# Interpretation of p-values in ARIMA model

I am starting my journey in time series analysis and forecasting and I'm currently facing some doubts.

If I fit an ARIMA model to my data and I get for example an ARIMA(2,1,2) in which the p-value for the second AR term is >0.05, should I re-estimate the model without that term?

Also, I must note that I'm using an automatic algorithm that estimates the parameters through maximum likelihood. Could this mean that I misconfigured the algorithm?

Thanks in advance for any help.

• Since you are getting a model, that means you are using some automatic model selection. Otherwise you chose a model ad then estimate it. Now usually the model selection is based on AIC or BIC or some other information criteria. If the purpose os forecasting, p-valuea shouldn't matter much (read Hyndman's blog for great details). Model selection based on AIC is sufficient. If the purpose is to interpret coefficients, the diagnostics is important. As you said you are using MLE then you must be making an assumption about the distribution of errors (default every where is normal)..cont.. – Dayne Sep 18 at 4:27
• ..cont..So first thing you should do is check your residuals for no autocorrelation and normality. All the problems of outliers (as discussed in ans below) will give a non-normal residuals. If residuals are normal then try a simpler (parsimonious) model and recheck. If they are also normal then prefer the parsimonious one. If residuals are not normal in either case then check for outliers and as last resort try some other distribution than normal. – Dayne Sep 18 at 4:30

If you have an over-parameterized model ( seems to me that this is a likely case ) with possibly self-cancelling structure you can often get a clue about this when you have p values that are not significant AND unfortunately even with p values that are small..

Make sure that your model identification procedure takes into account pulses , level/step shifts , seasonal pulses and or local time trends (auto.arima does not) . Perhaps you should follow the guidelines as to how to examine the acf and pacf yourself in order to identify a useful model. https://www.analyticsvidhya.com/blog/2015/12/complete-tutorial-time-series-modeling/ may be initially helpful BUT it also does not concern itself when you have untreated latent deterministic structure.

See @Adamo's words excerpted from here Interrupted Time Series Analysis - ARIMAX for High Frequency Biological Data? pinpointing the need to consider latent deterministic effects while identifying arima structure.

"The correlogram should be calculated from residuals using a model that controls for intervention administration, otherwise the intervention effects are taken to be Gaussian noise, underestimating the actual autoregressive effect." AND I would like to add possibly over-identifying the possible model .

If you post your data in a csv format , I may be able to help further.

Alternatively see Non sensical results from auto_arima for a detailed walk-through of an iterative approach to forming a useful model.

• I believe the issue here is how to initially identify the model form . After that you have choices as to how to estimate parameters in one way or the other. – IrishStat Sep 14 at 20:11
• i think deletion would be appropriate. – IrishStat Sep 15 at 8:09
• I would very much like to see a canonical answer from you on general time series modeling. I see a lot of your answers on this site using terminology such as pulses, intervention, level/step shifts, seasonal pulses and or local time trends, and I do not know what you mean by that, since that is unknown terminology to me. For ex., the first two links you provided make no mention of what a pulse is. – user2974951 Sep 16 at 11:21
• faculty.chicagobooth.edu/ruey.tsay/teaching/uts/lec10-08.pdf will be of help ... – IrishStat Sep 16 at 11:58
• Unfortunately that does not help me, it's a very short - 3 page incomplete lecture without clear definitions or explanations about what they are doing or why, or why it works, or why we should use this. – user2974951 Sep 16 at 12:06