Does possible non-stationarity matter if the model is OLS? I am working on an assignment where the current model is an OLS model that models the percent change in a variable, X, by regressing it against a bunch of economic variables such as unemployment rate, nominal GDP, HPI, etc.
My questions are:  


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*Should we consider the possibility of non-stationarity and seasonality for the variables, given that the model is OLS? If so, why?

*What are the alternatives for modeling such a relationship, time series, or any other?
The data are available for each independent variable and the dependent variable for 10 years for each quarter.  
 A: First of all, OLS is an estimation technique, not a model. I will assume you have a linear regression model that you would like to estimate using OLS.

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*Regarding non-stationarity, it is not covered under the OLS assumptions, so OLS estimates will no longer be BLUE if your data are non-stationary. In short, you do not want that. Also, it does not make sense to have a stationary variable explained by a random walk, or vice versa. A stationary process will revert to its mean while an integrated process may wonder off and away, hence the two are no match for each other. This situation is known as an unbalanced regression. (Although having variables of different orders of integration in the same regression equation can make sense when there is cointegration.)
Regarding seasonality, it is also a form of non-stationarity and you should model it explicitly. When ignored, seasonality may result in undesirable outcomes and misinterpretations. For example, you may find a statistically significant relationship between two variables when the only common underlying relationship between them is seasonality; think about modelling how weather depends on ice-cream sales.


*You should care about specifying the model correctly (or, more realistically, as well as you can) first and then choosing an estimation method. Perhaps your dependent variable is stationary while GDP is not stationary; then you normally cannot model how the first depends on the second; but perhaps it makes sense to ask how your dependent variable depends on changes in GDP (the first differences of DGP). Also, if you have seasonality, include some terms to account for it or adjust the data for seasonality before putting the variables into the model.
Also, keep in mind that you are working with a pretty small sample (40 observations). The asymptotic properties of your estimators may not be that relevant yet; there is little room for constructing a rich model. I'm not sure if you can do much about it, but that's a different topic.
A: First and most importantly, do NOT attempt to apply OLS techniques to summarize time series data.  OLS techniques were developed for and are the tool to use if the data is cross-sectional. In the distant past and absent time series techniques, OLS was adopted to summarize time series. We have 1000s of articles and 100s of textbooks written in an attempt to fix the failures resulting from the mis-application of OLS.
The first step in time series modeling is to insure the series is or are stationary.
