Page 411 - DMTH404_STATISTICS

Unit 30: Method of Least Square

Notes
Example 9:
Given the following trend equations:

(a) Y = 50 + 3X, with 1985 as the year of origin and unit of X = 1 year. Shift the origin to 1991.
1
(b) Y = 100 + 2.5 X, with origin at the middle of 1987 and 1988 and unit of X = year. Shift the
2
origin to (i) 1988 and (ii) 1992.
Solution.
(a) Replacing X by X + 6, in the trend equation, we get

Y = 50 + 3(X + 6) = 68 + 3X, the required trend equation.
(b) (i) For shifting origin to 1988 (i.e., middle of 1988), we have to replace X by X + 1. (note
1
that X = 1 for year)
2
 Y = 100 + 2.5(X + 1) = 102.5 + 2.5X
(ii) Replace X by X + 9, to get the required equation
 Y = 100 + 2.5(X + 9) = 122.5 + 2.5X
Conversion of Annual Trend Equation into Monthly trend Equation

Usually a trend is fitted to the annual figures because the fitting of a monthly trend is time
consuming. However, monthly trend equations are often obtained from annual trend equations.
Let the annual trend equation be Y = a + bX, where Y denotes annual figures and the unit of X =
1 year.
To obtain the monthly trend equation, we have to convert the constants a and b into monthly
values.
a
Thus, when a denotes an annual value, would give the value of the corresponding constant
12
for the monthly equation.
b
Further, the value of b denotes the annual change in Y per unit of X, i.e., per year. Therefore
12
a b
would be the monthly (average) change in Y per year. Thus, the equation Y = + X , denotes
12 12
a monthly average equation, where Y denotes monthly average for the year and unit of X = 1
year.

b b
In a similar way, the value = would denote the monthly change in Y per month.

12 12 144
a b
Thus, Y = + X , is the monthly trend equation, where Y denotes monthly figures and the
12 144
unit of X = 1 month.

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