Question

Fit a parabolic trend of the form Y = a + bX + cX² for the following Time Series.

Answer 1

Let x and y be the year and production

From the normal equation na + bΣx + cΣx² 

= Σy; 5a + b.0 + c. 10 = 115

5a + 10c = 115 … (1) From: aΣx + bΣx² + cΣx³ = Σxy

a.0 + b. 10 + c.0 = -25; b = -2.5 and From: aΣx² + bΣx³ + cΣx⁴ = Σx²y ;

a. 10 + b.0 + c.34 = 237; 10a + 34c = 237 … (2)

from(1) and (2) 5a + 10c =115 × 2

0 – 14c = -7; 

∴ c = 0.5; Put c = 0.5 in (1)

5a + 10(0.5)= 115; a = 22

The fitted parabolic trend equation is:-

y = a + bx + cx² ; y =22 – 2.5x + 0.5x²