# No lags outside confidence bands. How to build ARIMA in this case?

While working with a data on sp500 I have encountered one problem concerning the interpretation of ac and pac graphs. The problem is that none of them shows a significant lag outside of confidence band. I do not know how to choose p and q in this case in order to build my ARIMA model. Any help would be highly appreciated. I am also attaching the graphs.

• If all ACF and PACF values are within the confidence bounds, then what about ARIMA(0,0,0)? (PACF value #2 seems to be borderline significant, though.) Commented Oct 17, 2017 at 16:51

These functions inform us that there is no significant autoregressive component in your data, so p = 0.

You don't need to take differences of master data as well since you do not see declining autogression from lag 1 to 40 (referring to your charts), existence of which would be an indication of long memory, artifact of having not stationary (trendy data). So, d = 0.

Given that you cannot expect that your residuals contain any autoregressive dependency in the absence of any significant dependencies to model (and to produce residuals), the q component seems to be zero as well. So, q = 0.

ARIMA(0,0,0)


It looks like a good approximation of ARIMA parameters to your data.

Update: I just saw @Richard Hardy's comment while I was typing, so we came in simultaneously.

• Thank you for the explanation. Differencing of the variable was not my choice but rather a demand of a professor. Commented Oct 17, 2017 at 16:56
• I understand you had already differenced raw SP500 prior to using ac/pac. So I mean you need no higher order differencing. Commented Oct 17, 2017 at 16:57
• Differencing makes sense from a theoretical point of view. Stock prices should not tend towards an overall mean as an ARIMA(0,0,0) process. To a first approximation, the current price should include all currently known information, so only increments over today's price should be ARMA(0,0). That is, you should have a random walk, which is an ARIMA(0,1,0) process. The next step in modeling would be to model variance using an ARCH or GARCH process. Commented Oct 17, 2017 at 20:33