Parabolic Method of the Least Square

This method of least square is used only when the trend of a series is not linear, but curvilinear. Under this method, a curve of parabolic type is fitted to the data to obtain their trend values and to obtain such a curve, an equation of power series is determined in the following model:

Y_c = a + bX +cX² + dX³ + ….+ mXn

It may be noted that the above equation can be carried to any power of X according to the nature of the series. If the equation is carried only up to the second power of X, (i.e. X²), it is called the parabola of second degree, and if it is carried up to the 3rd power of X, (i.e. X³) it is called the parabola of 3rd degree. However, in actual practice the parabolic curve of second degree is obtained in most of the cases to study the non-linear trend of a time series. For this, the following equation is used.

Y_c = a + bX + cX²

Where, Yc represents the computed trend value of the Y variable, a’ the intercept of Y, b’ the slope of the curve at the origin of X and c’, the rate of change in the slope.

In the above equation, a, b, and c are the three constants, the values of which are determined by solving simultaneously the following three normal equations:

∑Y = Na + b∑X + c ∑X²

 ∑XY = a∑X + b ∑X² + c ∑X³

 ∑X²Y = a∑X² + b ∑X³ + c ∑X⁴

It may be noted here that the first normal equation has been derived by multiplying each set of the observed relationship by the respective coefficients of a, and getting them all totaled ; the second normal equation has been derived by multiplying each set of the observed relationship by the respective coefficients of b, and getting them al totaled ; and the third equation has been derived by multiplying each set of the observed relationship by the respective coefficients of c and getting them all totaled.

Further, it may be noted that by taking the time deviations from the midpoint of the time variable, if   ∑X and ∑X³ could be made zero, the above three normal equations can be reduced to the simplified forms to find the values of the relevant constants as follows:

∑Y = Na +  c ∑X², where a = ∑Y – c ∑X²/ N

∑XY = b∑X² , where b = ∑XY/ ∑X²

And ∑X²Y = a∑X² +  c ∑X⁴, where c = ∑X² – a∑X²/ ∑X⁴

From the above, it must be noticed that the value of b can be directly obtained as b = ∑XY/∑X², and the values of the other two constants a and c can be obtained by solving simultaneously the rest of the following two normal equations:

A = ∑Y – c ∑X²/N

C = ∑X²Y – a∑X²/∑X⁴

Once, the values of the three constants a, b and c are determined in the above manner, the trend line equation can be fitted to obtain the trend values of the given time series by simply substituting the respective values of X therein.