Time series and forecasting · insight-1235
In a regression with ARIMA errors, test the ARIMA innovations and not the regression residuals
- Evidence type
- documented fact
- Difficulty
- hard
- Roles
- Data scientist
- Industries
- retail, energy, cross-industry
Summary: situation, action, reason, boundary
Situation
A model with a promotion regressor and ARIMA(1,0,2) errors is checked by running a Ljung-Box test on the regression residuals, and the test fails.
Action
Extract the innovation residuals, which are the estimated ARIMA errors, and run the white-noise test and the ACF on those. Use the regression residuals only to plot the unexplained pattern against the predictors.
Reason
In this model structure only the ARIMA errors are assumed to be white noise, while the regression errors carry the autocorrelation that the ARIMA part is there to model.
Boundary
SARIMAX-style models with exogenous regressors, and R regression-with-ARIMA-errors workflows.
Symptom or risk
The regression residuals are allowed to be autocorrelated, so the failure sends the analyst to add terms that are not needed, and a pass on the wrong series would prove nothing.
Exceptions
Libraries differ in which residual they return by default, so confirm what the attribute you test actually contains.
Trade-offs
A clean innovation check does not guarantee accurate forecasts; it still needs an out-of-sample comparison.
Pitfalls
Adding AR terms until the regression residuals look white, which double counts the dynamics and inflates the fit.
Lessons and concepts
Claims and their sources
FPP3 states that only the ARIMA model errors are assumed to be white noise in a regression with ARIMA errors, the regression errors being allowed to be autocorrelated, and that the innovation residuals are the ones to check.
The statsmodels SARIMAX class takes exogenous regressors and exposes residuals and information criteria on its results object.