Page 228 - DMGT404 RESEARCH_METHODOLOGY

Research Methodology

Notes
Example: Find seasonal variations by the ratio to trend method, from the following data:
Year I -Qr II -Qr III -Qr IV -Qr
1975 30 40 36 34
1976 34 52 50 44
1977 40 58 54 48
1978 54 76 68 62
1979 80 92 86 82
Solution:
First we fit a linear trend to the annual totals.
Annual Totals 2
Years X XY X
(Y)
1975 140 - 2 - 280 4
1976 180 - 1 - 180 1
1977 200 0 0 0
1978 260 1 260 1
1979 340 2 680 4
Total 1120 0 480 10
1120 480
Now a = = 224 and b = = 48
5 10
 The trend equation is Y = 224 + 48X, origin : 1st July 1977, unit of X = 1 year.
224 48
The quarterly trend equation is Y =  X = 56 + 3X, origin : 1st July 1977, unit of
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X = 1 quarter.
Shifting the origin to III quarter of 1977, we get

1
Y = 56 + 3(X + ) = 57.5 + 3X
2
Table of Quarterly Trend Values
Year I II III IV
1975 27.5 30.5 33.5 36.5
1976 39.5 42.5 45.5 48.5
1977 51.5 54.5 57.5 60.5
1978 63.5 66.5 69.5 72.5
1979 75.5 78.5 81.5 84.5

Ratio to Trend Values
Year I II III IV
1975 109.1 131.1 107.5 93.2
1976 86.1 122.4 109.9 90.7
1977 77.7 106.4 93.9 79.3
1978 85.0 114.3 97.8 85.5
1979 106.0 117.2 105.5 97.0
Total 463.9 591.4 514.6 445.7
A 92.78 118.28 102.92 89.14
i
S. I. 92.10 117.35 102.11 88.44
403.12
Note that the Grand Average G = = 100.78. Also check that the sum of indices is 400.
4
Remarks: If instead of multiplicative model we have an additive model, then Y = T + S + R or
S + R = Y – T. Thus, the trend values are to be subtracted from the Y values. Random component
is then eliminated by the method of simple averages.

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