10 Time Series & Index Numbers · 10.1 Time series and index numbers
10.1.1 Time series: components, measuring trend (graphic, semi-averages, moving averages, least squares) and seasonal variation
A time series is data arranged in order of time; its movement is split into trend (T), seasonal (S), cyclical (C) and irregular (I) parts, and the trend is measured by the graphic, semi-averages, moving averages or least squares method.
- APPSC Paper I
- Advanced level
- 11 min read
- Free
Core facts
- Time series: values of a variable recorded at successive points or periods of time (yearly output, monthly prices, daily sales).
- Four components:
- Secular trend (T): smooth, long-term rise or fall (population growth, falling death rate).
- Seasonal variation (S): regular, repeating pattern within a year (weekly, monthly, quarterly); caused by climate, festivals, customs.
- Cyclical variation (C): up-and-down swings over more than a year (business cycle: prosperity, recession, depression, recovery); period not fixed.
- Irregular (random, erratic) variation (I): unpredictable effects of floods, wars, strikes, epidemics.
- Additive model: Y = T + S + C + I (components independent). Multiplicative model: Y = T × S × C × I (components related; most used).
- Methods of measuring trend: (1) freehand / graphic, (2) semi-averages, (3) moving averages, (4) least squares (most used, gives a mathematical equation).
- Moving average period should equal the period of the cycle; an odd period is centred directly, an even period must be centred again by a 2-item moving average.
- Least squares straight line: Y = a + bX. Normal equations: ΣY = na + bΣX, ΣXY = aΣX + bΣX².
- If X is measured from the middle so that ΣX = 0: a = ΣY / n, b = ΣXY / ΣX².
- Least squares line passes through the means; Σ(Y − Yc) = 0 and Σ(Y − Yc)² is minimum.
- Methods of measuring seasonal variation: (1) simple averages, (2) ratio to trend, (3) ratio to moving average (most widely used), (4) link relatives.
- Seasonal indices total 400 for quarters and 1,200 for months (average 100).
Example 1. Sales of umbrellas rise every June–August. Which component is this? Solution. It repeats every year within the year. Answer: seasonal variation
Example 2. Fit a straight line by least squares to 7 years when ΣY = 126, ΣXY = 84, ΣX² = 28 (X from the middle year). Find a and b. Solution. a = 126 ÷ 7 = 18; b = 84 ÷ 28 = 3. Answer: Y = 18 + 3X
Explained
1. Graphic (freehand) method. Plot the data and draw a smooth line through the middle of the fluctuations. Simple and flexible, but subjective: two people draw different lines, and it cannot be relied on for forecasting.
2. Method of semi-averages. Divide the series into two equal halves (drop the middle year if the number of years is odd), find the average of each half, plot each average against the middle of its half, and join the two points.
Example 3. Output (thousand tonnes): 2018: 40, 2019: 44, 2020: 42, 2021: 48, 2022: 52, 2023: 50, 2024: 56, 2025: 60. Find the trend by semi-averages and estimate 2026. Solution.
- First half (2018–21): (40 + 44 + 42 + 48) ÷ 4 = 174 ÷ 4 = 43.5, placed at mid-2019 and 2020 (2019.5).
- Second half (2022–25): (52 + 50 + 56 + 60) ÷ 4 = 218 ÷ 4 = 54.5, placed at 2023.5.
- Annual change = (54.5 − 43.5) ÷ 4 = 2.75.
- 2026 is 2.5 years after 2023.5: 54.5 + 2.5 × 2.75 = 61.375.
Answer: trend rises 2.75 a year; 2026 estimate about 61.4 thousand tonnes
Merits: objective, easy. Limits: assumes a straight-line trend; the arithmetic mean is affected by extreme values.
3. Method of moving averages. Average the first m values, then drop the first and add the next, and so on; place each average at the centre of its period.
Example 4. Find the 3-yearly moving averages: 2019: 20, 2020: 24, 2021: 22, 2022: 26, 2023: 30, 2024: 28, 2025: 34. Solution.
| Year | Value | 3-year total | 3-year MA |
|---|---|---|---|
| 2019 | 20 | – | – |
| 2020 | 24 | 66 | 22.00 |
| 2021 | 22 | 72 | 24.00 |
| 2022 | 26 | 78 | 26.00 |
| 2023 | 30 | 84 | 28.00 |
| 2024 | 28 | 92 | 30.67 |
| 2025 | 34 | – | – |
Answer: 22, 24, 26, 28, 30.67 for 2020–2024 (no trend values for the first and last year)
Example 5. Using the output data of Example 3, find the 4-yearly centred moving averages. Solution.
- 4-year totals: 174, 186, 192, 206, 218, centred at 2019.5, 2020.5, 2021.5, 2022.5, 2023.5.
- Centre them: add pairs of totals and divide by 8.
- 2020: (174 + 186) ÷ 8 = 45.00
- 2021: (186 + 192) ÷ 8 = 47.25
- 2022: (192 + 206) ÷ 8 = 49.75
- 2023: (206 + 218) ÷ 8 = 53.00
Answer: 45, 47.25, 49.75, 53 for 2020–2023 (two years lost at each end)
Merits: simple, flexible, removes cyclical swings if the period equals the cycle. Limits: trend values are lost at both ends; no equation, so no forecasting; choice of period matters.
4. Method of least squares (straight line). Choose the line Y = a + bX that makes the sum of squared deviations of actual values from the line a minimum. Take the middle year as origin so ΣX = 0.
Example 6. Fit a straight-line trend and estimate 2027: 2019: 8, 2020: 13, 2021: 16, 2022: 17, 2023: 21, 2024: 24, 2025: 27. Solution.
- Origin 2022, X = −3, −2, −1, 0, 1, 2, 3; ΣX = 0, ΣX² = 9 + 4 + 1 + 0 + 1 + 4 + 9 = 28.
- ΣY = 126. ΣXY = −24 − 26 − 16 + 0 + 21 + 48 + 81 = 84.
- a = 126 ÷ 7 = 18; b = 84 ÷ 28 = 3. Trend: Y = 18 + 3X (origin 2022, X in years, Y in the data's units).
- Trend values 2019–2025: 9, 12, 15, 18, 21, 24, 27.
- 2027: X = 5, Y = 18 + 15 = 33.
Answer: Y = 18 + 3X; 2027 estimate = 33
Merits: objective, gives trend for every year and a formula for forecasting. Limits: heavy arithmetic; adding one new year changes all values; the chosen type of curve (line, parabola) must fit the data.
5. Seasonal variation by simple averages. Average each season over the years; divide each seasonal average by the grand average of the seasonal averages; multiply by 100.
Example 7. Quarterly sales (Rs lakh) for three years. Find the seasonal indices.
| Year | Q1 | Q2 | Q3 | Q4 |
|---|---|---|---|---|
| 2023 | 60 | 48 | 52 | 72 |
| 2024 | 64 | 52 | 58 | 78 |
| 2025 | 68 | 56 | 61 | 84 |
Solution.
- Quarter averages: Q1 = 192 ÷ 3 = 64; Q2 = 156 ÷ 3 = 52; Q3 = 171 ÷ 3 = 57; Q4 = 234 ÷ 3 = 78.
- Grand average = (64 + 52 + 57 + 78) ÷ 4 = 251 ÷ 4 = 62.75.
- Indices: Q1 = 64 ÷ 62.75 × 100 = 101.99; Q2 = 82.87; Q3 = 90.84; Q4 = 124.30. Total = 400.
Answer: 101.99, 82.87, 90.84, 124.30 (Q4 is the peak season)
The simple average method assumes there is no trend; when the data rise over time (as here), later quarters are overstated, so the other methods are preferred.
Deeper
1. Even number of years in least squares. With an even n there is no middle year. Take the origin midway between the two middle years and measure X in half-years: X = −5, −3, −1, 1, 3, 5 for six years. Then b is the change per half-year; double it for the yearly change.
Example 8. Fit a straight line: 2020: 30, 2021: 34, 2022: 37, 2023: 43, 2024: 47, 2025: 49. Estimate 2026. Solution.
- Origin between 2022 and 2023; X = −5, −3, −1, 1, 3, 5; ΣX² = 25 + 9 + 1 + 1 + 9 + 25 = 70.
- ΣY = 240; ΣXY = −150 − 102 − 37 + 43 + 141 + 245 = 140.
- a = 240 ÷ 6 = 40; b = 140 ÷ 70 = 2 (per half-year), so 4 per year.
- Y = 40 + 2X. For 2026, X = 7: Y = 40 + 14 = 54.
Answer: Y = 40 + 2X (X in half-years); 2026 estimate = 54
2. Second-degree (parabolic) trend. Y = a + bX + cX². Normal equations:
- ΣY = na + bΣX + cΣX²
- ΣXY = aΣX + bΣX² + cΣX³
- ΣX²Y = aΣX² + bΣX³ + cΣX⁴
With ΣX = ΣX³ = 0: ΣY = na + cΣX², ΣXY = bΣX², ΣX²Y = aΣX² + cΣX⁴.
Example 9. Fit a parabola to 5 years: 6, 7, 10, 15, 22 (X = −2, −1, 0, 1, 2). Solution.
- ΣX² = 10, ΣX⁴ = 16 + 1 + 0 + 1 + 16 = 34, ΣY = 60, ΣXY = −12 − 7 + 0 + 15 + 44 = 40, ΣX²Y = 24 + 7 + 0 + 15 + 88 = 134.
- b = 40 ÷ 10 = 4.
- 60 = 5a + 10c and 134 = 10a + 34c. Doubling the first: 120 = 10a + 20c; subtracting, 14 = 14c, so c = 1, a = 10.
Answer: Y = 10 + 4X + X²
3. Other seasonal methods.
- Ratio to trend: fit a trend (least squares), express each original value as a % of its trend value, average these percentages season by season, adjust to total 400 (or 1,200). Removes trend; cycles are not removed.
- Ratio to moving average: compute a 12-month (or 4-quarter) centred moving average (it contains T and C), divide each original value by it (leaving S × I), average season-wise to remove I, then adjust. Most satisfactory and widely used.
- Link relatives (Pearson's method): link relative = current season ÷ previous season × 100; average them, chain them, correct for trend, adjust. Laborious.
4. Deseasonalising. Under the multiplicative model, deseasonalised value = actual value ÷ seasonal index × 100. Example: Q4 sales of 84 with index 124.30 → 84 ÷ 1.2430 = 67.6 (about).
5. Choosing the moving average period. If the data have a regular cycle of k years, a k-year moving average removes it completely; a wrong period leaves part of the cycle in the trend.
| Method | Equation? | Ends lost? | Objective? |
|---|---|---|---|
| Freehand | no | no | no |
| Semi-averages | line only | no | yes |
| Moving averages | no | yes | yes |
| Least squares | yes | no | yes |
Common traps
- ✗ "Business cycles repeat within a year." ✓ That is seasonal variation; cycles last more than a year.
- ✗ "A 4-year moving average is placed against the 2nd year." ✓ It falls between two years; it must be centred again.
- ✗ "Moving averages give trend values for every year." ✓ Values are lost at both ends.
- ✗ "With 6 years, X = −3 to 2 makes ΣX = 0." ✓ Use −5, −3, −1, 1, 3, 5 (half-year units), then double b for the yearly change.
- ✗ "Seasonal indices of four quarters total 100." ✓ They total 400 (1,200 for months).
- ✗ "Least squares minimises the sum of deviations." ✓ It minimises the sum of squared deviations; the plain sum is zero.
- ✗ "The simple average method works for data with a strong trend." ✓ It assumes no trend; use ratio to moving average.
Quick revision
- Components: trend, seasonal (within a year), cyclical (over a year), irregular.
- Additive Y = T + S + C + I; multiplicative Y = T × S × C × I.
- Trend methods: freehand, semi-averages, moving averages, least squares.
- Semi-averages: two halves, average at the middle of each half.
- Even moving-average period: centre it again.
- Least squares with ΣX = 0: a = ΣY/n, b = ΣXY/ΣX².
- Even n: X = …, −3, −1, 1, 3, …; yearly change = 2b.
- Seasonal methods: simple averages, ratio to trend, ratio to moving average (best), link relatives.
- Seasonal indices total 400 (quarters) or 1,200 (months).
- Deseasonalised = actual ÷ seasonal index × 100.
Practise this topic. Questions on Time series: components, trend & seasonal variation, with answers explained.
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