Time Series and Index Numbers
What you must understand
A time series is data recorded at successive points in time. Its four components are the trend (long-term direction), seasonal variation (regular repeating pattern within a year), cyclical variation (longer, irregular waves) and random/irregular variation.
Know how to calculate a moving average to smooth data and estimate the trend. For quarterly data use a 4-point moving average; because 4 is even, centre it with a two-stage average (add pairs of consecutive 4-point totals and divide by 8). For monthly data use a 12-point moving average, centred by dividing consecutive totals by 24. Seasonal variation is found by subtracting the trend from the actual value; average these differences for each season to get the average seasonal effect, adjusting so they sum to about zero.
For index numbers, know the simple price relative (p₁ ÷ p₀ × 100) and the weighted aggregate index (Σw × price relative ÷ Σw). Understand base year, current year, weights and chain base indices, and how to deflate a value to real terms.
Traps that appear in questions
- Forgetting to centre an even-point moving average.
- Losing the first and last values of the series when the moving average is taken.
- Using the wrong base year or forgetting to multiply by 100.
- Assuming a rising trend means every period rises; seasonal dips can coexist with an upward trend.
Worked example
Quarterly sales are 40, 50, 30, 60. Find the first centred 4-point moving average.
Answer: Total = 40 + 50 + 30 + 60 = 180. The trend is 180 ÷ 4 = 45, placed at the midpoint between quarters 2 and 3. With more data, centre by averaging consecutive totals and dividing by 8.
Rule of thumb: a moving average of n points removes a pattern of length n; choose n to match the season length.