# How can I determine the ARIMA orders ($p$,$d$,$q$) from this correlogram?

I need help for understanding how can I interpret this correlogram in order to determine the $p$, $d$ and $q$ orders for ARIMA model. I use Stata, and I am analysing a time series with really few data.

• What is this data? I suspect it's a difference series. – Aksakal Jul 23 '16 at 0:04

Since the PACF has more significant structure than the ACF the initially suggested model might be an MA model. The suggested order of the MA model would be 1 since there is only 1 significant ACF. For a longer/detailed discussion of model selection identification you might look at http://www.autobox.com/cms/index.php/blog/entry/build-or-make-your-own-arima-forecasting-model or basic model identification material from other available web sites . The important idea that untreated anomalies/deterministic structure can obfuscate the initial model identification. Over-modelling also known as kitchen-sink modelling such as a (3,0,3) frequently (read: nearly always) can creates/inject unreliable redundant ARIMA structure. If you post your data ( even a coded version) I will try and help specifically.

• Thumbs up this, it is really easy to over-fit ARIMA models, especially if there is only a low number of data. I think it is only with a lot of points (>100) that we can confidently model high interaction order like (2,0,2), which in my experience is rare phenomenon. – YCR Aug 24 '16 at 8:48

As I understand it, there is no objectively correct order, and the orders of ARMA/ARIMA you select may differ depending on which criterion you choose to optimise, e.g. whether you choose BIC or AIC, for instance. It is more an art than a science.

Given this, here is my 2c.

• The fact that the 1st lag ACF is negative and that the ACF dies out quickly suggests a low order of differencing is required. The order of differencing may even be zero, i.e. the series is stationary. This is easy to check using DF / Phillips Perron tests.
• The sharp cut-off of the ACF (notwithstanding point 8, which could be an anomaly) suggests the order of MA >0. Try three? The sharp cut off of the ACF also suggests a relatively low (<3) order for the AR part.

If it were me, what I would suggest is following the Box-Jenkins approach; - Using the intuition above, specify an ARIMA(3,0,3) (assuming stationarity). - Check for residual autocorrelation. - Assuming no residual autocorrelation, add and remove AR/MA lags iteratively in order to optimise your selection criterion. - Re-check your final model for residual serial correlation.

• We are looking for high-quality answers rather than one-liners. Could describe in greater details how did you arrived at "3,3", since the question is about it? – Tim Dec 21 '15 at 10:39