I have a dependent variable that are correlated with the passage of days (1, 2, 3, ...).
But I fitted a linear model without days/time, because I am more interested in other things I can influence somehow. Seems to me that a model with time "steal" influence of other coefficients, and I don't want that.
So I got a reasonable R², low p-values (~0) and a good residuals plot (y = residuals, x = fitted values). As far as I can see, my scatter plot of residuals means that the assumption E(u|X)=0 (OLS/Gauss-Markov) is ok.
After that, I decided to plot fitted values of variations of the original model (more or less variables, transformations of the variables or dummies included) in a time perspective (y = fitted val., x = days).
Even though I have not included time in my models, fitted values "dancing" around the real values, without any visual correlation or bias, proves that I can use that without major concerns?