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Time Series — Topic Review

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This review covers all four lessons in Time Series: Time Series Graphs and Trends, Smoothing Time Series, Seasonal Adjustment, and Forecasting.

Review Questions

  1. Fluency

    Q1 — Identifying Variation Components

    The monthly electricity demand for a Queensland suburb over two years shows the following pattern: demand peaks in summer (January–February) and is lowest in autumn (April–May). There is also a gradual increase in the overall average from Year 1 to Year 2. In one month, demand was unusually high due to a heatwave.

    1. Identify each component of variation present in this time series.
    2. Which component is the heatwave?
    3. Which component makes comparisons between individual months (e.g. comparing July Year 1 with August Year 1) potentially misleading?
  2. Fluency

    Q2 — Reading a Time Series Graph

    The graph below shows quarterly ice-cream sales (thousands of units) for a Queensland factory over 3 years. The trend is upward, but sales are consistently highest in Q1 and lowest in Q3 each year.

    1. Describe the overall trend in ice-cream sales.
    2. Why is it incorrect to say “sales fell from Q1 Year 2 to Q3 Year 2” and conclude the business is in decline?
    3. Sketch the shape you would expect if you plotted the 3-point moving average for this data.
  3. Fluency

    Q3 — 3-Point Moving Average

    The monthly sales figures ($000) for a hardware store are: 42, 38, 45, 51, 48, 55, 62, 58, 64, 70, 66, 74.

    1. Calculate the 3-point moving averages for months 2 through 11.
    2. What does the moving average reveal about the trend?
  4. Understanding

    Q4 — Centred Moving Average for Quarterly Data

    Quarterly visitor numbers (thousands) to a museum are: Q1 Y1 = 28, Q2 Y1 = 45, Q3 Y1 = 52, Q4 Y1 = 31, Q1 Y2 = 33, Q2 Y2 = 49, Q3 Y2 = 57.

    1. Calculate the first two 4-point moving totals.
    2. Calculate the first centred moving average (CMA) using these totals.
    3. Explain why centring is necessary for even-period moving averages.
  5. Understanding

    Q5 — Computing Seasonal Indices

    The table shows quarterly electricity costs ($000) for a childcare centre over 3 years:

    YearQ1Q2Q3Q4
    128181422
    230201524
    332211625
    1. Calculate the seasonal index for each quarter. Verify the indices sum to 4.
    2. Which quarter has the highest electricity costs? Is this consistent with the SI you calculated?
    3. Deseasonalise the Q1 Year 3 value of 32.
  6. Understanding

    Q6 — Deseasonalising and Interpreting

    The August retail sales for a homewares store were $142,000. The August seasonal index is SI = 0.78.

    1. Calculate the deseasonalised August sales.
    2. The September SI = 0.91 and actual September sales were $158,000. Calculate the deseasonalised September sales.
    3. A manager says: “September was a much better month than August because sales were higher.” Evaluate this claim using the deseasonalised values.
  7. Understanding

    Q7 — Using the Trend Equation to Forecast

    A least-squares line fitted to quarterly CMA data for a surf school gives: ŷ = 85 + 4.2t ($000), where t = 1 is Q1 Year 1. Seasonal indices: SIQ1 = 1.45, SIQ2 = 1.10, SIQ3 = 0.55, SIQ4 = 0.90.

    1. Find the trend value at t = 13 (Q1 Year 4) and t = 15 (Q3 Year 4).
    2. Calculate the seasonally adjusted forecast for each of these periods.
    3. Explain why the Q1 Year 4 forecast is so much higher than Q3 Year 4, even though the trend value is similar.
  8. Understanding

    Q8 — Reliability of Forecasts

    A business analyst has quarterly data from Years 1–3 (t = 1 to 12) and wants to forecast for Year 5 (t = 17 to 20).

    1. Is this interpolation or extrapolation? Explain.
    2. List three factors that could make this forecast unreliable.
    3. The analyst checks the residuals (actual − forecast) for the last 4 observed periods and finds they are all positive. What does this suggest about the model?
  9. Problem Solving

    Q9 — Full Analysis: Smoothing to Forecasting

    Quarterly revenue ($000) for an outdoor adventure company over 2 years is: Q1 Y1 = 95, Q2 Y1 = 140, Q3 Y1 = 180, Q4 Y1 = 110, Q1 Y2 = 105, Q2 Y2 = 155, Q3 Y2 = 200, Q4 Y2 = 125. Seasonal indices (given): SIQ1 = 0.69, SIQ2 = 1.01, SIQ3 = 1.30, SIQ4 = 1.00.

    1. Deseasonalise all 8 values.
    2. Using CAS, the trend line fitted to deseasonalised data is ŷ = 128.5 + 3.8t. Forecast the revenue for Q1 and Q3 of Year 3 (t = 9 and 11).
    3. Calculate the sum of all seasonal indices and verify they sum to 4. What would it mean if they did not?
  10. Problem Solving

    Q10 — Evaluating a Forecast

    An ice-cream manufacturer uses the trend equation ŷ = 2400 + 180t (cases) with seasonal indices SIQ1 = 1.65, SIQ2 = 1.10, SIQ3 = 0.52, SIQ4 = 0.73 to forecast production. The actual Q3 Year 3 production (t = 11) was 3,800 cases.

    1. Calculate the forecast for Q3 Year 3 (t = 11).
    2. Find the residual. Is the model over- or underestimating?
    3. The manufacturer’s accountant says the model is good because the forecast was “close to the actual.” How could you quantify how close the forecast is as a percentage?
    4. If the actual Q1 Year 4 production is 9,200 cases and the forecast for t = 13 gives 9,540, calculate the residual and state what it means for inventory planning.
  11. Problem Solving

    Q11 — Complete Time Series Investigation

    Monthly cafe revenue ($000) data shows a 3-point moving average (MA3) value of $48.2k at Month 6. The actual Month 6 value is $44.5k and actual Month 7 value is $52.1k.

    1. Using the MA3 = 48.2 at Month 6 and actual Month 7 = 52.1, work out the MA3 for Month 7 (given that Month 5 actual = $48.0k). Show your working.
    2. Month 7 has a seasonal index of 1.08. Deseasonalise the Month 7 actual value.
    3. The trend line fitted to deseasonalised data is ŷ = 42 + 0.85t. Predict the deseasonalised revenue for Month 14.
    4. The July seasonal index is 1.08. Forecast the actual July revenue for Month 14 (assuming July corresponds to Month 14 in a future year).
  12. Understanding

    Q12 — Interpreting Seasonal Indices

    A seasonal index of SI = 0.62 is calculated for Q3 (July–September) for a pool cleaning business in Queensland.

    1. Interpret this seasonal index in plain language.
    2. If the trend predicts $120,000 revenue for Q3 next year, what is the seasonally adjusted forecast?
    3. The actual Q3 revenue was $78,000. Calculate the residual and interpret it.
  13. Understanding

    Q13 — 5-Point Moving Average

    Monthly rainfall (mm) for a coastal town is: 85, 92, 78, 65, 58, 70, 88, 95, 110, 98, 86, 74.

    1. Calculate the 5-point moving averages for months 3 through 10.
    2. Compare the 5-point moving average with a 3-point moving average in terms of how much smoothing is achieved.
  14. Fluency

    Q14 — Describing Trends

    A time series graph of monthly online sales shows the following pattern: from Month 1–6 the trend is roughly flat; from Month 7–12 there is a steep upward trend; from Month 13–18 the trend flattens again.

    1. Describe the trend in each phase.
    2. Would a single straight-line trend fit this data well? Explain.
    3. What should an analyst do before fitting a linear trend line?
  15. Problem Solving

    Q15 — Extended Problem: Full Time Series Analysis

    A Queensland irrigation equipment retailer records quarterly sales ($000) over 3 years. Seasonal indices are: SIQ1 = 0.88, SIQ2 = 1.32, SIQ3 = 1.18, SIQ4 = 0.62. The trend line fitted to CMA data is ŷ = 215 + 7.5t (t = 1 is Q1 Year 1).

    1. Show that the seasonal indices are consistent.
    2. Forecast the revenue for each quarter of Year 4 (t = 13, 14, 15, 16).
    3. In Year 4, the actual Q2 revenue was $512,000. Calculate the residual for Q2 Year 4.
    4. The owner wants to plan staffing for Year 4. Based on the forecasts, in which quarter should they expect the most business? Which quarter should they minimise staffing?
    5. Comment on any limitations of using this model to plan Year 4 staffing levels.
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