Solutions: Forecasting

(a) n = 1 → tŷ = 120 + 8(1) = 128; n = 2 → tŷ = 120 + 8(2) = 136; n = 3 → tŷ = 120 + 8(3) = 144

(b) For period 9 (one year ahead): tŷ = 120 + 8(9) = 192 units

(c) The prediction for period 9 is extrapolation — it lies beyond the observed data range. Reliability decreases the further we forecast, because the linear trend may not continue indefinitely (sales could plateau, market saturation, etc.).
(a) Seasonally adjusted forecast = trend × seasonal index.

Period 9 is Q1 (quarter 1 of the next year). Trend prediction: tŷ = 240 + 12(9) = 348.

Adjusted forecast = 348 × 1.25 = $435,000

(b) Period 10 is Q2: tŷ = 240 + 12(10) = 360. Adjusted = 360 × 0.90 = $324,000

(c) Q1 has a seasonal index of 1.25, meaning Q1 sales are typically 25% above the trend. Q2 has SI = 0.90, meaning Q2 sales are typically 10% below trend. Despite the trend increasing, Q2 has a lower seasonal boost, so the adjusted forecast is lower.
(a) Deseasonalised values = actual ÷ seasonal index:

Month 1 (Jan, SI = 1.40): 280 ÷ 1.40 = 200

Month 4 (Apr, SI = 0.95): 190 ÷ 0.95 = 200

Month 7 (Jul, SI = 0.60): 132 ÷ 0.60 = 220

Month 10 (Oct, SI = 1.05): 231 ÷ 1.05 = 220

(b) The deseasonalised values show a gentle upward trend (200, 200, 220, 220), indicating the underlying trend is growing slowly. Without deseasonalising, the seasonal variation (summer peak, winter trough) would obscure this growth.

(c) Month 13 is January again (SI = 1.40). Using the trend: deseasonalised forecast ≈ 220 + 5(3) = 235 (rough extrapolation). Adjusted = 235 × 1.40 ≈ 329 visits.
(a) The least-squares line is fitted to deseasonalised (smoothed) data, not raw data, to capture the true underlying trend without seasonal distortion. If fitted to raw data, the seasonal peaks and troughs would affect the slope and intercept.

(b) Period 13 is Q1 again. Trend: Dŷ = 85 + 3.5(13) = 85 + 45.5 = 130.5. Seasonal index for Q1 = 1.30.

Forecast = 130.5 × 1.30 = 169.7 ≈ 170 units

(c) Period 14 (Q2, SI = 1.10): Dŷ = 85 + 3.5(14) = 134. Forecast = 134 × 1.10 = 147.4 ≈ 147 units
(a) Seasonal indices must sum to 4.00 for quarterly data (one for each quarter). Check: 1.30 + 0.85 + 0.70 + x = 4.00 → x = 4.00 − 2.85 = 1.15

(b) Q4 index = 1.15, meaning Q4 sales are typically 15% above the seasonal average. This is consistent with Christmas shopping boosting retail sales in Q4.

(c) Period 17 is Q1 of year 5. Trend: Dŷ = 50 + 4(17) = 118. Forecast = 118 × 1.30 = 153.4 ≈ 153 units
(a) Forecast = trend × seasonal index. Period 9 (Q1): tŷ = 1200 + 45(9) = 1605. SI for Q1 = 1.18. Forecast = 1605 × 1.18 = 1893.9 ≈ $1,894

(b) Period 10 (Q2, SI = 0.94): tŷ = 1200 + 45(10) = 1650. Forecast = 1650 × 0.94 = $1,551

(c) The revenue manager should use the seasonally adjusted forecasts for budgeting, as they account for predictable seasonal variation. However, they should note that forecasts further into the future are less reliable and external factors (economic changes, new competitors) are not captured in the model.
(a) Three factors that reduce the reliability of a time series forecast: (i) extrapolating far beyond the data range; (ii) irregular (random) events that are unpredictable (e.g., COVID-19, natural disasters); (iii) the trend or seasonal pattern changing over time (structural breaks).

(b) A forecast 5 years into the future is far less reliable than a forecast for next quarter. The linear trend assumption may not hold over 5 years; economic conditions, consumer behaviour, and market structure can all change significantly. Short-term forecasts rely on stable recent patterns; long-term forecasts require much stronger assumptions.

(c) To evaluate the model, the analyst should: compare forecasts to actual values (calculate forecast error = actual − forecast); check r² of the trend line; ensure the residual plot of the trend is random; and assess whether the seasonal indices remain stable across years.
(a) Deseasonalise each value: divide actual by its seasonal index.

Q1 Yr1: 320 ÷ 1.15 ≈ 278.3; Q2 Yr1: 210 ÷ 0.82 ≈ 256.1; Q3 Yr1: 185 ÷ 0.75 ≈ 246.7; Q4 Yr1: 345 ÷ 1.28 ≈ 269.5

Q1 Yr2: 355 ÷ 1.15 ≈ 308.7; Q2 Yr2: 238 ÷ 0.82 ≈ 290.2; Q3 Yr2: 210 ÷ 0.75 = 280.0; Q4 Yr2: 398 ÷ 1.28 ≈ 310.9

(b) The deseasonalised values show a clear upward trend from ~247–278 in Year 1 to ~281–311 in Year 2, confirming genuine underlying growth in demand. Without deseasonalising, the Q3 trough and Q4 peak would dominate and obscure this trend.

(c) Using the trend fitted to these 8 deseasonalised values (approximate): Dŷ ≈ 250 + 8(n). For period 9 (Q1 Yr3): Dŷ ≈ 250 + 8(9) = 322. Forecast = 322 × 1.15 ≈ 370 units
(a) Forecast error = actual − forecast. Period 7: 415 − 392 = +23 (actual exceeded forecast by 23). Period 8: 380 − 401 = −21 (actual fell short of forecast by 21).

(b) A pattern of consistently positive errors (actual always above forecast) would suggest the trend line has too shallow a slope — the model is systematically underestimating growth. Random positive and negative errors indicate a good model.

(c) Mean absolute error (MAE) = (|+23| + |−21|) / 2 = 44 / 2 = 22. This means on average the forecast is off by 22 units, which the analyst should compare to the typical sales volume to determine if this is acceptable.
(a) Step 1: Deseasonalise historical data by dividing by seasonal indices. Step 2: Fit a least-squares line to the deseasonalised data. Step 3: Use the trend equation to forecast the deseasonalised value for the future period. Step 4: Multiply the trend forecast by the seasonal index for that period to get the final forecast.

(b) Deseasonalised values: Jan = 95 ÷ 1.35 = 70.4; Apr = 58 ÷ 0.88 = 65.9; Jul = 45 ÷ 0.65 = 69.2; Oct = 72 ÷ 1.12 = 64.3. Approximate trend equation: Dŷ ≈ 68 − 0.5n (slight downward trend).

(c) January next year is period 13. Dŷ = 68 − 0.5(13) = 61.5. Forecast = 61.5 × 1.35 ≈ 83 guests. The trend is slightly downward so the summer peak is expected to be lower than last year despite the high SI.