# Using non-stationary time series data in OLS regression

I am using 1983-2008 annual data to test if both Gini coefficients and gross national saving in China and the US can affect the US current account balance. The data seem to be non-stationary, but I am a beginner and only know the basic multiple regression model and autoregressive distributed lag model, can I still use them to these models to test the effects? I know the models would be biased and not accurate, but do they give any useful information? My chosen control variables are real GDP, interest rate, dollar index and maybe some other national income components.

• With only 26 cases to study you might need to worry about overfitting when using more than two or three control variables.
– whuber
Apr 22 '14 at 16:29
• Yeah. I know. I am just wondering if the basic models would give some useful results on the variables of interest. I can reduce the number of control variables. Apr 22 '14 at 16:35

You can do anything you want, especially if it's a term paper or something of that nature.

To obtain useful results you can't use nonstationary data with OLS and time series. There are other more advanced methods where nonstationarity is a non issue. With OLS you have to difference real GDP and indices, and also apply log transform in many cases.

UPDATE: when using non stationary variables in OLS you run into the potentially fatal issue of spurious regression, there's a ton of literature on this subject. Basically, your regression results will turn out garbage in most cases. You may see very significant coefficients, but the significance is artificial, and disappears when you run a proper regression.

There's even more subtle phenomenon called "cointegration", but since you're working on undergrad paper, I would not worry about it. As a matter of fact, if your major is not statistics or econometrics, I would imagine your instructor will not penalize you for improper use of regressions.

Clarification: you can use non-stationary data with OLS if the series are cointegrated. However, when doing so you better show that the series are cointegrated indeed, then adjust the parameter covariance matrix accordingly if you need inference. The parameters themselves would be fine. As I mentioned in original answer this is advanced concepts that are usually outside undegrad courses.

• This is my undergraduate dissertation and what you said is just something that I am worrying about. I didn't learn much about econometric models, so I only can stick with the basics. But still thank you for your answer! Apr 22 '14 at 17:03
• If it's a thesis, then make sure you discuss the prelim results early on with your advisor, before writing anything. Apr 22 '14 at 17:25
• My advisor went on vacation and this is the problem. Only two weeks left before the submission of my thesis. I didn't plan to do econometric models. Now I just want to ask if I take the log and the first difference of my variables, and make them become more stationary, will OLS regression give a more useful results? Apr 22 '14 at 17:47
• Yes, you do the first difference of the long when it comes to real GDP, and similar items, which grow exponentially. Apr 22 '14 at 17:50
• Can they be interpreted just as a normal variables before transformation, or is there any special interpretation for them? Apr 22 '14 at 18:14

I have a graduate degree in econometrics specializing in times eries and survival analyis. I'll try to give you short undergrad advice instad of a proof.

1. You should never use OLS for time-series data (the only exception is SOMETIMES it is appropriate to use this technique for panel data). OLS results will be garbage - it will result in a spurious regression in which the results look good, but are void of econometric interpretation. MLE should be used instead. The short answer to why is that the covariance between your dependent variable and your error term will never be zero, one of the foundational assumptions of OLS. Instead of fitting a linear line, we have to fit a process to the data (AR(p), MA(q), ARMA(p,q), ARIMA(p,d,q), ARIMAX(p,q,x), VAR(p), etc.)

2. Differencing your data to make it stationary will not allow you to use OLS. Stationary data still follows a process, and your model specification should allow for this. If the error term of a two time series are stationary, then it is appropriate to use cointegration techniques, but this isn't something you should try to do on your own.

3. If you have access to a statistial package, research how to perform a Dickey-Fuller test for stationarity to determine if your data is stationary. If not, difference the data, and (assuming stationarity after first or second differencing), fit the appropriate process using MLE to your series.

4. Forwarning: from the description you provided of your data, it sounds like it would best be modelled using vector autoregression with a transfer function (VARMAX(p,q,x)). This is also known as a recursive VAR. This is appropriate if you're trying to determine if (assumed to be stationary) multiple time series are predicting another time series. These are very accurate when modelled correctly, but they are not parsimonous (you need a large number of degrees of freedom, and you're already using a estimation prodecure that is only asymptotically unbiased, so hopefully you have plenty of observations), and it isn't something I would recommmend an undergrad trying to do.

Hope that helps,

Keegan

• I am not sure I understand the part "one of the foundational assumptions of OLS is that the covariance of the dependent variable and the error term is zero". Could you perhaps elaborate a little? Oct 6 '14 at 20:36