Problem 6-05: Consider The Following Time Series Data ✓ Solved
Problem 6-05 Consider the following time series data. Choose
Problem 6-05 Consider the following time series data. Choose the correct time series plot. Which plot? What type of pattern exists in the data? (Horizontal or Trend Pattern?) Develop a three-week moving average for this time series. Compute MSE and a forecast for week 7. If required, round your answers to two decimal places. Week Time Series Forecast Value ? 5 17 ? 6 14 ? MSE: ? The forecast for week 7: ?
Use α = 0.2 to compute the exponential smoothing values for the time series. Compute MSE and a forecast for week 7. If required, round your answers to two decimal places. Week Time Series Forecast Value ? 3 16 ? 4 11 ? 5 17 ? 6 14 ? MSE: ? The forecast for week 7: ?
Compare the three-week moving average forecast with the exponential smoothing forecast using α = 0.2. Which appears to provide the better forecast based on MSE? Will it be (Three-week moving average or Exponential smoothing?) Explain.
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Paper For Above Instructions
The assignment centers on evaluating short-horizon forecasts for a small time series excerpt and comparing two common forecasting methods: a three-week moving average and single exponential smoothing with a fixed smoothing parameter α. Time-series forecasting literature establishes a spectrum of approaches, from simple averages to state-space exponential smoothing models, with guidance on when each method tends to perform well (Box & Jenkins, 1970; Hyndman & Athanasopoulos, 2018). In this context, the raw data provided include observed values for weeks 3 through 6 (Y3 = 16, Y4 = 11, Y5 = 17, Y6 = 14). The analysis below uses these values to illustrate method application and to report forecast quantities and error metrics consistent with standard forecasting practice (Montgomery, Jennings, & Kulahci, 2015).
1) Pattern assessment and plotting choice
Based on the presented subset, the series fluctuates around the mid-teens with no clear monotonic trend. There is no obvious upward or downward trajectory across weeks 3–6; rather, the values oscillate (16 → 11 → 17 → 14). With such behavior, a horizontal pattern (no strong trend) is the most reasonable characterization for this short excerpt, acknowledging that a longer series could reveal additional structure. This aligns with time-series forecasting guidance that distinguishes stable, flat behavior from trending or seasonal patterns (Box & Jenkins, 1970; Hyndman & Athanasopoulos, 2018).
2) Three-week moving average (MA3) forecast for week 7
The MA3 forecast for week 7 uses the three most recent observed weeks, weeks 4–6: Y4 = 11, Y5 = 17, Y6 = 14. The forecast for week 7 is the average of these three values: (11 + 17 + 14) / 3 = 42 / 3 = 14.00. This follows the standard MA3 forecasting rule, which smooths short-term fluctuations by averaging a fixed window of lagged observations (Chatfield, 2003; Hyndman & Athanasopoulos, 2018).
3) Exponential smoothing with α = 0.2
Single exponential smoothing (SES) updates are given by S_t = α Y_t + (1 − α) S_{t−1}, and the one-step-ahead forecast for week t is F_t = S_{t−1}. An initial smoothed value S_3 is needed. A common and reasonable choice is to set S_3 = Y_3 = 16. Using α = 0.2, the computations proceed as follows:
- S_3 = 16 (initial)
- S_4 = 0.2 × Y_4 + 0.8 × S_3 = 0.2 × 11 + 0.8 × 16 = 2.2 + 12.8 = 15.0; F_4 = S_3 = 16; error e_4 = Y_4 − F_4 = 11 − 16 = −5; e_4^2 = 25
- S_5 = 0.2 × Y_5 + 0.8 × S_4 = 0.2 × 17 + 0.8 × 15.0 = 3.4 + 12.0 = 15.4; F_5 = S_4 = 15.0; e_5 = 17 − 15.0 = 2; e_5^2 = 4
- S_6 = 0.2 × Y_6 + 0.8 × S_5 = 0.2 × 14 + 0.8 × 15.4 = 2.8 + 12.32 = 15.12; F_6 = S_5 = 15.4; e_6 = 14 − 15.4 = −1.4; e_6^2 = 1.96
Forecast for week 7: F_7 = S_6 = 15.12.
Forecast accuracy (MSE) based on weeks 4–6 is:
MSE = (e_4^2 + e_5^2 + e_6^2) / 3 = (25 + 4 + 1.96) / 3 ≈ 10.32. In practice this uses one-step-ahead forecasts for weeks 4–6, which are then compared against the observed Y_4, Y_5, and Y_6 (Gardner, 1985; Hyndman & Athanasopoulos, 2018). The forecast for week 7 under SES with α = 0.2 is 15.12.
4) Comparison and interpretation
Forecasts:
- MA3 forecast for week 7: 14.00
- SES forecast for week 7 (α = 0.2): 15.12
MSE considerations: The MA3 approach, as presented here, cannot be fully evaluated for weeks 4–7 with the given data because the MA3 forecast for week 4 and week 5 would require earlier observations (Y1 and Y2) that are not provided. A partial evaluation using a one-step-ahead MA3 forecast for week 6 (based on Y3–Y5) yields F_6^MA3 ≈ (16 + 11 + 17) / 3 = 14.67 with e_6^2 ≈ 0.44, suggesting strong short-term accuracy for that single point, but this does not reflect a complete MA3 forecast error profile over a multi-period horizon. In contrast, SES with α = 0.2 provides a complete forecast sequence for the available horizon (including F_7 = 15.12) and a calculable MSE of ≈ 10.32 over weeks 4–6. Across these calculations, SES with α = 0.2 shows a lower and more stable MSE than the partial MA3 evaluation, indicating better predictive performance for this limited data segment when comparing the two methods on the same evaluation window (Box & Jenkins, 1970; Makridakis, Wheelwright, & Hyndman, 1998; Hyndman & Athanasopoulos, 2018).
5) Trial-and-error exploration of α
A quick small-scale sensitivity check was performed for α ∈ {0.1, 0.2, 0.3} using the same initial setup (S_3 = 16) and the three data points Y_4 = 11, Y_5 = 17, Y_6 = 14:
- α = 0.1:
S_4 = 0.1×11 + 0.9×16 = 15.5; S_5 = 0.1×17 + 0.9×15.5 = 15.65; S_6 = 0.1×14 + 0.9×15.65 = 15.485;
F_4 = 16; F_5 = 15.5; F_6 = 15.65;
e_4^2 = 25, e_5^2 = 2.25, e_6^2 ≈ 2.7225; MSE ≈ 10.57
- α = 0.2 (above): MSE ≈ 10.32
- α = 0.3:
S_4 = 14.5; S_5 = 15.25; S_6 = 14.875;
F_4 = 16; F_5 = 14.5; F_6 = 15.25;
e_4^2 = 25, e_5^2 = 6.25, e_6^2 = 1.5625; MSE ≈ 10.94
These results suggest that, over this small data window, α = 0.2 provides the lowest mean squared error among the three tested values. While this pattern can be informative, it is important to note that a broader search over α and more data would be needed to make a robust recommendation (Montgomery, Jennings, & Kulahci, 2015; Hyndman & Athanasopoulos, 2018).
6) Summary and practical guidance
For the given short series, the 3-week moving average forecast for week 7 is 14.00, and the SES forecast for week 7 with α = 0.2 is 15.12. The SES-based MSE over weeks 4–6 is approximately 10.32, and a small α-sensitivity check indicates α = 0.2 yields the best fit among α ∈ {0.1, 0.2, 0.3} for this data subset. In practice, when data are sparse, SES often provides more robust forecasts than simple moving averages because it uses all available information with a tunable smoothing parameter (Box & Jenkins, 1970; Hyndman & Athanasopoulos, 2018). If the analyst can access more historical data (weeks 1–2 and beyond), computing full MA3 forecast errors and a more precise MSE comparison would be straightforward, and the decision between MA3 and SES could be refined further (Montgomery, Jennings, & Kulahci, 2015).
7) Practical note on reporting
When presenting results in coursework, clearly state which values are observed, which are forecasts, and which are errors. Include the formulas used, the exact parameter values, and the computed numbers to the requested precision. Also discuss data limitations (e.g., missing early weeks) and how they affect the interpretation of MSE and forecast comparison. This transparency aligns with forecasting best practices described in standard texts (Box & Jenkins, 1970; Hyndman & Athanasopoulos, 2018) and supports reproducibility in applied settings (Montgomery, Jennings, & Kulahci, 2015).