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I tried to fit an AR(1) model and was examining the estimates of the model. I had a question on the output (ran in SAS - Proc ARIMA):

The residual auto-correlation up to lag 6 was non-significant (in other words - there is no auto-correlation); however, after lag 6 it is significant.

  • What does that imply?
  • Does that mean the model needs improvement?
  • Also, why would it be significant after lag 6?
Autocorrelation Check of Residuals  (Highlighted values are the auto-correlation 
    values and significant values are italicized)
[To Lag] [Chi-Square]  [DF] [Pr > ChiSq]    [Autocorrelations]  
  6          9.46       5     0.0922     ** 0.023 0.146  0.092 -0.01   0.089  0.127**    
 12         24.68      11    *0.0101*    ** 0.17  0.178 -0.095 -0.042 -0.056 -0.103**    
 18         34.42      17    *0.0074*    ** 0.126 0.105  0.05  -0.014 -0.151 -0.005**  
 24         38.86      23    *0.0206*    **-0.046 0.042  0.133 -0.008 -0.029 -0.017**
 30         51.21      29    *0.0067*    **-0.067 0.094  0.104  0.183  0.054  0.015**
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  • $\begingroup$ Would those be a series of Ljung-Box tests on the residuals, each from lag 1 up to [To lag]? $\endgroup$
    – Scortchi ♦
    Commented Apr 30, 2013 at 14:43

2 Answers 2

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The Ljung-Box statistic is not significant calculated over lags one to six lag six simply because you have no particularly large autocorrelations until lags seven & eight. Plot & examine the auto-correlation function of the residuals. There's some evidence for lack of fit, so perhaps you can come up with a better model: higher autocorrelations seem to repeat at intervals of seven without dropping off, suggesting seasonal differencing with a period of seven might be worth a try; possibly a seasonal moving average term. All the same, with about 150 observations your current model doesn't look awful.

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try fitting a seasonal ARIMA, with a seasonal difference of 6.

Your model has significant autocorrellations at lags in multiples of 6. It's just a guess on my end, but I'd think the seasonal ARIMA with difference of 6 would help.

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  • $\begingroup$ Correlations at lags 6, 12, 18, 24, 30: 0.127, -0.103, -0.005, -0.017, & 0.015. Correlations at lags 7, 14, 21, 28: 0.17, 0.105, 0.133, 0.183. Which are bigger? $\endgroup$
    – Scortchi ♦
    Commented May 2, 2013 at 8:13
  • $\begingroup$ Oh, I see, I wasn't reading the results right. (never done arima in SAS, only R. results look different.) $\endgroup$
    – Faydey
    Commented May 2, 2013 at 18:24
  • $\begingroup$ Yes, too many stars all over the place. $\endgroup$
    – Scortchi ♦
    Commented May 2, 2013 at 21:28

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