Time series and forecasting · insight-1234
Difference the response and every predictor together in a dynamic regression
- Evidence type
- documented fact
- Difficulty
- medium
- Roles
- Data scientist, Data analyst
- Industries
- retail, finance, telecom
Summary: situation, action, reason, boundary
Situation
Weekly ticket sales are regressed on a price index, search volume and a trend; the response needs one difference to be stationary, but the predictors are left in levels.
Action
If any variable needs differencing, difference all of them, fit a regression with ARIMA errors by estimating on the innovations, and read each coefficient as an effect on changes rather than levels.
Reason
Differencing everything preserves the form of the relationship, and a regression with ARIMA errors is equivalent to a regression in differences with ARMA errors. Fitting on innovations avoids the spurious-regression problem.
Boundary
Demand models with trending drivers such as price indices, search volumes or macro series.
Symptom or risk
Mixing levels and differences changes what the coefficients mean, and fitting by ordinary least squares while ignoring error autocorrelation makes p-values too small, so unrelated predictors look important.
Exceptions
If predictor and response are cointegrated, a levels model can be valid; test that explicitly rather than assume it.
Trade-offs
Coefficients on changes are less intuitive for stakeholders who think in levels.
Pitfalls
Reporting a high R-squared and small p-values from a levels regression on trending series as evidence of a relationship.
Lessons and concepts
Claims and their sources
FPP3 recommends differencing all variables when any needs differencing, notes that a regression with ARIMA errors is equivalent to a regression in differences with ARMA errors, and warns that ignoring error autocorrelation makes p-values too small, a case it calls spurious regression.