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Statistics

Notes 5. Since the choice of a particular trend is arbitrary, the method is not, strictly, objective.

6. This method cannot be used to fit growth curves, the pattern followed by the most of the
economic and business time series.

30.2 Summary

 Given the data (Y , t) for n periods, where t denotes time period such as year, month, day,
t
etc., we have to find the values of the two constants, a and b, of the linear trend equation
Y = a + bt.
t
Using the least square method, the normal equation for obtaining the values of a and b
are:

Y = na + bt and
t
tY = at + bt 2
t
Let X = t - A, such that X = 0, where A denotes the year of origin.
The above equations can also be written as

Y = na + bX
XY = aX + bX 2
(Dropping the subscript t for convenience).

å Y å XY
Since SX = 0, we can write a = and b = 2
n å X

 Unlike the moving average method, it is possible to compute trend values for all the
periods and predict the value for a period lying outside the observed data.
 The results of the method of least squares are most satisfactory because the fitted trend
2
satisfies the two important properties, i.e., (i) S(Y - Y ) = 0 and (ii) S(Y - Y ) is minimum.
o t o t
Here Y denotes the observed value and Y denotes the calculated trend value.
o t
The first property implies that the position of fitted trend equation is such that the sum of
deviations of observations above and below this is equal to zero. The second property
implies that the sum of squares of deviations of observations, about the trend equation,
are minimum.
 It is not flexible like the moving average method. If some observations are added, then the
entire calculations are to be done once again.
 It can predict or estimate values only in the immediate future or past.
 The computation of trend values, on the basis of this method, doesn't take into account the
other components of a time series and hence not reliable.

30.3 Keywords

The fitted trend is termed as the best in the sense that the sum of squares of deviations of
observations, from it, are minimised.

Parabolic trend: The mathematical form of a parabolic trend is given by Y = a + bt + ct or
2
t
Y = a + bt + ct (dropping the subscript for convenience). Here a, b and c are constants to be
2
determined from the given data.

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