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Over a year, I have some daily response $y$ (suicides per day in US). It's data for one year only.

Other than seasons, what time covariate should I include? A month-covariate? A week-covariate? A day-covariate?

I have plotted the data, and cannot see an affect other than the seasons-effect. So, if I am to include the one of the other three possible covariates, which one should it be, and why? How do I "defend" its inclusion given that I can't see any clear effect via plots?

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When it comes to working with covariates, you are seeking to find out the effects through the use of control groups.

While more information on your model would be useful, an ANCOVA (Analysis of Covariance) would examine the effects using control variables.

e.g. If your regression model is:

Ysuicidesperday ~ βseasons

then you will want to include your control variables of month, week and day to examine how the results of your regression change when these interaction terms are included.

i.e. if you wanted to determine whether to include the monthly covariate for instance, you could compare the model when there is an interaction (Ysuicidesperday ~ βseasonsmonths) versus main effects (Ysuicidesperday ~ βseasons + βmonths).

Seeing how your model output differs (or not, as the case may be) when an interaction effect is included will give you a good indication as to the effect of the particular covariate on suicide rates.

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