MGMT90280 · Managerial Decision Analytics
Time Series Analysis & Forecasting
Time series analysis and forecasting is the Week-8 topic of MGMT90280 Managerial Decision Analytics at the University of Melbourne, and the basis for the exam's forecasting question. It covers the four data patterns (horizontal, trend, seasonal, cyclical), the forecast-accuracy measures MAE, MSE, MAPE and MFE, smoothing methods (moving average and exponential smoothing) and regression forecasting with a trend term and seasonal dummy variables. The workflow is always the same: plot the series, pick the method that fits its pattern, forecast, and prove which model is better on its errors.
What this chapter covers
- 01Patterns: horizontal/stationary (constant mean), trend (long-term drift), seasonal (repeats within ≤ 1 year), cyclical (swings over runs of years, no fixed period)
- 02Forecast error e_t = y_t − ŷ_t (actual minus forecast); the first k periods have no forecast so averages divide by n − k
- 03MAE = Σ|e_t|/(n−k) and MSE = Σe_t²/(n−k) — MSE punishes big misses harder
- 04MAPE = [Σ|e_t/y_t|/(n−k)] × 100 (unit-free %); MFE = Σe_t/(n−k) measures bias, not accuracy
- 05k-period moving average: F(t+1) = mean of the last k actuals — for a horizontal series only
- 06Simple exponential smoothing: F(t+1) = α·y_t + (1−α)·F_t, set F2 = y1, optimal α minimises MSE
- 07Regression forecasting: linear trend ŷ_t = b0 + b1·t; k seasons need k−1 seasonal dummies (one baseline)
- 08Seasonal + trend model: ŷ = b0 + b1·Q1 + b2·Q2 + b3·Q3 + b4·t; forecast by plugging in the dummy pattern and t
- 09Coefficient significance: two-tailed t = b/SE(b), df = n − p − 1, reject H0 if p-value < α
- 10Model comparison: prefer the model with a significant trend, lower MSE and higher adjusted R²
Seasonal + trend forecast, a trend-significance test, and a model choice
- +2(a) Year-4 quarters are t = 13, 14, 15, 16. Set the matching dummy to 1 and include the trend. Q1(13): 120 + 40 + 6·13 = 238. Q2(14): 120 + 85 + 6·14 = 289. Q3(15): 120 + 30 + 6·15 = 240. Q4(16): 120 + 6·16 = 216 (baseline, no dummy).
- +1(b) Hypotheses (two-tailed). H0: β_trend = 0 versus H_A: β_trend ≠ 0.
- +1Test statistic. t = b4/SE(b4) = 6/1.8 = 3.33, with df = n − p − 1 = 12 − 4 − 1 = 7 (p = 4 predictors: Q1, Q2, Q3, t).
- +2Critical value and decision. Two-tailed, α = 0.05, df = 7 gives t_crit = 2.365. Since |t| = 3.33 > 2.365 (two-tailed p-value ≈ 0.0125 < 0.05), reject H0: the trend term is statistically significant.
- +2(c) Model choice. The trend is significant and adding it lowers MSE and raises adjusted R² versus the seasonal-only model, so the seasonal + trend model forecasts better — report the seasonal + trend model.
Key terms
- Time series
- A sequence of observations recorded in time order (e.g. quarterly sales). You plot it first to identify the pattern, which decides the forecasting method.
- Horizontal / stationary pattern
- A series that fluctuates around a constant mean with no drift. Suited to smoothing (moving average or exponential smoothing).
- Trend vs seasonal vs cyclical
- Trend = steady long-term rise or fall; seasonal = a cycle that repeats within one year (quarters, months); cyclical = swings over runs of years with no fixed period. Seasonal is modelled with dummies; cyclical is much harder.
- Forecast error (e_t)
- e_t = y_t − ŷ_t, the actual minus the forecast. Early periods with no forecast are excluded, so accuracy measures divide by n − k.
- MAE / MSE / MAPE
- Accuracy measures: MAE = mean |e_t| (units); MSE = mean e_t² (punishes big misses); MAPE = mean |e_t/y_t| × 100 (unit-free %). For all three, lower is better.
- MFE (mean forecast error)
- The mean signed error Σe_t/(n−k). It measures bias, not accuracy: MFE < 0 means forecasts sat above actuals (over-forecasting); MFE > 0 means under-forecasting; near 0 = unbiased.
- Moving average
- F(t+1) = average of the last k actual values. A larger k smooths more but reacts slower; it assumes a horizontal series and cannot follow a trend or season.
- Exponential smoothing
- F(t+1) = α·y_t + (1−α)·F_t with 0 ≤ α ≤ 1 and F2 = y1. A small α smooths noise, a large α reacts fast; the optimal α minimises MSE.
- Seasonal dummy variables
- For k seasons you use k − 1 (0/1) dummies with one season as the omitted baseline; each dummy coefficient is the mean difference of that season versus the baseline. Using all k dummies plus an intercept is the dummy-variable trap.
- Seasonal + trend model
- ŷ = b0 + b1·Q1 + b2·Q2 + b3·Q3 + b4·t; the dummies capture the repeating cycle and b4·t captures the drift. Forecast by plugging in the future dummy pattern and its t value.
Time Series Analysis & Forecasting FAQ
When do I use smoothing versus regression forecasting?
Plot the series first. If it is roughly horizontal — fluctuating around a constant mean with no drift — use a moving average or exponential smoothing. If it drifts (trend) and/or repeats within the year (seasonal), fit a regression with a trend term and seasonal dummies, because a moving average cannot climb a trend or bend with a season.
What is the difference between MFE and MAE, MSE, MAPE?
MFE = Σe_t/(n−k) keeps the sign of each error, so positive and negative errors cancel — it measures bias (direction), where a negative MFE signals over-forecasting. MAE, MSE and MAPE take absolute or squared errors, so nothing cancels and they measure size (accuracy). A model can have MFE near zero yet large offsetting errors, so report both a size measure and MFE.
Can AI help me with time series forecasting in MGMT90280?
Yes, for understanding. Sia can explain step by step how to read a pattern, define every symbol, and rebuild a moving-average, exponential-smoothing or seasonal + trend calculation so you can reproduce the working yourself — including how to set up the trend t-test and state the reject / fail-to-reject decision. It will not sit an assessment for you or promise a grade, and note that Assignment 1 runs in the Respondus LockDown Browser with no internet or generative AI, so use Sia only to learn beforehand and confirm the rules on the LMS.
Exam move
Anchor everything on the plot-then-method habit: name the pattern (horizontal, trend, seasonal or cyclical) before choosing a technique. Build a one-line card for each accuracy measure — MAE and MSE in the data's units (MSE punishes big misses), MAPE as a unit-free percentage, and MFE as a signed bias check — and remember every average divides by n − k, not n. Drill one moving-average error table by hand (columns for y_t, F_t, e_t, |e_t|, e_t², |e_t/y_t|) and one exponential-smoothing recursion starting from F2 = y1. For the regression side, practise forecasting from a seasonal + trend equation by plugging in both the dummy pattern and the trend index t, then running the two-tailed t-test on the trend coefficient (t = b/SE, df = n − p − 1) and stating the explicit reject / fail-to-reject decision. This is the exam's forecasting question in the compulsory 5-question section; budget time in proportion to marks (about 1.2 minutes per mark on the 2-hour paper) and confirm the exact exam date and rules on the LMS.
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