# Is there a way to allow seasonality in regression coefficients?

Say I have a time series, Gt, and a covariate Bt. I want to find the relationship between them by the ARMA model:

Gt = Zt + β0 + β1Bt

where the residual Zt follows some ARMA process.

The problem is: I know for sure that β0 and β1 varies with the time of the year. Yet I do not want to fit a separate model to each month because that introduces discontinuity into my time series, which means I cannot calculate the autocorrelation function of the final residuals.

So, is there a time series model (or family of models, I wonder) that allows the correlation coefficients of its covariates to change seasonally?

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Edit: Thank you for those who replied here. I decided to just use seasonal dummies, but got busy so failed to reply in time.

Edit
(The same idea was proposed by Stephan Kolassa a few minutes before I posted my answer. The answer below can still give you some relevant details.)

You could use seasonal dummies. For simplicity I illustrate this for a quarterly time series. Seasonal dummies are indicator variables for each season. The $i$-th seasonal dummy takes on the value 1 for those observations related to season $i$ and 0 otherwise. For a quarterly series the seasonal dummies, $SD$, are defined as follows:

You can multiply each column in $SD$ by your explanatory variable $B_t$ and get the matrix $SDB$ defined above.

Then, you can specify your model as follows:

where the index $s$ indicates the season. Observe that we now have four coefficients (12 in your monthly series) $\beta_{1,s}$, one for each column in $SDB$.

The same for the intercept $\beta_0$ except that we must remove one column in $SD$ in order to avoid perfect collinearity. In a monthly series you would include for example the first 11 seasonal intercepts in $SD$.

Fitting the model for example by maximum likelihood will give you one coefficient estimate for each season. You could also test whether $\beta_{0,s}$ are the same for all $s$ or similarly if $\beta_{1,s}$ are constant across seasons.