# 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:

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

2. 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.

• The discussion and links here are fairly directly relevant. Also see Yule (1926) "Why do we Sometimes get Nonsense-Correlations between Time-Series?" J.Roy.Stat.Soc., 89, 1, pp. 1-63 ... and nearly 90 years of papers since on spurious regression in time series. – Glen_b Feb 6 '15 at 3:30
• thanks. Also, any suggestions for modeling techniques besides ols ? – Freewill Feb 6 '15 at 3:45
• OLS is an estimation method, so it's not really a modelling technique or modelling strategy. Examples of modelling strategies are, say, the LSE methodology and Box-Jenkins methodology. Examples of other estimation methods are maximum likelihood estimation and generalized method of moments. – Graeme Walsh Feb 19 '15 at 20:25

## 2 Answers

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.

1. 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.

2. 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.

• Thanks Richard. This was really helpful. I'm trying to base my understanding on the logic you provided : since the OLS assumptions don't include stationarity (or lack thereof) we should be checking for non-stationarity for both independent and dependent variable(s). Second, you recommend we specify the model correctly - but to specify the model correctly you'd have to fit one (e.g. using OLS) and see if given the assumptions and expectations of the model, does the model fits to the data appropriately and also whether it makes business sense.? – Freewill Feb 7 '15 at 16:21
• Roughly so. To make it clear, stationary data is OK for OLS, while non-stationary data is not OK. – Richard Hardy Feb 7 '15 at 16:37
• @user3007275 In relation to the first point about the order of integration of both dependent and independent variables, basically, you want to have so-called "balanced" equations, which means that the variables in your regression are of the same order. There are more advanced techniques that you'll encounter which will allow combinations of non-stationary variables to be modelled. Search for error-correction and cointegration. There are whole books on these topics. – Graeme Walsh Feb 19 '15 at 20:33

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.

• False. Full vector autoregression (VAR) models are optimally (consistently, efficiently) estimated using equation-by-equation OLS. Univariate autoregressions (AR models) can also be estimated consistently using OLS. Also, your use of word summarize may be improved upon. OLS is an estimation technique, so you can estimate a regression equation using OLS. Summarizing is something else. – Richard Hardy Feb 19 '15 at 20:39
• I can kind of see where you're coming from, but this is not correct. For example, a class of time-series models called unrestricted mixed data sampling (U-MIDAS) models can be estimated by OLS (see here), so too can Vector Autoregressions (VARs) in certain circumstances. To correct me, please cite some of the textbooks and articles. – Graeme Walsh Feb 19 '15 at 20:43
• Stationarity is not essential to time series analysis, even from a very narrow Box-Jenkins perspective. – Nick Cox Apr 7 '15 at 16:57
• Care to elaborate Nick? – Zach Goldberg Dec 4 '20 at 6:29