Question

For the following time series fit a curve of the type y = a + bjc + cx². Estimate the value for 1997.

Answer 1

The quadratic equation is:

y = a + bx + cx²

The normal equations are 

na + bΣx + cΣx² = Σy ………..(1) 

aΣx + bΣ²+cΣx³ = Σxy………..(2) 

aΣx² + bΣ3+cΣx⁴ = Σx²y………..(3) 

Here x = X – X̄ 

Let x and y be the year and the value

From (1) 7a + b.0 + c.28 = 261

7a + 28c = 261 ………….. (4)

From (2) a.0 + b.28 + c.0 = 310 

b= 11.07

From (3) a.28 + b.0 + c.196 

= 1216

28a + 196c = 1216 ………….(5)

Solving (4) and (5) we get a and c

4 x (4) – (5)

∴ c = 2.05

Put c = 2.05 in (4) we get

7a + 28(2.05) = 261

7a = 261 – 57.4

∴ a = 29.09

The trend equation is:

 ŷ = 29.09 +13.57x+2.05x²

Estimation for the trend value for 1997

Here x = 4 ; x² = 16

ŷ₁₉₉₇ = 29.09 +11.07 (4) + 2.05(16)

ŷ₁₉₉₇ = 106.17.