Page 312 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Unit 24: Time Series Methods—Principle of Least Square and Its Application

To obtain the value of the trend line for any given period, we simply substitute the Notes
value of X for that year in the equation Y = 18.78 – 0.21 X, where Y is the calculated
cal
cal
value of Y. As an example, for the year 1973, X = 0.
Thus
Y (1973) = 18.78 – 0.21 0
×
cal
= 18.78
As has been stated earlier, our interest in trend analysis is not only confined to
determining the growth pattern of a series in the past but also concerned with
forecasting future trend values. To forecast the trend values, we use the same technique
that was used to compute the trend values for the periods covered by the series. For
example, if we are interested in forecasting the trend value of annual sale of electronic
calculators for 1980, we observe that 1980 is 7 years ahead of the origin of the trend
equation we have established above, and thus
Y (1980) = 18.78 – 0.21×7
est
= 17.31
We may, thus, state that, under the assumption that the same trend factors that
produced the trend equation for 1968 and 1978 will remain operative, the average
annual sales of electronic calculators by an Indian manufacturer will have a trend
value of Rs. 17.31 lakhs during 1980.

The straight line trend studied earlier is appropriate when there is reason to believe that
the time series is changing, on the average, by equal absolute amounts in each time period.

However, in many situations the growth is such that the absolute amounts of the Y
variable increases more rapidly in later time periods than in earlier ones. When this is
the case, if a trend is to be fitted to the original data on the natural graph, a curvilinear
instead of the simple linear description would be necessary. However, the same data
when plotted on a semilogarithmic scale, may very often reveal linear average
relationship. In other words, a straight line would seem to describe the trend of the
series plotted on a semilogarithmic chart, such a growth pattern in the series can be
expressed by a geometric straight line.
To fit a geometric trend line, trend values are computed from the logarithms of the data
(Y) instead of the original data. Thus, the trend values obtained by this method will be
logarithms of trend values and could be related to logarithms of the data. However, in
practice, it is more useful to take the antilogs which will give the trend values in natural
numbers and that can be compared well with the original values of the series.
In fitting a straight line to the logarithms of the data by the least squares method, the
procedure used is identical with that for an arithmetic straight line studied earlier.
The only difference is that logarithms of the original data instead of the natural
numbers are used throughout. The normal equations now are:

∑ ( )log Y = n + log a (∑ log b X )

∑ (X log Y ) = log a ( )∑ +X ( log b X 2 ) ∑
By setting the origin at the middle of the series, the formulae for the Y intercept and
the slope become
∑log Y
log a = ... (3)
n

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