Method of Moving Averages: In the moving average method, the trend is described by smoothing out the fluctuations of the data by means of a moving average.. The p period moving totals or
Trang 1400 350 300 250 200 150 100 50 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
FIG 10.1
10.3 COMPONENT OF TIME SERIES
The analysis of Time Series consists of the description and measurement of various changes or movements as they appear in the series during a period of time These changes or movements are called the components of elements of time series Fluctuations in a time series are mainly due to four basic types of variations (or movements) These four types of component are:
1 Secular Trend or Long Term Movement (T)
2 Seasonal Variation or Seasonal Movement (S)
3 Cyclical Fluctuation or Cyclic Variation (C)
4 Residual, Irregular of Random Movement (I)
(1) Secular Trend: In Business, Economics and in our daily conversation the term Secular Trend or simply trend is popularly used Where we speak of rising trend of population
or prices, we mean the gradual increase in population or prices over a period of Time Similarly, by declining trend of production or sales, we mean gradual decrease in production or sales over a period of time The concept of trend does not include short range Oscillations, but refers to the steady movement over a long period time
“Secular trend is the smooth, regular and long term movement of a series showing continuous growth stagnation or decline over a long period of time Graphically it exhibits general direction and shape of time series” The trend movement of an economic time series may be upward or downward The upward trend may be due to population growth, technological advances, improved methods of Business Organization and Management, etc Similarly, the downward trend may be due to lack of demand for the product, storage of raw materials to be used in production, decline in death rate due to advance in medical sciences, etc
(2) Seasonal Variation: Seasonal variation is a short-term periodic movement, which occurs more or less regularly within a stipulated period of one year or shorter The major factors that cause seasonal variations are climate and weather conditions, customs and habits of people, religious festivals, etc For instance, the demand for electric fans goes up in summer season, the sale of Ice-cream increases very much in summer and the sale of woolen cloths goes up in winter Also the sales of jewelleries and ornaments go up in
Trang 2marriage seasons, the sales and profits of departmental stores go up considerably during festivals like Id, Christmas, etc
Although the period of seasonal variations refers to a year in business and economics, it can also be taken as a month, week, day, hour, etc depending on the type of data available Seasonal variation gives a clear idea about the relative position of each season and on this basis, it is possibe to plan for the season
(3) Cyclical Fluctuations: These refer to the long term oscillations, or swings about a trend line or curve These cycles, as they are some times called, may or may not be periodic that is they may or may not follow exactly similar patterns after equal intervals of time
In business and economic activities, moments are considered cyclic only if they recur after intervals of more than one year The ups and downs in business, recurring at intervals of times are the effects of cyclical variations A business cycle showing the swing from prosperity through recession, depression, recovery and back again to prosperity This movement varies in time, length and intensity
(4) Residual Irregular or Random Movement: Random movements are the variations in a time series which are caused by chance factors or unforeseen factors which cannot be predicted in advance For example, natural calamities like flood, earthquake etc, may occur at any movement and at any time They can be neither predicted nor controlled But the occurrence of these events influences business activities to a great extent and causes irregular or random variations in time series data
10.4 ANALYSIS OF TIME SERIES
Time series analysis consists of a description (generally mathematical) of the component movements present To understand the procedures involved in such a description consider graph (A), which shows Ideal Time Series, Graph (A1) shows the graph of long-term, or secular, trend line Graph (A2) shows this long-term trend line with a superimposed cyclic movement (assumed to be periodic) and graph (A3) shows a seasonal movement superimposed of graph (A2) The concept in graph (A) suggests a technique for analyzing time series
Y
t Long Term Trend
FIG 10.2 (Graph A1)
In Traditional or classical time series analysis, it is normally assumed that there is multiplicative relationship between the four components Symoblically
=
y=Result of the Four Components (or original data)
Trang 3It is assumed that trend has no effect on seasonal component Also it is assumed that the business cycle has no effect on the seasonal component Instead of Multiplicative model (1) some statisticians may prefer an additive model
+ +
=
where y is the sum of the four components
Y
t Long Term Trend and Cyclical Movement
Y
t Long Term Trend, Cyclical
and Seasonal Movement FIG 10.3
10.4.1 Analysis of Trend or Secular Trend
In time series analysis the analysis of secular trend is very important It helps us to predict or forecast future results There are four methods used in analyzing trend in time series analysis They are:
(a) Method of Free Hand Curve (or graphic)
(b) Method of Semi Averages
(c) Method of Moving Averages
(d) Method of Least Square
(=) Free Hand Method: It is simplest method for studying trend In this graphic method,
the time series data are first plotted on the graph paper taking time on the x-axis and observed values of the other variable on y-axis Then points obtained are joined by a free hand smooth
curve of first degree The line so obtained is called the trend curve and it shows the direction of
the trend The vertical distical of this line from x-axis gives the trend value for each time period.
This method should be used only when a quick approximate idea of the trend is required
Example 1 Fit a trend line to the following data by the free hand grpahical method.
Y Year ear 2000 2001 2002 2003 2004 2005 2006
Sales 52 54 56 53.5 57 54.5 59
Trang 460 59 58 57 56 55 54 53 52 51
1998 2000 2002 2004 2006 2008
Years FIG 10.4 (>) Method of Semi Averages: This method is very simple and gives greater accuracy than the method of free hand or graphical In this method, the given data is first divided into two parts and an average for each part is found Then these two averages are plotted on a graph paper with respect to the midpoint of the two respective time intervals The line obtained
on joining these two points is the required trend line and may be extend both ways to estimate intermediate values
Remark: If given data is in odd number, then divide the whole series into two equal parts ignoring the middle period
Example 2 Fit a trend line to the following data by the method of semi averages.
Year 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003
Clearance
Sol Here n=13 i.e., odd no of data
Now divide the given data into two equal parts (by omitting 1997)
Year Clearenc e Semi Total Semi Avarage
Trang 5130 120 110 100 90 80 70 60 50
1 2 3 4 5 6 7 8 9 10 11 12 13
Year FIG 10.5 Now the two semi averages 72.833 and 93.333 are plotted against the middle of their respective periods
Example 3 Draw the trend line by semi-average method using the given data
Production
(In Tons) Sol Here n = 6
Pr
Trang 6
265
260
255
250
245
Year
FIG 10.6 (?) Method of Moving Averages: In the moving average method, the trend is described
by smoothing out the fluctuations of the data by means of a moving average
Let (t y1, 1) (, t y2, 2), ,(t y n, n) be the given time series, where t t t1, , , ,2 3 t n denote the time periods and y y y1, 2, 3, ,y n denote the corresponding values of the variable The p-period moving totals (or sums) are defined as y y y1, 2, 3, ,y p,
2, 3, , p 1.
y y y + y y y3, 4, 5, ,y p+2, and so on.
The p-period moving averages are defined as
1 2 2 3 1 3 4 2
+ + + + + + + + +
etc
The p period moving totals (or sums) and moving averages be also called moving totals
(or sums) of order p and moving averages of order p respectively These moving
averages are also called the trend values
When we estimate the trend, we should select the order or period of the moving average (such as 3 yearly moving average, 5 yearly moving average, 4 yearly moving average, 8 yearly moving average, etc.) This order should be equal to the length of cycles in the time series The method of moving averages is the most frequently used approach for determining the trend because it is definitely simpler process of fitting a polynomial (1) Calculation of Moving Averages when the Period is Odd: In the case of odd period we would obtain the trend values and trend line as follows:
(=) In the case of 3–yearly period, first of all calculate the following moving totals (or
sums) y1 + y2 + y3, y2 + y3 + y4, y3 + y4 + y5, etc
In the case of 5 yearly period, calculate the following moving total (or sums)
y1 + y2 + y3 + y4 + y5, y2 + y3+ y4 + y5 + y6, y3 + y4 + y5 + y6 + y7, etc
A period may be a year, a week, a day, etc
(>) Place the moving totals at the centres of three respective time
Trang 7(? ) Calculte the corresponding moving averages for 3 yearly or 5 yearly periods by dividing the moving totals by 3 or 5 respectively Place these at the centre of the respective time
(@) If required we can plot these moving averages or trend values against the periods and obtain the trend line (or curve) from which we can determine the increasing or decreasing trend of the data
(A) It is more convenient to calculate the moving averages when the period is odd than when it is even, because there is only one middle period when the peirod is odd so and the moving average can be easily centred
(2) Calculation of Moving Averages when the Period is Even: When an even number
of data is included in the moving averages (as 4 years), the centre point of the group will
be between two years It is therefore necessary to adjust or shift (known technically as centre) these averages so that they concide with the years The 4 yearly moving total and the 4 yearly moving average may be obtained by the methods already outlined for the odd period average To centre the values, a 2 yearly moving average is taken of the even period moving average
A 2 yearly moving average is taken of the 4 yearly moving average The resulting average
is located between the two 4 yearly moving average values and, therefore, coincides with
the years The end results (i.e., a 2 yearly moving average of 4 yearly moving average) are
known as the 4 yearly moving average centred
We shall follow the steps given below in calculating the moving average when the order
is even, say 4
(i) We calculate the following moving totals:
+ + +
1 2 3 4
y y y y , y2+y3+y4+y5, y3+ + +y4 y5 y6, etc
(ii) Place these moving totals at the centres of the respective time spans In the case or
4 yearly time period, there are two middle terms viz 2nd and 3rd Hence, place this moving total against the centre of these two middle terms Similarly, place other moving totals at the centres of 3rd and 4th periods, 4th and 5th periods and so on
(iii) Calculate the corresponding moving averages for 4 yearly periods by dividing the
moving totals by 4 Place these at the centres of the time spans i.e., against the
corresponding moving totals (Note that the moving averages so placed do not coincide with the original time period.)
(iv) We take the total of 4 yearly moving averages taking two terms at a time starting
from the first and place the sum at the middle of these two terms The same procedure is repeated for other averages
(v) Finally, we take the two-period averages of the above moving averages by dividing
each by 2 These are the required trend values This process is called centreing of moving averages If required, we can plot these moving averages or trend values and obtain trend line or curve
The major disadvantage of this method is that some trend values at the beginning and end of the series cannot be determined
Example 4 Calculate 3 yearly moving averages or trend values for the following data.
Year (t) 1998 1999 2000 2001 2002 2003 2004 2005
Value(y) 3 5 7 10 12 14 15 16
Trang 8Sol 3 yearly moving averages means that there are three values induced in a group.
Calculation of 3 yearly moving averages
Year Value 3 Yearly 3 Yearly Moving Total
( )t ( )y Moving Total (Trend Valuea)
Hence the trend values are 5.00, 7.33, 9.67, 12.00, 13.67, 15.00
Example 5 Compute the 4 yearly moving averages from the following data:
Year 1991 1992 1993 1994 1995 1996 1997 1998
Annual sales 36 43 43 34 44 54 34 24
(Rs In crores)
Sol Calculation of 4 yearly moving averages
Annual Sales 4 Yearly 4 Yearly Total of Centred Moving Average (Rs in Crores) Moving Totals Moving Average col 4 (Centred) (Trend values)
Hence the trend values are 40, 42.375, 42.625, 40.25
Example 6 Assuming 5 yearly moving averages, calculate trend value from the data given below
and plot the results on a graph paper.
Year 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 Production
105 107 109 112 114 116 118 121 123 124 125 127 129 (Thousand)
Trang 9Sol Calculation of 5 yearly moving averages
Moving Total Averages (trend value)
Hence the trend values are 109.4, 111.6, 113.8, 116.2, 118.4, 120.4, 122.2, 124.0, and 125.6
Years
135 130 125 120 115 110 105 100
FIG 10.7 (@) Method of Least Square: This method is widely used for the measurement of trend
In method of least square we minimize the sum of the squares of the deviation of observed values from their expected values with respect to the constants
Let T= +a bx be the required trend line
By the principal of least square, the line of the best fit is obtained when the sum of the squares of the differences, S i is minimum i.e.
( )2
S =∑ T − −a bx is minimum
Trang 10When S i is minimum, we obtain normal equations as
∑T=na b+ ∑x
∑Tx=a∑ ∑x b+ x2
On solving these two equations, we get
a = T
n
∑ , b = Tx
x
∑
Remark: If we take the midpoint in time as the origin, the negative values in the first half
of the series balance out the positive values in the second half so
∑x=0
Example 7 Determine the equation of a straight-line which best fits the following data
(in Rs 000)
Compute the trend values for all the years from 1974 to 1978
Sol Let the equation of the straight-line of best fit, with the origin at the middle year 1976 and unit of x as 1 year, be
y= +a bx
By the method of least squares, the values of a and b given by
a y
N
=∑
and b 2xy
x
=∑
Here N = number of years = 5
Calculations for the line of best fit