Inclusion of year and seasons as variable for regression with non-stationary response? The common knowledge is that OLS only makes sense if both the response and explanatory variables are stationary (ignoring exceptions like cointegration), as otherwise, there could be effects of spurious regression.  It seems most posts I've come across suggest making Y and Xs stationary first (by differencing, removing trend/seasonality, etc) before running OLS. However, what I'm wondering is that, if I have a response Y that clearly exhibit trending with time and seasonality, can I simply add time and season as variables in the regression model, and run everything as usual?
For example, suppose the variable I want to predict is the quarterly GDP from tourism in the area. Does it make sense to run the model below?
$$
GDP_t = \beta_tt + \beta_1I_{spring} + \beta_2I_{summer} + \beta_3I_{fall} + \text{...other explanatory variables}
$$
This seems like something commonly done by practitioners. But is this theoretically sound?
 A: Regarding OLS only makes sense if both the response and explanatory variables are non-stationary, actually, it is the opposite: non-stationary should be replaced by stationary. Though as Chris Haug suggests in the comments, that can be relaxed further.
Regarding your example, a lot depends on what you assume about the GDP and other explanatory variables.

*

*If none of them is assumed to have a unit root and some of them are assumed to have a linear time trend, you may be fine.

*If only the GDP is assumed to have a unit root, you have an unbalanced regression where the left hand side (LHS) is diverging while the right hand side (RHS) is stationary.

*The same problem arises if you assume the GDP to be stationary but one or more variables on the right hand side to be nonstationary and not cointegrated.

*If both the GDP and some right hand side variable(s) contain unit roots and they are not cointegrated, the LHS will diverge from the RHS. If they are cointegrated, you may be fine (fine in the sense of obtaining consistent estimates, even though the distributions of estimators will be nonstandard due to the unit roots).

