSxC = I

12. SEMI-AVERAGE METHOD : In this method, first of all time-series is divided into two equal parts and thereafter, separate arithmetic mean is calculated for each part. It can be applied in cases of two situations : (1). When the number of years in a series is even – when the given number of years in a series is even like 4,6,8 etc. then the series can be easily divided into two equal parts. 20002001200220032004200520062007 412438444454470482490500 YEAR : SALES (LAKHS) YEARSALES 2000 2001 2002 2003 412 438 444 454 2004 2005 2006 2007 470 482 490 500 SEMI-TOTALSEMI-AVERAGEMIDDLE YEAR 1748 1942 X=1748/4=437 X=1942/4=485.5 2001.5 2005.5

13. (2). When the number of years in a series is odd : When the number of years in a series is odd like 5,7,9 etc. Then there will be a problem in dividing the series into equal parts. In such a case, the mid-year figure is to be dropped. For eg. If 1981 to 1989 (i.e. 9 years ) figures are given, then we will delete 1985, i.e. 5 th year and its corresponding figure and we will make 4-4 years’ parts i.e. 1981-1984 and 1986-1989. The remaining process will be the same as before. 1981198219831984198519861987 20222726302940 YEAR PROFIT (000) 198120 198222 198327 198426 198530 198629 198740 YEARPROFIT(000)SEMI-TOTALSEMI-AVGMIDDLE YR = 69 X = 69 /3 = 23 = 1982 = 99X = 99/3 = 33= 1986

14. Merits and demerits of semi-average method : Merits : (1). It is an easy method (2). It is free from bias (3). Trend values thus obtained are definite. (4). Less time and effort is involved in drawing the trend line. Demerits : (1). It is based on straight line trend assumption which does not always hold true. (2). It is affected by extreme values. (3). It ignores the effect of cyclical fluctuations.

15. Moving Average Method

16. MOVING AVERAGE METHOD : This method is very widely used in practice. Under this method, moving averages are calculated. In moving averages, one has to decide what moving year average – 3 year, 4 year, 5 year, 7 year should be taken up. It is studied in two different situations : (1). Odd period average method - when period of moving average is odd, then the following steps need to be taken ; (a) add up the values of corresponding to first 3 years in the time-series and put the sum before the middle year (i.e. 2 nd year) (b) leaving the 1 st year value, add up 2 nd, 3 rd and 4 th year value and put the sum in front of the middle year (i.e. 3 rd year). Carry this process further till we reach the last value of the series. (c) moving totals thus obtained are to be divided by the period of the moving average and show the trend values of different years.

17. Moving Average Graph Year Sales Actual

18. 1981198219831984198519861987 412438446454470483490 YEAR PRODUCTION YEARPRODUCTION3 YEARLY MOVING TOTALS 3 YEARLY MOVING AVERAGE 19 1981412 1982438 1983446 1984454 1985470 1986483 1987490 ----- 412+438+446=1296 1296/3 = 432 438+446+454 =1338 1338/3 = 446 446+454+470 =1370 1370/3 = 457 454+470+483 =1407 1407/3 = 469 470+483+490 =1443 1443/3 = 481 ----- ------

19. EVEN PERIOD MOVING AVERAGE : When moving average period is even, say 4 years, then moving averages have to be centered. It can be computed by two methods : (a). First method – the computation procedure of 4 yearly moving averages is as follows : (1). Add up first 4 values corresponding to the first 4 years and put the sum in between 2 nd and 3 rd year. The next total (i.e. from 2 nd to the 5 th year total) is to be put in between 34d and 4 th year. Carry on this process till the last value of the series. (2). Now add up the 1 st and 2 nd 4 year totals and put them in front of 3 rd year. Similarly, add up 2 nd and 3rd 4 year total and put them in front of the 4 th year. Carry on this process till the last value (3). 8 years’ totals thus obtained are to be divided by 8. These values are 4 yearly moving averages and show the trend values for different years. YEAR SALES (CRORES) 1970197119721973197419751976197719781979 78911101286510

20. YEARSALES (CRORES)4 YRLY MVNG TOTALS 2 PERIOD MVNG TTLS OF 4 YRLY 4 YRLY MVNG AVGS CENTERED 19707 ----- ------ 19718 ----- ------ 7+8+9+11=35 19729 35+38=73 73/8 = 9.125 8+9+11+10=38 197311 38+42=80 80/8 = 10 9+11+10+12=42 197410 42+41=83 83/8 = 10.375 11+10+12+8=41 197512 41+36=77 77/8 = 9.625 10+12+8+6=36 19768 36+31=67 67/8 = 8.375 12+8+6+5=31 19776 31+29=60 60/8 = 7.5 8+6+5+10=29 19785 ----- -------- 197910 197910 ----- --------

21. Freehand Curve Method

22. Freehand Curve Freehand Curve [Graph] Year Production Actual

23. METHODS OF MEASURING TREND : (1). Free-hand curve method – this is the simplest method of trend fitting. This method is also called trend fitting by inspection. Merits and demerits of freehand curve method are : Merits : (a). It is simple. (b). It is flexible. (c). No mathematical formula is used in this method. (d). It is also used for forecasting about future. Demerits : (a). It is based on subjective judgements. So bias may affect its findings. (b). There is lack of accuracy. (c). Long-run movement obtained from this method is not definite. 1971197219731974197519761977197819791980 150155165152174150174175160180 YEAR : PRODUCTION

25. LEAST SQUARE METHOD This is the best method of trend fitting in a time series and is most used in practice. This is a mathematical method and a trend line in this method is fitted or obtained in such a way that the following two conditions are fulfilled: 1.(y- )=0, i.e., the sum of the deviations of the actual values of Y and computed trend values ( ) is zero. 2.(y- is least, i.e. the sum of square of deviations of the actual and computed trend values from this line is least. The trend line thus fitted under this method is called as the line of best fit.

26. (A)FITTING OF STRAIGHT LINE TREND A straight line trend can be expressed by the following equation Y=a+bX Where Y= trend values X=unit of time a is the Y-intercept and b is the slope of line Y= Na+ b x XY= a X + b

27. There are two methods of computing Straight line trend by using Least square method: 1. Direct method: The procedure to compute straight line trend in this method is as follows- (i) Any year is taken as the year of origin. Usually first year or before that is taken as zero, deviations of other years are marked on 1,2,3…etc. Time deviations are denoted by X. (ii) Then X, Y, XY and XY and are computed. (iii) the values computed are put in the following normal equations: Y= Na+b X XY- a X+b The values of A and B are determined by solving the above set two normal equations (iv) Finally, the calculated values of a and b are put in Y = a + bX and trend values are computed.

28. Example 1 Fit a straight line trend by the method of least square to following data: Also obtain the trend values. 197919801982198319841985 579101217 YEAR PRODUCTION LAKH TONS

29. Solution : YEARPRODUCTION LAKH TONS (Y) DEVIATIONS FROM 1978 (X) XY 19795151 198072144 198193279 19821044016 19831256025 198417610236 N = 6 Y = 60 X = 21 XY = 248 = 91

30. The straight line trend is ddefined by the equation Y= a+bX Two normal equations are Y= Na+b X XY- a X+b Substituting the vales we get 60=6a+21b..(i) 248=21a+91b..(ii) Solving the two equations (i) and (ii) Multiplying (i) by 7 and (ii) by 2 and subtracting 420= 42a+147b 496= 42a+182b - - -. -76= -35b b=-76/-35=2.17

31. By stubstituting the values of b 60=6a+21(2.17) 6a=14.43 a=2.40 Hence, the trend equation is Y= 2.40+2.17X; Origin= 1978,X unit= 1 year. Computing he trend values For 1979, X= 1,Y= 2.40+2.17(1)= 4.57 For 1980, X= 2,Y= 2.40+2.17(2)= 6.74 For 1981, X= 3,Y= 2.40+2.17(3)= 8.91 For 1982, X= 4,Y= 2.40+2.17(4)= 11.08 For 1983, X= 5,Y= 2.40+2.17(5)= 13.25 For 1984, X= 6,Y= 2.40+2.17(6)= 15.42

32. (B) Fitting Of Second Degree Parabolic Trend Or Quadratic Trend There may be many such situations in economics and business fields in which a straight line trend may not represent the long-term tendency of the time series data. In such cases, a second degree parabolic trend or quadratic trend is fitted. The equation is: Y = a + b X + c X 2 Where, a is the Y- intercept, b is the slope of the curve at the origin and c is the rate of change in slope. Under the method of least square the values of the constants a, b and c are obtained by solving the following three normal equations: Y = N a + b X + c X 2 XY = a X + b X 2 + c X 3 X 2 Y = a X 2 + b X 3 + c X 4

34. Short-cut Method: If the time deviations are taken from the middle year (or arithmetic mean of two middle years), the values of X and X 3 would be zero. TYPE 1: ODD Number of Years EXAMPLE:- Fit a second degree parabolic trend to the following data. YearProduction (Y) Deviations from 1985 (X) XYX 2 YX2X2 X3X3 X4X4 19835-2-10204-816 19847-7711 19854000000 19869199111 198712224484816 N = 5270168410034

35. The second degree parabolic trend is given by the equation Y = a + b X + c X 2 The three normal equations are: Y = N a + b X + c X 2 XY = a X + b X 2 + c X 3 X 2 Y = a X 2 + b X 3 + c X 4 Substituting the values, we get 37 = 5a + b(0) + c(10)  37 = 5a + 10c …(i) 16 = a(0) + b(10) + c(0)  16 = 10b …(ii) 84 = a(10)+ b(0) + c(34)  84 = 10a + 34c...(iii)

36. Solving (i), (ii) & (iii), we get a = 5.98 = 6 (approx.) b = 1.6 c = 0.71 Thus, the second degree parabolic trend is Y = 6 + 1.6X + 0.71 X 2, Origin = 1985, X unit = 1 year Computation of Trend Values For 1983, X = -2, Y = 6 + 1.6(-2) + 0.7(-2) 2 = 5.64 For 1984, X = -1, Y = 6 + 1.6(-1) + 0.7(-1) 2 = 5.11 For 1985, X = 0, Y = 6 + 1.6(0) + 0.7(0) 2 = 6 For 1986, X = +1, Y = 6 + 1.6(+1) + 0.7(+1) 2 = 8.31 For 1987, X = +2, Y = 6 + 1.6(+2) + 0.7(+2) 2 = 12.04

37. TYPE 2 : EVEN Number of Years EXAMPLE:- Fit a second degree parabola to the following data. YearPrice (Y) Deviations from 1966.5 Deviations multiplied by 2 (X) XYX2YX2YX2X2 X3X3 X4X4 1994100-2.5-5-500250025-125625 1995170-1.5-3-3219639-2781 1996128-0.5-12812811 1997140+0.5+1+1401401+11 1998181+1.5+3+54316290+2981 1999192+2.5+5+960480025+125625 N = 68480694101607001414

38. Substituting the values in the above mentioned three normal equations, we get 848 = 6a + b(0) + 70c  848 = 6a + 70c …(i) 694 = a(0) + b(70) + c(0)  694 = 70b …(ii) 10160 = 70a + b(0) + 1414c  10160 = 70a + 1414c …(iii) Solving (i), (ii) & (iii), we get a = 136.125 b = 9.914 c = 0.4464 Thus, second degree parabola is Y= 136.125 + 9.914X + 0.4464X 2, Origin:1996.5, X unit= ½ year

39. (B) Fitting Of Exponential Trend If the time series is increasing or decreasing by a constant percentage rather than constant absolute amount, the fitting of exponential trend is considered appropriate. Such tendency is found in many economic and business data. The equation is Y = ab X Where a is Y- intercept and b is the slope of the curve at the origin of X. In the logarithmic form, the above equation is written as under: log Y = log a + X log b When plotted on a semi logarithmic graph, the curve gives a straight line (or called logarithmic straight line). However, on an arithmetic scale chart the curve gives a non-linear trend. Under the method of least square, the values of the constants a & b are obtained by solving the following two normal equations: log Y = N log a + log b X (X log Y) = log a X + log b X 2

40. The equation of the exponential trend is Y = ab X ….(i) Taking log both sides, we have log Y = log a + X log b.…(ii) Since, X = 0. Therefore, log a = 1.2341 & log b = 0.075 Thus, the exponential trend equation in log form is: log Y = 1.2341 + 0.075X Here, Origin = 1993, X unit = 1 year

41. TYPE 2: EVEN number of years EXAMPLE: Fit an exponential trend to the following data. YearPopulation (X) Xlog YX log Y X2X2 194131.9-51.5038-7.519025 195136.1-31.5575-4.67259 196143.91.6425-1.64251 197154.8+11.7388 1 198168.3+31.83445.50329 199184.4+51.92639.631525 N = 6010.20333.039570

42. The equation of the exponential trend is Y = ab X ….(i) In log form the equation is written as log Y = log a + X log b ….(ii) Since, X = 0. Therefore log a = 1.70 & log b = 0.043 Thus, the exponential trend equation in log form is: log Y = 1.70 + 0.043X Here, Origin = 1996,X unit = 5 years

43. Merits and Demerits of Least Square Method: MERITS: (i)This method is far better than moving average method because the trend values for all the years are obtained. Not even a single initial or terminal trend values is left over in this method. (ii)It results in a mathematical equation which may be used for forecasting. (iii)It is widely used method of fitting a curve to the given data. The results obtained are reliable and appropriate.

44. DEMERITS: (i)The computation process in this method is complex which is not easily understandable. (ii)This method does not have the attribute of flexibility. If some figures are added to or subtracted from the original data, all computations have to be redone. (iii)It is difficult to select an appropriate type of equation in this method. Results based on inappropriate selection of equation are likely to be misleading.