Time-Based Forecasting and Index Numbers

Learning objectives

By the end of this chapter you should be able to:

• Explain how a time series may be viewed as a combination of trend, seasonal, cyclical and irregular components, and what each represents.
• Calculate moving averages (including centred moving averages) to smooth data and estimate the underlying trend.
• Construct and apply seasonal measures using both additive and multiplicative approaches, including appropriate normalisation.
• Forecast future values by extending a trend estimate and adjusting for seasonality.
• Construct and interpret simple and weighted index numbers, including Laspeyres and Paasche indices.
• Explain key limitations of forecasting and index numbers (for example outliers, structural change and base-year effects) and state assumptions clearly.

Overview & key concepts

Time-based forecasting uses historical patterns to estimate future activity (for example sales volumes, demand, or workload). Index numbers measure relative change over time (for example changes in prices or quantities), often by comparing a current period to a base period.

Time series components

A time series is a sequence of observations recorded over time (monthly, quarterly, etc.). It is commonly analysed in terms of four influences:

• Trend (T):the longer-term underlying direction of the series.
• Seasonal (S):a repeating within-year pattern linked to the calendar (e.g. quarter-by-quarter).
• Cyclical (C):medium-to-longer-term rises and falls that typically extend beyond one year (often linked to economic cycles).
• Irregular (I):one-off or unpredictable movements (e.g. exceptional events) that do not follow a stable pattern.

Two widely used representations are:

• Additive model:Observed = T + S + C + I
• Multiplicative model:Observed = T × S × C × I

In many practical exam-style questions, the cyclical element is not estimated explicitly because:

• datasets are short (e.g. two to four years), and
• cycles usually span longer than one year, making them difficult to isolate reliably.

Where cyclical effects are not modelled separately, they are commonly absorbed into the trend estimate or treated as part of the irregular component. If you do this, state it as an assumption.

Core theory and frameworks

1) Choosing additive vs multiplicative seasonality

Use an additive approach when seasonal movement is roughly the same absolute amount each cycle (e.g. “about +150 units every Q4, regardless of the level”).

Use a multiplicative approach when seasonal movement is roughly the same proportion each cycle (e.g. “Q4 is about 25% above trend, whatever the level”). Multiplicative seasonal indices are factors around 1.00 (such as 0.85, 1.10, 1.30).

Operational decision rule:

• If theabsolute seasonal swing widens as the level rises, amultiplicativemodel is usually more appropriate.

2) Moving averages and trend estimation

A moving average smooths a time series by averaging consecutive observations. This reduces short-term noise and reveals the underlying direction more clearly.

• Ak-period moving averageuses k consecutive observations.
• The window length is usually aligned to the seasonal cycle (e.g. 4 quarters, 12 months).

Centred moving averages (CMA) for even-length windows

With an even window length (e.g. 4 quarters), each moving average lies between two periods. To align the trend estimate to an actual period, calculate a centred moving average:

• CMA = (MA₁ + MA₂) / 2

This produces a trend estimate that is positioned on a specific time period.

3) Seasonal measures: multiplicative and additive methods

A convenient umbrella term is seasonal measures:

• In amultiplicativeapproach, the seasonal measure is aseasonal index(afactor).
• In anadditiveapproach, the seasonal measure is aseasonal adjustment(anamount).

Seasonality is estimated by comparing actual results to a trend estimate (often the CMA).

Multiplicative seasonal indices (ratio-to-moving-average method)

1. Compute CMAs (trend estimate).
2. For each period with a CMA, compute theseasonal ratio:
3. Seasonal ratio = Actual / CMA
4. Group ratios by season (e.g. all Q1 ratios) and average them to get a preliminary index for each season.
5. Normaliseso that the average seasonal index over a full cycle equals 1.000.

Normalisation (multiplicative):

• If the average of the preliminary indices is m, then each index is adjusted as:
• Normalised index = Preliminary index / m

Caution: Multiplicative methods assume values are positive; if zeros or negatives occur, ratios may not be meaningful and an additive approach (or data treatment) may be more appropriate.

Additive seasonal adjustments (difference-to-moving-average method)

1. Compute CMAs (trend estimate).
2. For each period with a CMA, compute theseasonal difference:
3. Seasonal difference = Actual − CMA
4. Group differences by season (e.g. all Q1 differences) and average them to get a preliminary adjustment for each season.
5. Normaliseso that the seasonal adjustments sum to 0 across a full cycle.

Normalisation (additive):

• If the preliminary seasonal adjustments sum to k over the cycle, adjust each season by subtracting k ÷ number of seasons.

4) Forecasting using trend and seasonality

A standard forecasting flow is:

1. Estimate the trend (often using CMAs or a fitted straight line).
2. Extend the trend into future periods.
3. Apply seasonality:
   • Multiplicative:Forecast = Trend × Seasonal index
   • Additive:Forecast = Trend + Seasonal adjustment

Always state assumptions (for example that seasonality remains stable and that there is no structural break).

5) Index numbers

Index numbers express change relative to a base period (often set to 100).

Simple indices

• Simple price index for one item:
• (Current price / Base price) × 100
• Simple aggregate index (unweighted):
• (Σ current prices / Σ base prices) × 100
• This can be misleading if items differ in importance.

Weighted indices: Laspeyres and Paasche

Let:

• p₀, q₀ = base period price and quantity
• p₁, q₁ = current period price and quantity

Laspeyres price index (base quantities as weights):
Laspeyres = [Σ(p₁ q₀) / Σ(p₀ q₀)] × 100

Paasche price index (current quantities as weights):
Paasche = [Σ(p₁ q₁) / Σ(p₀ q₁)] × 100

Interpretation:

• Laspeyres keeps the basket fixed at base quantities.
• Paasche uses the current basket.

Fisher’s Ideal (useful extension)

A commonly cited compromise is Fisher’s Ideal index, which combines both weighting approaches:

• Fisher’s Ideal = √(Laspeyres × Paasche)

It is not always required, but it is a helpful way to discuss how different weighting choices can be blended.

6) Limitations and assumptions

Key limitations include:

• Outliers:can distort moving averages and seasonal measures.
• Structural change:changes in strategy, products, pricing, competition, or the economic environment can break historical patterns.
• Short datasets:seasonal measures are more reliable when averaged across several cycles.
• Cyclical movements:may be hard to distinguish from trend unless long time horizons are available.
• Base-year effects (index numbers):results can depend heavily on the chosen base period and weighting.
• Changing mix and quality:indices can mislead if items are not comparable over time.

Worked example

Narrative scenario

A retail company, Trendy Apparel, tracks quarterly sales volumes (units) over two years. Management suspects a steady upward trend, with a consistent uplift in the fourth quarter linked to holiday demand.

Quarterly sales are:

• Year 1:Q1 820, Q2 900, Q3 980, Q4 1,300
• Year 2:Q1 860, Q2 940, Q3 1,020, Q4 1,360

The company wants a Year 3 quarterly forecast using a trend-and-seasonality approach.

Required

1. Compute a 4-quarter moving average for the sales data.
2. Calculate centred moving averages (CMA) to estimate the trend.
3. Estimate multiplicative seasonal indices using the ratio-to-moving-average method.
4. Forecast sales for Year 3 by extending the trend and applying seasonal indices.
5. Interpret the results and discuss limitations.

Solution

Rounding policy (apply consistently)

• Moving averages and CMAs:1 decimal place
• Ratios and indices:4 decimal places
• Forecast units:nearest whole unit

Step 1: 4-quarter moving averages

Compute each moving average from four consecutive quarters:

• MA (Y1 Q1–Q4) = (820 + 900 + 980 + 1,300) / 4 =1,000.0
• MA (Y1 Q2–Y2 Q1) = (900 + 980 + 1,300 + 860) / 4 =1,010.0
• MA (Y1 Q3–Y2 Q2) = (980 + 1,300 + 860 + 940) / 4 =1,020.0
• MA (Y1 Q4–Y2 Q3) = (1,300 + 860 + 940 + 1,020) / 4 =1,030.0
• MA (Y2 Q1–Q4) = (860 + 940 + 1,020 + 1,360) / 4 =1,045.0

Step 2: Centre the moving averages (CMA)

Because 4 is an even window length, each MA sits between quarters. Centre by averaging consecutive MAs:

• CMA aligned toY1 Q3= (1,000.0 + 1,010.0) / 2 =1,005.0
• CMA aligned toY1 Q4= (1,010.0 + 1,020.0) / 2 =1,015.0
• CMA aligned toY2 Q1= (1,020.0 + 1,030.0) / 2 =1,025.0
• CMA aligned toY2 Q2= (1,030.0 + 1,045.0) / 2 =1,037.5

Alignment note: CMAs start at Y1 Q3 because you need two consecutive 4-quarter moving averages to centre. Therefore Y1 Q1 and Y1 Q2 have no CMA (insufficient surrounding data), and the last available CMA in this dataset is Y2 Q2.

Step 3: Seasonal ratios and multiplicative seasonal indices

Compute seasonal ratios (Actual / CMA) for quarters where a CMA exists:

Notes:

• The MA column shown above is the first of the two moving averages used to centre the CMA for that period, which helps make the workings easier to follow.
• With only two years of data, there is only one ratio per quarter. In longer datasets, you would average multiple Q1 ratios, multiple Q2 ratios, and so on.

Normalise indices (average = 1.000)

Mean of the four ratios:
(0.9751 + 1.2808 + 0.8390 + 0.9060) / 4 = 1.0002

Normalised indices = ratio / 1.0002:

• Q10.8388, Q20.9058, Q30.9749, Q41.2805
• (These indices average to 1.000 over the four quarters.)

Step 4: Extend the trend and forecast Year 3

4.1 Estimate quarterly trend increase (approximation)

CMA rises from 1,005.0 (Y1 Q3) to 1,037.5 (Y2 Q2):
Increase = 32.5 over 3 quarter-steps ⇒ 32.5 / 3 = 10.83 units per quarter (approx.)

Method note: This “slope-from-CMAs” approach is a practical approximation. An equally acceptable alternative is to deseasonalise the data and fit a straight-line trend using least squares.

4.2 Project trend values from the last CMA (Y2 Q2)

Starting from Y2 Q2 CMA = 1,037.5, add 10.83 each quarter:

4.3 Apply seasonal indices (multiplicative)

Forecast = Trend × Seasonal index (rounded to whole units):

Interpretation of the results

• The trend implies steady underlying growth of about11 units per quarter.
• The seasonal uplift is strongest inQ4, where the index1.2805indicates sales around28% abovetrend for that quarter.
• Forecasts reflect both effects: gradual growth through the year and a pronounced Q4 peak.

Limitations to state

• Onlytwo yearsof data are available and there is onlyone seasonal ratio per quarter, so indices are sensitive to any unusual quarter.
• The forecast assumesstable seasonalityandno structural change(for example in product range, pricing, marketing intensity, competition, or customer behaviour).
• Any cyclical influence has not been estimated explicitly and is assumed to be either negligible or absorbed into the trend/irregular movement for this short dataset.

Method marks and assumptions

Where method marks are typically earned

When presenting a solution under time pressure, aim to show the steps that demonstrate the method:

• Correct moving-average setup and calculations.
• Correct centring logic and period alignment.
• Correct seasonal ratio (or difference) calculations.
• Correct normalisation.
• Clear trend extension approach.
• Correct seasonal adjustment applied to forecasts.
• Brief interpretation and stated limitations.

Layout matters

Workings score better when they are easy to follow. Present calculations in a compact table where possible (for example Period / Actual / MA / CMA / Ratio or Difference / Index or Adjustment) and apply a consistent rounding policy.

Assumptions checklist (state explicitly)

• Measurements are consistent over time (same definition of “units”, same sales capture point).
• Seasonal pattern remains stable in shape and relative strength.
• No major structural change (strategy, products, capacity, pricing, competition, or market).
• Dataset is representative (no exceptional outliers dominating the seasonal measures).
• If cyclical effects are not modelled separately, explain how they are treated (ignored, absorbed into trend, or considered irregular).

Common pitfalls and misunderstandings

• Confusing trend with seasonality:trend is the underlying direction; seasonality repeats on a calendar cycle.
• Wrong moving-average window:use the seasonal cycle length (4 for quarterly, 12 for monthly).
• Forgetting to centre even-length moving averages:without centring, trend estimates are misaligned to periods.
• Not normalising seasonal measures:multiplicative indices must average to 1.000; additive adjustments must sum to 0 over the cycle.
• Misapplying model choice:additive suits constant absolute swings; multiplicative suits proportional swings.
• Over-trusting the output:a neat forecast can still be unreliable if the business environment has changed.
• Index number confusion:Laspeyres and Paasche can differ because weights differ; explain what each index represents.

Summary and further reading

Time series forecasting often separates observed values into trend, seasonal pattern, cyclical influence and irregular movement. Moving averages (and centred moving averages for even windows) provide a practical way to estimate the trend. Seasonal indices (multiplicative) or seasonal adjustments (additive) then refine forecasts to reflect within-year patterns.

Index numbers convert changes across time into a relative measure, typically with a base set to 100. Weighted indices such as Laspeyres and Paasche incorporate item importance and can diverge when consumption patterns shift. Fisher’s Ideal provides a useful compromise measure for stronger discussion answers.

FAQ

What is the difference between additive and multiplicative models in forecasting?

Additive models treat seasonality as a fixed increase or decrease added to trend. Multiplicative models treat seasonality as a factor applied to trend. If the absolute seasonal swing widens as the level rises, multiplicative is usually a better fit.

How do moving averages help in time series analysis?

Moving averages smooth short-term volatility by averaging adjacent observations. This highlights the underlying trend and reduces the risk of reacting to temporary spikes or dips.

How are additive seasonal adjustments calculated?

Using a trend estimate (often a CMA), compute Actual − CMA for each period, average these differences by season, and then normalise the seasonal adjustments so they sum to zero across a full cycle.

Why is it important to normalise seasonal measures?

Normalisation prevents the seasonal component from inflating or deflating the overall level of the forecast. Multiplicative indices must average to 1.000 across the cycle; additive adjustments must sum to 0.

What are the limitations of using index numbers?

Index numbers depend on the base period and weights and can be distorted by changes in product mix, changes in quality/specification, and substitution effects.

How do Laspeyres and Paasche indices differ?

Laspeyres uses base-period quantities as weights (a fixed base basket). Paasche uses current-period quantities (a current basket). They can diverge when consumption patterns and relative prices change.

Summary (Recap)

• A time series can be viewed as trend, seasonal, cyclical and irregular influences.
• Moving averages smooth data; centred moving averages align trend estimates to periods when the window length is even.
• Multiplicative seasonality uses Actual/CMA ratios; additive seasonality uses Actual − CMA differences.
• Normalisation is essential: indices average to 1.000 (multiplicative) or adjustments sum to 0 (additive).
• Forecasts extend the trend and then apply seasonal measures.
• Index numbers compare periods; Laspeyres and Paasche are weighted approaches and may differ.
• Always state assumptions and recognise limitations such as outliers, short datasets, structural change and base-year effects.

Glossary

Time series
A set of observations recorded in time order (e.g. monthly demand, quarterly sales).

Trend (T)
The underlying longer-term direction of the series after smoothing out short-term fluctuations.

Seasonality (S)
A repeating within-cycle pattern linked to the calendar (e.g. quarters or months).

Cyclical (C)
A longer-term wave-like movement that typically spans more than one year and may reflect broader economic conditions.

Irregular (I)
Unpredictable movements that do not follow a stable pattern and may reflect one-off events.

Moving average
A smoothing method that replaces each observation with an average of neighbouring observations.

Centred moving average (CMA)
A moving average adjusted so that a trend estimate aligns with a specific time period (used when the moving-average window is even).

Seasonal index (multiplicative)
A factor showing how a season typically compares with the underlying trend (indices average to 1.000 over a full cycle).

Seasonal adjustment (additive)
A seasonal amount added to or subtracted from trend (adjustments sum to 0 over a full cycle).

Additive model
A model where components combine by addition: Observed = T + S + C + I.

Multiplicative model
A model where components combine by multiplication: Observed = T × S × C × I.

Index number
A measure expressing change relative to a base period, often with the base set to 100.

Weighted index
An index that reflects item importance through weights (commonly quantities or expenditure shares).

Laspeyres price index
A weighted price index using base-period quantities as weights: Σ(p₁ q₀) / Σ(p₀ q₀) × 100.

Paasche price index
A weighted price index using current-period quantities as weights: Σ(p₁ q₁) / Σ(p₀ q₁) × 100.

Fisher’s Ideal index
A combined index defined as the square root of (Laspeyres × Paasche), often used as a compromise measure.