Time Series Analysis: Trend, Seasonal, Cyclical and Irregular Components, Moving Average and Least Squares for SSC Statistics

A time series is a set of observations of a variable taken at regular intervals of time, such as yearly sales, monthly rainfall or daily share prices. Time series analysis finds the pattern in such data and is used to forecast the future. SSC CGL Statistics regularly asks about the components of a time series, the moving average method and the least squares trend line. This guide covers all of them with solved examples and practice questions.

Components of a Time Series

Models of a Time Series

Measurement of Trend

Moving Average

For a 3-year moving average, add the values of three successive years and divide by 3; place the result against the middle year. A period equal to the length of the cycle removes the cyclical fluctuations. For an even period (such as 4 years), the averages are centred by taking a 2-point average of successive moving averages.

Least Squares Trend Line

The straight-line trend is Y = a + bX. When the time variable X is coded so that ΣX = 0 (the origin is placed in the middle year), the normal equations simplify to:

a = ΣY / n and b = ΣXY / ΣX².

Solved Examples

Example 1. The sales (in thousands) in 2019 to 2023 are 10, 12, 14, 16 and 18. Find the trend line and forecast for 2025.
Take 2021 as the origin: X = −2, −1, 0, 1, 2. ΣY = 70, so a = 70/5 = 14. ΣXY = (−20) + (−12) + 0 + 16 + 36 = 20 and ΣX² = 10, so b = 2. The trend line is Y = 14 + 2X. For 2025, X = 4 and Y = 14 + 8 = 22.

Example 2. Find the 3-year moving averages of 10, 12, 14, 16 and 18.
(10 + 12 + 14)/3 = 12; (12 + 14 + 16)/3 = 14; (14 + 16 + 18)/3 = 16.

Example 3. In a multiplicative model the trend value is 200, the seasonal index is 120% and other components are ignored. Find the expected value.
200 × 120/100 = 240.

Example 4. Which component of the time series explains the rise in sales during Diwali every year?
Seasonal variation.

Seasonal Index

• The simple average method finds the average for each month (or quarter) over the years and divides it by the grand average, then multiplies by 100.
• The seasonal indices of the 12 months add up to 1,200 (their average is 100); for 4 quarters they add up to 400.
• Other methods are the ratio to trend and the ratio to moving average methods.

Uses of Time Series

• Understanding the past behaviour of a variable.
• Forecasting future values for planning.
• Comparing the changes in different series.
• Separating seasonal and trend effects for decisions in business and government.

Tricks

• Code the years around the middle year to get ΣX = 0.
• For an odd number of years, X is …, −2, −1, 0, 1, 2, …; for an even number, use half-year steps such as −3, −1, 1, 3.
• The cyclical component is not the same as the seasonal component; seasonal repeats within a year.
• Random variation cannot be forecast.

Quick Tips

• Code the time variable around the middle to simplify the least squares method.
• Learn the four components with their symbols.
• Remember the sum of seasonal indices (1,200 or 400).
• Distinguish seasonal from cyclical variation.

Quick Revision Points

• Components: T, S, C and I.
• Additive: Y = T + S + C + I; multiplicative: Y = T × S × C × I.
• Least squares: a = ΣY/n and b = ΣXY/ΣX² when ΣX = 0.
• Moving averages smooth out short-term variations.
• Seasonal indices average 100.

Practice Questions With Answers

1. What are the four components of a time series?
   Answer: Secular trend, seasonal, cyclical and irregular variations
2. Which model is Y = T × S × C × I?
   Answer: Multiplicative model
3. Find the 3-year moving average of 6, 9 and 12.
   Answer: 9
4. Which component is caused by floods and strikes?
   Answer: Irregular variation
5. What is the sum of seasonal indices for 12 months?
   Answer: 1,200
6. Which method of trend fitting is the best?
   Answer: Method of least squares
7. If ΣY = 60 for 5 years with ΣX = 0, find a.
   Answer: 12
8. If ΣXY = 30 and ΣX² = 10, find b.
   Answer: 3
9. Which variation shows a long-term rise or fall?
   Answer: Secular trend
10. Business cycles are an example of which component?
    Answer: Cyclical variation

How to Practise This Chapter

Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Statistics notes page, read the previous topic Index Numbers and the next topic Sampling Theory and Methods.

Frequently Asked Questions

What is a time series?

A set of observations of a variable recorded at regular time intervals.

What are the components of a time series?

Secular trend, seasonal variation, cyclical variation and irregular variation.

What is the secular trend?

The long-term upward or downward movement of the data.

What is the seasonal variation?

The regular pattern that repeats within a year.

What is the cyclical variation?

The ups and downs over a period of more than one year, as in business cycles.

What is the additive model?

Y = T + S + C + I.

What is the moving average method?

A method that smooths the data using the averages of successive groups of values.

What is the least squares method?

A method that fits a trend line by minimising the sum of squares of the deviations.

What is the seasonal index?

A number showing the seasonal level of each period relative to the average, with an average of 100.

Why do we code the time variable?

To make ΣX = 0 so that the calculations become simple.

What are the uses of time series?

Understanding the past, forecasting and planning.

Is it asked in SSC?

Yes. It is a part of the SSC CGL Statistics syllabus.