Step-by-Step Solution
Topic: Method of Least Squares
1. Coding the Time Variable (X)
There are 5 years (Odd number of years). We take the middle year (2018) as the origin.
$$ u = Year – 2018 $$
| Year | u (X) | u² |
|---|---|---|
| 2016 | -2 | 4 |
| 2017 | -1 | 1 |
| 2018 | 0 | 0 |
| 2019 | 1 | 1 |
| 2020 | 2 | 4 |
| Sum | Σu = 0 | Σu² = 10 |
The equation of the straight line trend is Y = a + bu.
$$ a = \frac{\sum Y}{N} $$ and $$ b = \frac{\sum uY}{\sum u^2} $$
Here, N = 5.
(A)(I) TREND LINE FOR RURAL (YR)
| Year | u | Rural (Y) | uY |
|---|---|---|---|
| 2016 | -2 | 3 | -6 |
| 2017 | -1 | 6 | -6 |
| 2018 | 0 | 9 | 0 |
| 2019 | 1 | 16 | 16 |
| 2020 | 2 | 24 | 48 |
| Sum | 0 | ΣY = 58 | ΣuY = 52 |
Calculate constants:
$$ a = \frac{58}{5} = 11.6 $$
$$ b = \frac{52}{10} = 5.2 $$
(A)(II) TREND LINE FOR URBAN (YU)
| Year | u | Urban (Y) | uY |
|---|---|---|---|
| 2016 | -2 | 9 | -18 |
| 2017 | -1 | 18 | -18 |
| 2018 | 0 | 21 | 0 |
| 2019 | 1 | 29 | 29 |
| 2020 | 2 | 38 | 76 |
| Sum | 0 | ΣY = 115 | ΣuY = 69 |
Calculate constants:
$$ a = \frac{115}{5} = 23 $$
$$ b = \frac{69}{10} = 6.9 $$
(B) FORECAST 2021 (URBAN)
For the year 2021:
$$ u = 2021 – 2018 = 3 $$
Substitute u = 3 into the Urban trend equation:
Y = 23 + 6.9(3)
Y = 23 + 20.7
(C) FORECAST 2021 (RURAL)
For the year 2021:
$$ u = 3 $$
Substitute u = 3 into the Rural trend equation:
Y = 11.6 + 5.2(3)
Y = 11.6 + 15.6
⚠️ Silly Mistake Audit: Calculating ‘u’
Check the Origin Year!
A common mistake when forecasting is using the wrong value for ‘u’. Always calculate u = Target Year – Base Year. In this case, the base year is 2018. For 2021, u is 3 (2021 – 2018), not 4 or 5.