Time Series Analysis- Lecture note 4

Measurement of Trend using Method of Least Square

This method is most widely used in practice. It is a mathematical method and with its

help a trend line is fitted to the data in such a manner that the following two conditions are satisfied.

1. The sum of the deviations of the actual values of Y and the computed values of Y is zero.

2. The sum of the squares of the deviations of the actual values and the computed values is the least.

The line obtained by this method is called the “line of best fit”.

This method of least squares may be used either to fit a straight line trend or a parabolic

trend.

FITTING OF LINEAR TREND

Given the data (Y(t), t) for n periods, where t denotes time period such as year, month, day, etc. We have to the values of the two constants, ‘a’ and ‘b’ of the linear trend equation: Y(t) = a + bt

Where the value of ‘a’ is merely the Y-intercept or the height of the line above the origin. That is, when t=0, Y= a.

The other constant ‘b’ represents the slope of the trend line. When b is positive, the slope is upwards, and when b is negative, the slope is downward.

This line is termed as the line of best fit because it is so fitted that the total distance of deviations of the given data from the line is minimum. The total of deviations is calculated by squaring the difference in trend value and actual value of the variable.

using the least square method, the normal equation for obtaining the values of a and b are:

Σ Y(t) = na + bΣt

ΣtY(t) =aΣt + bΣt^2

Let X = t – A, such that ΣX = 0, where A denotes the year of origin.

The above equations can also be written as

ΣY = na + bΣX

ΣXY =aΣX + b ΣX2

Since Σx = 0 i.e. deviation from actual mean is zero

We can write

FITTING OF PARABOLIC TREND

The mathematical form of a parabolic trend is given by: Y(t) =a + bt+ct^2

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 :

Σ Y = na + bΣt +cΣt^2

ΣtY =aΣt + bΣt^2 + cΣt^3

Σt^2Y =aΣt^2 + bΣt^3 + cΣt^4

By selecting a suitable year of origin, i.e. define X = t –origin such that ΣX = 0, the computation work can be considerably simplified. Also note that if ΣX = 0, then ΣX^3 will also be equal to zero. Thus, the above equations can be rewritten as:

Σ Y = na +cΣX^2 ……….(1)

ΣXY = bΣX^2 ……….(2)

ΣX^2Y = aΣX^2 + cΣX^4……….(3)

These are the three equations to find the value of constants a, b and c.