Forecasting Is Everywhere in Business

Every one of these is a decision waiting on a number. A decision is only as good as the forecast it rests on.

The Basic Setup

Notation for the whole course:

• \(y_t\) — observed value at time \(t\)
• \(h\) — forecast horizon: harder as \(h\) grows, as uncertainty compounds
• \(\hat{y}_{t+h|t}\) — forecast of \(y_{t+h}\) made using information at time \(t\)
• \(e_t = y_t - \hat{y}_{t|t-1}\) — forecast error

Under squared-error loss the best forecast is the conditional expectation, \(\hat{y}_{t+h|t} = \mathbb{E}[y_{t+h} \mid \mathcal{F}_t]\), where \(\mathcal{F}_t\) is everything known at time \(t\).

Components of a Time Series

Any series can be decomposed into four parts (additive form): \(y_t = T_t + S_t + C_t + I_t\).

• \(T_t\) Trend — long-run direction
• \(S_t\) Seasonality — regular, calendar-driven pattern
• \(C_t\) Cycle — medium-run business-cycle swings
• \(I_t\) Irregular — random shocks

Which parts matter depends on the series: retail sales are trend + seasonality, quarterly GDP is trend + cycle, daily stock returns are mostly irregular.

Always Beat a Benchmark First

Before presenting any model, check that it beats a naive benchmark: a model that cannot beat the baseline has zero value. Each forecast (red) is built from a line (blue):

CA_1 FOODS; RMSE over the 52 shaded test weeks. In the M4 Competition many machine learning models failed to beat simple exponential smoothing (Makridakis et al. 2020).

The Problem with Equal Weights

Most classical methods weight every observation equally — but what if the world changes? After a store renovation, equal weights keep forecasting the old store for months.

Exponential weighting: an observation \(k\) weeks old gets weight \(\alpha(1-\alpha)^k\). High \(\alpha\) = short memory; low \(\alpha\) = long memory:

Fitted \(\alpha\) on two aisles of one store. Log scale: each week back is the same step down, \(\times(1-\alpha)\).

Simple Exponential Smoothing (SES)

Large \(\alpha\) chases every wiggle; small \(\alpha\) is smooth but slow. SES nests both benchmarks: \(\alpha = 1\) is naïve, \(\alpha \to 0\) the historical mean. CA_1 FOODS. No trend, no season.

Holt’s Method: Adding Trend

SES is flat. For a trending series, track the level \(\ell_t\) and the slope \(b_t\):

Holt-Winters: Level, Trend, and Seasonality

US retail’s December spike grows as overall sales grow: multiplicative.

Needs at least \(2m\) observations to initialize (\(m=12\) for monthly data).

The ETS Framework: A Unified View

Hyndman and Athanasopoulos (2021) unify every exponential smoothing method as a state space model: Error × Trend × Seasonal. Trend and season set the forecast’s shape:

The error, A or M, changes the intervals, not the shape. Additive error with a multiplicative season is unstable, so 9 × 2 − 3 = 15 models. Automatic selection: fit all 15, keep the minimum AIC (ETSModel in statsmodels). Schematic shapes, not data.

How Do We Measure Forecast Accuracy?

Draw each out-of-sample error \(e_{T+h} = y_{T+h} - \hat{y}_{T+h|T}\) as a square. CA_1 FOODS, the 52 test weeks:

A tie on MAE, not on RMSE: five big misses carry 40% of SES’s squared error. When big misses are costly — a stockout — prefer RMSE.

Comparing Across Series: MAPE and MASE

RMSE and MAE come in the units of \(y_t\), so they compare models on the same series. To compare across series, scale the errors:

\[ \text{MAPE} = \frac{100\%}{H}\sum_{h=1}^{H}\left|\frac{e_{T+h}}{y_{T+h}}\right| \qquad\qquad \text{MASE} = \frac{\text{MAE}}{\displaystyle\frac{1}{T-m}\sum_{t=m+1}^{T} |y_t - y_{t-m}|} \]

The MASE denominator is seasonal naïve’s average miss on the training data. \(\text{MASE} < 1\): smaller errors than that; \(\text{MASE} > 1\): worse. It is one of the M4 Competition’s two accuracy measures (Makridakis et al. 2020).

MAPE breaks down when \(y_t\) is near zero; MASE does not. MASE \(< 1\) is no win over seasonal naïve on the test weeks: on CA_1 FOODS both score 0.59.

The Golden Rule: Always Evaluate Out-of-Sample

Never evaluate a forecast model on the data used to fit it. In-sample fit measures how well the model memorized the past, not how well it predicts the future.

flowchart LR
    A["Training set<br/>t = 1 … T"] -->|forecast origin| B["Test set<br/>t = T+1 … T+H"]

ECON 8310: Four-Part Structure

Lecture 13 closes the course with a synthesis of all four parts. Python throughout: statsmodels, scikit-learn, XGBoost, PyTorch, Prophet, pyGAM, PyMC.

Tools and What You Will Be Able to Do

The Python stack: statsmodels (ETS, ARIMA, VAR) · scikit-learn (models, pipelines, cross-validation) · xgboost (gradient boosting) · torch (neural networks) · prophet and pygam (GAMs) · pymc (Bayesian models).

By the end of the semester you will be able to:

• Choose the right method for a given business problem
• Implement and tune it in Python
• Evaluate it rigorously out-of-sample
• Interpret the result for a non-technical stakeholder
• Use AI tools effectively to accelerate the analysis
• Avoid the most common forecasting mistakes

Lecture 1: Key Takeaways

Next: ARIMA & VAR — a regression-based approach to time series with rich autocorrelation structure.