Question 4 (a)
Straight-line trend by least squares, monthly conversion, shift of origin and detrending
14 marks
- Time to write
- About 28 minutes
- Topic
- Trend fitting
Question
Question 4 (a) · 14 marksEvergreen Electronics Ltd. has recorded its revenue for each year (in crores) from 2017 to 2024, as shown below:
| Year | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | 2024 |
|---|---|---|---|---|---|---|---|---|
| Revenue (crores ₹) | 118 | 132 | 149 | 140 | 155 | 169 | 158 | 176 |
You have to:
Fit a straight-line trend to the revenue data by the method of least squares.
Compute the trend values for each of the years 2017 through 2024.
Convert the annual trend values to monthly trends, and also give the reason for dividing the slope coefficient by 144.
Move the origin to the year 2024 when calculating the trend.
Eliminate the trend using the additive model.
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Read from the question
- Compute…Give…Two asks in one question: answer both.
- ComputeFormula first, then the working, then the answer with its unit (₹, %, units).
Write it in this order
- Read the data; note exactly what is asked 2 min
- 1 Working table and the trend equation 4 marks 7 min
- 2 Trend values for 2017 to 2024 2 marks 4 min
- 3 Monthly trend and the reason for dividing by 144 3 marks 5 min
- 4 Shifting the origin to 2024 2 marks 3 min
- 5 Eliminating the trend additive model 3 marks 5 min
- Check: arithmetic, units (₹, %), final answer underlined 2 min
Try it first. About 28 minutes for 14 marks. Write it out, then open the solution.
Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
Method. The syllabus asks for trend fitting by least squares, shifting of origin and conversion of an annual linear trend to a monthly basis; the DU guideline limits conversion to linear trends. The trend Y = a + bX is fitted from the least-squares normal equations ΣY = na + bΣX and ΣXY = aΣX + bΣX². With an even number of years (8), time is measured from the middle of 2020 and 2021 in half-year units, X = 2(year − 2020.5), so that ΣX = 0 and the equations reduce to a = ΣY / n and b = ΣXY / ΣX².
Working table (revenue Y in ₹ crore)
| Year | Y | X | XY | X² | Trend T | Y − T |
|---|---|---|---|---|---|---|
| 2017 | 118 | −7 | −826 | 49 | 124.17 | −6.17 |
| 2018 | 132 | −5 | −660 | 25 | 131.44 | 0.56 |
| 2019 | 149 | −3 | −447 | 9 | 138.71 | 10.29 |
| 2020 | 140 | −1 | −140 | 1 | 145.99 | −5.99 |
| 2021 | 155 | 1 | 155 | 1 | 153.26 | 1.74 |
| 2022 | 169 | 3 | 507 | 9 | 160.54 | 8.46 |
| 2023 | 158 | 5 | 790 | 25 | 167.81 | −9.81 |
| 2024 | 176 | 7 | 1,232 | 49 | 175.08 | 0.92 |
| Total | 1,197 | 0 | 611 | 168 | 1,197 | 0 |
a = ΣY / n = 1,197 / 8 = 149.625; b = ΣXY / ΣX² = 611 / 168 = 3.6369.
Y = 149.625 + 3.6369X, origin: middle of 2020 and 2021 (1 January 2021); X unit: half a year; Y: annual revenue (₹ crore).
In whole years (one year = 2 half-years, so the slope doubles): Y = 149.625 + 7.2738X′, X′ in years from the same origin. Revenue rises by about ₹7.27 crore a year on trend.
(ii) Trend values, 2017 to 2024
From T = 149.625 + 3.6369X (column Trend T above): 2017 ₹124.17 crore; 2018 ₹131.44; 2019 ₹138.71; 2020 ₹145.99; 2021 ₹153.26; 2022 ₹160.54; 2023 ₹167.81; 2024 ₹175.08 crore. Each year adds 2b = ₹7.27 crore, and the trend values add up to ΣY, as they must.
The annual equation (X′ in years) is converted in two steps:
- Divide a by 12: the annual equation gives revenue for a whole year; dividing by 12 gives the average revenue of a month: 149.625 / 12 = 12.4688.
- Divide b by 144: the slope 7.2738 is a change in annual revenue per year. Dividing it by 12 converts the annual revenue into monthly revenue, and dividing by 12 again converts the time unit from a year into a month: 7.2738 / (12 × 12) = 7.2738 / 144 = 0.0505. This double conversion, of the variable and of the time unit, is the reason the slope is divided by 144 while the intercept is divided only by 12.
Monthly trend: Ym = 12.4688 + 0.0505x, origin 1 January 2021, x unit one month, Ym monthly revenue (₹ crore). To centre the origin on the middle of January 2021, add half a month's slope: Ym = 12.4940 + 0.0505x.
Moving the origin forward by 3.5 years (from 1 January 2021 to the middle of 2024) changes only the intercept: a′ = 149.625 + 7.2738 × 3.5 = 175.083, the trend value of 2024.
Y = 175.083 + 7.2738X″, origin 2024, X″ in years (or, in half-year units, Y = 175.083 + 3.6369X).
(v) Trend eliminated (additive model)
Under the additive model Y = T + S + C + I; annual data contain no seasonal part, so subtracting the trend leaves the cyclical and irregular movements: Y − T = C + I (last column of the table). The deviations sum to zero; revenue ran above trend in 2019 and 2022 (by ₹10.29 and ₹8.46 crore) and below it in 2017, 2020 and 2023.
Where the 14 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Working table and the trend equation | 4 |
| 2 | Trend values for 2017 to 2024 | 2 |
| 3 | Monthly trend and the reason for dividing by 144 | 3 |
| 4 | Shifting the origin to 2024 | 2 |
| 5 | Eliminating the trend (additive model) | 3 |
| Total | 14 |
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