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I have fit an auto-regressive model to a data series. The model seems to explain the given data relatively well given that the total $R^2$ is 0.95. However, SAS has a diagnostic plot called white noise test in proc autoreg. Past lag 10, the null hypothesis of white noise is rejected in this plot. i.e the bands go past the 0.05 level. (this seems bad) The acf and pacf plots for the residuals seem to be fine as all bars are inside the error bands. (this is as expected)

I'm wondering if anyone can explain the impact that a white noise test like this will have on my model.

Particularly, I am interested in how valid it makes the $R^2$ value. I am only interested in explaining my current data - not in prediction. As such, I am mostly just interested in the validity of the $R^2$ value.

Due to privacy considerations, I cannot post my diagnostic plots, but they are essentially the same as the figure in this question : ARIMA modeling white noise probabilities vs. residual autocorrelation/PACF

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If the data set is seasonal say monthly it is possible that if this is left untreated in your model one could get rejection beyond lag 10. In general the bands around a sample acf/pacf are VERY APPROXIMATE and should be looked at with skepticism as they are based upon a VERY APPROXIMATE standard error prewsumption.

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  • $\begingroup$ Thanks for your answer IrishStat. I'm not sure I understand what you mean by if it is left untreated, I could get rejection beyond lag 10. Currently my model consists of 11 indicator variables for month as well as 12 auto-regressive error terms that are eliminated using step-wise backward elimination. The final model has all 11 month indicator variables (they are not necessarily all significant) and a subset of the 13 auto-regressive error terms. $\endgroup$ – DanRoDuq Sep 2 '15 at 14:14
  • $\begingroup$ What I meant by untreated was if you had omitted seasonal; structure and it was needed. Without seeing your data I can't really identify the flaw on your modelling approach. Please post tour data and all the statistics of your model and I will take a look. Also include the acf of your residuals. Your data set might need transforming or segmentation and/or level shifts/local time trends in order to fully pass tests of sufficiency. By the way incorporating unnecessary monthly indicators might/could also play a role.. $\endgroup$ – IrishStat Sep 2 '15 at 14:59

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