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Unit 9: Regression Analysis

9.2.2 Fitting of Parabolic Trend Notes

2
The mathematical form of a parabolic trend is given by Y = a + bt + ct or Y = a + bt + ct 2
t
(dropping the subscript for convenience). Here a, b and c are constants to be determined from
the given data.
Using the method of least squares, the normal equations for the simultaneous solution of a, b,
and c are:
2
Y = na + b t + c t
2 3
tY = a t + b t + c t
2 2 3 4
t Y = a t + b t + c t
By selecting a suitable year of origin, i.e., define X = t - origin such that SX = 0, the computation
work can be considerably simplified. Also note that if SX = 0, then SX3 will also be equal to zero.
Thus, the above equations can be rewritten as:
2
Y = na + c X ..(i)
XY = b X 2 ...(ii)

2
2
X Y = a X + c X 4 ...(iii)
XY
b
From equation (ii), we get 2 .... (iv)
X
Y c X 2
Further, from equation (i), we get a .... (v)
n

2
n X Y X 2 Y
And from equation (iii), we get c 2 .... (vi)
n X 4 X 2

Thus, equations (iv), (v) and (vi) can be used to determine the values of the constants a, b and c.

9.2.3 Fitting of Exponential Trend

The general form of an exponential trend is Y = a.bt, where a and b are constants to be determined
from the observed data.
Taking logarithms of both sides, we have logY = log a + t log b.

This is a linear equation in log Y and t and can be fitted in a similar way as done in case of linear
trend. Let A = log a and B = log b, then the above equation can be written as log Y = A + Bt.
The normal equations, based on the principle of least squares are:

log Y = nA + B t
2
and tlog Y = A t + B t .

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