When is the average of all historical data expected to provide the best results?

The average of all historical data is expected to provide the best results when the underlying time series is stationary. A stationary time series means that the statistical properties of the data, such as the mean and variance, remain constant over time. In such cases, past values can be good indicators of future values since there are no trends or seasonal patterns that could distort predictions.

When a time series is stationary, the average serves as a reliable estimator because it captures the inherent fluctuations in the data without being influenced by trends or seasonal shifts. This allows analysts to use the historical mean effectively as a forecast for future values, as the underlying behavior of the data remains consistent.

In contrast, if the data is random, it does not follow a predictable pattern, making the average less reliable for forecasting. For normally distributed data, while the average is a useful measure of central tendency, it does not guarantee the best results for forecasting in non-stationary series. Additionally, when seasonal effects are present, the average might be affected by those fluctuations, leading to inaccurate forecasts. Thus, the concept of stationarity is crucial for ensuring that the average of historical data can provide the best results.