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?

1 Answer

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).
• That first "common knowledge" sentence in the question is still false with the correction you suggest. OLS can make sense even if neither is stationary, even if you discount cointegration for whatever reason (e.g. both have a deterministic time trend). It's really all about the error term. May 13 at 11:53
• @ChrisHaug, thanks, edited. May 13 at 12:12
• Thank you. Yea I meant "stationary" instead of "non-stationary" in my original question (now fixed). So basically, the time and seasonality variables on the RHS will be enough to address the trend/seasonality component of the non-stationarity on the LHS. However, we still need GDP and the rest of variables on RHS to not have unit root (unless there's cointegration) in order for OLS to make sense? May 13 at 14:47
• @wwyws, more or less so. It is not impossible that a linear time trend and seasonal dummies will not adequately account for the seasonal patterns, as these patterns can be quite intricate. Also, if the variables on the RHS are seasonal while the one on the LHS is not, having a time trend and seasonal dummies is still beneficial; they helps make the residuals free of seasonality, and that is something we want. May 13 at 15:09