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# Stats 5

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### Stats 5

1. 1. Time Series Generally, planning of economic and business activities is based on predictions of production, demand, sales etc. The future can be predicted by a detailed study of the past variations. Thus, future demand can be predicted by studying the variations in the demand for last few years. A time series may be defined as a collection of readings belonging to different time periods, of some economic variable or composite of variables. A series of observations of a phenomenon recorded at successive points of time is called Time Series. It is a chronological arrangement of statistical data regarding the phenomenon. Generally, time series are those of production, demand, sales, price, imports, exports, bank rate, value of money, etc. Usually in time series equidistant points of time are considered. There may be weekly, monthly, yearly, etc recordings. A graphical presentation of a time series is called Historigram. COMPONENTS OF A TIME SERIES In a time series, the observations vary with time. The variation occurring in any period is the result of many factors. The effects of these factors may be summed up as four components. They are – a. Trend. ( Secular trend, Long Term Movement) b. Seasonal Variation. Cyclical variation ( Business Cycle) c. Irregular variation ( Random Fluctuation, Erratic Variation) d. Cyclical Variation An analytical Study of different components of a time series, the effects of these components, etc is called analysis of time series. The utility of such analysis is – a. Understanding the past behaviour of the variable b. Knowing the existing nature of variation
2. 2. c. Predicting the future trend d. Comparison with other similar variables. Trend (Secular Trend) Trend is the overall change taking place in the time series over a long period of time. It is the change taking place in a period of many years. Most of the time series show a general tendency to increase, decrease or to remain constant over a long period of time. Such an overall change occurring is the trend. Examples a. Steady increase in the population of India in the past many years is an upward trend. b. Steady increase in the price of gold in last many years is an upward trend. c. Due to availability of greater medical facilities, death rate is decreasing. Thus, death rate shows a downward trend. d. Atmospheric temperature at a place, though show short time variation, does not show significant upward or downward trend. The root cause of trend is technological advancement, growth of population change in tastes etc. Trend is measured, mainly by the method of moving averages and by the method of least squares. Seasonal Variation The regular and periodic variation in a time series is called seasonal variation. Generally, the period of seasonal variation would generally, the period of seasonal variation would be within one year. The factors causing seasonal variation are (1) weather condition, (2) customs, tradition and habits of people. Seasonal variation is predictable. Examples a. An increase in the sales of woolen cloths during winter. b. An increase in the sales of note – books during the month of June, July and August.
3. 3. c. An increase in atmospheric temperature during summer. Cyclical Variation (Business Cycle) Cyclical Variation is an oscillatory variation which occurs in four stages viz – prosperity, recession, depression and recovery. Generally, such variation occurs in economic and business activities. They occur in a gap of more than one year. One cycle consisting of four stages occurs in a period of few years. The period is not definite. Generally, the period is 5 to 10 years. Many Economists have explained the causes of cyclical variation. Each of them is significant. Irregular variation (Random Fluctuation) Apart from the regular variations, most of the time series show variations which are totally unexpected. Irregular variations occur as a result of unexpected happenings such as wars, famines, strikes, floods etc. they are unpredictable. Generally, the effect of such variation lasts for a short period. Examples a. An increase in the price of vegetables due to a strike by the railway employees. b. A decrease in the number of passengers in the city buses, occurring as a result of strike by public sector employees. c. An increase in the number of deaths due to earthquakes. Measurement Of Trend o Graphic (or Free-hand Curve fitting) Method o Method of Semi-Averages o Method of Curve Fitting by the Principle of Least Squares o Method of Moving Averages
4. 4. METHOD OF SEMI-AVERAGES Problem 1: Estimate value for 2000. If the actual sales figures are 35000 units, how do you account for the difference between the figures obtained? Years 1993 1994 1995 1996 1997 1998 Sales 20 24 22 30 28 32 Answer: Sales 3 yearly Semi- Year Trend Values (‘000s) Year (‘000s) Semi Avg average 1993 22 – 2.667 19.333 1993 20 1994 22 22 1994 24 66 22 1995 22 + 2.667 24.667 1995 22 1996 30 - 2.667 27.333 1997 30 30 1996 30 1998 30 + 2.667 32.667 1997 28 90 30 1999 32.667 + 2.667 35.334 1998 32 2000 35.334 + 2.667 38 (30-22) = 8 8/3 = 2.667 The difference is because of the assumption that there is a linear relationship between the given time series values. Moreover, the effects of seasonal, cyclical and irregular variations have been completely neglected.
5. 5. Problem 2: From the following series find the Trend by Semi Average method. Estimate the value for the year 1999. Year 90 91 92 93 94 95 96 97 98 Value 170 231 261 267 278 302 299 298 340 Answer: Year Values 4 yearly Semi- Semi-Totals Average 1990 170 1991 231 929 232 1992 261 1993 267 1994 278 1995 302 1996 299 1239 310 1997 298 1998 340 (310 – 232) = 78 78 / 5. Estimate of the year 1999: 310+(5/2)*(78/5) = 349 METHOD OF CURVE FITTING: PRINCIPLE OF LEAST SQUARES
6. 6. Fitting of Linear Trend: y = a + bx To find a & b: (i) ∑y = na + b∑x; (ii) ∑xy = a ∑x + b ∑x2 Fitting of a Second Degree (Parabolic) Trend: y = a + bx + cx2 To find a, b & c: (i) ∑y = na + b∑x + c∑x2 (ii) ∑xy = a∑x + b∑x2 + c∑x3 (iii) ∑x2y = a ∑x2 + b∑x3 + c∑x4 Problem 3: Fit a linear trend from the following data. Estimate the production for the year 1999. Verify ∑(y- ye) = 0 where ye is the corresponding trend value of y. Year 1990 1992 1994 1996 1998 Production 18 21 23 27 16 Answer: Let us consider the year 1994 to be the mid point (It would be nice to take this as the mid point as there are odd number of years). Year Production x x2 xy Trend Values (y-ye) 1990 18 -4 16 -72 20.6 -2.6 1992 21 -2 4 -42 20.8 0.2 1994 23 0 0 0 21 2 1996 27 2 4 54 21.2 5.8 1998 16 4 16 64 21.4 -5.4 105 40 4 0 Fitting of Linear Trend: y = a + b x
7. 7. To find a & b: ∑y = n a + b∑ x 105 = a*5 + b*0 a = 21 ∑xy = a ∑x + b ∑x2 4 = a*0 + b*40 b = 0.1 Therefore the equation will be given by: y = 21 + 0.1x Estimated production of 1999: y = 21 + 0.1*5 y=21.5 thousands of units. Problem 4: Calculate the quarterly trend values by the method of least squares for the following quarterly data for the last 5 years given below: Year I Quarter II Quarter III Quarter IV Quarter 1994 60 80 72 68 1995 68 104 100 88 1996 80 116 108 96 1997 108 152 136 124 1998 160 184 172 164 Answer: Year Total Average U U2 Uy Trend Values 1994 280 70 -2 4 140 64 1995 360 90 -1 1 -90 88 1996 400 100 0 0 0 112 1997 520 130 1 1 130 136
8. 8. 1998 680 170 2 4 340 160 560 0 10 240 Fitting of Linear Trend: y = a + b U To find a & b: ∑y = n a + b∑ U 560 = a*5 + b*0 a = 112 ∑Uy = a ∑U + b ∑U2 240 = a*0 + b*10 b = 24 Therefore the equation will be given by: y = 112 + 24x Therefore the quarterly increment is : (24/4)=6 By the calculations we come to know that the quarterly increment is 6. Therefore the values for second & third Quarters of 1994 are: 64 - (6/2) & 64 + (6/2) respectively. Year I Quarter II Quarter III Quarter IV Quarter 1994 55 61 67 73 1995 79 85 91 97 1996 103 109 115 121 1997 127 133 139 145 1998 151 157 163 169