6 Business Forecasting

Forecasting foundations

A Forecast is an estimate of an unknown future value made using information available at the time. Businesses use forecasts to plan inventory, staffing, production, cash flow, and capacity. A forecast is not a goal, meaning a desired outcome, or a plan, meaning actions chosen in response to forecasts and goals. Judge a forecast by how well it anticipates outcomes, not by whether it matches a target.

A Time series is a sequence of observations recorded over time, usually at regular intervals, such as weekly orders or monthly revenue. Before forecasting, plot the observations and check that time periods are consistent. Look for missing values, unusual events, and changes in how data were recorded; a graph can reveal patterns that a single summary statistic would hide.

Write the actual value at time $t$ as $y_t$. A forecast made at time $T$ for $h$ periods ahead is $\hat{y}_{T+h\mid T}$; it uses information available through time $T$. Forecasts farther into the future are generally more uncertain. A Prediction interval is a range intended to reflect plausible future outcomes and can be considered alongside a point forecast.

Patterns in time-series data

A series can display several patterns at once. Level is the typical value around which observations fluctuate; for example, a store may sell about 500 units per week without a persistent increase or repeating calendar pattern. Trend is sustained upward or downward movement, such as a subscription service adding customers.

Seasonality is a pattern that repeats at a known, fixed interval. Retail sales might rise each December; for monthly data, the yearly seasonal period is $12$. A Cycle is a longer-term rise and fall without a fixed, reliably repeating period, such as demand changing with broader economic conditions. A recurring calendar pattern is seasonality, while a fluctuation with less predictable timing is a cycle.

Irregular variation consists of unpredictable movements, including random noise and one-off events such as a temporary closure. A single spike or a short run of increases does not prove a lasting pattern. Forecasting aims to carry forward repeatable signals, not random fluctuations or exceptional events that are unlikely to recur.

Baseline forecasting methods

A Baseline, also called a benchmark, is a simple forecast used as a reference point. A more elaborate method is useful only if it performs better on data that were not used to construct the forecast. Let $y_1, \ldots, y_T$ be the observed values through time $T$.

Mean method: Forecast each future period as the historical average. This can be a reasonable benchmark when the series fluctuates around a relatively stable level.

Naive method: Use the most recent observation for every future period. It is useful when recent values are informative and can be hard to beat for some economic or financial series.

Seasonal naive method: Use the most recent observation from the same season. For monthly data, each forecast for a future March equals the latest observed March, and each forecast for a future April equals the latest observed April. This is a natural benchmark when seasonal patterns persist.

Drift method: Extend the average change between the first and last observations. This is a simple trend-based benchmark that assumes the average historical change continues.

For example, if the last three monthly sales values are 100, 110, and 120 units, the naive forecast for each of the next months is 120. If the data are seasonal, a seasonal-naive forecast instead uses the most recent value from the matching month in the prior seasonal cycle. Match the benchmark to the pattern being assessed, and compare it with alternative methods using the same forecast dates and horizons.

Testing forecast accuracy

For a forecast and its later observed value, the Forecast error is the actual value minus the forecast:

A positive error means the actual value exceeded the forecast; a negative error means the forecast was too high. Evaluate forecast errors on observations that were not used to make the forecasts. Residuals are calculated on data used to fit a model, so small residuals alone do not show that a method forecasts well.

Test in time order

Train on earlier observations and test forecasts against later ones, keeping the time order intact. Do not randomly shuffle time-series data: doing so can let future information leak into model development. Choose test periods that reflect the business's actual forecast horizon and, where possible, include enough observations to evaluate the seasons that matter.

Rolling-origin evaluation repeats the test at several points: fit using the data available at each origin, forecast ahead, and compare with subsequent actual values. Repeating the process helps show whether results depend on one unusually favorable test period.

Choosing accuracy measures

For $n$ test observations, common summaries of forecast accuracy include:

• Mean absolute error (MAE): $\mathrm{MAE}=\operatorname{mean}(|e_t|)$. It reports the typical absolute miss in the series' original units, such as units sold; lower is better.

• Root mean squared error (RMSE): $\mathrm{RMSE}=\sqrt{\operatorname{mean}(e_t^2)}$. It uses the original units like MAE, but squaring makes large misses count more heavily; lower is better.

• Mean absolute percentage error (MAPE): $\mathrm{MAPE}=\operatorname{mean}(|100e_t/y_t|)$. It expresses errors as percentages, but is undefined when an actual value is zero and can become very large when actual values are near zero.

• Mean absolute scaled error (MASE): the forecast's MAE divided by the in-sample MAE of a simple naive benchmark, or of a seasonal-naive benchmark for seasonal data. It supports comparisons across series with different units. A MASE below $1$ means the forecast's average absolute error is smaller than that benchmark's scaled error.

For example, if actual sales are 100, 120, and 110 units and forecasts are 90, 130, and 105, the errors are $10$, $-10$, and $5$ units. Thus, $\mathrm{MAE}=(|10|+|-10|+|5|)/3=8.3$ units, and $\mathrm{RMSE}=\sqrt{(10^2+(-10)^2+5^2)/3}\approx 8.7$ units. RMSE exceeds MAE because it gives relatively more weight to larger errors.

Using results for business decisions

No single metric answers every business question. Report an interpretable measure such as MAE in business units, and add another measure when it addresses a real need, such as using MASE to compare differently scaled series. Consider whether over-forecasting and under-forecasting have different costs: an average error near zero can conceal large misses in both directions.

Compare forecasts against an appropriate baseline, examine results by forecast horizon or season, and select measures that reflect the decision being supported.