Practice Maths

Forecasting

Key Terms

Trend line
A line ŷ = a + bt fitted to smoothed (CMA or deseasonalised) data; t is numbered from t = 1.
Trend forecast
Substitute the future time period t into the trend equation to obtain a deseasonalised (trend) forecast.
Seasonal forecast
Trend value × SIseason; re-introduces the seasonal effect to give the final predicted value.
Two-step forecasting
Step 1: find the trend value using the equation. Step 2: multiply by the relevant seasonal index.
Extrapolation risk
Forecasts assume the current trend and seasonal patterns continue; accuracy decreases further into the future.
Numbering time periods
Number t = 1, 2, 3, … from the start of the dataset; be consistent when counting to a future period.

Using the Trend Line to Forecast

Once a trend line ŷ = a + bt is fitted to smoothed (deseasonalised or CMA) data, it can be used to forecast future values. Number the time periods consecutively (t = 1, 2, 3, …) from the start of the dataset.

Trend-only forecast at time t: ŷ = a + bt

Seasonally adjusted forecast: Forecast = (a + bt) × SIseason

Two-Step Forecasting Process

  1. Use the trend line equation to predict the deseasonalised (trend) value at the desired future time period.
  2. Multiply by the seasonal index for that season to re-introduce the seasonal effect and obtain the actual forecast.

Worked Example

A trend line fitted to quarterly CMA data gives: ŷ = 120 + 4.5t ($ 000), where t = 1 is Q1 Year 1. Seasonal indices: SIQ1 = 1.20, SIQ2 = 0.85, SIQ3 = 0.75, SIQ4 = 1.20.

Forecast for Q3 of Year 4 (t = 15):

Trend value: ŷ = 120 + 4.5(15) = 120 + 67.5 = $187.5k

Season adjusted forecast = 187.5 × 0.75 = $140.6k

This forecast says Q3 Year 4 revenue is expected to be about $140,600, reflecting both the upward trend and the typically quiet Q3 season.

Limitations of Forecasting

  • Forecasts assume the trend and seasonal patterns continue unchanged into the future.
  • Accuracy decreases as you forecast further into the future (extrapolation risk).
  • Irregular events (recessions, pandemics, natural disasters) cannot be predicted.
  • Always state your assumptions when presenting a forecast.
    • Hot Tip: When forecasting with seasonal adjustment, do it in two clear steps: (1) calculate the trend value using the trend equation; (2) multiply by the seasonal index for that specific season. Write both steps in your working — forgetting to re-introduce the seasonal effect is one of the most common errors in this topic.

Full Lesson: Forecasting

1. Why Forecast?

Forecasting is one of the primary purposes of time series analysis. Businesses need revenue forecasts to set budgets, plan staffing, and manage inventory. Governments use forecasts to plan infrastructure spending. Forecasting cannot be perfectly accurate — the future is unknown — but a well-constructed model provides a scientifically-grounded estimate that is better than guessing.

2. The Role of the Trend Line

After computing centred moving averages (or deseasonalising data), we fit a least-squares linear regression line to the smoothed values. This trend line models the long-term direction of the data and can be extrapolated beyond the observed data range to generate forecasts. In General Maths, CAS (ClassPad) is used to compute the regression equation from the smoothed data.

Label time periods sequentially: t = 1 for the first data point, t = 2 for the second, and so on. When forecasting for a future period, continue the numbering (e.g. if observed data covers t = 1 to 12, then the next quarter is t = 13, and so on).

3. Adding Back the Seasonal Effect

The trend line predicts deseasonalised (trend) values. To convert these to realistic forecasts of actual values, multiply by the appropriate seasonal index:

Forecastactual = Trendt × SIseason

This step re-introduces the predictable seasonal pattern that was removed during smoothing. Without it, the forecast would underestimate values in high seasons and overestimate in low seasons.

4. Evaluating Forecast Accuracy

After a forecast has been made and the actual value is observed, the accuracy can be assessed using the residual:

Residual = Actual − Forecast

A positive residual means the actual value exceeded the forecast (the model underestimated). A negative residual means the actual was below the forecast (the model overestimated). Tracking residuals over multiple forecasts helps assess the model’s reliability.

5. Practical Considerations

How far ahead can we forecast? Generally, the further ahead the forecast, the less reliable it becomes. For quarterly data, forecasting 2–4 quarters ahead is reasonable. Forecasting 10 years ahead from a 3-year dataset would be highly speculative.

What can invalidate a forecast? Any structural change in the underlying process: a new competitor entering the market, a change in government policy, a technological disruption, or a global event. These create the “irregular variation” in actual outcomes that models cannot anticipate.

Confidence in forecasts: Professional forecasters always provide a range (prediction interval) around the point forecast, acknowledging that the actual value will differ from the prediction. In General Maths, single-value (point) forecasts are expected.

Mastery Practice

  1. Fluency

    A trend line fitted to quarterly CMA data gives ŷ = 80 + 3.2t ($000), where t = 1 is Q1 Year 1. Use this equation to find the trend value at:

    1. t = 8 (Q4 Year 2)
    2. t = 13 (Q1 Year 4)
  2. Fluency

    A time series has seasonal indices: SIQ1 = 1.15, SIQ2 = 0.90, SIQ3 = 0.78, SIQ4 = 1.17. The trend line gives a value of $310,000 at t = 17.

    1. If t = 17 corresponds to Q1, find the seasonally adjusted forecast.
    2. If t = 17 corresponds to Q3, find the seasonally adjusted forecast.
  3. Fluency

    The actual value in a particular quarter is $248,000. The forecast for that quarter was $235,000.

    1. Calculate the residual.
    2. What does a positive residual tell you about the forecast?
  4. Fluency

    A trend line is ŷ = 55 + 2.8t (units: thousands of items). What does the value 2.8 represent in terms of the data?

  5. Understanding

    The quarterly revenue data ($000) for a fishing charter company over 3 years gives the following trend line (fitted to CMA data, t = 1 is Q1 Y1): ŷ = 145 + 5.6t. Seasonal indices: SIQ1 = 1.28, SIQ2 = 1.05, SIQ3 = 0.65, SIQ4 = 1.02.

    1. Forecast the revenue for Q2 Year 4 (t = 14).
    2. Forecast the revenue for Q3 Year 4 (t = 15).
    3. Why is the Q3 forecast so much lower than Q2 despite being close in time?
  6. Understanding

    A student uses a trend line ŷ = 200 + 8t to forecast quarterly sales. She forecasts for t = 20 (two years beyond the last observed data point at t = 12).

    1. Calculate the trend value at t = 20.
    2. Explain two reasons why this forecast may be unreliable.
    3. Is forecasting at t = 14 (one period beyond the data) more reliable than at t = 20? Explain.
  7. Understanding

    An analyst forecasts quarterly hotel bookings (thousands) using ŷ = 22 + 1.4t with SIQ1 = 0.72, SIQ2 = 0.88, SIQ3 = 1.55, SIQ4 = 0.85.

    1. Forecast actual Q3 Year 3 bookings (t = 11).
    2. Forecast actual Q1 Year 4 bookings (t = 13).
    3. Comment on why Q3 bookings are expected to be so much higher than Q1 bookings despite a similar trend value.
  8. Understanding

    A trend line ŷ = 360 − 5.2t is fitted to deseasonalised monthly data. t = 1 is January Year 1.

    1. Is this an upward or downward trend? Describe what this means in context.
    2. Find the trend value for t = 24 (December Year 2).
    3. If the December SI = 1.22, forecast the actual December Year 2 value.
  9. Problem Solving

    The quarterly visitor numbers (thousands) to a national park are modelled by the trend equation ŷ = 48 + 2.2t (fitted to CMA data, t = 1 is Q1 Year 1). Seasonal indices: SIQ1 = 1.35, SIQ2 = 0.90, SIQ3 = 0.58, SIQ4 = 1.17.

    1. Forecast the actual visitor numbers for each quarter of Year 4 (t = 13, 14, 15, 16).
    2. In Q2 Year 4, the actual visitor count was 104,300. Calculate the residual.
    3. Suggest one possible reason for the residual being positive.
    4. Calculate the total forecast visitors for Year 4.
  10. Problem Solving

    A business analyst has the following information:

    • Quarterly data observed for 3 years (t = 1 to 12)
    • Trend line: ŷ = 420 + 12.5t ($000)
    • Seasonal indices: Q1 = 0.82, Q2 = 1.14, Q3 = 1.28, Q4 = 0.76
    • The actual Q4 Year 3 revenue (t = 12) was $480,000
    1. Calculate the forecast for Q4 Year 3 using the trend line and seasonal index.
    2. Find the residual for Q4 Year 3. Is the model underestimating or overestimating?
    3. Forecast the revenue for Q1 and Q2 of Year 4 (t = 13 and 14).
    4. A manager wants to know the forecast for Q3 Year 5 (t = 23). Calculate this forecast and comment on its reliability.
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