Assuming that a seasonal ARIMA model could offer a sensible approximation to the process, you could try using function
auto.arima from the "forecast" package in R. It will first select the order of integration (by default using the KPSS test, optionally using augmented Dickey-Fuller or Phillips-Perron unit root tests) and seasonal integration (by default using OCSB test, optionally using Canova-Hansen test). Then it will select the SARIMA order based on the an information criterion (by default using AICc, optionally using AIC or BIC) from a pool of models defined by a local search procedure explained in Hyndman & Khandakar (2008).
Regarding seasonality: when dealing with seasonal data, make sure to supply a seasonal time series (e.g. a
ts object with a specified frequency which is 12 for monthly data) to
auto.arima. Seasonal integration will be determined by testing (OCSB or Canova-Hansen), while seasonal AR and MA components will be determined by minimization of an information criterion. Since it makes sense to use AIC for model selection when the goal is forecasting (see e.g. Rob J. Hyndman's blog post "Facts and fallacies of the AIC"), you may not worry too much if no seasonal part is selected but model residuals show mild patterns of remaining seasonality; AIC should select some seasonal components if the seasonality can be estimated precisely enough to improve forecasting but would leave them out if that cannot be done (so that inclusion of the seasonal components could do more harm through increased model variance than they help through reduced model bias).
Also note that the values of information criteria are not comparable between non-differenced and differenced data (again, see Rob J. Hyndman's blog post "Facts and fallacies of the AIC"), so you cannot use, say, AIC to compare between ARIMA(1,0,0) and ARIMA(0,1,0).
- Hyndman, Rob J., and Yeasmin Khandakar. Automatic time series for forecasting: the forecast package for R. No. 6/07. Monash University, Department of Econometrics and Business Statistics, 2007.