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Obtain the trend values for the data in using 4-yearly centered moving averages.

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Obtain the trend values for the data in using 4-yearly centered moving averages.

Year 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985
Index 0 2 3 3 2 4 5 6 7 10

The following table shows the index of industrial production for the period from 1976 to 1985. using the year 1976 as the base year. Obtain the trend values for the following data using 4 yearly centered moving averages:

Year 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985
Index 0 2 3 3 2 4 5 6 7 10
Chart
Sum
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Solution

Construct the following table for finding 4 yearly centered moving average

Year t Index yt 4–yearly moving total 4–yearly moving averages 2 unit
moving total
4 yearly centred moving average (trend values)
1976 0        
           
1977 2 8 2    
           
1978 3 10 2.5 4.5 2.25
           
1979 3 12 3 5.5 2.75
           
1980 2 14 3.5 6.5 3.25
           
1981 4 17 4.25 7.75 3.875
           
1982 5 22 5.5 9.75 4.875
           
1983 6 28 7 12.5 6.25
           
1984 7        
           
1985 10        
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Measurement of Secular Trend
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Chapter 4: Time Series - Exercise 4.1 [Page 66]

RELATED QUESTIONS

Obtain the trend line for the above data using 5 yearly moving averages.


The following table shows the production of gasoline in U.S.A. for the years 1962 to 1976.

Year 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976
Production
(Million Barrels)
0 0 1 1 2 3 4 5 6 7 8 9 8 9 10

i. Obtain trend values for the above data using 5-yearly moving averages.
ii. Plot the original time series and trend values obtained above on the same graph.


The simplest method of measuring trend of time series is ______.


Fill in the blank :

The method of measuring trend of time series using only averages is _______


Fill in the blank :

The complicated but efficient method of measuring trend of time series is _______.


State whether the following is True or False :

All the three methods of measuring trend will always give the same results.


Solve the following problem :

The following table shows the production of pig-iron and ferro- alloys (‘000 metric tonnes)

Year 1974 1975 1976 1977 1978 1979 1980 1981 1982
Production 0 4 9 9 8 5 4 8 10

Fit a trend line to the above data by graphical method.


Fit a trend line to the following data by the method of least squares.

Year 1974 1975 1976 1977 1978 1979 1980 1981 1982
Production 0 4 9 9 8 5 4 8 10

Solve the following problem :

The percentage of girls’ enrollment in total enrollment for years 1960-2005 is shown in the following table.

Year 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
Percentage 0 3 3 4 4 5 6 8 8 10

Fit a trend line to the above data by graphical method.


Solve the following problem :

Fit a trend line to the data in Problem 7 by the method of least squares.


Following data shows the number of boxes of cereal sold in years 1977 to 1984.

Year 1977 1978 1979 1980 1981 1982 1983 1984
No. of boxes in ten thousand 1 0 3 8 10 4 5 8

Fit a trend line to the above data by graphical method.


Solve the following problem :

Fit a trend line to data in Problem 13 by the method of least squares.


Solve the following problem :

Obtain trend values for data in Problem 13 using 4-yearly moving averages.


Solve the following problem :

Following table shows the all India infant mortality rates (per ‘000) for years 1980 to 2010.

Year 1980 1985 1990 1995 2000 2005 2010
IMR 10 7 5 4 3 1 0

Fit a trend line to the above data by graphical method.


Solve the following problem :

Fit a trend line to data in Problem 16 by the method of least squares.


Solve the following problem :

Obtain trend values for data in Problem 16 using 3-yearly moving averages.


Solve the following problem :

Following tables shows the wheat yield (‘000 tonnes) in India for years 1959 to 1968.

Year 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968
Yield 0 1 2 3 1 0 4 1 2 10

Fit a trend line to the above data by the method of least squares.


The complicated but efficient method of measuring trend of time series is ______


The simplest method of measuring trend of time series is ______


The method of measuring trend of time series using only averages is ______


Obtain trend values for data, using 4-yearly centred moving averages

Year 1971 1972 1973 1974 1975 1976
Production 1 0 1 2 3 2
Year 1977 1978 1979 1980 1981 1982
Production 4 6 5 1 4 10

Obtain the trend values for the data, using 3-yearly moving averages

Year 1976 1977 1978 1979 1980 1981
Production 0 4 4 2 6 8
Year 1982 1983 1984 1985 1986  
Production 5 9 4 10 10  

Following table shows the all India infant mortality rates (per ‘000) for years 1980 to 2010

Year 1980 1985 1990 1995
IMR 10 7 5 4
Year 2000 2005 2010  
IMR 3 1 0  

Fit a trend line by the method of least squares

Solution: Let us fit equation of trend line for above data.

Let the equation of trend line be y = a + bx   .....(i)

Here n = 7(odd), middle year is `square` and h = 5

Year IMR (y) x x2 x.y
1980 10 – 3 9 – 30
1985 7 – 2 4 – 14
1990 5 – 1 1 – 5
1995 4 0 0 0
2000 3 1 1 3
2005 1 2 4 2
2010 0 3 9 0
Total 30 0 28 – 44

The normal equations are

Σy = na + bΣx

As, Σx = 0, a = `square`

Also, Σxy = aΣx + bΣx2

As, Σx = 0, b =`square`

∴ The equation of trend line is y = `square`


Obtain the trend values for the following data using 5 yearly moving averages:

Year 2000 2001 2002 2003 2004
Production
xi
10 15 20 25 30
Year 2005 2006 2007 2008 2009
Production
xi
35 40 45 50 55

The complicated but efficient method of measuring trend of time series is ______.


The following table shows gross capital information (in Crore ₹) for years 1966 to 1975:

Years 1966 1967 1968 1969 1970
Gross Capital information 20 25 25 30 35
Years 1971 1972 1973 1974 1975
Gross Capital information 30 45 40 55 65

Obtain trend values using 5-yearly moving values.


Fit a trend line to the following data by the method of least square :

Year 1980 1985 1990 1995 2000 2005 2010
IMR 10 7 5 4 3 1 0

Complete the following activity to fit a trend line to the following data by the method of least squares.

Year 1975 1976 1977 1978 1979 1980 1981 1982 1983
Number of deaths 0 6 3 8 2 9 4 5 10

Solution:

Here n = 9. We transform year t to u by taking u = t - 1979. We construct the following table for calculation :

Year t Number of deaths xt u = t - 1979 u2 uxt
1975 0 - 4 16 0
1976 6 - 3 9 - 18
1977 3 - 2 4 - 6
1978 8 - 1 1 - 8
1979 2 0 0 0
1980 9 1 1 9
1981 4 2 4 8
1982 5 3 9 15
1983 10 4 16 40
  `sumx_t` =47 `sumu`=0 `sumu^2=60` `square`

The equation of trend line is xt= a' + b'u.

The normal equations are,

`sumx_t = na^' + b^' sumu`              ...(1)

`sumux_t = a^'sumu + b^'sumu^2`      ...(2)

Here, n = 9, `sumx_t = 47, sumu= 0, sumu^2 = 60`

By putting these values in normal equations, we get

47 = 9a' + b' (0)       ...(3)

40 = a'(0) + b'(60)      ...(4)

From equation (3), we get a' = `square`

From equation (4), we get b' = `square`

∴ the equation of trend line is xt = `square`


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