Page 320 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

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

24.3 Summary Notes

• The device for getting an objective fit of a straight line to a series of data is the least squares
method. It is perhaps the most commonly employed and a very satisfactory method to describe
the trend. The Mark off theorem states that for a given condition, the line fitted by the method
of least squares is the line of “best” fit in a well-defined sense. The term “best” is used to mean
that the estimates of the constants a and b are the best linear unbiased estimates of those constants.
• In the least squares method, the sum of the vertical deviations of the observed values from the
fitted straight line is zero. Secondly, the sum of the squares of all these deviations is less than
the sum of the squared vertical deviations from any other straight line. The method of least
squares can be used for fitting linear and non-linear trends as well.
• It should be noted that in computing the trend it is convenient to use the middle of the series as
the origin. If the series contains an odd number of periods, the origin is the middle of the given
period. If an even number of periods is involved, the origin is set between the two middle
periods.
• It is important to recognize that the least squares technique requires that the type of line desired
be specified. Once this has been done, the technique generates the line of best fit of that type,
that is, the least squares line.
• The line obtained by this method is called the line of best fit because it is this line from where the
sum of the positive and negative deviations is zero and sum of the squares of the deviations
least, i.e., (Y – Y ) = 0 and (Y – Y ) least.
2
c c
• It is seldom possible to justify on theoretical grounds any real dependence of a variable on the
passage of time. Variables do change in a more or less systematic manner over time, but this
can usually be attributed to the operation of other explanatory variables. Thus many economic
time series show persistent upward trends over time due to a growth of population or to a
general rise in prices, i.e., national income and the trend element can to a considerable extent be
eliminated by expressing these series per capita or in terms of constant purchasing power.
• Hence, mathematical methods of fitting trend are not foolproof. In fact, they can be the source
of some of the most serious errors that are made in statistical work. They should never be used
unless rigidly controlled by a separate logical analysis. Trend fitting depends upon the judgement
of the statistician, and a skilfully made free-hand sketch is often more practical than a refined
mathematical formula.
24.4 Key-Words

1. Homogeneity of regression : The assumption that the regression line expressing the dependent
variable as a function of a covariate is constant across several groups
or conditions.
2. Homogeneity of variance : The situation in which two or more populations have equal
variances.
24.5 Review Questions

1. Discuss the method of least squares for the measurement of trend.
2. Write the normal equations to determine the values of a and b in the trend equation y = a + bx,
given the n observations.
3. Explain the principle of least square method and its application.
4. What are the limitations of least square method ?
5. Discuss the merits and demerits of least square method.

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