8.1 Point forecasts and prediction intervals

Time series forecasting combines point estimates with prediction intervals to provide a comprehensive view of future outcomes. Point forecasts offer specific predictions, while intervals quantify uncertainty around these estimates, giving decision-makers a range of potential scenarios to consider.

Understanding both point forecasts and prediction intervals is crucial for effective planning and risk management. These tools help businesses and analysts make informed decisions by providing concrete estimates alongside measures of uncertainty, enabling more robust strategies in various fields.

Point Forecasts and Prediction Intervals

Point forecasts in time series

• Single-valued estimates of future observations in a time series represent the most likely or expected future value at a specific time point (sales forecast for next quarter)
• Obtained using various forecasting methods such as exponential smoothing or ARIMA models (Holt-Winters method, ARIMA(1,1,1))
• Provide a concrete estimate of future values, helping in decision-making and planning (inventory management, resource allocation)
• Serve as a basis for comparing and evaluating different forecasting models (mean squared error, mean absolute percentage error)
• Used as a starting point for constructing prediction intervals around the point estimate

Concept of prediction intervals

• Ranges of values likely to contain the true future observation with a specified probability, consisting of a lower and upper bound around the point forecast (90% prediction interval: [100, 150])
• Reflect the uncertainty associated with the point forecast, providing a measure of the uncertainty surrounding it
• Indicate the range of plausible future values based on the chosen probability level (95% prediction interval, 80% prediction interval)
• Wider intervals imply greater uncertainty, while narrower intervals suggest more precise forecasts (wider intervals for long-term forecasts, narrower for short-term)

Construction of prediction intervals

• Parametric methods assume a specific distribution for the forecast errors, commonly normal distribution, allowing the use of standard normal quantiles (±1.96 for 95% intervals)
  • Interval width determined by the standard deviation of the forecast errors ($\pm 1.96 \times \sigma$)
• Non-parametric methods do not rely on distributional assumptions
  • Empirical quantiles of the forecast errors can be used to construct intervals (2.5th and 97.5th percentiles for 95% intervals)
  • Bootstrap methods involve resampling the residuals to generate multiple forecast paths and obtain interval estimates
• Factors influencing prediction interval width:
  1. Forecast horizon: Intervals typically widen as the forecast horizon increases due to growing uncertainty (wider intervals for yearly forecasts compared to monthly)
  2. Model complexity: More complex models may have narrower intervals due to better fit, but overfitting should be avoided (ARIMA vs. simple exponential smoothing)
  3. Variability in the time series: Higher variability leads to wider prediction intervals (stock prices vs. stable demand for essential goods)

Uncertainty in point forecasts

• Prediction intervals quantify and communicate the uncertainty surrounding point forecasts, emphasizing that the point forecast is an estimate and not a guarantee
  • Highlight the range of plausible future values indicated by the prediction intervals (point forecast: 120, 95% prediction interval: [90, 150])
• Considerations when interpreting prediction intervals:
  • Choice of probability level affects the width of the intervals and the level of confidence (80% vs. 95% intervals)
  • Prediction intervals do not account for model uncertainty or structural changes in the time series (regime shifts, external shocks)
  • Unusual or extreme future events may fall outside the prediction intervals (black swan events, outliers)
• Communicating uncertainty to stakeholders:
  • Present prediction intervals alongside point forecasts to provide a more complete picture (point forecast and interval in a table or graph)
  • Explain the meaning and limitations of prediction intervals in clear, non-technical terms (probability, range of likely outcomes)
  • Discuss the implications of uncertainty for decision-making and risk management (contingency planning, scenario analysis)