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.

 A: 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.
A: 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. 
