Assuming I already have an existing linear regression model: $y(t) = a + bx(t) + e(t)$,


  • $a$ & $b$ are the constant coefficients

  • $e$ is the model residual error

  • $t$ denotes time

You've noticed that:

  • There are trends in residuals and as a result the assumption of normality with constant variance is violated.
  • There is "seasonality" in the model residuals showing repeating pattern in prediction errors over time (e.g. follows an annual cycle).

Question: How do we improve or adjust the model but without changing the constant coefficients?

This is only a hypothetical question. I am only looking for the approach that can be taken.


1 Answer 1


If you know the length of the seasonal cycle in your residuals, you can fit an ARIMA model to the residuals, yielding a so-called regression with ARIMA errors. This will work only on residuals and leave your regression coefficient estimates for $a$ and $b$ as they are.

In R, you can do this using auto.arima in the forecast package. Feed your regressors into the xreg parameter.


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