Management Accounting · Summarising and analysing data

Time Series Analysis and Forecasting for ACCA MA

Updated 11 October 2026 · Fact-checked

A time series is a set of figures recorded at regular intervals. You split it into trend, seasonal variation, cyclical and random parts. Use moving averages to find the trend, subtract (additive) or divide (multiplicative) to get seasonal variations, average them, then add or multiply them onto the projected trend to forecast.

Understand Time Series Analysis and Forecasting

A time series is a set of values recorded at regular intervals, such as quarterly sales over several years. Time series analysis looks for patterns in the past so you can forecast the future.

The series has four possible components:

Trend (T): the long-term underlying direction, up or down.
Seasonal variation (S): a short-term, regular pattern that repeats within a year or other fixed period, such as higher sales every December.
Cyclical variation (C): a longer swing around the trend, often linked to the economy, lasting several years. It is hard to predict and rarely tested numerically.
Random variation (R): unpredictable one-off effects such as a strike or bad weather.

To see the trend, you need to remove the seasonal effect. A moving average does this. It averages one full cycle of data (four quarters, or seven days) and moves along one period at a time. Seasonal ups and downs cancel out, leaving the trend. If the number of periods is even, such as four quarters, the average falls between two periods. You then take a centred average of two consecutive averages so it lines up with an actual period.

The additive model says Actual = Trend + Seasonal (+ random). Seasonal variations are fixed amounts, such as +$20 or -$15. The multiplicative model says Actual = Trend × Seasonal. Seasonal variations are proportions, such as 1.10 or 0.90, and the seasonal effect grows as the trend grows. Use additive when the seasonal swings stay about the same size. Use multiplicative when they grow with the trend.

To forecast, you extend the trend line, then adjust it with the average seasonal variation for the period you want. Forecasts are more reliable for the short term, and they assume past patterns continue.

Key formulas to remember

Additive model: Y = T + S + R; Seasonal variation is an absolute amount. Seasonal variations should sum to zero over a full cycle.
Multiplicative model: Y = T × S × R; Seasonal variation is a factor or percentage. Factors should average 1 (sum to the number of periods in the cycle).
Seasonal variation, additive: S = Actual - Trend; Calculate for each period, then average the figures for the same season across years.
Seasonal variation, multiplicative: S = Actual ÷ Trend; Average the ratios for the same season across years.
Moving average, odd number of periods: Sum of n values ÷ n; Placed against the middle period. Example: 7-day average for daily data.
Centred moving average, even number of periods: (Average 1 + Average 2) ÷ 2, or (½ first + middle values + ½ last) ÷ 4 for quarters; Places the trend against an actual period.
Adjusting seasonal variations (additive): Adjustment = Total of averages ÷ number of seasons; subtract it from each average; Makes the variations sum to zero.
Forecast, additive: Forecast = Projected trend + Seasonal variation; Project the trend first, using the average change per period.
Forecast, multiplicative: Forecast = Projected trend × Seasonal factor; Use the factor for the correct season.
Seasonally adjusted figure: Additive: Actual - S. Multiplicative: Actual ÷ S; Removes the seasonal effect so you can see the underlying trend.

How to solve Time Series Analysis and Forecasting questions

Use this method for any time series question. Read the question first to see which step it actually asks for.

1Identify the cycle length (4 for quarters, 7 for days, 12 for months) and whether the model is additive or multiplicative.
2Calculate the moving average over one full cycle. If the cycle length is even, centre it by averaging two consecutive moving averages.
3Match each trend value with its actual figure for the same period.
4Find the seasonal variation: Actual - Trend (additive) or Actual ÷ Trend (multiplicative).
5Average the variations for each season across the years. Adjust so additive variations sum to zero, or multiplicative factors sum to the number of seasons.
6Project the trend forward using the average change per period, or the trend line given.
7Combine trend and seasonal variation for the target period: add or multiply.
8Check the answer is sensible: right season, right sign, and similar in size to the actual data.

Quickest way: Shortcut for number entry and multiple choice questions

When to use it: Use when the question gives you trend values or seasonal variations and asks for one forecast, or one missing figure.

Read what is given. If the trend and seasonal variation are already supplied, skip the moving averages.
For a missing seasonal variation, use the rule that additive variations sum to zero (or multiplicative factors average 1). Solve for the unknown.
For a forecast, count how many periods the target is beyond the last trend value. Add that many times the average trend change.
Apply the seasonal variation for the target season only.
Round only at the end and check the sign, since a minus variation lowers the forecast.

Common mistakes in Time Series Analysis and Forecasting

Forgetting to centre a four-quarter moving average.
The first average falls between quarters 2 and 3, and students place it against one quarter.
Fix: Average two consecutive moving averages, or use half the first and last values plus the middle three, divided by 4.
Using Actual - Trend in a multiplicative model, or Actual ÷ Trend in an additive one.
Students do not check which model the question states.
Fix: Underline the model first. Additive means subtract, multiplicative means divide.
Not adjusting seasonal variations so they sum to zero.
Students treat the raw averages as final.
Fix: Add the averages. If the total is not zero, divide it by the number of seasons and subtract that from each average.
Forecasting with the wrong season or wrong number of periods.
Students lose count when projecting the trend forward.
Fix: Write the period numbers down. Extend the trend period by period, then pick the seasonal variation for that period's quarter.
Applying a seasonal variation to the actual figure instead of the trend.
Students mix up seasonally adjusted data with forecasting.
Fix: A forecast is projected trend plus or times seasonal variation. Seasonal adjustment is the reverse: actual minus or divided by seasonal variation.
Treating cyclical variation as seasonal.
Both look like repeating swings.
Fix: Seasonal repeats within a year or fixed short period. Cyclical lasts several years and has no fixed length.

Worked examples

Example 1

Quarterly sales (units) are: Year 1 Q1 100, Q2 140, Q3 180, Q4 120; Year 2 Q1 112, Q2 152, Q3 192, Q4 132. A four-quarter moving average (uncentred) for Y1 Q1 to Q4 is 135, and for Y1 Q2 to Y2 Q1 is 138. Find the centred trend for Y1 Q3 and the additive seasonal variation for that quarter.

Show the solution

Check the first average: (100 + 140 + 180 + 120) ÷ 4 = 540 ÷ 4 = 135. This sits between Q2 and Q3.
Check the second: (140 + 180 + 120 + 112) ÷ 4 = 552 ÷ 4 = 138. This sits between Q3 and Q4.
The centred trend for Y1 Q3 is the average of the two: (135 + 138) ÷ 2 = 136.5.
Actual for Y1 Q3 is 180.
Additive seasonal variation = Actual - Trend = 180 - 136.5 = +43.5.

Answer: Centred trend for Y1 Q3 is 136.5 units and the seasonal variation is +43.5 units.

Example 2

Using an additive model, the trend for a business is T = 200 + 5x, where x is the quarter number (Y1 Q1 is x = 1). Average seasonal variations are Q1 -12, Q2 +4, Q3 +20, Q4 -12. Forecast sales for Year 3 Q2.

Show the solution

First check the variations sum to zero: -12 + 4 + 20 - 12 = 0. No adjustment is needed.
Year 3 Q2 is quarter number (2 × 4) + 2 = 10, so x = 10.
Trend = 200 + 5 × 10 = 250.
Seasonal variation for Q2 is +4.
Forecast = 250 + 4 = 254.

Answer: Forecast sales for Year 3 Q2 are 254 units.

Exam tips

Objective test questions often give the trend and seasonal variations and ask for one forecast. Check the model, count the periods, and pick the right season.
Check whether the question wants a trend value, a seasonal variation, a seasonally adjusted figure or a forecast. They use different formulas.
Additive variations must sum to zero and multiplicative factors must sum to the number of seasons. Use this to find a missing value fast.
In multiple response questions, remember forecasts assume past patterns continue and are less reliable the further ahead you go.
Keep your calculator workings in order on scrap paper. With centred averages, one slipped period ruins every later figure.

Practice questions from Summarising and analysing data

Time Series Analysis and Forecasting in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Time Series Analysis and Forecasting: frequently asked questions

What is the difference between additive and multiplicative models?

The additive model adds a fixed seasonal amount to the trend: Y = T + S. The multiplicative model multiplies the trend by a seasonal factor: Y = T × S. Use multiplicative when seasonal swings grow as the trend grows.

How do I calculate seasonal variation in an additive model?

Find the trend with a centred moving average, then subtract it from the actual figure for each period. Average the results for each season across years. Adjust the averages so they sum to zero.

Why do I need to centre a moving average?

With an even cycle, such as four quarters, the average falls halfway between two periods. Centring averages two consecutive values so the trend lines up with an actual period and can be compared to actual data.

How reliable are time series forecasts?

They are most reliable in the short term. They assume the trend and seasonal pattern continue, so they can be wrong if conditions change, if random events occur, or if the forecast goes far beyond the data.