Say I have a time series,

G, and a covariate_{t}B. I want to find the relationship between them by the ARMA model:_{t}

G_{t}= Z_{t}+ β_{0}+ β_{1}B_{t}where the residual

Zfollows some ARMA process._{t}The problem is: I know for sure that

βand_{0}βvaries 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._{1}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.

**Answer**

**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:

SD=[10000100001000011000⋮⋮⋮⋮1000010000100001]SDB=[B10000B20000B30000B4B5000⋮⋮⋮⋮Bn−30000Bn−20000Bn−10000Bn]

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

Then, you can specify your model as follows:

Gt=Zt+β0,sSDt+β1,sSDBt,

where the index s indicates the season. Observe that we now have four coefficients (12 in your monthly series) β1,s, one for each column in SDB.

The same for the intercept β0 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 β0,s are the same for all s or similarly if β1,s are constant across seasons.

**Attribution***Source : Link , Question Author : eddieisnutty , Answer Author : javlacalle*