Time Series Analysis
A time series is a set of observations recorded at successive points in time (daily stock prices, monthly sales, annual GDP). Time series analysis identifies patterns in historical data to forecast future values — a critical business planning tool.
Components of Time Series
Four components: Trend (T): long-term upward or downward movement over years (population growth, GDP). Seasonal variation (S): regular fluctuations within a year (ice cream in summer, retail in Dashain/Tihar). Cyclical variation (C): fluctuations over several years linked to business cycles (boom-recession-recovery). Irregular variation (I): unpredictable, random (earthquakes, pandemics). Models: Additive: Y = T + S + C + I. Multiplicative: Y = T × S × C × I.
Measurement of Trend
Freehand curve: draw smooth curve visually — quick but subjective. Semi-averages: divide data in half, calculate mean of each, draw line — simple but only linear. Moving averages: each point is average of surrounding values, smoothing fluctuations. Choice of period m depends on data periodicity (m=4 for quarterly, m=12 for monthly). Moving averages eliminate seasonal variation when m equals seasonal period.
Least Squares Method
Fits trend line Y = a + bX minimising squared deviations. Using coding (middle period as origin): b = ΣXY/ΣX² and a = ΣY/n. Gives best-fitting straight line. For parabolic trend: Y = a + bX + cX² (three normal equations). Can be extended for forecasting, though accuracy decreases for distant periods.
Seasonal Variation
Simple averages: average for each season across years, express as percentage of grand average. Ratio-to-trend: actual/trend values, average ratios per season. Ratio-to-moving-average (most common): actual/centred moving average, average ratios per season. Indices above 100 = above-average periods; below 100 = below-average. Sum of seasonal indices = 400 (quarterly) or 1200 (monthly).
Deseasonalisation
Seasonally adjusted data removes seasonal effects to reveal underlying trends. Multiplicative: deseasonalised = actual / (seasonal index/100). Essential for meaningful comparison across periods — comparing December vs March retail sales without adjustment is misleading.
Forecasting
Trend projection: extend trend line into future. Seasonal adjustment: apply indices to trend forecast. Example: trend forecast Q3 = Rs 100,000, Q3 seasonal index = 120, seasonal forecast = Rs 100,000 × 1.20 = Rs 120,000. Accuracy decreases over longer horizons.
Summary
Time series analysis decomposes data into trend, seasonal, cyclical, and irregular components. Moving averages and least squares measure trends. Seasonal indices quantify recurring patterns. Together, these tools enable business forecasting for planning, budgeting, and strategic decision-making.
Worked Example: Least Squares Trend Line
The annual sales (Rs lakhs) of a company for 5 years are:
| Year | Sales (Y) | X (coded) | XY | X² |
|---|---|---|---|---|
| 2020 | 50 | −2 | −100 | 4 |
| 2021 | 55 | −1 | −55 | 1 |
| 2022 | 65 | 0 | 0 | 0 |
| 2023 | 70 | 1 | 70 | 1 |
| 2024 | 80 | 2 | 160 | 4 |
| Total | ΣY=320 | ΣX=0 | ΣXY=75 | ΣX²=10 |
Solution (using coding method with 2022 as origin):
a = ΣY/n = 320/5 = 64
b = ΣXY/ΣX² = 75/10 = 7.5
Trend equation: Y = 64 + 7.5X (origin: 2022, unit: 1 year)
Interpretation: Base sales (2022) are Rs 64 lakhs. Sales increase by Rs 7.5 lakhs per year. This upward trend indicates steady business growth.
Forecast for 2026: X = 4 (2026 is 4 years from 2022). Y = 64 + 7.5(4) = 64 + 30 = Rs 94 lakhs. Note: this extrapolation assumes the linear trend continues, which may not hold if market conditions change.
Worked Example: Seasonal Indices
A shop’s quarterly sales and seasonal indices are:
| Quarter | Seasonal Index | Interpretation |
|---|---|---|
| Q1 (Jan-Mar) | 85 | 15% below average — post-Dashain/Tihar slowdown |
| Q2 (Apr-Jun) | 95 | 5% below average — moderate season |
| Q3 (Jul-Sep) | 105 | 5% above average — festival preparation begins |
| Q4 (Oct-Dec) | 115 | 15% above average — Dashain/Tihar peak season |
| Total | 400 | (Must equal 400 for quarterly data) |
If the trend forecast for Q1 2025 is Rs 100 lakhs, the seasonally adjusted forecast = 100 × (85/100) = Rs 85 lakhs. The business should prepare for lower sales in Q1 and stock up for Q4.
Exam Tips
Tip 1: For least squares, always code X values with the middle year as origin (makes ΣX = 0, simplifying calculations). Tip 2: For moving averages with even period (e.g., 4-quarter), you need centred moving averages — average two consecutive 4-period averages. Tip 3: Seasonal indices must sum to 400 (quarterly) or 1200 (monthly) — adjust if they don’t. Tip 4: Deseasonalised value = Actual / (Seasonal Index/100) for multiplicative model. Tip 5: Clearly state the origin and unit when writing a trend equation.