Sales Forecasting (Exponential Smoothing and ARIMA)

Forecasting with exponential smoothing and ARIMA

Sales, demand and traffic forecasts drive budgets, inventory, staffing and capacity plans. For regularly spaced series with a reasonable history, two families of statistical methods have proven hard to beat in forecasting competitions such as the M3 and M4: exponential smoothing and ARIMA. This tool fits several models from both families, compares their accuracy on the most recent periods, and forecasts with the best one, with 80% and 95% prediction intervals.

The exponential smoothing models

• Simple exponential smoothing (SES): the forecast is a weighted average of past values, with weights that decline exponentially into the past. The smoothing parameter α (0–1) controls how fast: high α reacts quickly to recent changes, low α smooths heavily. Suitable for series without trend or seasonality.
• Holt’s linear trend: adds a smoothed trend, updated with parameter β, so forecasts continue in a straight line.
• Holt’s damped trend: multiplies the trend by a damping factor φ each period, so long-range forecasts level off. Damped trends are among the most accurate methods in practice because trends rarely continue forever.
• Holt-Winters additive: adds seasonal factors that are added to the level and trend, updated with γ. Use it when seasonal swings are roughly constant in size.
• Holt-Winters multiplicative: seasonal factors multiply the level, so seasonal swings grow with the level of the series, common for growing sales.

Initial level, trend and seasonal factors are taken from the first season (the standard heuristic of Hyndman et al., 2008), and α, β, γ and φ are chosen by minimising the sum of squared one-step-ahead errors. The estimates match statsmodels’ ExponentialSmoothing with the same initialisation.

ARIMA

ARIMA(p, d, q) models describe a series by its own past values (autoregressive terms, p), past forecast errors (moving average terms, q) and differencing (d) to remove trends. The tool chooses d with the augmented Dickey-Fuller test (difference until the series is stationary), then fits all models with p and q from 0 to 2 by conditional sum of squares and keeps the one with the lowest AICc, including a mean or drift term. Forecast intervals come from the model’s ψ-weights. ARIMA models in this tool are non-seasonal, so for strongly seasonal series Holt-Winters usually wins the comparison.

How the best model is chosen

In-sample fit always favours more complex models, so the automatic mode uses a holdout: each model is fitted to all but the last h periods (by default one season, or 20% of the series), forecasts those h periods, and is scored by RMSE, MAE and MAPE. The model with the lowest holdout RMSE is then refitted to the full series to produce the forecast. You can also choose a method yourself, set a different holdout length, or compare models in the table.

Prediction intervals

A point forecast is only half the answer; plans should allow for uncertainty. For exponential smoothing, the tool simulates 1,000 future paths by running the model forward with randomly resampled in-sample errors and takes the 10th/90th and 2.5th/97.5th percentiles as the 80% and 95% intervals. For ARIMA, intervals are analytic. Intervals widen with the horizon. If they are too wide to be useful, a shorter horizon, aggregation (monthly instead of weekly) or additional information is needed.

Worked example with the built-in data

The example contains 72 months of sales with a rising trend and a strong seasonal peak in November and December. Holding out the last 12 months, Holt-Winters additive forecasts them most accurately (MAPE 1.8%, RMSE 40), far better than the non-seasonal methods (MAPE 9–25%) and slightly better than the multiplicative version (2.5%). The ARIMA model chosen automatically, ARIMA(2,2,2), cannot capture seasonality and has a MAPE of 8.9%. Refitted on all data, Holt-Winters forecasts about 28,150 units for the next 12 months, about 10% more than the last 12, with a December peak of roughly 2,930 (95% interval 2,680 to 3,190). The Ljung-Box test finds no remaining autocorrelation in the residuals.

Preparing your data

• One row per period in time order, with no gaps. Fill missing periods (for example with zero sales, if that is what happened) before forecasting.
• Set the season length to the number of periods per cycle: 12 for monthly data, 4 for quarterly, 52 for weekly data with yearly seasonality, 7 for daily data with weekly seasonality, or 1 for none.
• Seasonal models need at least two full seasons plus a few periods; three or more seasons give better seasonal factors.
• Add a period column (for example 2025-03) and the forecast table will continue the labels automatically.
• Consider cleaning one-off outliers (a stock-out, a one-time order) that would otherwise distort the level or seasonality.

Measuring accuracy

• RMSE (root mean squared error) is in the units of the series and penalises large errors.
• MAE (mean absolute error) is in the same units but less sensitive to occasional large errors.
• MAPE (mean absolute percentage error) is unit-free and easy to communicate, but undefined when actual values are zero and inflated when they are small.
• The Ljung-Box test checks whether the residuals are white noise; significant autocorrelation means some pattern is left in the data.

Limitations

These methods extrapolate the patterns in the history. They cannot anticipate events that have not happened before, such as a new competitor, a price change or a pandemic, and they do not use explanatory variables such as advertising or prices. For those, use regression-based approaches such as the marketing mix model or the interrupted time series tool. For new products without history, the Bass diffusion model is more appropriate.

Reporting

Report the series, the forecast method and how it was chosen, its holdout accuracy, and the forecasts with intervals. Example: “Monthly sales were forecast with an additive Holt-Winters model, selected by accuracy over a 12-month holdout (MAPE = 1.8%). The model forecast 28,148 units for the next 12 months.”

Glossary of forecasting terms

• Horizon: how many periods ahead you forecast; accuracy usually falls as the horizon grows.
• Holdout (test set): the most recent periods kept aside to measure out-of-sample accuracy.
• Level, trend and season: the three components that exponential smoothing tracks and updates each period.
• Smoothing parameter: a weight between 0 and 1 that controls how fast a component reacts to new data.
• Damping: gradually flattening the trend so long-range forecasts do not grow without limit.
• Stationarity: a constant mean and variance over time; ARIMA achieves it by differencing.
• AICc: Akaike’s information criterion corrected for small samples, used to compare models fitted to the same data.
• Prediction interval: the range within which future values are expected to fall with a given probability, wider than a confidence interval for the mean.

Key references: Hyndman and Athanasopoulos (2021), Forecasting: Principles and Practice (3rd ed.); Hyndman, Koehler, Ord and Snyder (2008), Forecasting with Exponential Smoothing; Gardner (2006), International Journal of Forecasting; Box and Jenkins (1970).