Residuals in time series modeling refer to the differences between the observed values of a time series and the values predicted by a model. In simpler terms, when a model forecasts or predicts future values, it is based on existing data patterns. The residual for each point in the time series is calculated by subtracting the model's predicted value from the actual observed value for that same time point. These residuals are crucial for assessing how well a model is performing.
For example, imagine you have a time series of monthly sales data for a retail store. After fitting a linear regression model to this data, you might predict that sales for a given month will be $10,000. If the actual sales turn out to be $12,000, the residual for that month would be $2,000 (i.e., $12,000 - $10,000). If your model accurately captures the underlying trend and seasonality, the residuals should exhibit no systematic pattern; they should be randomly distributed around zero.
Analyzing residuals is important for model diagnostics. If residuals show a pattern, it indicates that the model may not be adequately capturing the complexities of the data. For instance, if the residuals display a cyclic pattern, it might suggest that there are seasonal effects not accounted for in the model. By examining residuals, developers can make informed decisions about refining their models, such as incorporating additional variables or using different modeling techniques to improve forecasting accuracy.