# How to determine the AR and MA from these correlograms?

I've been trying to ARIMA models for the UK nominal GDP. I've determined the I to be not stationary - I(1). And now am having trouble identifying what the AR(p) and MA(q) would be from the linked correlograms:

ACF for differenced variable (https://imgur.com/qZp6nWF)

PACF for differenced variable (https://imgur.com/IybguSy)

My initial thoughts were: since there no pattern for PACF, MA is 0. And because of the 2 significant spikes in PACF, the AR would be 2. But I'm confused about relevant lags.

The acf and pacf suggest possible non-ivertibility perhaps due to the wrong fixup for non-stationarity. See Box & Jenkins Table A , Chart B & C for invertiblity requirements. I have seen similar plots ( an I have seen a lot of plots ! ) often suggesting incorrect differencing . Rather than differencing there might be a need for de-meaning based upon either a change in trend in the original series or a change in level (intercept change). If you post your data I will try and help you further. Note that if a series changes level the acf suggests non-stationarity not necessarily remedied by differencing. To prove this to yourself simulate a wn series with a mean shift .. then examine it's acf

Those seem like reasonable possibilities based on a visual inspection of the plots, though the fact that there is slight oscillating behaviour in the bars indicates that your model might improve with a fixed seasonal term. In any case, it is generally worth testing these things formally by using standard statistical tests comparing nested models.

When you fit an ARIMA model, lower-order forms of the model (e.g., with lower orders for the MA and/or AR parts) are nested within higher-order forms. Thus, you can determine the appropriate order for the model using standard statistical techniques for comparing nested models. Specifically, you can easily fit all ARIMA models up to some sufficiently high order (based on observation of the ACF and PACF), and then use partial F-tests to test whether there is a sufficient reduction in the residual sum-of-squares to justify the additional parameters in the higher-order models. This is all very similar to testing nested forms in regression models.

• I wonder if significance testing is relevant here. Testing some restrictions as a means to test a theory could make sense, but if the problem is that of model selection, significance testing is probably not the way to go. (Of course, this has been discussed multiple times, but I would still like to reference a nice blog post by Rob J. Hyndman, "Statistical tests for variable selection".) – Richard Hardy Mar 28 at 12:37