Stationarity, Differencing, and Regression Relevance

# Problem - stationarity and relevance


# Problem - stationarity and relevance












I am doing my Bachelor's thesis at the moment and I ran into a problem I was hoping you could help me out with.

While running my data (in Eviews) I had relevant variables. However, when turning to a unit root test, it turned out I had a unit root problem (non stationarity, random walks). So I corrected for this using first differences and it worked to solve the unit root problem. But: now that I have these first differences, my variables stopped showing relevance.

So my questions are the following:

- Is it a problem when variables to stop showing relevance to the model when applying first differences, because without these they are relevant?

- If so, is there a way to solve for this?

- When applying first differences, are you supposed to do this with all variables or only with the ones who show the problem of nonstationarity?

Thank you!

## Answer by Barnaby (score 2)

https://quant.stackexchange.com/a/17397

Any data transformation to assure stationarity eliminates part of the signal in many cases the signal is not completely eliminated so you can still perform the required analyses but in some others as may the your case the signal is erased and the results seem to indicate your variables have lost predictible power although its predictive power may have been erroneously asserted if potential errors due to common regression effects of spurious regression serial correlation lack of linearity or additivity (for linear) or normality of the data were overlooked. therefore your problem is data dependant one.

There are also requirements as to have a valid result in regression. Weak sense stationarity (WSS) is necesary to avoid spourious regression and other regression non valid results. For this you need that the 1st moment ( -a) Stationarity of variance -b) stationarity of mean and -c) and also stationarity of the autocovariance. Depending on the number of times you diferenciate the resulting data will be equivalent to (one) removing the trend, (2,3) times equal to aplying a 2 or 3 degree polynomial to the signal. Diferenciation however is only one way of solving the issue of non stationarity and it effects on regression as mentioned by the equivalent procedures. Thus you can remove the trend, apply a low degree polynomial loess regression (in accordance with the influence that you evaluate prior data have on later data). This should take care in most cases of the MA part). Then you are left with the AR and other portions of the signal. You can identify or remove AR terms by determining the AR effect on the data. (for example to remove certain colored noises)

What is important is to determine what is the part of the signal you are interested in. It is important to diferenciate between deterministic and stockastic parts of the data. The deterministic part (moving average and trend) and stokastic part which includes non AR noise, AR, AR noise and other ((e.g) fractional integration).

There are several aspects to stationarity and there are several tests for stationarity (none of them is perfect - google "R stationarity tests"). Unit root tests are used to ascertain stationarity of a signal for this specific issue, however there are non staionarity issues that could affect your data such us long term memory processes which may not be asserted by the tests. In this regard higher order unit root or other non stationary producing factors such us long range dependence, fractional integration, pink noise if ignored, the system dynamics become systematic errors in regression equations and thus regression will have no meaning (e.g. changes in variance invalidating second-order stationarity)

- All variables should be stationary both response and independant variables.

- Also to consider is to segment the data as per when changes in the variance ocurr by using change point analysis and a similar variance then is stable across some time, this will take care of the stationarity of the variance without removing any information). Also make sure the segment is sufficient as per to have valid regression results statistically (training sample is sufficient to provide a test with valid results).

- When the results of the regression are obtained do not forget to reverse them into the original terms (remember the regression was done on differences so your results will also be in those terms)

- Check for homocedasticity, serial correlation, normality,

Take a look at this link http://www.econ.ku.dk/metrics/Econometrics2_05_II/Slides/08_unitroottests_2pp.pdf

Hope this helps