Non-constant variance in ARIMA I have been trying to understand financial time series data. When I run the 1-d difference to obtain stationarity, two things happen. First of all, the test statistic fails to reject the null hypothesis, suggesting that the data is stationary. However, I can clearly observe volatility clustering (specially around 2008) and there is no way that I can fit an ARIMA model to that data set (not even the difference of the difference is stationary).
To me this suggests that a GARCH model would be more appropriate for this kind of data. If I follow what the stationarity test says and use that "stationary data", I obtain an ARIMA (0,0,1) model of which log-likelihood is -10000 with an AICc value of 25000. To me it looks like R understands that there is no information from previous values that can be helpful in predicting future values. What shall I do here? Is ARIMA failing here because of the lack of stationarity and should I use GARCH instead?
This is the link to my script in R on Github : https://github.com/rsotogar/Time-series/blob/master/log-crude-prices.pdf
 A: In order to answer your question as to what steps should be routinely taken, I thought that it might be good to consider alternative remedies for non-constant variance of model's residuals.
The residuals from a useful model should have an expectation that is zero for all data points and an variance (locally pooled) that is homogeneous. In order to proceed one needs to consider possible data transformation options. Not all transformations are useful and some can gave negative consequences. Transformations are like drugs .. some are good for you and some not so !
One needs to deal this by adjusting the expectation for identified pulses  , seasonal pulses ,level/step shifts and local time trends. This is done by performing Intervention Detection. https://cran.r-project.org/web/packages/tsoutliers/index.html and AUTOBOX or SAS provide possible solutions thus avoiding unwarranted variance transformations.
After adjusting possible errors to have a common expected value of 0. , the errors now must be examined to verify constant error variance through time . I am aware of three distinct ways to do this . 
1)  Examine the variance of the residuals to ascertain any linkage between the expected value and the error variance by looking at different groups. This is done via the Box-Cox test (power transforms ) http://stats.stackexchange.com/questions/18844/when-and-why-to-take-the-log-of-a-distribution-of-numbers. This is the oldest remedy . Early researchers ignored/assumed that the errors had constant expected value thus they inadvertently proposed unwarranted power transforms when simple pulses could explain the heterogeneity. https://autobox.com/pdfs/vegas_ibf_09a.pdf presents the case for not POWER transforming the AIRLINE SERIES.
2) examine the variance of the residuals to assess whether or not there are discrete points in time where the error variance changes deterministically  . This leads directly to Weighted Estimation by evolving a weight for each observation thus effectively normaling . See http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html for a much-overlooked proposal to deal with non-constant error variance in the financial markets.
When the occasion arises that neither proposal 1 or 2 is minimally sufficient to render the error process homogeneous , one of my personal statistical heroes and friend Clive Granger proposed number 3
3) The Garch (Generalized Autoregressive Conditional Heteroscedasticity approach ) where it is assumed that the model's error variance changes stochastically over time according to some arima model that needs to be identified and parameterized.
As to what combination of these "medecines" or remedies should be used for a particular data set ...only the data knows for sure . “primum non nocere.”   .... "first do no harm" comes to mind . In all things complications/transformations should be introduced carefully because they all have unintended consequences . 
