Using ACF and PACF to Select ARIMA Orders

Summary

This answer outlines a visual approach to choosing autoregressive and moving-average orders in an ARIMA model. It recommends using the autocorrelation function (ACF) and partial autocorrelation function (PACF) to identify candidate AR and MA terms, with sharp cutoffs serving as clues about which component to try first. After fitting a candidate model, the patterns in its residual correlations can guide whether another term may be needed.

For differencing, the response suggests comparing models with several differencing orders using only a constant, then considering the standard deviation and ACF behavior. It is a short heuristic, not a complete model-selection procedure: it does not discuss information criteria, out-of-sample validation, parameter significance, or how to handle ambiguous plots. The recommendation to select differencing by the lowest standard deviation is not fully justified, so it should be treated as an initial suggestion rather than a reliable standalone rule.

Key ideas

• Use ACF and PACF patterns to propose AR and MA orders after differencing.
• A sharp PACF cutoff can suggest an AR component, while a sharp ACF cutoff can suggest an MA component.
• Compare candidate differencing orders and inspect the resulting series' correlation patterns.
• Residual correlation spikes can motivate adding an AR or MA term.
• The suggested standard-deviation comparison is a heuristic and does not replace broader model validation.

Tags

# How to identify the orders p and q for ARIMA model using least squares method?


# How to identify the orders p and q for ARIMA model using least squares method?












I would like to identify the orders p and q for ARIMA model using least squares method in Matlab. I have got also two data files (one with noise and one without)

Previously I identified p and q for AR and MA using ACF function and PACF, but now I have mixed model (ARIMA).

Could you give any tip hint how to do this?

## Answer by Celeste (score 3)

https://quant.stackexchange.com/a/17979

To identify the number of AR and MA terms you still need to look at the ACF and PACF.

To identify the orders of differencing, the easiest way is run an ARIMA model on the data with different orders of differencing (0,1,2) and with only a constant (no AR or MA term). Look at the standard deviation of these models, as well as the ACF plot - the optimal model is most likely the one with the lowest standard of deviation.

Once the identified order of differencing is taken, look at the ACF and PACF of the stationary series. In particular, look at whether the PACF or ACF cuts off sharply - if the PACF cuts off sharply start with an AR model, and vice versa. From there, look at the ACF and PACF again to determine if you need to add another term of either variety. Specifically, if there is a spike at a lower-order lag in the ACF then you should increase the MA term by 1, and vice versa.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.