ETX5900 · Business Statistics
Time Series Analysis & Forecasting
Time Series Analysis & Forecasting is Module 11 / Week 11 of ETX5900 Business Statistics at Monash University, the unit's final topic (Berenson Ch 14). It models a variable recorded over time against a time index t = 1, 2, 3, … — a linear trend Ŷt = b0 + b1t for steady movement or a quadratic trend for curvature — then measures the fit with MAD and MSFE (both smaller = better). Seasonal series are handled by the multiplicative model Y = T×S×C×I, removing season by dividing each value by its seasonal index.
What this chapter covers
- 01Time series vs an ordinary sample: order matters because value depends on time
- 02The four components: Trend, Seasonal, Cyclical, Irregular (T, S, C, I)
- 03Linear trend model Ŷ_t = b₀ + b₁t, with t = 1, 2, 3, … as the time index
- 04Quadratic trend model Ŷ_t = b₀ + b₁t + b₂t² for accelerating/decelerating series
- 05Fitting the trend line by least squares (same b₀, b₁ mechanics as regression, with t as predictor)
- 06Forecasting by substituting the next t into the fitted equation
- 07Mean Absolute Deviation MAD = Σ|Yₜ−Fₜ|/n (average error size, in original units)
- 08Mean Square Forecast Error MSFE = Σ(Yₜ−Fₜ)²/n (squared, penalises large errors)
- 09Choosing a model: lower MAD / MSFE wins, and how the two can disagree
- 10Multiplicative model Yₜ = Tₜ×Sₜ×Cₜ×Iₜ and deseasonalising by dividing by the seasonal index
Fit a linear trend, forecast the next period, and measure the fit with MAD
- +1Set up the sums. t̄ = (1+2+3+4+5)/5 = 3; Ȳ = (40.5+43.5+48+51.5+56.5)/5 = 240/5 = 48. Deviations t−t̄ = −2,−1,0,1,2 and Y−Ȳ = −7.5,−4.5,0,3.5,8.5.
- +1Slope. Σ(t−t̄)(Y−Ȳ) = (−2)(−7.5)+(−1)(−4.5)+0+(1)(3.5)+(2)(8.5) = 15+4.5+0+3.5+17 = 40; Σ(t−t̄)² = 4+1+0+1+4 = 10. So b₁ = 40/10 = 4 ($00 per week).
- +1Intercept and equation. b₀ = Ȳ − b₁·t̄ = 48 − 4×3 = 36, so Ŷ_t = 36 + 4t.
- +1Forecast week 6. Substitute t = 6: Ŷ₆ = 36 + 4×6 = 36 + 24 = 60 ($00), i.e. about $6,000.
- +1Fitted values and errors. Ŷ at t=1…5 = 40, 44, 48, 52, 56; errors eₜ = Yₜ−Ŷₜ = +0.5, −0.5, 0, −0.5, +0.5.
- +1MAD. Σ|eₜ| = 0.5+0.5+0+0.5+0.5 = 2.0 → MAD = 2.0/5 = 0.40 ($00). (For comparison MSFE = Σeₜ²/n = 1.0/5 = 0.20, in squared units.)
Key terms
- Time series
- Data recorded on the same variable at regular intervals over time (yearly, quarterly, monthly). Unlike an ordinary sample, the order of the observations matters because the value depends on time.
- Trend (T)
- The long-run upward or downward drift in a series over many periods, captured by fitting a linear or quadratic model to the time index t.
- Seasonal component (S)
- A regular pattern that repeats within a year (e.g. a Q4 retail peak). In the multiplicative model it is an index: 1.10 means a period runs 10% above the yearly average, 0.85 means 15% below.
- Linear trend model
- Ŷ_t = b₀ + b₁t, fitted by least squares with the time index t = 1, 2, 3, … as the predictor. b₁ is the average change in Y per period; forecast by substituting the next t.
- Quadratic trend model
- Ŷ_t = b₀ + b₁t + b₂t², used when the series curves — growth that accelerates or decelerates. The b₂ term bends the trend line; a plot against t shows whether curvature is present.
- MAD (Mean Absolute Deviation)
- MAD = Σ|Yₜ − Fₜ| / n — the average size of the forecast errors, in the original units of Y. Lower is a better-fitting model.
- MSFE (Mean Square Forecast Error)
- MSFE = Σ(Yₜ − Fₜ)² / n — the average squared forecast error, so large misses are penalised heavily. In squared units, and lower is better; it can rank models differently from MAD.
- Multiplicative model
- Yₜ = Tₜ × Sₜ × Cₜ × Iₜ, decomposing a series into Trend, Seasonal, Cyclical and Irregular ratios. Deseasonalise by dividing each value by its seasonal index (Yₜ / Sₜ).
Time Series Analysis & Forecasting FAQ
How do I know whether to use a linear or a quadratic trend?
Plot the series against the time index t. If it rises or falls by a roughly constant amount each period, the linear trend Ŷ_t = b₀ + b₁t fits. If it curves — growth that speeds up or slows down — add the b₂t² term and use the quadratic trend. If you fit both, compare them on MAD and MSFE and keep the model with the lower values, remembering the two measures can occasionally disagree.
What is the difference between MAD and MSFE, and which should I report?
Both measure forecast accuracy and both are lower-is-better. MAD is the average absolute error, in the original units of Y. MSFE squares each error before averaging, so it is in squared units and punishes a single large miss far more than several small ones. Report MSFE when big errors are especially costly and MAD when a typical error matters more; because they can rank two models differently, quoting both and stating your criterion is safest.
Can AI help me with time-series forecasting in ETX5900?
Yes — Sia can explain the method step by step: how the time index t = 1, 2, 3, … drives the trend line, how to read b₀ and b₁ from Excel or R output and substitute the next t for a forecast, how to compute MAD and MSFE from the list of errors, and why you divide by a seasonal index to deseasonalise a multiplicative series. Work your own practice series through it to build the routine. Sia explains and coaches; it does not sit your exam or promise guaranteed answers, grades or a pass.
Exam move
Treat every forecasting question as one routine: eyeball the series against the time index t to pick a linear or quadratic shape; read b₀ and b₁ (and b₂) from the least-squares output, write the equation Ŷ_t = b₀ + b₁t, and forecast by substituting the next t. Then always attach an accuracy measure — list the signed errors eₜ = Yₜ − Fₜ, and compute MAD = Σ|e|/n and MSFE = Σe²/n, remembering MSFE is in squared units and cannot be compared head-to-head with a MAD. For seasonal quarterly or monthly data, check the seasonal indices average to 1, then deseasonalise by dividing (Yₜ / Sₜ) and multiply back to re-seasonalise a forecast. Because the exam supplies the formula sheet and statistical tables, practise choosing the right model and computing cleanly rather than memorising formulae, and finish every answer with one interpretation sentence naming the trend direction or the season removed.
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