Method of Least Squares

Time Series — Chapter 5, GSEB Class 12 Statistics

Fitting a Trend Line

The method of least squares fits a straight trend line through the time series data. We assign natural numbers (t = 1, 2, 3, ...) to the time periods and use the least squares formulas to find the best-fit line.

Trend Line Equation

\hat{Y} = a + bt

Slope (b)

b = \frac{n \sum ty - \sum t \sum y}{n \sum t^2 - (\sum t)^2}

Intercept (a)

a = \bar{y} - b\bar{t}

t = time period (natural number: 1, 2, 3, ...), y = observed value, n = number of observations, ȳ = mean of y, t̄ = mean of t.

Worked Example

Fit a trend line for the following sales data

Year	t	Sales (y)	ty	t²
2017	1	35	35	1
2018	2	40	80	4
2019	3	38	114	9
2020	4	46	184	16
2021	5	50	250	25
2022	6	54	324	36
2023	7	52	364	49
Total		315	1351	140

t̄ = Σt / n = 28 / 7 = 4

ȳ = Σy / n = 315 / 7 = 45.00

b = (nΣty − ΣtΣy) / (nΣt² − (Σt)²) = (7 × 1351 − 28 × 315) / (7 × 140 − 28²)

= (9457 − 8820) / (980 − 784) = 3.25

a = ȳ − b·t̄ = 45.00 − 3.25 × 4 = 32.00

Ŷ = 32.00 + 3.25t

Forecasting

To forecast future values, assign the next natural number to t and substitute:

Forecast for 2024 and 2025:

2024: t = 8 (next natural number after 7)

Ŷ = 32.00 + 3.25 × 8 = 58.00

2025: t = 9

Ŷ = 32.00 + 3.25 × 9 = 61.25

Remember: t for future years continues the natural number sequence. If data has 7 years (t = 1 to 7), then 2024 gets t = 8, 2025 gets t = 9.

Key Takeaways

Assign natural numbers to time periods: t = 1, 2, 3, ...
b = (nΣty − ΣtΣy) / (nΣt² − (Σt)²) — the slope of the trend line.
a = ȳ − b·t̄ — the intercept where the line crosses the Y-axis.
Trend line: Ŷ = a + bt — substitute t for any year to forecast.
For future years, continue the natural number sequence (t = n+1, n+2, ...).

Method of Least Squares

Notes

Method of Least Squares

Fitting a Trend Line

Worked Example

Forecasting

Key Takeaways