# What is the largest n root transformation I should consider for making a time series stationary?

Currently, I am working with multiple time series and not all of them are stationary. In order to make them stationary I am considering different transformations and checking the augmented dickey fuller test. I have considered $$log(x), \sqrt{x}$$, and $$x^\frac{1}{3}$$ as well as box cox. Some of the transforms have not made my series stationary but higher order roots have such as $$\frac{1}{4}, \frac{1}{5}, etc$$.

My questions are:

1. Should I consider higher order n root transformations for making a series stationary?

2. If yes, then at what point is too high? So far, the highest I have used is $$x^\frac{1}{7}$$.

• There is little difference among these "higher order" roots in terms of their ability to modify statistical properties of the series. A seventh root would rarely be used (except in some special applications of physics where the theory indicates its use); you would ordinarily choose the logarithm in its stead. See stats.stackexchange.com/questions/60431 for further explanation and stats.stackexchange.com/questions/292417 for why the log closely approximates a seventh root. – whuber Jun 17 '19 at 17:03

SOME TRANSFORMATIONS ARE GOOD FOR YOU AND OTHERS NOT SO GOOD

Detrending , Power Transformations .Differencing AND ARMA are all forms of transformations. Determining the minimally sufficient (parsimonious) combination requires a combination of skillful people and skillful techniques .

Simple scripts i.e. hard and fast rules are to be studiously avoided as they limit the scope of the solution and often obfuscate like taking the nth root of anything.

Power transforms are discussed When (and why) should you take the log of a distribution (of numbers)? and some more material on transforms here : optimal Box-Cox transforms should be based upon the residuals from a useful model not necessarily the original series. .

I should also add there are two other forms of transformations often suggested by the data ...

Due to changes in parameters over time SEGMENT the data Due to deterministic error variance change(s) over time employ Weighted Least Squares (GLS)

A very good/priceless discussion of transformations is here BOX-COX TRANSFORMATION always stabilize variance where a veritable pantheon of heavy/knowledgeable SE hitters chime in on the topic. Lots to learn here , if you are willing to follow all the threads .

• How does this answer the question? – whuber Jun 17 '19 at 17:00
• it warns against arbitrarily using power transforms like nth roots as they can have negative side effects. Did you not get that message ? Is there any positive suggestions that you can make so I can improve the clarity ?. – IrishStat Jun 17 '19 at 17:01
• I didn't get that message because it's not part of the question. If you wish to comment on the question, then please do that in a comment box. Reserve answers for responding to the question. The most positive suggestion I can make is to read every question on its own terms and respond constructively to it. – whuber Jun 17 '19 at 17:06
• Regardless, your post--however correct or generally useful it might be--is still just commentary on the question. An actual answer would explicitly, clearly, and constructively address the two points asked at the end of the post. – whuber Jun 17 '19 at 17:12
• I'm sorry, but "no" to the second question doesn't even make any sense. Regardless, DO NOT ANSWER QUESTIONS IN COMMENTS and DO NOT COMMENT IN ANSWERS. – whuber Jun 17 '19 at 17:18