# What to do with a linear regression model with non-normal residuals and evidence of seasonality?

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

where:

• $a$ & $b$ 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.

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.