Thomas is right, ARIMA should be estimated with non-stationary series.

In this case, you need to run ARIMA(p,2,q), but also need to be sure that the variables are I(2): second difference of the series.  

Note that I in ARIMA is for integrated. So, first you difference until you get a stationary series, and then, you fit an ARMA model to the stationary series. Many series should be stationary after you take first differences. However, some series, such as capital stocks, money and probably prices seem to have two unit roots, I(2), i.e. you have to take first differences twice. You should also check your specification by looking at deterministic terms, and lag selection needed for the tests. 

Hello Debora, you can use various methodologies in order to obtain stationarity on your data: 

1. transforming your data using square roots. You have already tried the LN transformation maybe (it depends on your series) you can obtain a stationary time series by considering the square roots transformation.

2. detrending or de-seasonalizing your data. If you can eliminate the trend or the seasonality hopefully you can obtain a stationary time series.

3. differencing two times your series or more..

4. considering eventual other problems like outliers (I assume you have already considered the problem but maybe they can be a possibility to take in to account).

In general you have to consider at the same time points in your model construction like structural changes. I think for an ARIMA model it is necessary to take in to account not too many observation (parameter drift could be frequent) and surely eventual breaks.

In general differencing a time series could be not enough in order to obtain a stationary time series.

I hope these considerations are useful for you.

Kind Regards,

Carlo

My concern was how to reach stationarity when several methods (i.e. difference, Box-Cox, log, square-root..) don't work. What's more, my data are non-seasonal ones, which means that outliers are totally random: I was looking for different methods than the ones I already evaluated.

So, I think that only Carlo Drago and Juehui Shi hit the mark, actually (despite this, some series are still non-stationary, I tried everything you suggested me).

Thank you all for your answers.

Some time series variables may have a deterministic component that is a function of other variables and no reasonable amount of transformation or differencing will make them stationary.  In such a case some for of ARMAX or related modelling will improve your forecasting. The functional forms, differencing, and X-variables to be included should be guided by economic and other theory.

Best Regards

John

Many time series consist of two components one of which is deterministic and the other stochastic. It may be the case that if the deterministic component is removed the stochastic component can be modeled using methods appropriate to to stationary or difference stationary processes. If values for the deterministic components can be projected this will also improve the forecasts.    With a "trend stationary" process" the deterministic trend is treated in this way. 

Without any knowledge of the nature of the time series and the nature of the non-stationarities and possible deterministic effects  it is very hard to make  definite recommendations.

John

Well, my time series are actually spare parts requests, so the main problem within my series is that I may have both many requests in quantity 1 and then, maybe, 430, and that's why I can't reach stationarity.

Suppose I want to eliminate these ouliers from my time series: is there any particular path I should follow? That is, can I simply remove them or should I - maybe - remove the highest and the lowest ones without misrepresenting too much?

Ate there twp types of orders in your series - one from wholesalers or other large units and one from smaller units. If so it might be better to work on each series separately. Is there another explanation. Are your data daily, monthly, hourly etc. If you have explanatory data can you try some kind of poisson regression. These are just suggestions that might be useful.

John

Hello Debora,

having many requests in quantity 1 and few with a large quantity doesn't mean that the time series is instationary. However, a linear autoregressive model is perhaps not the right approach in this case. You could consider, for example, a pure statistical approach with an appropriate distribution function, having a large probability for quantity 1 and a low probability for quantity 430. If the distribution function does not change over time, this would be a stationary system, even if there are "sudden changes" from 1 to 430 in the time series, because the probability of this to happen does not change over time - as it is likely to be in your case as well.

You can use different method for your model !!!

Depending on which software you are using, you will need to get the unit root tests using Augmented dickey-fuller unit root test , set the lag length  by using Shwartz information criteria, then perform ADF tests on levels and first difference, compare your output  with critical values given either at 1% or 5% 

use difference method or take Ln 

 First difference will make your data stationery

there are two method very popular:

1/ differentiation : take the differences between values in the series.

2/ transformation :First applying the log on the series, and take the differences between values in the result series.

It would have been better if you upload a screenshot showing the autocorrelation graph of the series, in many orders of differencing.

you can difference the series many times and see how the autocorrelation plot looks like in each time. and he can help to capture the right order of differencing and is the minimum differencing required to get a near-stationary series which roams around a defined mean and the ACF plot reaches to zero fairly quick.

ref:

https://www.machinelearningplus.com/time-series/arima-model-time-series-forecasting-python/

Try using ndiffs() on your time series to see how many time you need to difference your data to make it stationary.